pdfs.semanticscholar.org · 2019-03-07 · Aix-Marseille Université école doctorale 184 UFR...

Aix-Marseille Universitéécole doctorale 184

UFR sciences

Institut de Mathématiques de Marseille

Thèse présentée pour obtenir le grade universitaire de docteur

Spécialité: Mathématiques

Types in LudicsEugenia Sironi

Rapporteurs:

Christian Retoré Université de MontpellierKazushige Terui Kyoto University

Jury:

V. Michele Abrusci Università di Roma 3Claudia Faggian Université Paris-DiderotChristophe Fouqueré Université de Paris 13 (co-directeur)Myriam Quatrini Aix-Marseille Université (co-directeur)Laurent Regnier Aix-Marseille UniversitéChristian Retoré Université de Montpellier (rapporteur)

Soutenue le 15/01/2015 À Marseille.

This thesis is licensed under a http://creativecommons.org/licenses/by-nc-sa/4.0/Attribution-NonCommercial-ShareAlike 4.0 International licence.

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Abstract

This thesis proposes a representation of the notion of type, with a particularinterest on dependent types, in Ludics.Ludics is a theory introduced by Girard [13]. It comes from a fine analysisof the multiplicative, additive fragment of polarized Linear Logic (MALLp).One of its aim is to reconstruct logic from the notion of interaction.A type is a class of objects that behave in the same way with respect to otherobjects. The notion of type is common to several domains as ComputationTheory, Game Semantics and Martin-Löf’s Intuitionistic Type Theory. Us-ing the terminology of Martin-Löf, the canonical terms of a type are theprimitive elements of the type, that is the objects that characterize it. Thenon-canonical terms are the terms obtained by applying some operations oncanonical terms and that once computed give a canonical term. Terms areseen as programs and two terms are equal when their computation gives thesame result, that is the same canonical term.We introduce the notion of principal behaviour, that is well-suited to repre-sent canonical terms. We introduce also the notion of separable behaviour,that gives us a tool to define functions in a simple way.We represent natural numbers, lists, records, dependent functions, pairs anddiscuss dependent record types.We focus then on Martin-Löf’s Type Theory and propose a representationfor some basic types and constructions.

Résumé

Cette thèse propose une représentation de la notion de type, avec un intérêtparticulier pour les types dépendants, en Ludique.La Ludique est une théorie introduite par Girard [13]. Elle vient d’une fineanalyse du fragment multiplicative, additive polarisé de la Logique Linéaire(MALLp). Un des ses buts est de reconstruire la logique à partir de la notiond’intéraction.Un type est une classe d’objets qui se comportent de la même façon par rap-port aux autres objets. La notion de type est commune à plusieurs domainescomme la Théorie de la Calculabilité, la Sémantique des Jeux et la ThéorieIntuitioniste de Types de Martin-Löf. Avec la terminologie de Martin-Löf, letermes canoniques d’un type sont les éléments primitives du type, c’est à direles objets qui le caractérisent. Les termes non canoniques sont les termesobtenu appliquant une opération aux termes canoniques et une fois calculésdonnent un terme canonique. Les termes sont vu comme des programmeset deux termes sont égaux quand leur calcul donne le même résultat, c’est àdire le même terme canonique.On introduit la notion de comportement principal, qui est bien adapté à la

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représentation des termes canoniques. On introduit aussi la notion de com-portement séparable, qui nous donne un outil pour définir les fonctions demanière simple.On représente les nombres naturelles, les listes, les records, les fonctionsdépendantes, les couples et on discutes les records dépendantes.On se focalise après sur la Théorie de Martin-Löf pour proposer une représen-tation de certaines types de base et constructions.

Remerciements

Je remercie mes directeurs de thèse Christophe Fouqueré et Myriam Qua-trini pour le soutien et la patience avec lesquels ils m’ont guidé pendant cestrois ans.Je remercie Christian Retoré et Kazushige Terui pour avoir accepté le rôlede rapporteur de ce texte et tous les membres du jury.Merci à Alain Lecomte et tous les membres du projet LOCI, qui a permit ledéveloppement de cette thèse.Merci aux membres de l’équipe LDP qui m’ont accueilli d’abord pour monmaster puis pendant ma thèse: Emmanuel Beffara, Yves Lafont, MyriamQuatrini, Laurent Regnier et Lionel Vaux.Ringrazio Michele Basaldella per l’incoraggiamento, le interessanti chiac-chierate e le utilissime correzioni.Merci à tous mes collègues (et assimilés): Anna, Clément, Emilie, Etienne,Fabio, Florent, Francesca, Irene, Jacky, JB, Joël, Jordi, Marcelo, Marc B.,Marc M., Matteo, Michele, Paolo, Pascale, Pierre O., Sarah, Virgile. Pourles pauses, les ragots, les soirés les longues journées à passées ensemble à Lu-miny et la patience avec laquelle vous m’avez soutenue pendant la redaction.Grazie a Sara, Laura, Ioio, Jacopo ed Ernesto perchè senza di loro il liceosarebbe stato insopportabile.Grazie ad Ardy e Chiar per aver resto divertente perfino il Castelnuovo.Grazie a Giorgio, Lorenzo e Nadia, senza di voi Roma 3 sarebbe stata tris-tissima.Grazie a tutto il "gruppo": Alessandro, Andrea, Eugenia, Federica, Flavia,Giulietta, Ioio, Peter, Silvia, Smarta. Nonostante il passare degli anni e ladistanza so che continueremo a trovarci alle 21,30 a ottaviano.Grazie a Irene per gli aperitivi di chiacchiere al Treebar ed il nostro esserelentiiiiisssiiiimeeee.Grazie a Lavy, per tutti i "momenti di vita quotidianen" e perchè è semprecome se non ci vedessimo solo da 5 minuti.Grazie a Ioio per le risate (Baden Baden!), le manifestazioni, i subsonica e ilunghi abbracci.Grazie a Peter per le fughe estive a Torvaianica, le ore passate a spettegolaree il fatto di aspettarmi sempre a braccia aperte.Merci à Cris-Cros, Gab, Léa, Lo, Max, Sara e Sarah pour m’avoir fait sentir

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chez moi à Marseille et avoir organisé le plus bel anniversaire surprise.Merci à Valeria pour l’energie et l’envie de danser qu’on a partagè.Grazie a Mela, che mi ha tenuto compagnia durante tutti questi anni di stu-dio e continua ad aspettarmi davanti alla porta.Grazie a tutta la mia famiglia perchè quando non sono a Gallarate per menon è davvero Natale.Grazie a mia madre e mio padre per avermi insegnato l’importanza di seguirele proprie passioni.Grazie a Rosa perchè non smetteremo mai di essere un punto fisso l’una perl’altra.Grazie a Carlo per tutte le volte che abbiamo rivisto insieme i video di Guz-zanti e per i silenzi nei tuoi film.Grazie a Bernardo e Francesca Romana, perchè non pensavo fosse possibilevolere bene a qualcuno anche prima di conoscerlo.Grazie a Marc di esserci.

Contents

1 Ludics: Original Setting and New Developments 131.1 A Presentation of Original Ludics . . . . . . . . . . . . . . . . 13

1.1.1 Normalization and Orthogonality . . . . . . . . . . . . 181.1.2 Behaviours, Incarnation . . . . . . . . . . . . . . . . . 21

1.2 Developments and Subsequent Works . . . . . . . . . . . . . . 231.2.1 From Chronicles to Paths . . . . . . . . . . . . . . . . 231.2.2 Non-Linear Ludics . . . . . . . . . . . . . . . . . . . . 33

1.3 New Notions: Principality and Separability . . . . . . . . . . 341.3.1 Principality . . . . . . . . . . . . . . . . . . . . . . . . 341.3.2 Separability . . . . . . . . . . . . . . . . . . . . . . . . 38

1.4 Summary and Comments . . . . . . . . . . . . . . . . . . . . 43

2 Types and terms in Ludics 452.1 Basic Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.1.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . 472.1.2 Booleans . . . . . . . . . . . . . . . . . . . . . . . . . . 522.1.3 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.1.4 Records . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.2 Constructions and Dependency . . . . . . . . . . . . . . . . . 762.2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 772.2.2 Dependent Functions . . . . . . . . . . . . . . . . . . . 812.2.3 Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.2.4 Dependent Record Types . . . . . . . . . . . . . . . . 99

2.3 Summary and Comments . . . . . . . . . . . . . . . . . . . . 105

3 Martin-Löf’s Type Theory and Ludics 1073.1 Martin- Löf’s Type Theory . . . . . . . . . . . . . . . . . . . 107

3.1.1 Categorical Judgements . . . . . . . . . . . . . . . . . 1083.1.2 Hypothetical Judgements . . . . . . . . . . . . . . . . 1093.1.3 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.1.4 Universes . . . . . . . . . . . . . . . . . . . . . . . . . 1103.1.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.2 Martin-Löf’s Type Theory in Ludics . . . . . . . . . . . . . . 111

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3.2.1 Context and Judgements . . . . . . . . . . . . . . . . . 1123.2.2 The Rules of N . . . . . . . . . . . . . . . . . . . . . . 1143.2.3 The Rules of ListpNq . . . . . . . . . . . . . . . . . . . 1183.2.4 The Rules of Π . . . . . . . . . . . . . . . . . . . . . . 1223.2.5 The Rules of Σ . . . . . . . . . . . . . . . . . . . . . . 1253.2.6 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Introduction

The aim of this thesis1 is to introduce the notion of type in Ludics, in thesense of type theory, with a particular interest on dependent types.In the literature, several works investigated how to use the objects of Lu-dics for an application in Linguistics [10], [18]. In [10] Ludics is proposedas a framework to formalize dialogues, focusing on the elements of dialoguesthat are supports of the interaction. A dialogue between two interlocutorsis deconstructed into a sequence of interventions and for each speaker, thesequence of interventions and expectations is then sufficient for recoveringan underlying design. On the other side, the possibilities for formulatinglinguistic semantics in terms of records and record types were explored byCooper in [7]. A representation of dependent types in Ludics looks theninteresting to fill the gap between Ludics and Type Theory.

A type, as for instance natural numbers, is a class of objects that behave inthe same way with respect to other objects. The notion of type is common toseveral domains as Computation Theory, Game Semantics and Martin-Löf’sIntuitionistic Type Theory. Using the terminology of Martin-Löf, the canon-ical terms of a type are the primitive elements of the type, that is the objectsthat characterize it. The non-canonical terms are the terms obtained by ap-plying some operations on canonical terms and that once computed give acanonical term. Terms are seen as programs and two terms are equal whentheir computation gives the same result, that is the same canonical term. Toestablish that A is a type we need to know what an object of type A is andwhat it means for two objects of type A to be equal.

Ludics is a theory introduced by Girard [13]. One of its aim is to reconstructlogic from the notion of interaction. The central notion is no more the notionof truth or proof, but interaction between objects called designs. It comesfrom a fine analysis of the multiplicative, additive fragment of polarized Lin-ear Logic (MALLp). Another aim of Ludics is to overcome the distinctionbetween syntax and semantics. These two worlds, usually distinct, become a

1The research work for this thesis has been developed in the frame of the ANR projectLOCI ("Locativité et Interaction en Linguistique, Logique et Informatique").

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10 CONTENTS

unique universe, in which an object is completely determined by the objectsit interacts with. This means that properties are expressed and tested in-ternally (and interactively), because the objects themselves test each other.The main object of Ludics, called design, represents a cut-free (para)-proof(not a proof because there is a special rule, called daimon, which ends theinteraction) of a certain proof where everything is erased, but locations. Bythe word location, we mean the "place" occupied by a subformula of a for-mula. The importance of the notion of location can be explained by anintuition given by computer science: proofs do not manipulate the "idea"of a formula, but the address in the memory where it is stored: its location[8]. Designs represent both an abstraction of a formal proof and its semanticinterpretation, therefore syntax and semantics meet in this notion. A be-haviour is a set of designs that "behave" in the same way with respect tointeraction. Such a behaviour is characterized by its minimal designs.

Since a type is a class of terms that behave in the same way and it is com-pletely characterized by its canonical terms and computation, then we pro-pose to represent types by behaviours and canonical terms by their materialdesigns. To avoid the possibility to have the z as result of the normaliza-tion of a set of designs (that represents the computation of a set of terms),we only consider z´free material designs, i.e., the z´free incarnation of thebehaviour.We introduce the notion of principal set that is strictly connected to thenotion of z´free incarnation. This notion is well-suited for representingfaithfully canonical terms. We also introduce the notion of separable be-haviour, that intuitively tells us that we can "separate" the designs of itsz´free incarnation and gives us a tool to define functions is a simple way.We propose a representation in Ludics of basic types: natural numbers, listsof natural numbers and records. In our representation of basic types, prin-cipality holds for all of them, whereas separability holds for all but records.We show that principality is too strict for being stable under dependentproduct, and even for function types. This means that given two principalbehaviours A and B that represent the types A and B, then the behaviourthat represents the type of functions from A to B is not always principal.In order to bypass this failure, we restrain ourself to separable behaviours.This gives us a natural way to define a representation of functions. We alsopropose a representation for pairs and discuss dependent record types.

Dependent type theories started in the early 1970’s, when Martin-Löf de-veloped his Intuitionistic Theory of Types (ITT) ( see e.g. [19],[20]). Typeshave been introduced from the initial motivation to improve the paradoxicalstructure of sets. Later, they were found to be much closer to the notionof computation, thanks to the Curry-Howard isomorphism (see e.g. [16]) :a one-to-one correspondence between some logical systems and some type

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systems such that propositions are mapped to types and proofs of a propo-sition are mapped to terms of the corresponding type. Martin-Löf’s TypeTheory is based on Curry-Howard isomorphism: proofs are mapped into pro-grams and the execution of these programs corresponds to the applicationof the cut-elimination procedure to the proofs. The corresponding of thecut-elimination procedure in Ludics is the notion of interaction (normaliza-tion, Chapter 1Ludics: Original Setting and New Developmentschapter.1).Interaction is a primitive notion in Ludics, in the sense that, the objects ofLudics are characterized by how they interact with other objects. Finally werecall the basic types and constructions of Martin-Löf’s Type Theory andwe propose a representation for them in Ludics.

The plan of the thesis is the following:

• In Section 1.1A Presentation of Original Ludicssection.1.1 we recallthe basic notions of Ludics introduced by Girard in [13]. In Section1.2Developments and Subsequent Workssection.1.2 we consider somesubsequent works and new developments. In Section 1.2.1From Chroni-cles to Pathssubsection.1.2.1 we recall some notions and result from thework by Fouqueré and Quatrini [11]. The notion of visitable path [11]is particularly important, since it permits to recover the elements of in-carnation. We also recall the relation betweenMALLp connectives andoperations on visitable paths established in another work by Fouqueréand Quatrini [12]. In Section 1.2.2Non-Linear Ludicssubsection.1.2.2we shortly recall the works by Maurel [21], Terui [26], Basaldella andFaggian [2], Basaldella and Terui [3], that propose a non-linear exten-sion of Ludics.In Section 1.3New Notions: Principality and Separabilitysection.1.3we introduce two new notions: principal behaviour and separable be-haviour that permits us to characterize and manipulate the z-free, ma-terial designs of a behaviour. These notions will be central in Chapter2Types and terms in Ludicschapter.2 for our representation of typesin Ludics. We use some notions introduced in [11] to prove a propertyof principal behaviours and that principality is closed with respect toMALLp connectives ‘,b, Ó, Ò, &.

• In Chapter 2Types and terms in Ludicschapter.2 we propose a repre-sentation for basic types and constructions in Ludics. In Section 2.1Ba-sic Typessection.2.1 we represent some basic types: natural numbers,booleans, lists of natural numbers and records. In our representationof basic types, principality holds for all of them, whereas separabilityholds for all but records. Principality is not stable under functions, butrestricting ourself to separable behaviours, in Section 2.2Constructionsand Dependencysection.2.2 we can give a satisfying representation forfunctions and pairs. We finally discuss dependent record types.

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• In Chapter 3Martin-Löf’s Type Theory and Ludicschapter.3 we recallthe types and constructions of Martin-Löf’s type theory and then wepropose a representation of the rules of N, ListpNq, pΠx : AqBpxq,pΣx : AqBpxq and intensional equality in Ludics.

Chapter 1

Ludics: Original Setting andNew Developments

In Section 1.1A Presentation of Original Ludicssection.1.1 we define thefundamental notions of Ludics [13]. In Section 1.2.1From Chronicles toPathssubsection.1.2.1 we recall some notions and results from subsequentworks, in particular the notion of visitable path [11] and the relation be-tween MALLp connectives and operations on visitable paths established in[12].In Section 1.2.2Non-Linear Ludicssubsection.1.2.2 we recall some works thatinvestigate non-linear extensions of Ludics [21], [26], [2], [3].In Section 1.3New Notions: Principality and Separabilitysection.1.3 we intro-duce two new notions: principal behaviour and separable behaviour. Thesenotions will be central in Chapter 2Types and terms in Ludicschapter.2 forour representation of types in Ludics. Then we use some notions introducedin [11] to prove a property of principal behaviours and the closure of princi-pality w.r.t. MALLp connectives ‘,b, Ó, Ò,&.

1.1 A Presentation of Original Ludics

In this section we recall some notions of Ludics, we refer the reader to [13]for a formal thorough presentation. Ludics is a theory introduced by Girard[13] to reconstruct logic starting from the notion of interaction. The centralobject is no more truth or proof, but interaction defined on designs. To definedesigns we first reformulate the definitions of address, action and chroniclefrom [13].

Definition 1. An address (also called locus in [13]), often denoted by agreek letter ξ, is a finite (maybe empty) sequence of natural numbers. Anaddress α is a subaddress of ξ when α is a prefix of ξ. For instance 1.0.2is a subaddress of 1.0.2.4.0.An action κ is

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14CHAPTER 1. LUDICS: ORIGINAL SETTING AND NEW DEVELOPMENTS

• either a positive proper action p`, ξ, Iq or a negative proper actionp´, ξ, Iq where ξ is called the address of κ. I is a finite set of naturalnumbers calleds ramification.

• or the positive (non proper) action daimon denoted by z.

We say that an action κ is justified by the action κ1 when the address of κ isbuilt from the address of κ1. This means that if the address of κ1 is σ and itsramification is I, then there exists i P I such that the address of κ is σ.i. Theactions κ and κ1 must have opposite polarity. For instance p`, ξ.0.2, t0uq isjustified by p´, ξ.0, t2uq, and p´, α.3, t2uq is justified by p`, α, t3uq.

Definition 2. A chronicle c is a non empty, finite alternate sequence ofactions such that :

• Each action κ of c is either initial (if κ is negative then it is the firstaction of c) or justified by a previous action of opposite polarity. Inparticular, negative actions are justified by the positive action whichimmediately precede them.

• Actions have distinct addresses.

• If present, a daimon is the last action of the chronicle.

We reformulate the definition of coherence between chronicles from [13].

Definition 3. Two chronicles c1 and c2 are coherent, written c1 ¨ c2,when the two following conditions are satisfied:

• Comparability: Either one extends the other, or they first differ onnegative actions. That is if wκ1 ¨ wκ2 then either κ1 “ κ2 or κ1 andκ2 are negative actions.

• Propagation: When they first differ on negative actions and these neg-ative actions have distinct addresses, then the addresses of followingactions in c1 and c2 are pairwise distinct.

Example 1. The chronicles c “ p`, ξ, t0, 2uqp´, ξ.0, t1uqp`, ξ.0.1,Hq andc1 “ p`, ξ, t0, 2uqp´, ξ.2, t1, 2uq are coherent, while c and c1 “ p`, ξ, t0, 2uqp´, ξ.0, t1uqz are not coherent, because they first differ on a positive action.

We consider chronicles based on a sequent Γ $ ∆, where ∆ is a finite set ofaddresses, Γ contains at most one address and the addresses of Γ Y ∆ arepairwise disjoint, i.e., no address is a subaddress of another one. The set ∆contains the addresses of the initial positive actions of the chronicle. If Γ isempty, then the base is said to be positive, otherwise the base is said to benegative and Γ contains the address of the initial negative actions.

1.1. A PRESENTATION OF ORIGINAL LUDICS 15

Definition 4. ([13])A design D based on Γ $ ∆, is a set of chronicles based on Γ $ ∆ suchthat the following conditions are satisfied:

• Forest: The set is prefix closed.

• Coherence: The set is a clique with respect to ¨. That is, the chroniclesof the set are pairwise coherent.

• Positivity: A chronicle without extension in D (also said maximal)ends with a positive action.

• Totality: D is non empty when the base is positive, in that case all thechronicles begin with a (unique) positive action.

We say that a design is positive when its base Γ $ ∆ is positive (Γ isempty), otherwise it is said negative.

A design can also be represented as a proof-like sequent structure: asso-ciating a positive rule to each positive action and a negative rule to allnegative actions with the same address.For instance we associate the rule p´, ξ.1, tt0u, t1uuq to the actions p´, ξ.1, t0uqand p´, ξ.1, t1uq and write it as

$ ξ.1.0 $ ξ.1.1

ξ.1 $ .

Given a negative base σ $ Γ, the action p´, σ,Hq is represented by the rule$ Γ

σ $ Γ

The proof-like presentation of a design is given by the following notion ofdesign as dessin.

Definition 5. ([13])A design (as dessin) based on Γ $ ∆ is a tree with root Γ $ ∆ built bymeans of the following rules:

• The daimon

z$ Λ

• Positive rule:

. . . ξ.i $ Λi . . .p`, ξ, Iq

$ ξ,Λ

where I is a ramification, for all i P I the Λi are pairwise disjoint andincluded in Λ.


• Negative rule :

¨ ¨ ¨ $ ξ ‹ I,ΛI . . .p´, ξ,N q

ξ $ Λ

where N is a set of ramifications, for all I P N ΛI Ď Λ and we denotewith ξ ‹ I the sequence ξ.i1, . . . , ξ.in where I “ ti1, . . . , inu.

A subdesign of a design D is a subtree of D, that is still a design.

Example 2. In the left the design D is represented as a tree-like structureof actions, while in the right as a proof-like sequent structure.

p`, ξ, t1, 3uq

p´, ξ.3, t0uq

p`, ξ.3.0,Hq

p´, ξ.1, t0uq

p`, ξ.1.0, t0uq

p´, ξ.1, t1uq

p`, ξ.1.1, t0uq

$ ξ.3.0H

ξ.3 $

ξ1.0.0 $

$ ξ.1.0

ξ.1.1.0 $

$ ξ.1.1

ξ.1 $

$ ξ

When we consider a design as a set of chronicles, we abusively only writemaximal chronicles. For instance the chronicles of D (Example 2example.2)are the following:

• p`, ξ, t1, 3uqp´, ξ.3, t0uqp`, ξ.3.0,Hq,

• p`, ξ, t1, 3uqp´, ξ.1, t0uqp`, ξ.1.0, t0uq,

• p`, ξ, t1, 3uqp´, ξ.1, t1uqp`, ξ.1.1, t0uq.

Example 3. The design whose only action is z is called Dai.

Dai “ tzu Dai “z

$ ξ.1, ..., ξ.n.

The base of a design is its first sequent in the bottom-up view, i.e., its root,when we consider it like a tree. For instance, in Example 2example.2 thebase is $ ξ, while in Example 3example.3 the base is $ ξ.1, ..., ξ.n.

Example 4. In [13] Girard introduces a design, called Fax, to representsthe identity function. More precisely, given the disjoint addresses ξ and σ,the design Faxξ$σ based on ξ $ σ is given as follows:

Faxξ$σ “

...

...

Faxσ.i$ξ.iσ.i $ ξ.i

$ σ, ξ ‹ I......

Pf pNqξ $ σ

...Faxξ$σ “ p´, ξ, Iq

p`, σ, Iq

Faxσ.i$ξ.i

...


@I P Pf pNq. Here Pf pNq denotes the set of finite subsets of N. Let I “t1, ..., nu, then we use ξ ‹ I to denote ξ.1, ..., ξ.n. We shorten Faxξ$σ by Fax,when the base is clear.

A Fax corresponds in terms of game semantics to a copycat strategy, in thesense that at each step we copy the last action of the opponent. To representthe partial identity function we consider a subset of Fax that only containsthe ramifications necessary to interact with the designs of a given set A ofz-free designs (based on $ ξ) and call this design IdA.

Formally, we define IdA, based on ξ $ σ, as follows:

• Let A P A be composed by just one action, i.e., A “ tp`, ξ, Iqu. Thenthe chronicle p´, ξ, Iqp`, σ, Iq belongs to IdA.

• For all A P A that contain strictly more that one action, the first actionof A is of the form p`, ξ, Iq. Let us call this action κ`ξ,I . For all i P Iκ`ξ,I generates the address ξ.i. Since A contains more than one action,then for all i P I the action κ`ξ,I is followed by a negative action basedon ξ.i and with a ramification R. Let us call this action κ´ξ.i,R. Sincemaximal chronicles end with a positive action, then the action κ´ξ.i,R isfollowed by at least one action. We denote by Aξ.i,R the subdesign of Abased on $ ξ.i ‹R that follows κ`ξ,Iκ

´ξ.i,R.

We recall that if R “ tr1, ..., rku, then ξ.i ‹R “ ξ.i.r1, ..., ξ.i.rk.

A “ κ`ξ,Itκ´ξ.i,R Aξ.i,R | i P I,Aξ.i,R is the subdesign of A based on $

ξ.i ‹Ru

For all action κ`α,I “ p`, α, Iq we write κ`α,I “ p´, α, Iq. Given an-other address β, we call κ`β{α,I the same action but with address β,i.e., p`, β, Iq.

Then for all A P A, IdA contains the chronicles

κ`ξ,I κ`

σ{ξ,I κ´

σ.i{ξ.i,R IdtAξ.i,Ru

such that i P I and Aξ.i,R is the subdesign of A based on $ ξ.i ‹R.

• IdA does not contain other chronicles.

For instance let A “ tE1,E2u

E1 “

A0

$ ξ.0.1

ξ.0 $

A1

$ ξ.1.0

ξ.1 $

$ ξ , E2 “

A2

$ ξ.2.1

ξ.2 $

$ ξ


E1 “p`, ξ, t0, 1uq

p´, ξ.0, t1uq

A0

p´, ξ.1, t0uq

A1

E2 “p`, ξ, t2uq

p´, ξ.2, t1uq

A2

Then IdA is the following design

IdA “

IdtA0u

ξ.0.1 $ σ.0.1

$ ξ.0, σ.0.1

σ.0 $ ξ.0

IdtA1u

ξ.1.0 $ σ.1.0

$ ξ.1, σ.1.0

σ.1 $ ξ.1

$ ξ.0, ξ.1, σ

IdtA2u

ξ.2.1 $ σ.2.1

$ ξ.2, σ.2.1

σ.2 $ ξ.2

$ ξ.2, σ

ξ $ σ

IdA “ p´, ξ, t0, 1uq

p`, σ, t0, 1uq

p´, σ.0, t1uq

p`, ξ.0, t1uq

IdtA0u

p´, σ.1, t0uq

p`, ξ.1, t0uq

IdtA1u

p´, ξ, t2uq

p`, σ, t2uq

p´, σ.2, t1uq

p`, ξ.2, t1uq

IdtA2u

A design Faxξ$σ intuitively corresponds to the identity function. In thefollowing chapter we will propose a representation of functions by designsthat are minimal w.r.t. inclusion in a set of designs. The designs IdA willthen be useful for this representation.

1.1.1 Normalization and Orthogonality

Definition 6. ([13])A cut is an address which appears with opposite polarity (on the left and onthe right of $) in the base of two designs.A net is a finite set of designs. A cut-net is a net where:

• all the addresses occurring in the bases are pairwise disjoint or equal,

• each address appears in at most two bases, in this case it is a cut,

• the graph whose vertices are the sequents and whose edges are the cutsis connected and acyclic.


Given a cut-net we can distinguish a particular design, called main design.It is the only positive design of the cut-net, if there is one. Otherwise it isthe only negative design whose base contains an address that is not part of acut. The first rule (in the bottom up view) of the main design is called themain rule.

A cut-net is closed when all addresses in bases are part of a cut.We remark that in the case of a closed cut-net, the main design is a positivedesign, then its main rule is positive.

Interaction, i.e., cut-elimination, is defined on cut-nets. First we consider thecase of a closed cut-net. In this case if the interaction ends (without failing)the result is tzu, while in the general case it can be a design D ‰ tzu.

Definition 7. ([13])Let R be a closed cut-net. The design resulting from the interaction, writtenJRK and called the normalization of R, is defined in the following way: letD be the main design of R, with first action κ,

• Daimon: if κ is the daimon, then JRK “ tzu

• otherwise κ is a proper positive action p`, σ, Iq such that σ is partof a cut with another design with last rule p´, σ,N q, (N aggregatesramifications of actions on the same address σ)

– Failure: If I R N , then the interaction fails.

– Conversion: otherwise, the interaction follows the connectedpart of subdesigns obtained from I with the rest of R.

Definition 8. ([13]) Now let us consider the general case, where thenet is not supposed to be closed. Thus, the main rule can be positive ornegative, and besides the cases of the precedent definition there are two newpossibilities:

• Positive commutation: the net is positive, with main rule p`, ξ, Iqbut ξ is not a cut. Let Di be as in the case of conversion above, anddefine R1 by replacing D with the Di. R1 splits into several connectedcomponents, and each Di lies in a component Ri, which is a net, andthe Ri are pairwise distinct. Let the Ei be the respective normal formsof the Ri (they exist because the Ri are negative). The normal form ofR is the design whose first rule is p`, ξ, Iq and which proceeds with Eiabove the premise of index i.

• Negative commutation: The net is negative, with main design Dand main rule p´, ξ,N q. For I P N let DI be the subdesign of D abovethe premise of index I of the last rule, and let us replace D with DI in


R, and let RI be the connected component of DI (we do not directlyget a net, as above, because of weakening). Let N 1 be the subset of Nmade of those I for which RI has a normal form EI . The normal formof R is defined as the design ending with p´, ξ,N q and which proceedswith EI above the premise of index I.

As Girard says: "In other terms, the positive commutation recopies thefirst rule (in the bottom-up view) and then proceeds separately above eachpremise. The negative commutation does the same, but some premises maydisappear" [13].

Example 5. Let E,F be the following designs.

E “

H

$ α.0.0

α.0 $

H

$ α.1.1

α.1 $$ α , F “

...G$ β

α.1.1 $ β

$ α.1, β

α.0.0 $ α.1, β

$ α.0, α.1, β

α $ β JE,FK “ G .

In terms of chronicles it corresponds to

E “ p`, α, t0, 1uq

p´, α.0, t0uq

p`, α.0.0,Hq

p´, α.1, t1uq

p`, α.1.1,Hq

“ Fp´, α, t0, 1uq

p`, α.0, t0uq

p´, α.0.0,Hq

p`, α.1, t1uq

p´, α.1.1,Hq

G

The red dashed line above represents the interaction between E and F.

Definition 9. ([13])Let E be a design and R a net such that tE,Ru is a closed cut-net. We saythat E and R are orthogonal, written E K R, when JE,RK “ tzu.

Given a set E of designs we denote by EK the set of designs orthogonal toall the elements of E, EK “ tD | @E P E,D K Eu.

Remark 1. From [13] given a set E of designs on the same base, we havethat E Ď EKK.


1.1.2 Behaviours, Incarnation

Definition 10. ([13])A set E of designs with the same base is called a behaviour when it is equalto its biorthogonal, i.e., E “ EKK.

Definition 11. ([13])Given a design D we define its incarnation in a behaviour G as

|D|G :“č

tD1 |D1 Ď D,D1 P Gu.

We say that D is material in G when it is equal to its incarnation in G,i.e., D “ |D|G.The incarnation of G, |G|, is then, by definition, the set of the materialdesigns in it, i.e.,

|G| :“ t|D|G |D P Gu.

1.1.2.1 z-Shortening

In this work, an important construction w.r.t. incarnation and generation ofbehaviours is the following notion of z-shortening of a set of designs.

Definition 12. A z-shorten of a chronicle c is either c or a chronicle ofthe form c1z, when c “ c1κ

`c2 and κ` is a proper, positive action.Given a set of designs E we define its z-shortening Ez as the set of designsobtained from E by z-shortening chronicles.

Example 6. Let E “ tDu where

D “

H

$ α.1.0

α.1 $

H

$ α.3.1

α.3 $$ α D “ p`, α, t1, 3uq

p´, α.1, t0uq

p`, α.1.0,Hq

p´, α.3, t1uq

p`, α.3.1,Hq

Then Ez contains D and the following designs:

z$ α.1.0

α.1 $

z$ α.3.1

α.3 $$ α ,

H

$ α.1.0

α.1 $

z$ α.3.1

α.3 $$ α ,

z$ α.1.0

α.1 $

H

$ α.3.1

α.3 $$ α ,

z$ α.

p`, α, t1, 3uq

p´, α.1, t0uq

z

p´, α.3, t1uq

z

, p`, α, t1, 3uq

p´, α.1, t0uq

p`, α.1.0,Hq

p´, α.3, t1uq

z


p`, α, t1, 3uq

p´, α.1, t0uq

z

p´, α.3, t1uq

p`, α.3.1,Hq

, Dai

Lemma 1. Given a set E of designs on the same base, Ez Ď EKK.

Proof. If D P Ez then there exists E P E such that D is obtained from Eby z-shortening chronicles, i.e., D P tEuz. Since E P E, then for all F P EK

E K F. D is a z-shortening of E, therefore for all F P EK D K F, i.e.,D P EKK. Thus Ez Ď EKK.

Lemma 2. Let D be a material design in a behaviour G, then all the designsin its z-shortening are material in G, i.e., if D P |G| then tDuz Ď |G|.

Proof. Let E P tDuz, then either E “ D (in this case there is nothing toprove) or E is obtained from D by z-shortening chronicles. We prove bycontradiction that E P |G|. Let F Ĺ E such that F P G, that is E R |G|.This means that there exist a negative action κ´ and a chronicle c P F, suchthat cκ´ P E and cκ´ R F. cκ´ P E and E P tDuz, then cκ´ P D. For allG P GK, G K E and G K F. This means that the computation of JE,GKdoes not use κ´. Let F1 be D without the chronicle cκ´ and its extensionsin D, then F1 Ĺ D and for all G P GK G K F1, i.e., F1 P G. Then D is notmaterial in G (contradiction).

Lemma 3. Let E be a subset of the incarnation of a behaviour G, then allthe designs in its z-shortening are material in G, i.e., if E Ď |G|, then Ez

Ď |G|.

Proof. Ez “Ť

EPEtEuz. From Lemma 2lemme.2, for all E P E tEuz Ď |G|.

Thus Ez Ď |G|.

Remark 2. Given a set E of designs on the same base, its orthogonal isequal to the orthogonal of its z-shortening, i.e., EK “ pEzqK.This property is easily shown: since E Ď Ez, then we have the first inclusionpEzqK Ď EK. Let D P EK, then by definition of Ez we have that D P pEzqK.Therefore EK “ pEzqK.

1.1.2.2 The Mistery of the Incarnation

Incarnation leads to some interesting properties when we consider connec-tives in Ludics. In particular when we consider the additive conjunction &we find a very important result called the"mystery of incarnation" [13]: Given two negative, disjoint behaviors Gand H, on the same base

1.2. DEVELOPMENTS AND SUBSEQUENT WORKS 23

|G&H| » |G| ˆ |H|.

where G&H :“ GXH.

This gives the possibility to graps in two different ways the elements ofG&H, as intersection or as cartesian product.We will see in Chapter 2Types and terms in Ludicschapter.2 that this prop-erty is very interesting w.r.t. the notion of subtyping when we representrecords in Ludics.

1.2 Developments and Subsequent Works

In this section we shortly recall some developments of Ludics and subse-quent works. In section 1.2.1From Chronicles to Pathssubsection.1.2.1 werecall a work by Fouqueré and Quatrini that proposes a generalization ofthe notion of chronicle to the notion of path and in particular the notionof visitable path [11]. They use this notion to capture the incarnation ofa behaviour. We use this result in the next chapter to investigate aboutthe principality of several sets of designs. We recall then the relation be-tween MALLp connectives and operations on visitable paths established inanother work by Fouqueré and Quatrini [12]. In Section 1.3New Notions:Principality and Separabilitysection.1.3 we prove that principality is stablew.r.t. MALLp connectives Ò, Ó,b,‘ and & and this will permit us to easilyprove the principality of some basic types in Chapter 2Types and terms inLudicschapter.2. Before proving such property we recall the presentation ofLudics as a Hyland-Ong style Game Semantics [15], [23] proposed in [9] and[2]. In section 1.2.2Non-Linear Ludicssubsection.1.2.2 we recall some worksof Maurel [21], Basaldella and Faggian [2], Terui [26], Basaldella and Terui[3], that propose a non-linear extension of Ludics.

1.2.1 From Chronicles to Paths

So far, we defined a design as a set of chronicles. Now we generalize thenotion of chronicle introducing the notions of view and path [11] to definea design as a set of paths. The notion of path is very close to the notionof play in Game Semantics. Intuitively a path of a design is a sequence ofactions that visits several chronicles and the view of a path is one of suchchronicles. There is a particular class of paths, called visitable paths that arethe paths of a set of designs E visited during the interaction with a net of EK.

The notion of visitable path is used to capture the incarnation of a behaviour[11]. In Chapter 2Types and terms in Ludicschapter.2 we will require thecapture of incarnation to prove that the sets that represent lists of naturalnumbers of fixed length and all lists of natural numbers are principal (Lemma


17lemme.17, Proposition 15prop.15, Proposition 19prop.19). Roughly speak-ing principality characterizes behaviours that are generated by a z-free setof designs. We also use the notion of visitable path to investigate someproperties of the notion of principal behaviour (Proposition 6prop.6).

Definition 13. ([11])A base of net β is a non-empty finite set of sequents of pairwise disjointaddresses: Γ1 $ ∆1, ...,Γn $ ∆n such that each Γi contains exactly oneaddress ξi, except at most one that may be empty, and the ∆j are finite sets.A sequence of actions s is based on β if an action of s either is hereditarilyjustified by an element of one of the sets Γi or ∆i, or is the daimon andin this case is the last action of s. An action is initial if its address is anelement of one of the sets Γi or ∆i.Let s be a finite sequence of actions based on β, such that there are notseveral actions of s with the same address. The view xsy is the subsequenceof s defined as follows: xεy “ ε; xκy “ κ; xwκ`y “ xwyκ`; xwκ´y “ xw0yκ´

where w0 either is empty if κ´ is initial or is the prefix of w ending with thepositive action which justifies κ´.

Definition 14. ([11])A path p based on β “ Γ1 $ ∆1, ...,Γn $ ∆n is a finite sequence of actionsbased on β such that

• Alternation: The polarity of actions alternates between positive andnegative.

• Justification: A proper action is either justified, i.e., its address is builtby one of the previous actions in the sequence, or it is called initial withan address in one of the Γi (resp. ∆i) if the action is negative (resp.positive).

• Negative jump (no jump on positive actions) : Let qκ be a prefix ofp. If κ is a positive proper action justified by a negative action κ1

then κ1 Px q y. If κ is an initial positive proper action then its addressbelongs to one ∆i and either κ is the first action of p and Γi is empty,or κ is immediately preceded in p by a negative action with an addresshereditarily justified by an element of Γi Y∆i.

• Linearity: Actions have distinct addresses.

• Daimon: If present, a daimon ends the path. If it is the first action inthe path p then one of the Γi is empty.

• Totality: If there exists an empty Γi, then p is non empty and beginseither with z or with a positive action with an address in ∆i.


When a path p starts with a negative action and ends with a positive one,we call it positive-ended, negative.

Remark 3. We remark from [11] that given a path p of a design D, the viewof p, xpy, is a chronicle of D.

Example 7. Let D be the following designH

$ ξ.0.1.0.1

ξ.0.1.0 $

$ ξ.0.1

ξ.0 $

H

$ ξ.1.1.0.0

ξ.1.1.0 $

$ ξ.1.1

ξ.1 $

$ ξ

p`, ξ, t0, 1uq

p´, ξ.0, t1uq

p`, ξ.0.1, t0uq

p´, ξ.0.1.0, t1uq

p`, ξ.0.1.0.1,Hq

p´, ξ.1, t1uq

p`, ξ.1.1, t0uq

p´, ξ.1.1.0, t0uq

p`, ξ.1.1.0.0,Hq

p`, ξ, t0, 1uq

p´, ξ.0, t1uq

p`, ξ.0.1, t0uq

p´, ξ.0.1.0, t1uq

p`, ξ.0.1.0.1,Hq

p´, ξ.1, t1uq

p`, ξ.1.1, t0uq

p´, ξ.1.1.0, t0uq

p`, ξ.1.1.0.0,Hqp xpy

The sequence p “ p`, ξ, t0, 1uqp´, ξ.0, t1uqp`, ξ.0.1, t0uqp´, ξ.1, t1uqp`, ξ.1.1, t0uqp´, ξ.0.1.0, t1uqp`, ξ.0.1.0.1,Hqp´, ξ.1.1.0, t0uqp`, ξ.1.1.0.0,Hq is a path ofD.

The view of p, xpy, is the chronicle p`, ξ, t0, 1uqp´, ξ.1, t1uqp`, ξ.1.1, t0uqp´, ξ.1.1.0, t0uqp`, ξ.1.1.0.0,Hq (the red dashed line above).

We remark that a chronicle c is a path such that each negative action isjustified by the immediately precedent action. We generalize the definitionof coherent chronicles and z-shortening of a chronicle to paths.


Definition 15. ([11])Two paths p1, p2 on the same base are coherent, noted p1 ¨ p2, when:

• their first action have same polarity: either positive and the first actionsare the same or negative;

• for all sequences w1κ`1 and w2κ

`2 respectively prefixes of p1 and p2: if

xw1y “ xw2y, then κ`1 “ κ`2 ;

• for all sequences w1κ´1 and w2κ

´2 respectively prefixes of p1 and p2, let

w01 (resp. w0

2) be either the empty sequence if κ´1 (resp. κ´2 ) is initialor the prefix of p1 (resp. p2) ending by the justification of κ´1 (resp.κ´2 ),

– if xw01y “ xw0

2y and κ´1 and κ´2 have distinct addresses then for all

actions σ1 and σ2 such that w1κ´1 w

11σ1 and w2κ

´2 w

12σ2 are respec-

tively prefixes of p1 and p2, and such that κ´1 P xw1κ´1 w

11σ1y and

κ´2 P xw2κ´2 w

12σ2y, we have that σ1 and σ2 have distinct addresses.

If p1 ¨ p2, then either one extends the other or they first differ on negativeactions.

Notation:Given a path p we denote by xxpyy the set of views of the (non empty) prefixesof p, that is xxpyy “ txqy | q is a non empty prefix of pu.

Remark 4. From [11] two paths p1 and p2 are coherent iff xxp1yyY xxp2yy isa design, i.e., a set of pairwise coherent chronicles.

Definition 16. Given a path p, a z-shorten of p is either p or a prefix of pended by z, i.e., p1z, when p “ p1κ

`p2 and κ` is a proper, positive action.

Example 8.Let p “ p`, ξ, t1uqp´, ξ.1, t0uqp`, ξ.1.0, t3uqp´, ξ.1.0.3, t2uqp`, ξ.1.0.3.2,Hq.Then the z-shortens of p are the following paths:

• p,

• p`, ξ, t1uqp´, ξ.1, t0uq p`, ξ.1.0, t3uqp´, ξ.1.0.3, t2uqz,

• p`, ξ, t1uqp´, ξ.1, t0uqz,

• z.

Definition 17. Given a z-free (or proper) path p of a certain design wedefine the opposite of p, p as the sequence of actions obtained from p bychanging polarity of each action:

• ε “ ε,


• pp`, ξ, Iq “ pp´, ξ, Iq,

• pp´, ξ, Iq “ pp`, ξ, Iq.

We define the dual of p, rp, as follows:

• Ăwz “ w, Ćwκ` “ wκ`z if κ` is positive and κ` ‰z,

• Ćwκ´ “ wκ´ for all negative action κ´.

Given a path p, rp is not always a path, as showed in the following example.However the dual of a chronicle is a path.

Example 9. Let p “ p`, ξ, t0uqp´, ξ.0, t1uqp`, σ, t1uq. Then rp “ p´, ξ, t0uqp`, ξ.0, t1uqp´, σ, t1uqz, which is not a path because of the action p´, σ, t1uq:it is a negative action but it is neither an initial action nor justified in p.

Definition 18. Given a set E of designs on the same base, a visitable pathin E is a sequence of actions in a design D P E which are visited during anormalization with a net of designs of EK.

Remark 5. We remark from [11] that visitable paths always end with apositive action.

Notation: Given a set E of designs on the same base, PE and VE respec-tively denote the set of paths and the set of visitable paths of E. Given aset C of paths, rC “ trp : p P Cu and xxCyy “ txxpyy : p P Cu, i.e., xxCyy is theset of views of the prefixes of paths of C.

A property of visitable paths is given in [11] as follows:

Proposition 1. (4.8 , [11])Let E be a set of designs and p a positive visitable path of a design of E,then

• rp is a path and

• for all prefix wκ´ of p, for all D P E, if w is a path of D, then wκ´ isa path of D.

We prove a further proposition that permits us to characterize the visitablepaths of a set in a restricted case. We will use it in Chapter 2Types andterms in Ludicschapter.2 to prove that all the paths of Nat, Ln and ListpNatqare visitable.

Proposition 2. Let E be a set of designs on the same base such that forall designs E,E1 P E, for all chronicles cκ` P E and c1κ` P E1, we have thatc “ c1.Let p be a positive-ended path of some design of E such that:




Then p is a visitable path of E.

To prove this Proposition we use the following Lemma.

Lemma 4. Let E be a set of designs on the same base such that for alldesigns E,E1 P E, for all chronicles cκ` P E and c1κ` P E1, we have thatc “ c1.Let p be a positive-ended path of some design of E such that:



Let R be the following set of paths:for all prefix wκ`0 of p, for all positive action κ` ‰ κ`0 such that xwκ`y is achronicle of some design of E, we set that wκ`z is a path of R. We add rpto R. Then R is a set of pairwise coherent paths.

Proof. Let wκ`0 be a prefix of p and xwκ`y a chronicle of D1 P E.We prove that xx

rpyyY xx wκ`z yy is a design.

• Either w is empty, then κ` and κ`0 are initial actions and xxκ`zyy “

tκ`, κ`z u. Therefore κ` can only appear as initial action in xxrpyy.

Since κ`0 is a prefix of rp, then all the chronicles of xxrpyy and κ`z first

differ on a negative action. Therefore xxrpyyY xxκ`z yy is a design.

• Or w is non empty.xx wκ`z yy “ xxwyy Y txwκ` yu Y tx wκ`z yu. Since w is a prefix of rp,then xxwyy Ď xx

rpyy. Moreover wκ` and wκ`0 differ only on their lastaction that is negative, then xwκ` y is coherent with all the chroniclesof xx

rpyy. Therefore xxrpyyY xxwκ` yy is a set of pairwise coherent chron-

icles.We prove by contradiction that xx

rpyyY xx wκ`z yy is a design.Let us suppose that xx

rpyyY xx wκ`z yy is not a design. Then there existtwo chronicles c1 P xx

rpyy and c2 P xxwκ`z yy such that c1 and c2 are notcoherent.Since the chronicles of xx

rpyy Y xx wκ` yy are pairwise coherent andxxwκ`z yy “ xxwκ` yyY txwκ`z yu, then c2 “ xwκ`z y.c1 is coherent with xwκ` y and c1 is not coherent with xwκ`z y. Ifκ` R c1, then c1 would be coherent with xwκ`z y. Therefore κ` P c1,i.e., c1 “ cκ`κ´.


Either κ´ “z or it is a proper positive action. In the first casec1 “ cκ`z, i.e., c1 “ x

rpy. Since c1 is coherent with xwκ` y and boththe chronicles c1 and xwκ`z y end with z, then c1 is coherent withxwκ`z y. But we supposed that these paths are not coherent.Therefore κ´ ‰z. This means that there exists a path w1κ

`κ´ thatis a prefix of p.Either w1κ

`κ´ is a prefix of w or wκ`0 is a prefix of w1. In both caseslet κ´1 and κ´2 be respectively the last action of w1 and w. Since xw1κ

`y

and xwκ`y are two chronicles of some designs of E, then by hypothesisxw1y “ xwy and in particular κ´1 “ κ´2 . Therefore this action appearstwice in p. But by definition an action can not appear twice in a path(contradiction).

To prove that all the paths of R are pairwise coherent we just have to provethat for all w1κ

`1 z ‰ w2κ

`2 z in R, w1κ

`1 z is coherent with w2κ

`2 z.

Since w1 and w2 are prefixes of p, then either they are equal or one is a prefixof the other.

• Let w1 “ w2, then w1κ`1 z and w2κ

`2 z are coherent between them.

• Let w1 be a strict prefix of w2, then there exists κ1` such thatw2 “ w1κ

1`w11.We prove by contradiction that w1κ

`1 z and w2κ

`2 z are coherent.

Let w1κ`1 z be not coherent with w2κ

`2 z. Since w2 “ w1κ

1`w11, thenκ`1 “ κ1`. By definition of R, w2 is a prefix of p, then w1κ

`1 is a prefix

of p. But by definition of R, we have that w1κ`1 is not a prefix of p

(contradiction).Therefore w1κ

`1 z and w2κ

`2 z are coherent between them.

Thus the paths of R are pairwise coherent and R is a net of designs.

Now we prove the Proposition enunciated above.

Proposition 3. Let E be a set of designs on the same base such that forall designs E,E1 P E, for all chronicles cκ` P E and c1κ` P E1, we have thatc “ c1.Let p be a positive-ended path of some design of E such that:



Then p is a visitable path of E.


Proof. Let E be a set of designs and p a positive path of some design D0 ofE as in the hypothesis. We build the set of paths R as in Lemma 4lemme.4.From Lemma 4lemme.4, R is a set of pairwise coherent paths and R “ xxRyy

is a net of designs. Moreover R is such that for each D P E, R K D, i.e.,R P EK. Finally p is visited during the normalization of tD0,Ru, thus p isvisitable in E.

We need in various cases to characterize the designs of the incarnation |EK|using the maximal cliques rC Ď VE , but we need to add two further propertieson the set C: it must be finite-stable and saturated.

Definition 19. ([11])Let E be a set of designs based on β and C a set of paths of designs of E. Wesay that C is finite-stable when for all strictly increasing (w.r.t. inclusion)sequence ppnq of elements of C, if

Ť

xxpnyy is included in a design of E, thenthe sequence ppnq is finite. C is saturated when for all prefix q of an elementof C such that qκ` P VE (κ` ‰z) we have that qκ` is a prefix of an elementof C.

Proposition 4. (5.16, [11]) Let E be a set of designs of the same base. LetR P |EK|, then there exists C Ď VE such that C is finite-stable and saturated,rC is a maximal clique of ĂVE and xx rCyy “ R.

In [11] there is a result that permits to recover the designs of |EKK| usingvisitable paths. We do not use this result, but we use Proposition 4prop.4to prove the principality of a set of designs in Chapter 2Types and terms inLudicschapter.2.

1.2.1.1 Visitable Paths and MALL Connectives

We recall the relation betweenMALLp logical connectives ‘,b, Ó, Ò and op-erations on visitable paths proposed in [12]. In Section 1.3.1Principalitysubsection.1.3.1we introduce the notion of principality and prove that it is stable w.r.t. con-nectives ‘,b, Ó, Ò and &. The structure of some basic types as lists andrecords is constructed on these connectives, therefore this property is usefulto prove their principality.

Definition 20. Given two positive designs A and B we say that they arealien when the ramification of the first action of A and the ramification ofthe first action of B are disjoint.Given two positive behaviours A and B they are alien when their designsare pairwise alien.

Definition 21. ([13])

• Let G be a negative behaviour of base ξ.i $,


Óξ.i G “ pp`, ξ, tiuqGqKK.

• Let C be a positive behaviour of base $ ξ.i,

Òξ.i C “ pp´, ξ, tiuqCqKK.

• Let pAkqkPK be a family of pairwise disjoint, positive behaviours,

À

kPK Ak “ pŤ

kPK AkqKK

• Let A and B be two positive, alien designs on the same base $ ξ:

– If A or B is Dai, then AbB “ Dai.– Otherwise A “ p`, ξ, IqA1 and B “ p`, ξ, JqB1.

Then AbB “ p`, ξ, I Y JqpA1 YB1q.

• Let A and B be two positive, alien behaviours,

AbB “ tAbB |A P A,B P BuKK.

Theorem 1. (Internal Completeness [13])

• Let K ‰ H and pAkqkPK be a family of pairwise disjoint, positivebehaviours, then

À

kPK Ak “Ť

kPK Ak

• Let A and B be two alien positive behaviours, then

AbB “ tAbB |A P A,B P Bu.

• Let G be a negative behaviour of base ξ.i $, then

Óξ.i G “ tzu Y p`, ξ, tiuqG.

• Let C be a positive behaviour of base $ ξ, then

Òξ.i C “ p´, ξ, tiuqC.

Definition 22. (2.7 [12])

• Let p and q be two positive-ended negative paths on disjoint bases β, γsuch that at least one of them does not end with a daimon. The shuffleof p and q, noted p�q, is the set of all sequences of the form p1q1...pnqnbased on β Y γ such that:


– each sequence pi and qi is either empty or a positive-ended nega-tive path,

– p1...pn “ p and q1...qn “ q,– if pn ends with a daimon then qn is empty.

• The definition is extended to paths pκ1z and qκ2z where pκ1z andqκ2z are two positive-ended, negative paths on disjoint bases:

pκ1z � qκ2z “ ppκ1z � qq Y pp� qκ2zq.

• The definition is extended to paths rp and rq where r is a positive-endedpath, p and q are positive-ended, negative paths on disjoint bases:

rp� rq “ rpp� qq.

Definition 23. (2.8 [12])Let P and Q be two sets of positive-ended paths, the shuffle of P and Q,noted P � Q, is the set

Ť

pp� qq, the union being taken on p P P , q P Qsuch that p� q is defined.

Definition 24. (3.2 [12])Let A and B be positive, alien behaviours of base $ ξ.The extension of A w.r.t. B, written ArBs, is the set of designs D suchthat either D “ tzu or there exist the designs A P A of first action p`, ξ, Iqand B P B of first action p`, ξ, Jq and D is obtained from A by replacing itsfirst action by p`, ξ, I Y Jq.

We recall that given a behaviour A, we denote by VA the set of visitablepaths of A (Definition 18defi.18).

Proposition 5. (3.6 [12])

• Let A be a negative behaviour of base ξ.i $,

VÓξ.iA “ tzu Y p`, ξ, tiuqVA.

• Let B be a positive behaviour of base $ ξ.i,

VÓξ.iB “ p´, ξ, tiuqVB Y tεu.

• Let K ‰ H and pAkqkPK be a family of pairwise disjoint, positivebehaviours,

VÀ

kPK Ak“

Ť

kPK VAk.

• Let A and B be alien positive behaviours,

VAbB “ tq |rq is a path and q P VArBs � VBrAsu.


1.2.1.2 Ludics as a Game Semantics

The connections between Linear Logic and Game Semantics has been inves-tigated by several works (see e.g. [22]). In particular the link between Ludicsand Game Semantics has been studied. First Faggian and Hyland [9] andthen Basaldella and Faggian [2] propose a presentation of Ludics in termsof HON Game Semantics (see e.g. [15], [23]), where actions correspond tomoves, chronicles to views and designs to innocent strategies. In Game Se-mantics an arena represents the interpretation of a type. A strategy is typed,in the sense that it is a strategy on a certain arena. The notion of interactionbetween strategies is defined only when they belong to compatible arenas,as for instance a : AÑ B and b : A. Their interaction always gives as resulta strategy of type B.In Ludics, designs with compatible bases can always interact between them.But differently from Game Semantics, their interaction can also diverge.There is another remarkable difference between the approach of Game Se-mantics and Ludics as observed in [26]: in Game Semantics we first have thenotion of type (arena) and then interaction (composition), while in Ludicsthe notion of interaction (normalization), comes at first. Then, from thisnotion, we can define interactive types (behaviours): a set of designs whichbehave well with respect to interaction.

1.2.2 Non-Linear Ludics

Ludics has got a big limitation: it is linear. In this section we recall severalworks that investigate about non-linear extensions of Ludics.

• Terui defines Computational Ludics, i.e., a non-linear reformulation ofLudics from a computational point of view [26].

• First Maurel [21] and then Basaldella and Faggian [2] propose a ex-tension of Ludics with exponentials. They add neutral actions whichallow them to use several times the same subdesign.

• Basaldella and Terui [3] study the traditional duality between proofsand models in the settings of Computational Ludics enriched with ex-ponentials (following the approach of [26]).

The main differences between Computational Ludics and original Ludics arethe following:

• Designs are no more built with absolute addresses (sequences of naturalnumbers), but they are similar to (infinitary) λ terms. In particular,their syntax include variables and variable bindings.


• Girard’s designs capture only cut-free and identity-free proofs and thislimits the computational power considerably. To overcome this lackof computational expressivity, designs are extended with explicit cutsand identities.

Terui represents data as natural numbers and lists with a class of designscalled data-designs. This class of designs enjoy a generalization of the sep-aration property [13] called strong separation [26]. In this work, Terui wasnot interested in record types and dependency, while our goal is to representdependent record types in Ludics.

In section 1.3.2Separabilitysubsection.1.3.2 we will introduce the notion ofseparable behaviour that is related to strong separation.

In next chapter we propose a representation of basic types, constructions anddiscuss dependent record types in original Ludics. We do not consider non-linear extensions of Ludics. We choose a linear world because it is enoughto model dialogues [10]. Several works (see e.g. [7], [6]) explored the pos-sibilities for formulating linguistic semantics in terms of records and recordtypes. Therefore, it looks interesting to investigate about a representationof dependent record types in the original formulation of Ludics.

1.3 New Notions: Principality and Separability

In this section we introduce the notions of principal behaviour and separablebehaviour.We say that a set of designs E generates a behaviour A, when EKK “ A. Abehaviour can be generated by several sets of designs. When a behaviour isgenerated by a z-free set of designs and moreover this set is equal to its z-free incarnation, then we say that the behaviour is principal. A behaviour isseparable when there is a set of visitable paths that characterize the designsof its z-free incarnation with some further properties.These notions will be fundamental in Chapter 2Types and terms in Ludicschapter.2for our representation of types in Ludics. Since we represent types by be-haviours, then we need to characterize and manipulate the designs that be-long to its z-free incarnation.

1.3.1 Principality

Now we introduce the notion of principal set of designs. Roughly speaking,a set E is principal when it contains enough z-free designs to recover thebehaviour EKK, i.e., z-free generators of EKK. This means that E is the setof z-free, material designs in EKK.

1.3. NEW NOTIONS: PRINCIPALITY AND SEPARABILITY 35

Remark 6. We remark that the notion of generator that we consider is notthe same notion of generator as in [26]. For us the generators of a behaviourA are the material designs in it, i.e., |A|.

We want to represent types by behaviours and in Ludics a behaviour iscompletely determined by its material designs. Moreover, z-free designscharacterize the representation of the proofs of the multiplicative additivefragment of polarized Linear Logic (MALLp) in [13]. We can then say thatthe generators of a behaviour that interest us are the z-free material designsin it. If a set E of designs is principal, then it is exactly the set of generatorsof EKK that interest us, i.e., z-free.

Furthermore, since in Type Theory a type is completely determined by itscanonical terms (and the rules of computation), then the notion of principalset looks like a good candidate to represent the notion of canonical terms.The idea is to represent the canonical terms of type A by a set E of designsand then to prove that this set is principal, i.e., it contains exactly the z-freegenerators of the behaviour EKK. We represent then the type with EKK andits canonical terms by E. In Chapter 2Types and terms in Ludicschapter.2we will use the notion of principality to represent the canonical terms of typeN, the lists of length n of natural numbers and the lists of natural numbers.

1.3.1.1 Definition

Definition 25. A non-empty set E of designs is said to be principal whenits elements are z-free and its z-shortening is the incarnation of its biorthog-onal, i.e., |EKK| “ Ez.We say that a behaviour A is principal if there exists a principal set EAsuch that pEAqKK “ A.

Notation: given a behaviour A, we denote by |A|z´free the set of its z-free, material designs. |A|z´free corresponds in C-Ludics to the set of pureI-designs of A [26].

Remark 7. A non-empty set E of designs is principal iff E “ |EKK|z´free.

We use the notion of visitable paths [11] (Section 1.2.1From Chronicles toPathssubsection.1.2.1) to prove a property of principal behaviours.

Proposition 6. Let A be a principal behaviour, then for all wz P VA, thereexists a positive proper action κ` such that wκ` P VA.

Proof. Let A be a principal behaviour and wz P VA. Since A is principalthere exists a z-free set EA such that VA “ V|A| “ V

pEAqz, then wz P

VpEAqz

. This means that there exist D P pEAqz and R P |AK| such that

w is visited during the normalization between D and R, wz is a path of D,


w P R, w P V|AK|. Since D P pEAqz, there exists a design D1 P EA such that

D P tD1uz. Since w is z-free then w is a path of D1. The last action of w isnegative and EA is z-free then there exists a positive, proper action κ` suchthat wκ` P D1. R P |AK| then @E P A R K E. D1 P EA, by Remark 1rem.1EA Ď EKKA and A “ EKKA , therefore R K D1. w is a z-free path visitedduring the normalization between D and R, w P D, w P D1, D P tD1uz,then w is visited during the normalization between D1 and R. w is z-free itends with a negative action, wκ` P D1 and D1 K R, then the normalizationbetween D1 and R visits wκ`, i.e., wκ` P R. This means that wκ` P VEA .VEA Ď V

pEAqzand V

pEAqz“ VA. Therefore wκ` P VA.

1.3.1.2 Principality and MALL Connectives

In the previous section, we recall the relation between MALLp connectivesand operations on visitable paths established in [12].Now we prove that principality is stable w.r.t. MALLp connectives‘,b, Ó, Ò,&.

To prove that principality is stable w.r.t. ‘ and b we use the followingLemma. It says that when A and B are principal, then AbB is generatedby tA b B |A P |A|z´free,B P |B|z´freeu and when for all k P K, Ak isprincipal, then

À

kPK Ak is generated byŤ

kPK |Ak|z´ free.

Lemma 5. 1. Let A and B be two alien, positive, principal behaviours,then tAbB |A P |A|z´free,B P |B|z´freeuKK “ AbB.

2. Let K ‰ H and pAkqkPK be a family of principal, positive, pairwisedisjoint behaviours, then p

Ť

kPK |Ak|z´ freeqKK “

À

kPK Ak.

Proof. 1. Let E “ tA b B |A P |A|z´free,B P |B|z´freeu. Since E Ď

AbB, then EKK Ď pAbBqKK, that is AbB.Now we prove that AbB Ď EKK.Let D P A bB, then by internal completeness [13] there exist E P Aand F P B such that D “ Eb F.There exist E1 P |A| and F1 P |B| such that E1 Ď E, F1 Ď F andE1 b F1 Ď D. Since A and B are principal, then |A| “ p|A|z´freeqz

and |B| “ p|B|z´freeqz. Therefore there exist E0 P |A|z´free andF0 P |B|z´free such that E1 P tE0u

z and F1 P tF0uz. By definition

E0 b F0 P E, then for all R P EK, R K E0 b F0. Since E1 P tE0uz and

F1 P tF0uz, then R K E1 b F1. Since E1 b F1 Ď D, then R K D, that is

D P EKK.Thus EKK “ AbB.

2. Let F “Ť

kPK |Ak|z´free. Since F ĎÀ

kPK Ak, then FKK Ď pÀ

kPK AkqKK,

that isÀ

kPK Ak.


We prove now the second inclusion,À

kPK Ak Ď FKK.Let F P

À

kPK Ak, sinceÀ

kPK Ak “Ť

kPK Ak, then there existsk0 P K such that F P Ak0 . Then there exists a design F1 Ď F such thatF1 P |Ak0 |. Since Ak0 is principal, then there exists E P |Ak0 |z´free

such that F1 P tEuz. Since E P |Ak0 |z´free, then E P F and for allR P FK we have that R K E. Since F1 P tEuz, then R K F1. SinceF1 Ď F, then R K F, that is F P FKK.Thus FKK “

À

kPK Ak.

Remark 8. Let A be a behaviour and A0 a z-free set such that |A| “ pA0qz,

then A0 generates A that is pA0qKK “ A.

From Remark 2rem.2 pA0qK “ ppA0q

zqK, then pA0qKK “ ppA0q

zqKK. More-over ppA0q

zqKK “ |A|KK and |A|KK “ A, then pA0qKK “ A.

Remark 9.

• Let G be a negative behaviour based on ξ.i $, then| Óξ.i G| “Óξ.i |G| YDai.

• Let A be a positive behaviour based on $ ξ.i, then | Òξ.i A| “Òξ.i |A|.

We easily prove these remarks: since Óξ.i G is a positive behaviour and Daiis material in every positive behaviour, then Dai P | Óξ.i G|. By definitionthe other designs of | Óξ.i G| are of the form p`, ξ, tiuqG where G P |G|, i.e.,they belong to Óξ.i |G|. For the same reason | Òξ.i A| “Òξ.i |A|.

Proposition 7. 1. Let G be a negative, principal behaviour based onξ.i $, then the behaviour Óξ.i G is principal.

2. Let A be a positive, principal behaviour based on $ ξ.i, then the be-haviour Òξ.i A is principal.

3. Let K ‰ H and pAkqkPK be a family of pairwise disjoint, positive,principal behaviours, then the behaviour

À

kPK Ak is principal.

4. Let A and B be alien, positive, principal behaviours, then the behaviourAbB is principal.

5. Let F and G be two negative, disjoint behaviours with the same base,then the behaviour F&G “ tFXG |F P F,G P Gu is principal.

Proof. 1. From Remark 9rem.9 | Óξ.i G| “Óξ.i |G| Y Dai. Since G isprincipal, then |G| “ p|G|z´freeqz and | Óξ.i G| “Óξ.i p|G|z´freeqz YDai, that is pÓξ.i |G|z´freeqz. Óξ.i |G|z´free is z-free and from Remark8rem.8 it generates Óξ.i G. Therefore Óξ.i G is a principal behaviour.


2. From Remark 9rem.9 | Òξ.i A| “Òξ.i |A|. Since A is principal, then|A| “ p|A|z´freeqz. Therefore | Òξ.i A| “Òξ.i p|A|z´freeqz that ispÒξ.i |A|z´freeqz. The set Òξ.i |A|z´free is z-free and from Remark8rem.8 it generates Òξ.i A. Therefore Òξ.i A is a principal behaviour.

3. By internal completeness [13], we have |À

kPK Ak| “Ť

kPK |Ak|. Sincefor all k P K, Ak is principal, then there exist a family of principalsets pEkqkPK such that |Ak| “ pEkq

z, i.e., Ek “ |Ak|z´free. Therefore|Ť

kPK Ak| “Ť

kPKpEkqz “ p

Ť

kPK Ekqz. From Lemma 5lemme.5, we

have thatŤ

kPK Ek generatesÀ

kPK Ak.Thus

À

kPK Ak is a principal behaviour.

4. By internal completeness [13], we have |A b B| “ |A| b |B|. SinceA and B are principal behaviours, then |A|z´free “ p|A|z´freeqz

and |B|z´free “ p|B|z´freeqz. Therefore |A b B| “ p|A|z´freeqz bp|B|z´freeqz, that is p|A|z´freeb|B|z´freeqz. |A|z´freeb|B|z´freeis z-free and from Lemma 5lemme.5 we have that |A|z´freeb|B|z´freegenerates AbB. Thus AbB is a principal behaviour.

5. By internal completeness |F&G| “ tF Y G |F P |F|,G P |G|u. SinceF and G are principal behaviours, then |F| “ p|F|z´freeqz and |G| “p|G|z´freeqz. Therefore |F&G| “ tFYG |F P |F|z´free,G P |G|z´freeuz.tF Y G |F P |F|z´free,G P |G|z´freeu is z-free and from Remark8rem.8 it generates F&G. Therefore F&G is a principal behaviour.

1.3.2 Separability

In this section we recall the Separation result given by Girard [13]. We definethen class of separable behaviours, for which holds a generalization of thisproperty. We first recall the precedence relation between designs definedin [13].

Definition 26. Let E,F be two designs. We define the relation ď overdesigns as follows.

E ď F iff tEuK Ď tFuK.

E ď F can be read as E is more defined than F (or equally F is less definedthan E).

The Separation Theorem [13] says that:two designs D and D1 are distinct if and only if there exists a net R suchthat D K R and D1 M R.This induces that ď is a partial order.


We remark that if D ę D1, then D ‰ D1. Therefore there exists a netR such that R K D and R M D1.

Given a behaviour A we wonder if for each design of A, there exists a netwhich separates it from all the other designs of |A|z´free. This would meanthat for all A P |A|z´free there exists a net RA such that RA is orthogonalto A and it is not orthogonal to all the other designs of |A|z´free. Thisproperty is not true for all behaviours (Example 10example.10). We showthat this property holds (Lemma 6lemme.6) for behaviours that we call sep-arable (Definition 28defi.28). This property reminds us the notion of strongseparation [26].Let us reformulate strong separation [26] in terms of Ludics as follows.

Definition 27. A design D admits strong separation if there exists a netR such that for all z-free design D1 such that D ę D1, we have that R K Dand R M D1.

Given a behaviour A, we have that the distinct designs of |A|z´free are notin relation between them, as explained in the following remark.

Remark 10. We remark that E ď F means that F is obtained from E bymeans of the enlargement of the negative rules (adding new negative ruleson the same address) and a replacement of some positive rules by z. Thismeans that if E and F are two distinct designs of |A|z´free, then E ę F andF ę E.

Strong separation is a property of designs. We define then a class of separablebehaviours which extends to a set of designs the notion of strong separation,but only w.r.t. the designs of |A|z´free.In Chapter 2Types and terms in Ludicschapter.2 we will see that separabilitygives us a tool to easily define functions. In fact separability gives us the toolto define a family of designs tDA |A P |A|z´freeu such that for all A ‰ A1 P|A|z´free we have that DA K A and DA M A1. Moreover

Ť

AP|A|z´freeDA

is a design that permits us to define the graph of a function on A.

1.3.2.1 Separable Behaviour

Definition 28. We say that a behaviour A is separable when there existsa set of visitable paths tpA |A P |A|z´freeu such that for all A P |A|z´free

• pA is a path of A,

• for all A1 ‰ A, A1 P |A|z´free, we have that pA and xpAy are not pathsof A1, pA1 and xpA1y are not paths of A, the chronicles xpAy and xpA1y

are distinct and coherent.


Separability is a strong property. Intuitively the properties of visitable pathstpA |A P |A|z´freeu tells us that for all A P |A|z´free there is a path pA

that is specific to A. Moreover the hypothesis on the path xpAy gives us thepossibility to construct a net DA that separates A from all the other designsof |A|z´free.This property is a sort of strong separation: each design of |A|z´free enjoysstrong separation, but only w.r.t. the designs of |A|z´free. The definitionof separability tells us how to construct such a family of nets tDA |A P

|A|z´freeu.

Lemma 6. Let A be a separable behaviour and A P |A|z´free. Then forall A1 P |A|z´free, A ‰ A1 there exists a net DA such that DA K A andDA M A1.

Proof. Let A P |A|z´free. Since A is separable, then the net DA “ xxpAyyz

is such that DA K A. For all A1 P |A|z´free, A1 ‰ A, since xpAy is not a pathof A1, then we have that DA M A1.

The properties on the paths xpAy gives us a very simple way to define a designthat represents the graph of a function, as we will see in Section 2.2.2Depen-dent Functionssubsection.2.2.2.

Being a separable behaviour means that we have a "test" such that forall designs of |A|z´free we can distinguish it from all the other designs of|A|z´free. Since we are linear, then we would like to have just one test thatworks for all designs of |A|z´free, but it looks difficult in general. We showin the following example that there exist behaviours that are not separa-ble. In particular the problem with the designs of the example is that theirchronicles differ on a negative action on the same address. We will find thesame problem with the representation of records in Ludics as we will see inChapter 2Types and terms in Ludicschapter.2.

Example 10. We define a behaviour A that is not separable.

Let A “ tA1,A2,A3,A4uKK where

A1 “

σ.1.1.0 $

$ σ.1.1

σ.1.2.0 $

$ σ.1.2

σ.1 $$ σ

p`, σ, t1uq

p´, σ.1, t1uq

p`, σ.1.1, t0uq

p´, σ.1, t2uq

p`, σ.1.2, t0uq


A2 “

σ.1.1.0 $

$ σ.1.1

σ.1.2.3 $

$ σ.1.2

σ.1 $$ σ

p`, σ, t1uq

p´, σ.1, t1uq

p`, σ.1.1, t0uq

p´, σ.1, t2uq

p`, σ.1.2, t3uq

A3 “

σ.1.1.1 $

$ σ.1.1

σ.1.2.0 $

$ σ.1.2

σ.1 $$ σ

p`, σ, t1uq

p´, σ.1, t1uq

p`, σ.1.1, t1uq

p´, σ.1, t2uq

p`, σ.1.2, t0uq

A4 “

σ.1.1.1 $

$ σ.1.1

σ.1.2.3 $

$ σ.1.2

σ.1 $$ σ

p`, σ, t1uq

p´, σ.1, t1uq

p`, σ.1.1, t1uq

p´, σ.1, t2uq

p`, σ.1.2, t3uq

The elements of AK are the following designs and their extensions on somenegative actions

z$ σ.1.1.0

z$ σ.1.1.1

σ.1.1 $

$ σ.1σ $

p´, σ, t1uq

p`, σ.1, t1uq

p´, σ.1.1, t0uq

z

p´, σ.1.1, t1uq

z

z$ σ.1.2.0

z$ σ.1.2.3

σ.1.2 $

$ σ.1σ $

p´, σ, t1uq

p`, σ.1, t2uq

p´, σ.1.2, t0uq

z

p´, σ.1.2, t3uq

z

z$ σ.1σ $

p´, σ, t1uq

z


|A|z´free “ tA1,A2,A3,A4u.

The actions p´, σ.1, t1uq and p´, σ.1, t2uq have the same address, then thepaths of A1,A2,A3 and A4 can not travel through several chronicles, i.e., theonly paths are their chronicles. There exist no chronicle c of A1 such thatc R A2, c R A3 and c R A4. Therefore we can not define a set of visitable pathsas desired.Thus, A is not separable.

Remark 11. We remark that the behaviourA defined in the Example 10example.10is principal but not separable. This means that

PRINCIPAL œ SEPARABLE.

In the following example we show that separability is not closed w.r.t. theconnective &, i.e,

F,G SEPARABLE œ F&G SEPARABLE.

Example 11. We define two negative, separable behaviours F and G suchthat F&G is not separable.

Let F “ tF1,F2uKK and G “ tG1,G2u

KK where

F1 “

σ.1.1.0 $

$ σ.1.1

σ.1 $p´, σ.1, t1uq

p`, σ.1.1, t0uq

F2 “

σ.1.1.1 $

$ σ.1.1

σ.1 $p´, σ.1, t1uq

p`, σ.1.1, t1uq

G1 “

σ.1.2.0 $

$ σ.1.2

σ.1 $p´, σ.1, t2uq

p`, σ.1.2, t0uq

G2 “

σ.1.2.3 $

$ σ.1.2

σ.1 $p´, σ.1, t2uq

p`, σ.1.2, t3uq

|F|z´free “ tF1,F2u. Let pF1 “ p´, σ.1, t1uqp`, σ.1.1, t0uq andpF2 “ p´, σ.1, t1uqp`, σ.1.1, t1uq, then pF1 R F2 and pF2 R F1.Moreover p1 and p2 are chronicles, then xp1y “ p1 and xp2y “ p2. xp1y “

p`, σ.1, t1uqp´, σ.1.1, t0uq and xp2y “ p`, σ.1, t1uqp´, σ.1.1, t1uq are distinctand coherent between them. Therefore F is a separable behaviour. Since theyhave the same structure, we remark that also G is separable.

|F&G|z´free “ tD1,D2,D3,D4u where

D1 “

σ.1.1.0 $

$ σ.1.1

σ.1.2.0 $

$ σ.1.2

σ.1 $ D1 “p´, σ.1, t1uq

p`, σ.1.1, t0uq

p´, σ.1, t2uq

p`, σ.1.2, t0uq

1.4. SUMMARY AND COMMENTS 43

D2 “

σ.1.1.0 $

$ σ.1.1

σ.1.2.3 $

$ σ.1.2

σ.1 $ D2 “p´, σ.1, t1uq

p`, σ.1.1, t0uq

p´, σ.1, t2uq

p`, σ.1.2, t3uq

D3 “

σ.1.1.1 $

$ σ.1.1

σ.1.2.0 $

$ σ.1.2

σ.1 $ D3 “p´, σ.1, t1uq

p`, σ.1.1, t1uq

p´, σ.1, t2uq

p`, σ.1.2, t0uq

D4 “

σ.1.1.1 $

$ σ.1.1

σ.1.2.3 $

$ σ.1.2

σ.1 $ D4 “p´, σ.1, t1uq

p`, σ.1.1, t1uq

p´, σ.1, t2uq

p`, σ.1.2, t3uq

That is D1 “ F1 YG1, D2 “ F1 YG2,D3 “ F2 YG1 and D4 “ F2 YG2.We remark that the paths of D1 are just its chronicles, and there exist nochronicle c of D1 such that c R D2, c R D3 and c R D4.Therefore F&G is not separable.

1.4 Summary and Comments

After having introduced the basic notions of Ludics [13] we recalled a re-cent work [11] where the notion of chronicle is generalized to the notion ofpath and the class of visitable paths gives us a tool to capture the incar-nation of a behaviour without computing it. Thanks to the notions andproperties recalled from [11] we proved a property of principal behavioursand the principality of some sets of designs in Chapter 2Types and termsin Ludicschapter.2. Several works [21], [2], [26], r3s propose a non-linear ex-tension of Ludics. As already pointed out, we choose to work in Ludics andnot in a non-linear extension of it. Our decision is motivated from the factthat Ludics is enough to model dialogues [10] and dependent record typeshave interesting applications in Linguistics [7]. Moreover the characteriza-tion of the incarnation using visitable paths proposed [11] is very useful torepresent canonical terms and for the moment we do not have such a char-acterization in non-linear Ludics. Our goal is to propose a first step in themore general setting of a representation of dependent record types in Ludics.

We remark that in Ludics there are behaviours that do not represent anyformula of MALL. This may give an intuition of dependency, as for instancethe following example taken from [12].Let F “ tE,FuKK where


E “

ξ.1.0.0 $

$ ξ.1.0

ξ.1 $

ξ.2.0.0 $

$ ξ.2.0

ξ.2 $

$ ξ F “

z$ ξ.1.0

ξ.1 $ ξ.2 $

$ ξ .

The visitable paths of F are the positive-ended prefixes and the z-shorteningsof the path p “ p`, ξ, t1, 2uqp´, ξ.1, t0uqp`, ξ1.0, t0uqp´, ξ.2, t0uqp`, ξ.2.0, t0uq.This means that we can not visit the chronicle of ξ.2 before the chronicle ofξ.1, and this is an intuitive idea of dependency.

To be able to represent basic types and constructions in Chapter 2Types andterms in Ludicschapter.2 we introduced the notions of principal and separa-ble behaviour. These notions are inspired from the notions of pure designand strong separation from [26]. We recalled the relation between MALLpconnectives and operations on visitable paths established in [12]. Then weproved that principality is closed w.r.t. MALLp connectives ‘,b, Ò, Ó and&. Since some basic types are constructed using these connectives, then thisresult will help us to prove their principality.

Chapter 2

Types and terms in Ludics

In this chapter we propose a representation in Ludics for types and terms.We consider the basic types natural numbers, booleans, lists, records, theconstructions functions and pairs and discuss dependent record types. Forall basic types our representation holds principality and for all but recordsit holds separability. Principality is not stable under Ñ, but when we re-strain ourself to separable behaviours we have a very natural representationof functions.

The notion of type is common to several domains as for instance compu-tation theory (data type) and grammatical classification. Using the termi-nology of Martin-Löf’s Type Theory, the canonical terms of a type are theprimitive elements of the type, that is the objects that characterize it. Thenon-canonical terms are the terms obtained by applying some operations oncanonical terms and that once computed give a canonical term. Terms areseen as programs and two terms are equal when their computation give thesame result, that is the same canonical term. To establish that A is a typewe need to know what an object of type A is and what it means for twoobjects of type A to be equal.

How to represent a type in Ludics? Since a type is a class of objects that be-have in the same way, then it looks natural to represent types by behavioursand canonical terms by designs. Computation corresponds in Ludics to cut-elimination (normalization). Since the normalization of a cut-net is a design,then we propose to represent non-canonical terms as cut-nets such that theirnormalization represents a canonical term. A type is completely determinedby its canonical terms and the computation rules as a behaviour is deter-mined by its material designs and normalization. Since we represent a typeA with a behaviour A, then the material designs of A look like a good candi-date to represent the canonical terms of type A. Computation is representedby normalization and when during a normalization we find a z the result

45

46 CHAPTER 2. TYPES AND TERMS IN LUDICS

of the normalization is z. This means that if we represent a function or itsargument by a design that is not z-free, then the result of the computationcould be represented by z. To avoid this possibility we choose to representcanonical terms by z-free designs.

Given a behaviour A that represents the type A, we represent the canonicalterms of type A by the material, z-free designs of A, written |A|z´free.Since a non-canonical term is just a term that after computation yields acanonical one, then we represent a non-canonical term by a cut-net R suchthat its normalization represents a canonical term, i.e., JRK P |A|z´free.Since two terms of type A are equal when they yield the same canonicalterm, then equal terms are represented by cut-nets such that their normal-ization gives as result the same design D P |A|z´free. Equal canonical termsare then represented by equal designs (the normalization of a design is thedesign itself).

A set of designs E is principal (Definition 25defi.25) when E “ |EKK|z´free.This notion represents faithfully canonical terms, i.e., we represent the canon-ical terms of type A by a principal set and A by the principal behaviourgenerated by it.

We represent the non-canonical terms of type A by cut-nets R, then they donot belong to A. But when we consider a non-canonical term we are onlyinterested in the canonical term that it generates. Therefore what interestsus of the cut-net R is its normalization, that is a design of |A|z´free. Thuswe can say that we represent the type A by the behaviour A.

In Section 2.1Basic Typessection.2.1 we propose a representation for basictypes N, Booleans, ListnpNq, ListpNq, ListnpAq and records that holds theproperty of principality. For all these basic types apart records our repre-sentation also holds separability. But the notion of principality is too strictfor being stable under dependent product, and even when built by functionsover principal behaviours. This means that given two principal behavioursAand B that represent the types A and B, then the behaviour that representsthe type of functions from A to B is not always principal. For bypassing thisfailure we use the notion of separable behaviour (Definition 28defi.28), thatintuitively means that we can distinguish with a set of visitable paths thez´free, material designs of the behaviour. This notion gives us a simple andnatural way to define a representation of functions. In Section 2.2Construc-tions and Dependencysection.2.2 we represent the constructions functions,pairs and discuss an example of dependent record types.

2.1. BASIC TYPES. 47

2.1 Basic Types.

In this section we propose a representation for basic types in Ludics: naturalnumbers, booleans, lists and records. Principality represents faithfully thenotion of canonical terms, while separability gives us a tool to define easilyfunctions over canonical terms (Section 2.2.2Dependent Functionssubsection.2.2.2).We prove that the for all basic types our representation holds principalityand for all all but records it holds separability.

2.1.1 Natural Numbers

In this section we define a set of designs, Nat, that represents natural num-bers. We prove that Nat is principal and NatKK is separable.

We represent a natural number n P N by a design nσ on a unary positivebase $ σ, in the following inductive way:

0σ “H

$ σ pn+1qσ “

nσ.0.1$ σ.0.1

σ.0 $$ σ .

In terms of chronicles:

0σ “ p`, σ,Hq pn +1qσ “ p`, σ, t0uq

p´, σ.0, t1uq

nσ.0.1

Abusively, we may write n instead of nσ. Furthermore we abbreviate the

design n as

H

$ σ.n

$ σ , where 0 :“ ε (the empty sequence) and n` 1 :“ n.0.1.

We denote with Nat the set of designs which represent natural numbers,i.e., Nat “ tn |n P Nu. This representation of natural numbers is very closeto Terui’s representation in Computational Ludics [26], they only differ onthe polarity.

To prove that Nat is principal, we prove first some preliminary results.Most of these properties will be generalized in next sections to the represen-tation of lists of natural numbers.


2.1.1.1 First Properties of Nat

We first focus on the duals of the chronicles of Nat, then we describe therelation between chronicles of distinct designs of Nat and finally we charac-terize |NatK|.

We remark in the Example 9example.9 that the dual of a path is not al-ways a path. Chronicles are particular paths such that their dual are paths,but not always chronicles.

Example 12. Let c be the chroniclep`, γ, t0uqp´, γ.0, t0, 1uqp`, γ.0.0, t0uqp´, γ.0.0.0, t1uqp`, γ.0.1, t2uq.Then rc “ p´, γ, t0uqp`, γ.0, t0, 1uqp´, γ.0.0, t0uqp`, γ.0.0.0, t1uqp´, γ.0.1, t2uqz.Since p´, γ.0.1, t2uq is not initial and it is not justified by the immediatelyprecedent action, then rc is not a chronicle.

The chronicles of the designs of Nat are such that their dual are still chron-icles.

Lemma 7. For all n P Nat if c is a chronicle of n, then rc is a chronicle.

Proof. Let n P Nat and c be a chronicle of n. For each action κ of c,the address of κ is determined from the immediately precedent action (inparticular all the negative actions in n give rise to only one possible addressfor the positive action which follows). By definition of Nat we have that c isz-free. Then rc “ cz, i.e., we obtain rc changing the polarity of all the actionsof c and adding z at the end. When we change the polarity of all the actionsof c, we obtain a sequence of proper actions where the address of each actionis determined by the action just before, i.e., a chronicle. Therefore rc is achronicle.

Given two designs i, j P Nat their chronicles are either the same (when theyrepresent the same natural number i “ j) or they differ on a positive actionon the same address.

Proposition 8. Let n,n1 P Nat, and c be a chronicle of n.

• If n “ n1, then c P n1.

• If n ą n1, then Dc1 such that c1p`, σ.n1,Hq P n1 and c1p`, σ.n1, t0uq is aprefix of c.

• If n ă n1, then Dc1 such that c “ c1p`, σ.n,Hq and c1p`, σ.n, t0uq P n1.

Proof. • If n “ n1, then n “ n1, so c P n1.

• If n ą n1.


– Either n1 “ 0. In this case n1 “ 0, whose only action is p`, σ,Hq.Since n ą n1, then the first action of n is p`, σ, t0uq. The thesisfollows taking c1 “ ε.

– Or n1 ‰ 0. Let c1 be the prefix of c which ends with the ac-tion p´, σ.n1 ´ 1.0, t1uq. By definition of Nat, it is such thatc1p`, σ.n1, t0uq is a prefix of c. Then c1p`, σ.n1,Hq P n1.

• If n ă n1.

– Either c is not maximal in n, then c P n1.

– Or c is maximal in n. Then its last action is p`, σ.n,Hq. Let c1 bec without its last action, then c “ c1p`, σ.n,Hq and c1p`, σ.n, t0uq Pn1.

The chronicles of distinct designs of Nat cannot start differ on a negativeaction, as showed in the following Lemma.

Lemma 8. For all n,n1 P Nat, let κ´1 , κ´2 be negative actions such that

cκ´1 P n and cκ´2 P n1, then we have that κ´1 “ κ´2 .

Proof. By definition all the negative actions in Nat are of the form p´, σ.i.0, t1uqfor some i P N. Then there can not be two distinct negative actions withthe same address. Since each action of a chronicle of Nat is justified by theimmediately precedent action, then κ´1 “ κ´2 .

Which designs are the elements of |NatK|?

Proposition 9. |NatK| “ tFσuz, where

Fσ “ tcnz | cn is the maximal chronicle of n P Natu.

We can define Fσ as Fσ.0 where Fσ.i is defined by induction

@i P N, Fσ.i :“

z$

Fσ.i`1

$ σ.i.0

σ.i $ ,Fσ.i “ p´, σ.i,Hq

z

p´, σ.i, t0uq

p`, σ.i.0, t1uq

Fσ.i`1

Proof. • We first prove that Fσ P NatK.For all n P Nat, Fσ contains the chronicle cnz where cn is the maximalchronicle of n. Therefore for all n P Nat, Fσ K n, i.e., Fσ P NatK.


• We prove by contradiction that Fσ P |NatK|. Let F Ĺ Fσ, such thatF P NatK, that is Fσ R |NatK|. Then there exists a positive-endedchronicle c such that c P Fσ and c R F.

– Either c “ p´, σ,Hqz, then JF,0K ‰ tzu. Therefore F R NatK(contradiction).

– Or there exists n P N such that c “ p´, σ, t0uq...p´, σ.n,Hqz.Since F Ĺ Fσ, then F does not contain any z-shorten of c. There-fore JF,nK ‰ tzu, i.e., F R NatK (contradiction).

Therefore Fσ P |NatK|.

• Furthermore an incarnation is closed by z-shortening (by Lemma 2lemme.2),thus tFσuz Ď |NatK|.

• Now we prove the second inclusion |NatK| Ď tFσuz.Let E P |NatK|, this means that for all n P Nat E K n and E is minimalw.r.t. inclusion. For all n P Nat let cn be the maximal chronicle of n.Since for all n P Nat, we have E K n, then E contains a z-shorteningof rcn. E does not contain other chronicles, otherwise it would not beminimal, then E P tFσu

z.Thus |NatK| Ď tFσuz.Therefore tFσuz “ |NatK|.

In the previous Proposition we characterized |NatK|, now we wonder whatare the elements of NatKK. We can partition NatKK into three subsets:

• |NatKK|z´free: we will prove that it is equal to Nat in the followingsection (Proposition 10prop.10).

• |NatKK|not z´free: it is the set of the z-shortenings of designs of Nat.They give a partial information about some design of Nat.

• NatKKz|NatKK|: in this case some of their path are not visitable, there-fore they are not interesting (Example 13example.13).

Example 13. The design

D “

H

$ σ.0.1

σ.0 $

H

$ σ.1.1

σ.1 $$ σ , D “ p`, σ, t0, 1uq

p´, σ.0, t1uq

p`, σ.0.1,Hq

p´, σ.1, t1uq

p`, σ.1.1,Hq


D is z-free and belongs to NatKK, but since it contains the chronicle p`, σ, t0, 1uqp´, σ.1, t1uqp`, σ.1.1,Hq it is not material in it. Then it does not belong toNat, i.e., it does not represent a canonical term of type N.

2.1.1.2 Nat is Principal

Proposition 10. Nat is principal, i.e., it is z-free and |NatKK| “ Natz.

Proof. • The fact that Nat is z-free follows from its definition.

• From Remark 1rem.1 we have Nat Ď NatKK.We prove by contradiction that Nat Ď |NatKK|.Let n P Nat, suppose that n R |NatKK|, i.e., there exists E Ĺ n suchthat E P NatKK. Since n contains only one maximal chronicle c, Econtains only one maximal chronicle c1 that is a prefix of c.

– If n “ 0, then n “ p`, σ,Hq and E “ n. But E Ĺ n (contradic-tion).

– Otherwise (n ‰ 0) c “ p`, σ, t0uqp´, σ.0, t1uq...p`, σ.n,Hq. Sincemaximal chronicles end with a positive action, then there existssome n1 ă n such that c1 “ p`, σ, t0uqp´, σ.0, t1uq...p`, σ.n1, t0uq.We consider the design Fσ defined in Proposition 9prop.9.Since Fσ does not contain the chronicle rc1 or any z-shorten of it,then Fσ M E. Since Fσ P NatK, then E R NatKK (contradiction).Then Nat Ď |NatKK|.

Therefore from Lemma 3lemme.3 we obtain Natz Ď |NatKK|.

• Now we prove the second inclusion |NatKK| Ď Natz. Let D P |NatKK|,then D is orthogonal to all the elements of NatK. From Proposi-tion 9prop.9 we have that Fz

σ “ |NatK|, then D K Fσ. This meansthat D contains a chronicle c1 that is a z-shorten of the chroniclep`, σ, t0uqp´, σ0, t1uq...p`, σ.n,Hq (for some n P N). Note that tc1u P NatKK. Moreover D ismaterial in NatKK , i.e., if there exists E Ď D such that E P NatKK, thenE “ D. Then D “ tc1u, i.e., D P Natz. Therefore |NatKK| Ď Natz.

Thus Natz “ |NatKK|.

2.1.1.3 NatKK is Separable

We prove now that NatKK is separable.The property of separability gives us a tool to manipulate the designs of|NatKK|z´free that will be particularly important in Section 2.2Construc-tions and Dependencysection.2.2 to define functions on natural numbers.To prove that NatKK is separable we use the following proposition which tellsus that all the positive-ended paths of Nat are visitable in NatKK.


Proposition 11. All the positive-ended paths of Nat are visitable in NatKK.

Proof. Let p be a positive-ended path of Nat. To prove that p is visitablewe use the Proposition 2prop.2.We first prove that rp is a path and for all prefix wκ´ of p, for all D P NatKK,if w P D, then wκ´ P D.By definition p is a chronicle, then rp is a path.Let wκ´ be a prefix of p and D P NatKK such that w P D. By definitionof Nat, there exists i P N such that the last action of w is p`, σ.i, t0uq andfor all j ą i we have that w P j. From Proposition 10prop.10 we have|NatKK| “ pNatqz. Since p is z-free, then there exists n P Nat such thatE P tnuz and E Ď D. Since w P D, then n ą i. Since wκ´ is a path of Nat,then κ´ “ p´, σ.i.0, t1uq and wκ´ P n. Therefore wκ´ P E and wκ´ P D.By definition Nat and p hold all the other hypothesis of Proposition 2prop.2,then p is visitable in NatKK.

Proposition 12. The behaviour NatKK is separable.

Proof. Let n P N. By definition of Nat, n has got only one maximal pathpn. From Proposition 11prop.11 follows that pn is visitable. pn differs onits last action with all the other maximal paths of Nat. Then for all n1 ‰ nin Nat, we have pn R n1. By definition of Nat pn is a chronicle and fromLemma 7lemme.7 Ăpn is a chronicle. Therefore xpny differs from all xpn1y on itslast action. Since pn and pn1 are chronicles, then xpny “ pn and xpn1y “ pn1 .Therefore xpny “ pn and xpn1y “ pn1 . Since pn is not a path of n1 and pn1 isnot a path of n, then xpny is not a path of n1 and xpn1y is not a path of n.By definition of Nat, Ăpn and Ăpn1 are chronicles that differ on a negative actionon the same address, then xpny and xpn1y are coherent.

2.1.2 Booleans

Now we consider the type Bool “ t0, 1u of booleans. We propose to rep-resent it by the behaviour BoolKK “ t0,1uKK and its canonical terms 0, 1respectively by the designs 0,1 P Nat. We remark that Bool is principal andBoolKK is separable.

Remark 12. Bool is principal and BoolKK is separable.

By construction BoolK is composed by the z-shortening and the negativeenlargements of the design


z$

z$

σ.1 $

$ σ.0σ $

p´, σ,Hq

z

p´, σ, t0uq

p`, σ.0, t1uq

p´, σ.1,Hq

z

Therefore |BoolKK|z´free “ Bool.Since |BoolKK|z´free “ t0,1u, that is a subset of Nat, then from Proposition12prop.12 follows that BoolKK is separable. In particular the paths that "sepa-rate" 0 and 1 are p0 “ p`, σ,Hq and p1 “ p`, σ, t0uqp´, σ.0, t1uqp`, σ.1,Hq.

2.1.3 Lists

In this section we represent the lists of terms of type A by a set of designs,written ListpAq. In particular we represent the lists of fixed length n by theset LnA. We shorten LnN by Ln. We prove that:

• Ln is principal (Proposition 15prop.15) and LKKn is separable (Propo-sition 16prop.16),

• ListpNatq is principal (Proposition 19prop.19) and pListpNatqqKK isseparable (Proposition 20prop.20),

• pLnAqKK is principal, when A is principal (Proposition 21prop.21).

These properties permit us to define easily functions on lists of natural num-bers (Section 2.2.2Dependent Functionssubsection.2.2.2).

Notation:

• nil denotes the empty list.

• 0 is the empty sequence ε and for all i P N, i` 1 :“ i.1.1.

• 0 “ ε and for all i P N, i` 1 “ i.0.1.

• ă a1, ..., an ą denotes the list a1.p...pan.nilq...q. In the following weomit the parentheses and write a.b for a.pbq.

Suppose A a type and A the design based on $ ξ.0.1, that represents theelement a : A. We define the design Dξ

l which represents the list l (of termsof type A), on the base $ ξ, as follows:

• if l “ nil


Dξnil “

H

$ ξ, Dξnil “ p`, ξ,Hq

in the following we denote it by Nil,

• if l “ a.b

Dξa.b “

A$ ξ.0.1

ξ.0 $

Dξ.1b

$ ξ.1

ξ.1 $

$ ξ

Dξa.b “ p`, ξ, t0, 1uq

p´, ξ.0, t1uq

A

p´, ξ.1, t1uq

Dξ.1b

We denote by ListpAq the set of designs that represent the lists of terms oftype A, ListpAq “ tDξ

l | l P ListpAqu.We denote by LnA the set of designs that represent the lists of length n ofelements of type A, i.e., LnA “ tD

ξl | l P ListnpAqu. We shorten LnN by Ln. In

the following we omit the base ξ and write directly Dl.

Example 14. The following design D2.1 P L2 represents the list 2.1.nil

D2.1 “

H

$ ξ.0.1.2

$ ξ.0.1

ξ.0 $

H

$ ξ.1.0.1.1

$ ξ.1.0.1

ξ.1.0 $

H

$ ξ.2

ξ.1.1 $

$ ξ.1

ξ.1 $

$ ξ

D2.1 “ p`, ξ, t0, 1uq

p´, ξ.0, t1uq

2

p´, ξ.1, t1uq

p`, ξ.1, t0, 1uq

p´, ξ.1.0, t1uq

1

p´, ξ.1.1, t1uq

p`, ξ.2,Hq


2.1.3.1 Lists of Length n of Natural Numbers: Ln.

In this section we prove some preliminary results on Ln. We start withsome properties on the chronicles and the paths of Ln. Then we characterizethe visitable paths of Ln to finally prove that Ln is principal (Proposition15prop.15) and LKKn is separable (Proposition 16prop.16).

In Lemma 7lemme.7 we proved that the dual of chronicles of Nat are stillchronicles. This property holds for chronicles of Ln.

Lemma 9. For all D P Ln if c is a chronicle of D, then rc is a chronicle.

Proof. Similar as for Lemma 7lemme.7.

The chronicles of two designs of Ln are either the same or they start differon a positive action on the same address.

Proposition 13. Let D,D1 be two designs of Ln and c a chronicle of D,then one of the following holds:

• c P D1

• Di, j P N, Dc1 such that

– either c1p`, ξ.i.0.1.j, t0uq is a prefix of c and c1p`, ξ.i.0.1.j,Hq PD1

– or c “ c1p`, ξ.i.0.1.j,Hq and c1p`, ξ.i.0.1.j, t0uq P D1.

Proof. Let D and D1 represent respectively the lists ă a0, ..., an´1 ą andă a10, ..., a

1n´1 ą and c P D.

1. If c is a prefix of the chronicle with last action p`, ξ.n,Hq, then thischronicle is common to all the elements of Ln. Therefore c P D1.

2. If c is a prefix of the chronicle which gives access to an element ai ofthe list.If ai “ a1i, then c P D1.If ai ‰ a1i, then there are two cases:

• ai ą a1i.

– Either a1i “ 0. Let c1 be the chronicle of D with last actionp´, ξ.i.0, t1uq, i.e., p`, ξ, t0, 1uq...p´, ξ.i.0, t1uq. Either c isa prefix of c1 (in this case c P D1), or it is an extension ofc1. In the latter case c1p`, ξ.i.0.1, t0uq is a prefix of c andc1p`, ξ.i.0.1,Hq P D1.


– Or a1i ‰ 0. Let c1 be the chronicle of D with last actionp´, ξ.i.0.1.a1i ´ 1.0, t1uq. Either c is a prefix of c1 (in thiscase c P D1) or c is an extension of c1. In the latter casec1p`, ξ.i.0.1.a1i, t0uq is a prefix of c and c1p`, ξ.i.0.1.a1i,Hq PD1.

• a1i ą ai.

– if c is not maximal in D, i.e., it does not end with the actionp`, ξ.i.0.1.ai,Hq, then c P D1 (from Proposition 8prop.8),

– otherwise c ends with the action p`, ξ.i.0.1.ai,Hq. Let c1

be c without its last action, i.e., c “ c1p`, ξ.i.0.1.ai,Hq andc1p`, ξ.i.0.1.ai, t0uq P D

1.

Since the designs of Nat only have one maximal chronicle, then their pathsare their chronicles. The structure of the designs of Ln is more complicated,indeed there are paths that are not chronicles.

We proved above that the chronicles of the elements of Ln have a partic-ular form, what about the paths of Ln?

Lemma 10. If c is a chronicle of E P Ln and a subsequence of a path ofanother design F P Ln, then c is a chronicle of F.

Proof. From Proposition 13prop.13 since c is a chronicle of E and E,F belongto Ln then one of the following holds:

• c P F

• Di, j P N, Dc1 such that

– either c1p`, ξ.i.0.1.j, t0uq is a prefix of c and c1p`, ξ.i.0.1.j,Hq P F

– or (c “ c1p`, ξ.i.0.1j,Hq and c1p`, ξ.i.0.1.j, t0uq P F.

In the latter case c contains an action that does not belong to F, then cwould not be a subsequence of a path of F (contradiction). Therefore c is achronicle of F.

Lemma 11. Let E,F P Ln, let q be a path of E and F, let κ an action suchthat qκ is a path of E but not of F. Then κ is positive.

Proof. Let q be a path of E and F, qκ a path of E and not of F. Weprove by contradiction that κ is positive. Suppose that κ is negative, thenxqκy “ xq1yκ`0 κ, where κ

`0 justifies κ, xq1κ

`0

y is a chronicle c1 of E, c1 is a


subsequence of q and q is a path of F. Then from Lemma 10lemme.10, c1 isa chronicle of F. The justifier of κ is the last action of c1. Since in Ln thereare never two distinct negative actions on the same address, then c1κ is achronicle of F. Moreover xqκy “ c1κ P F and q is a path of F, then qκ is apath of F (contradiction). Therefore κ is positive.

Given two designs E,F of Ln and a path p of E either it is a path of F or pstarts differ on a positive action on the same address with a path of F.

Proposition 14. Let E,F be two elements of Ln and p a path of E. Thenone of the following holds:

• p is a path of F

• there exists two positive actions κ, κ1 on the same address, i P N and aprefix q of p such that

– qκ is a prefix of p,

– qκ1 is a path of F,

∗ either κ “ p`, ξ.i.0.1.ai,Hq and κ1 “ p`, ξ.i.0.1.ai, t0uq∗ or κ “ p`, ξ.i.0.1.ai, t0uq and κ1 “ p`, ξ.i.0.1.ai,Hq.

Proof. If p is a path of F, then there is nothing to prove, so let p R F.If n “ 0, since L0 contains only one design, then E “ F and p is a path of F(as above).If n ‰ 0, E ‰ F (otherwise is like the first case) and p is not a path of F,then there exists an action κ such that wκ is a prefix of p, w is a path ofF and wκ is not a path of F. Since n ‰ 0, then all the chronicles of theelements of Ln start with the same positive action p`, ξ, t0, 1uq, so all pathsof E and F have at least their first action in common. Therefore w is notempty. Then κ is not an initial action. From Lemma 11lemme.11, κ is apositive proper (because the elements of Ln are z-free) action. xwκy is achronicle c1 of E which ends with a positive action, then from Proposition13prop.13 there exists c11 such that either (c11p`, ξ.i.0.1.ai, t0uq is a subse-quence of c1 and c11p`, ξ.i.0.1.ai,Hq P F) or (c1 “ c11p`, ξ.i.0.1.ai,Hq andc11p`, ξ.i.0.1.ai, t0uq P Fq. Since c11 is a prefix of c1 and c1 “ xwκy, then thereexists a prefix r of w such that xry “ c11. Since r is a prefix of w and w is aprefix of p, then r is a prefix of p. The thesis follows taking q “ r.

Even if the dual of a path is not always a path (Example 9example.9), theduals of the paths of Ln are still paths as proved in the following lemma.

Lemma 12. If p is a path of Ln, then rp is a chronicle, hence a path.


Proof. If p is a path of the empty list, then p “ p`, ξ,Hq, rp “ p´, ξ,Hqz andrp is still a path. Otherwise p has a particular form p “ p`, ξ, t0, 1uqκ11κ1κ

12κ2...

κ1jκj where κ1i is negative and justifies κi. When we change the polaritiesto obtain rp we find that each negative (non initial) action of rp is justifiedby the action which immediately precedes it in rp. This means that rp is achronicle.

Which paths of Ln are visitable in Ln? The following Lemma says that allthe positive-ended paths of Ln are visitable in it.

Lemma 13. All the positive-ended paths of Ln are visitable in Ln.

Proof. Let p be a positive-ended path of Ln. We use Proposition 2prop.2 toprove that p is visitable.From Lemma 12lemme.12 follows that rp is a path. We prove by contradictionthat for all prefix wκ´ of p, for all D P Ln, if w is a path of D, then wκ´ isa path of D.Let wκ´ be a prefix of p and D P Ln such that w is a path of D and wκ´

is not a path of D. Then from Lemma 11lemme.11, κ´ would be a positiveaction (contradiction). By definition, Ln and p hold all the other hypothesesof Proposition 2prop.2.Therefore from Proposition 2prop.2 follows that p is visitable in Ln.

For all design of Ln there exists a path which covers all its actions.

Remark 13. By definition of Ln to each element of a list l corresponds oneand only one maximal chronicle of Dl P Ln. Then an enumeration of themaximal chronicles of a design of Ln corresponds to an order on the elementsof the list and nil.

Lemma 14. Let E P Ln and p be a path of E. Then we can extend p to apath that covers all the actions of E

Proof. By definition, Ln is z-free and the only branches are additives. There-fore we can always extend p to a path that covers E.

We remark that in the following we will need to use the paths thatcovers the designs of Ln. We explicit then their construction in the followingLemma.

Lemma 15. Given a design E P Ln, there exists a path p which covers allthe actions of E.


Proof. Let C “ pc1, ..., cn`1q be an enumeration of the maximal chronicles ofE. Given two distinct elements ci ‰ cj (where i ă j) of C they start differon a negative action κij of cj . We denote by c1j the rest of cj after κij , i.e.,cj “ wκijc

1j . We define a sequence of actions p as c1κ12c

12κ23c

13....c

1n`1. The

idea is to jump from a chronicle to the other starting from the first actionthey differ on. By definition the sequence p covers all the actions of E. Nowwe prove that p is a path of E: by construction, it is alternated and it holdslinearity, daimon and totality (see Definition 14defi.14). Suppose that qκ`

is a prefix of p, then either κ` is initial or there exists a negative action κ´0which justifies κ`. In the latter case κ´0 is immediately before κ`, i.e., itis the last action of q. Then by definition of view, κ´0 P xqy. Thus p is apath.

Example 15. We consider the design D P L3 which represents the listă 3, 1, 0 ą.

D “

H

$ ξ.0.1.3

$ ξ.0.1

ξ.0 $

H

$ ξ.1.0.1.1

$ ξ.1.0.1

ξ.1.0 $

H

$ ξ.2.0.1

ξ.2.0 $

H

$ ξ.3

ξ.2.1 $

$ ξ.2

ξ.1.1 $

$ ξ.1

ξ.1 $

$ ξ .

D “ p`, ξ, t0, 1uq

p´, ξ.0, t1uq

3

p´, ξ.1, t1uq

p`, ξ.1, t0, 1uq

p´, ξ.1.0, t1uq

1

p´, ξ.1.1, t1uq

p`, ξ.2, t0, 1uq

p´, ξ.2.0, t1uq

0

p´, ξ.2.1, t1uq

p`, ξ.3,Hq

Its maximal chronicles are:

c1 “ p`, ξ, t0, 1uqp´, ξ.0, t1uqp`, ξ.0.1, t0uq...p`, ξ.0.1.3,Hq


c2 “ p`, ξ, t0, 1uqp´, ξ.1, t1uqp`, ξ.1, t0, 1uqp´, ξ.1.0, t1uqp`, ξ.1.0.1, t0uqp´, ξ.1.0.1.0, t1uqp`, ξ.1.0.1.1,Hqc3 “ p`, ξ, t0, 1uqp´, ξ.1, t1uqp`, ξ.1, t0, 1uqp´, ξ.1.1, t1uqp`, ξ.2, t0, 1uqp´, ξ.2.0, t1uqp`, ξ.2.0.1,Hqc4 “ p`, ξ, t0, 1uqp´, ξ.1, t1uqp`, ξ.1, t0, 1uqp´, ξ.1.1, t1uqp`, ξ.2, t0, 1uqp´, ξ.2.1, t1uqp`, ξ.3,Hq

We shorten them asc2 “ w2p´, ξ.1, t1uqc

12, c3 “ w3p´, ξ.1.1, t1uqc

13, c4 “ w4p´, ξ.2.1, t1uqp`, ξ.3,Hq.

One of the paths defined in Lemma 15lemme.15 which covers all the ac-tions of D is then

p “ c1p´, ξ.1, t1uqc12p´, ξ.1.1, t1uqc

13p´, ξ.2.1, t1uqp`, ξ.3,Hq.

For each D P Ln there are several paths which cover all the actions of D.These paths only differ on the order of their actions. Are they coherent?Yes, indeed all paths that belong to the same design are pairwise coherent.

Now we consider the paths that cover distinct designs of Ln following thesame order (see Lemma 15lemme.15). These paths are not pairwise coherent,while their dual are coherent, as showed in the following Lemma.

Lemma 16. Let E,F be two distinct designs of Ln, let p a path (resp. q)that covers E (resp. F) (following the same order to visit their chronicles,i.e., the elements of the list), then p and q are not coherent whereas rp and rqare coherent.

Proof. Let E and F respectively represent the distinct lists ă a1, ..., an ą andă a11, ..., a

1n ą. Since E ‰ F, then there exists i P t1, ..., nu such that ai ‰ a1i

(let ai ă a1i). Having seen the structure of the elements of Ln, there exists achronicle c P E,F such that cp`, ξ.i.0.1.ai,Hq P E and cp`, ξ.i.0.1.ai, t0uq PF. p and q cover all the actions of E and F, this means that there exists asubsequence w1κ

`1 of p and a prefix w2κ

`2 of q such that xw1y “ xw2y “ c and

κ`1 ‰ κ`2 , in particular κ`1 “ p`, ξ.i.0.1.ai,Hq and κ`2 “ p`, ξ.i.0.1.ai, t0uq.

Therefore p and q are not coherent. p and q start differ on a positive actionon the same address, then rp and rq start differ on a negative action on thesame address, moreover rp and rq are chronicles (from the proof of Lemma12lemme.12). Therefore rp and rq are coherent.

Remark 14. If p and q are two paths which cover all the actions of a designE visiting its chronicles following two different orders, then p and q startdiffer on a negative action. Then their duals start differ on a positive action,i.e., rp and rq are not coherent.


From Proposition 4prop.4 to recover the material designs of LKKn , we needto find which are the maximal cliques of ČVpLnqK . But what are the paths ofČVpLnqK? The following Lemma says that the paths of ČVpLnqK are the visitablepaths of Ln. This result really simplifies the proof of the principality of Ln.

Lemma 17. ČVpLnqK “ VLn.

Proof. Given an enumeration β of 1, ..., n ` 1 we define the set Pβ of pathsas

Pβ “ trp | DE P Ln such that p covers all the actions of E following the order

given by βu.

Let E P Ln, then E contains n ` 1 maximal chronicles. The order followedby the elements of Pβ is then an enumeration of p1, ..., n` 1q.From Lemma 16lemme.16, we have that the elements of Pβ are pairwise co-herent. Thus, as a set of pairwise coherent paths forms a design, we candefine the design Gβ “ xxPβyy, i.e., the set of views of prefixes of the ele-ments of Pβ . Let G “ tGβ |β is a enumeration of 1, ..., n ` 1u. By Lemma13lemme.13, we know that all the positive-ended paths of Ln are visitable.Furthermore, in Ln, the ramification of a negative action is always the sin-gleton t1u. Hence for all β, a positive action of Gβ is followed by at most onenegative action: thus it is not possible to “jump” from a chronicle to another,i.e., the only paths of G are its chronicles, which are visitable. The paths ofĄVLn are the z-shortenings of the chronicles of G. Therefore V

Gz “ ĄVLn .We want to show now that Gz “ |pLnqK|, from which follows that V

Gz “

VpLnqK :

• First we prove that G Ď |pLnqK|.Let Gβ P G. For all E P Ln there exists a path p that covers E such thatrp P Gβ , i.e., Gβ K E. Then Gβ P pLnqK. We prove by contradictionthat it is material in it. Suppose that there exists E Ĺ Gβ such thatE P pLnqK, then there exists a path rp which belongs to Gβ and rp R Esuch that p covers a design L P Ln. E Ĺ Gβ , then E cannot containa prefix of rp ended by z. This means that E M L, i.e., E R pLnqK(contradiction). Therefore G Ď |pLnqK|.Thus from Lemma 3lemme.3 we have Gz Ď |pLnqK|.

• Now we prove that |pLnqK| Ď Gz.Let F P |pLnqK|, then for all Dl P Ln there exists a path pl of Dl

such that rpl is a path of F. From Lemma 12lemme.12 all the rpl arechronicles. For all rpl ‰ Ăpl1 , since they are in the same design, thenthey are coherent between them. This means that pl and pl1 mustfollow the same order to visit the chronicles of Dl and Dl1 , otherwise


by construction of Ln, rpl and Ăpl1 would differ on a positive action andwould not be coherent. This means that there exists Gβ P G such thatthe rpl are z-shortenings of the chronicles of Gβ . Since F is minimal,then it does not have other chronicles apart the rpl. Then F P tGβu

z,i.e., F P Gz.

Therefore |pLnqK| “ Gz and VpLnqK “ VGz . Since V

Gz “ ĄVLn , then VLKn “ĄVLn .

Lemma 18. Let D P Ln, then the set of positive-ended paths of D is amaximal clique of visitable paths of Ln.

Proof. From Lemma 13lemme.13 all the positive-ended paths of D are vis-itable. Since D is a design, then all its paths are coherent. We prove that itis maximal, i.e., for all path p P VLn , p R D, there exists a path of D that isnot coherent with p. Let p P VLn such that p R D. Then there exists D1 ‰ Dsuch that D1 P Ln and p P D1. From Proposition 14prop.14 there exists apath qκ P D such that p and qκ start differ on a positive action, i.e., theyare not coherent.

We have now enough tools to prove that Ln is principal and LKKn is separable.

Proposition 15. Ln is principal, i.e., it is z-free and |pLnqKK| “ pLnqz.

Proof. • The fact that Ln is z-free follows from its definition.

• We start proving pLnqz Ď |pLnqKK|.Let D P Ln, then from Lemma 13lemme.13 all the positive-ended pathsof D are visitable. Let rC be the set of positive-ended paths of D. FromLemma 18lemme.18, rC is a maximal clique of visitable paths of Ln.Thus C Ď VpLnqK by means of Lemma 17lemme.17. C is finite-stablebecause it contains a finite number of paths, so each sequence of pathsis finite. It is saturated because in Ln there does not exist two negativeactions with the same address, then if q is a prefix of an element of Csuch that qκ` P VpLnqK (κ` ‰z) then qκ` is a prefix of an element of C(there is only one possible choice for κ`). Therefore from Proposition4prop.4 follows that xx rCyy “ D P |pLnqKK|.Thus Ln Ď |pLnqKK|.Therefore from Lemma 3lemme.3 follows pLnqz Ď |pLnqKK|.

• Now we prove |pLnqKK| Ď pLnqz.Let D P |pLnqKK|, from Proposition 4prop.4 there exists C Ď V KLnsuch that C is finite stable, saturated, rC is a maximal clique of ČVpLnqK

and xx rCyy “ D. We want to show that D P pLnqz. From Lemma


17lemme.17, ČVpLnqK “ VLn . rC is a maximal clique of ČVpLnqK , then it isa maximal clique of VLn . Moreover all positive-ended paths of Ln arevisitable (Lemma 13lemme.13). Which paths of Ln can form a maximalclique? rC cannot contain two distinct paths which respectively cover(all the actions of) two distinct elements of Ln, indeed by definitionthese paths differ on a positive action, therefore they are not coherentbetween them. Then the elements of rC are all the chronicles of anelement of Ln and their z´shortenings. This means that there existsE P pLnqz such that the elements of rC are all the paths of E. ThenD “ xx rCyy “ E P pLnqz.Therefore |pLnqKK| Ď pLnqz.

So Ln may represent the canonical terms of type ListnpNq ( lists of naturalnumbers of length n).Now we prove that LKKn is a separable behaviour.This property gives us a very simple tool to define functions on lists oflength n. In next section we generalize this result to pListpNatqqKK todefine functions on lists of natural numbers (Section 2.2Constructions andDependencysection.2.2).

Proposition 16. For all n P N the behaviour LKKn is separable.

Proof. From Proposition 15prop.15 |pLnqKK|z´free “ Ln. Let Dl P Ln bethe design that represents the list l. From Lemma 15lemme.15 there exists apath pl that covers Dl following the order of the elements of l. From Lemma13lemme.13 all the positive-ended paths of Ln are visitable, in particular plis visitable. For all Dl1 P Ln such that Dl1 ‰ Dl, since pl covers Dl, then pl isnot a path of Dl1 . From Lemma 12lemme.12 rpl is a chronicle. By definitionrpl “ plz, then xply “ pl and xpl1y “ pl1 . Therefore xply “ pl and xpl1y “ pl1 .Since pl is not a path of Dl1 and pl1 is not a path of Dl then xply is not apath of Dl1 and xpl1y is not a path of Dl. Since pl ‰ pl1 then pl ‰ pl1 andxply ‰ xpl1y. By definition of Ln, rpl and Ăpl1 differ on a negative action on thesame address, then xply and pl1 are coherent.

2.1.3.2 Lists of Natural Numbers: ListpNatq

Now we consider lists of natural numbers of any length, i.e., the set ListpNatq.In this section we generalize some of the results of the previous sectionto ListpNatq. Finally, we prove that ListpNatq is principal (Proposition19prop.19) and pListpNatqqKK is separable (Proposition 20prop.20).

Lemma 9lemme.9 holds for ListpNatq.


Lemma 19. For all D P ListpNatq if c is a chronicle of D, then rc is achronicle.

Proof. Since D P ListpNatq then there exists n0 P N such that D P Ln0 . Byhypothesis c is a chronicle of D, then from Lemma 9lemme.9 follows that rcis a chronicle.

We can generalize Proposition 13prop.13 to ListpNatq as follows.

Proposition 17. Let D,D1 be two elements of ListpNatq and c a chronicleof D, then one of the following holds:

• c P D1

• Dc1 such that

– either Di, j P N such that either (c1p`, ξ.i.0.1.j, t0uq is a prefixof c and c1p`, ξ.i.0.1.j,Hq P D1) or (c “ c1p`, ξ.i.0.1j,Hq andc1p`, ξ.i.0.1.j, t0uq P D1).

– or Dk P N, such that either (c1p`, ξ.k, t0, 1uq is a prefix of c andc1p`, ξ.k,Hq P D1) or (c “ c1p`, ξ.k,Hq and c1p`, ξ.k, t0, 1uq PD1).

Proof. Let D,D1 P ListpNatq and c a chronicle of D.

• Either there exists n0 P N such that D,D1 P Ln0 . In this case the thesisfollows from Proposition 13prop.13.

• Or there exist n ‰ m P N such that D P Ln and D1 P Lm. Let n ă m(if n ą m we just exchange them), D (resp. D1) represents the listă a1, ..., an ą (resp. ă a11, ..., a

1n, ..., a

1m ą).

– Either there exists i P t1, ..., nu such that c is a prefix of thechronicle that represent the term ai. In this case the thesis followsfrom Proposition 13prop.13.

– Or c is a prefix of the chronicle wp`, ξ.n,Hq P D. By definitionof Lm we have wp`, ξ.n, t0, 1uq P D1. Either c ‰ wp`, ξ.n,Hq orc “ wp`, ξ.n,Hq. In the first case, c P D1, in the latter case thethesis follows taking c1 “ w and k “ n.

Lemma 10lemme.10 and Lemma 11lemme.11 hold for ListpNatq.

Lemma 20. If c is a chronicle of E P ListpNatq and a subsequence of a pathof another design F P ListpNatq, then c is a chronicle of F.


Proof. Similar as for Lemma 10lemme.10, using Proposition 17prop.17 in-stead of Proposition 13prop.13.

Lemma 21. Let E,F P ListpNatq, let q be a path of E and F, let κ an actionsuch that qκ is a path of E but not of F. Then κ is positive.

Proof. Similar as for Lemma 11lemme.11, using Lemma 20lemme.20 insteadof Lemma 10lemme.10.

We can generalize Proposition 14prop.14 to ListpNatq as follows

Proposition 18. Let E,F be two elements of ListpNatq and p a path of E.Then one of the following holds:

• p is a path of F

• there exists two positive actions κ, κ1 on the same address, i, j P N anda sequence q such that

– qκ is a prefix of p,– qκ1 is a path of F,

∗ either (κ “ p`, ξ.i.0.1.ai,Hq and κ1 “ p`, ξ.i.0.1.ai, t0uq) or(κ “ p`, ξ.i.0.1.ai, t0uq and κ1 “ p`, ξ.i.0.1.ai,Hq).

∗ either (κ “ p`, ξ.j,Hq and κ1 “ p`, ξ.j, t0, 1uq) or (κ “p`, ξ.j, t0, 1uq and κ1 “ p`, ξ.j,Hq).

Proof. Let E,F P ListpNatq and p a path of E.If p is a path of F, then there is nothing to prove.If there exists n0 P N such that E,F P Ln0 then the thesis follows fromProposition 14prop.14.Let n ‰ m P N such that E P Ln, F P Lm and p is not a path of F.

• If n “ 0 then m ‰ 0 (if m “ 0 and n ‰ 0 we just exchange them)and E “ Nil, i.e., p “ p`, ξ,Hq. Since m ‰ 0 then the first action ofeach path of F is p`, ξ, t0, 1uq. The thesis follows taking q equal to theempty sequence ε.

• Otherwise n,m ‰ 0. Since p is not a path of F, then there exists anaction κ such that wκ is a prefix of p, w is a path of F and wκ is nota path of F.Sinc n ‰ 0, then all the chronicles of the elements of Lnand Lm start with the same positive action p`, ξ, t0, 1uq, so all pathsof E and F have at least their first action in common. Therefore w isnot empty and κ is not an initial action. From Lemma 21lemme.21,κ is a positive proper (because the elements of ListpNatq are z-free)action. xwκy is a chronicle c1 of E which ends with a positive action,then from Proposition 17prop.17 there exists c11 such that one of thefollowing holds:


– there exist i, j P N such that either (c11p`, ξ.i.0.1.j, t0uq is a prefixof c1 and c11p`, ξ.i.0.1.j,Hq P F) or (c1 “ c11p`, ξ.i.0.1j,Hq andc11p`, ξ.i.0.1.j, t0, 1uq P F).

– there exists k P N such that either (c11p`, ξ.k, t0, 1uq is a prefix ofc1 and c11p`, ξ.k,Hq P F) or (c1 “ c11p`, ξ.k,Hq and c11p`, ξ.k, t0, 1uq PF).

Since c11 is a prefix of c1 and c1 “ xwκy, then there exists a prefix r ofw such that xry “ c11. Since r is a prefix of w and w is a prefix of p,then r is a prefix of p. The thesis follows taking q “ r.

Lemma 12lemme.12 still holds for ListpNatq.

Lemma 22. If p is a path of ListpNatq, then rp is a chronicle, hence a path.

Proof. Let p be a path of ListpNatq, then there exists n0 P N such that p isa path of Ln0 . From Lemma 12lemme.12 follows that rp is a chronicle, hencea path.


Lemma 23. All the positive-ended paths of ListpNatq are visitable in ListpNatq.

Proof. Similar as for Lemma 13lemme.13, using Lemma 21lemme.21, 22lemme.22instead of Lemma 11lemme.11, 12lemme.12.


Lemma 24. Given a design E P ListpNatq, there exists a path p which coversall the actions of E.

Proof. Let E P ListpNatq, then there exists n0 P N such that E P Ln0 . FromLemma 15lemme.15 follows that there exists a path which covers all theaction of E following the order of the elements of the list represented byE.

Lemma 25. For all n P N VpLnqK Ď VpListpNatqqK.

Proof. Let q be a path in VpLnqK , then there exist F P pLnqK and G P pLnqKK

such that q is a path of F and rq is a path of G. Since pLnqKK Ď pListpNatqqKKthen G P pListpNatqqKK. We define F1 such that F1 P pListpNatqqK and q isa path of F1, from which follows that q P VpListpNatqqK .By hypothesis q P VpLnqK , VpLnqK “ ĄVLn (Lemma 17lemme.17) and ĄVLn “

ĄPLn

(Lemma 13lemme.13). Therefore q P ĄPLn . Then the negative actions of qare of one of the following forms:


• κl “ p´, ξ.l,Hq or κ1l “ p´, ξ.l, t0, 1uq for some l P N, l ď n

• κi,j “ p´, ξ.i.0.1.j,Hq or κ1i,j “ p´, ξ.i.0.1.j, t0, 1uq for some i, j P N,i, j ď n.

By definition of ĄPLn , in q there can not be two distinct negative actions onthe same address. Therefore κl P q iff κ1l R q and κi,j P q iff κ1i,j R q.

1. Either q ends with a z.Then we define the following set of paths:R “ tqu Y twκlz |wκ1l is a prefix of quYtwκ1lz |wκl is a prefix of quYtwκi,jz |wκ1i,j is a prefix of quYtwκ1i,jz |wκi,j is a prefix of qu.Let us prove that R is a set of pairwise coherent paths.If n “ 0, then the only path of R is q and there is nothing to prove.Let n ‰ 0 and p1 ‰ p2 be two paths of R. If p1 is a prefix of p2 (orviceversa), then they are coherent between them. Otherwise p1 and p2

differ on a negative action κ, κ1 on the same address.

• Either p1 “ wκlr and p2 “ wκ1lz (or viceversa). By definition κ1lcan not belong to r, then p1 ¨ p2.

• Or p1 “ wκi,jr and p2 “ wκ1i,jz (or viceversa). By definition κ1i,jcan not belong to r, then p1 ¨ p2.

Therefore p1 and p2 are coherent.Since R is a finite set of pairwise coherent paths then we can definethe design F1 “ xxRyy.We prove that for all D P ListpNatq there exists a path p of D suchthat rp is a path of F1.We said above that q P ĄPLn and q “ q0z. For all D P ListpNatq, eitherq0 is a path of D or there exists a path p of D such that q and rp firstdiffer on the last negative action of rp. This means that q “ w1κ

´w11zand rp “ w1κ

´0 z. Since in PLn each positive action is justified by the

immediately precedent action, then κ´ and κ´0 are justified by thesame action, i.e., they have the same address. Either there exists l ď nsuch that κ´ “ κl and κ´0 “ κ1l (or viceversa) or there exist i, j ď nsuch that κ´ “ κi,j and κ´0 “ κ1i,j (or viceversa). In both cases bydefinition of R, we have that rp is a path of F1.Therefore F1 P pListpNatqqK and q P F1.Thus q P VpListpNatqqK .

2. Or q does not end with a z. Since q is visitable, then from Remark5rem.5 it ends with a positive actions action κ` ‰z. Since q is z-free, then there exists an extension of q, q1 that ends with z such that


q1 P VpLnqK . We define the set of paths R asR “ tq1uYtwκlz |wκ

1l is a prefix of q1uYtwκ1lz |wκl is a prefix of q1uY

twκi,jz |wκ1i,j is a prefix of q1uYtwκ1i,jz |wκi,j is a prefix of q1u. We

prove as in p1q (replacing q by q1) that the paths of R are pairwisecoherent and that the design F1 “ xxRyy belongs to pListpNatqqK.By definition of R, q1 is a path of F1, then q is a path of F1. Thusq P VpListpNatqqK .


Lemma 26. Let E P ListpNatq and p be a path of E. Then we can extend pto a path that covers all the actions of E

Proof. Similar as for Lema 14lemme.14.

Lemma 27. Let Q “ trpl | DDl P ListpNatq, pl covers Dluz. Then

1. Q “ ČPListpNatq,

2. P|pListpNatqqK| Ď Q,

3. VpListpNatqqK “ Q.

Proof. 1. By definition Q Ď ČPListpNatq.We prove that ČPListpNatq Ď Q. Let rp P ČPListpNatq, then there existsDl P ListpNatq such that p is a path of Dl. From Lemma 26lemme.26we can extend p to a path q that covers Dl. The path rq belongs to Q.Since rp is a z-shortening of rq, then rp P Q.

2. Let q P P|pListpNatqqK|, then there exists B P |pListpNatqqK| such that qis a path of B. Since B P |pListpNatqqK|, then for all Dl P ListpNatqthere exists a path ql of Dl such that rql is a path of B and for allDl1 ‰ Dl, Dl1 P ListpNatq we have that rql is coherent with Ăql1 . ByLemma 22lemme.22, all the rql are chronicles and since B is minimal,then the rql are its only chronicles. By definition of ListpNatq, sincethe rql must be coherent between them, then they differ on a negativeaction on the same address. Therefore the paths of B are just itschronicles. The set of paths of B, PtBu is then a subset of ČPListpNatq.From p1q follows that ČPListpNatq “ Q. Therefore q P Q.

3. • pĚqLet rp P Q, then there exists Dl P ListpNatq and a path pl thatcovers Dl such that rp is a z-shorten of rpl and p is a prefix of pl.Since Dl P ListpNatq, then there exists n0 such that Dl P Ln0 .


From Lemma 13lemme.13 all the paths of Ln0 are visitable, thenp P VLn0 . From Lemma 17lemme.17, we have ĆVLn0 “ VpLn0 qK , thenrp P VpLn0 qK . From Lemma 25lemme.25, we have that VpLn0 qK ĎVpListpNatqqK . Therefore rp P VpListpNatqqK .

• pĎqLet q P VpListpNatqqK , then q P P|pListpNatqqK|. From (2) follows thatP|pListpNatqqK| Ď Q. Therefore q P Q.


Lemma 28. ČVpListpNatqqK “ VListpNatq.

Proof. From Lemma 27lemme.27 we have VpListpNatqqK “ Q andQ “ ČPListpNatq.From Lemma 23lemme.23 follows that the set of positive-ended paths ofListpNatq is equal to VListpNatq. For all negative-ended path q of ListpNatqthere exists a positive-ended path p of ListpNatq such that q is a prefix of p.Thus ČVpListpNatqqK “ VListpNatq.


Lemma 29. Let D P ListpNatq, then the set of positive-ended paths of D isa maximal clique of visitable paths of ListpNatq.

Proof. From Lemma 23lemme.23, all the positive-ended paths of D are vis-itable. Since D is a design ,then all its paths are coherent. We prove that itis maximal, i.e., for all path p P VListpNatq, if p R D, then there exists a pathof D that is not coherent with p. Let p P VListpNatq such that p R D. Thenthere exists D1 ‰ D such that D1 P ListpNatq and p P D1. From Proposition18prop.18 there exists a path qκ P D such that p and qκ start differ on apositive action, i.e., they are not coherent.

Remark 15. Let E,F be two distinct designs of ListpNatq, p (resp. q) apath that covers E (resp. F). Then they differ on a positive action, thereforethey are not coherent between them.

Now we have enough tools to prove that ListpNatq is principal and pListpNatqqKKis separable.

Proposition 19. ListpNatq is principal, i.e., it is z-free and |pListpNatqqKK| “pListpNatqqz.

Proof. Similar as for Proposition 15prop.15, using Lemmas 23lemme.23,28lemme.28, 29lemme.29 instead of Lemmas 13lemme.13, 17lemme.17, 18lemme.18.


We prove that pListpNatqqKK is a separable behaviour.Separability gives us a simple device to maniputale the designs of |pListpNatqqKK|z´freethat we use in Section 2.2Constructions and Dependencysection.2.2 to definefunctions on lists of natural numbers.

Proposition 20. The behaviour pListpNatqqKK is separable.

Proof. By Lemma 24lemme.24 we have that for all Dl P ListpNatq thereexists a path pl that covers Dl following the order of the elements of l. FromLemma 22lemme.22, for all Dl ‰ Dl1 P ListpNatq we have pl and pl1 are twochronicles, then xply “ pl and xpl1y “ pl1 . Therefore xply “ pl and xpl1y “ pl1 .Since pl is not a path of Dl1 and pl1 is not a path of Dl, then xply is nota path of Dl1 and xpl1y is not a path of Dl. By definition, pl and pl1 differon a negative action on the same address, therefore they are distinct andcoherent.

2.1.3.3 Lists of length n of terms of type A: pLnAqKK.

We recall that we represent lists of length n of terms of type A byLnA “ tD

ξl | l is a list of length n of terms of type Au.

We prove that when A is represented by a principal behaviour A, then forall n P N pLnAqKK is a principal behaviour (Proposition 21prop.21).

Proposition 21. Let A be represented by means of the principal behaviourA, then for all n P N pLnAqKK is a principal behaviour.

Proof. We prove the thesis by induction on n.

• By definition L0A “ tp`, ξ,Hqu, then pL0

AqKK is principal.

• LetA be the principal behaviour that representsA andB “ tp`, ξ.1,HquKK.Then pL1

AqKK “ pÓξ.0Òξ.0.1 Aq b pÓξ.1Òξ.1 Bq. By hypothesis A is prin-

cipal and by definition B is principal. Since principality is closed w.r.t.Ò, Ó and b (Proposition 7prop.7), then L1

A is principal.

• Let pLnAqKK be a principal behaviour, then pLn`1A qKK “ pÓξ.0Òξ.0.1 Aqb

pÓξ.1Òξ.1 pLnAqKKq. From Proposition 7prop.7 follows that pLn`1A q is

principal.

Thus for all n P N pLnAq is principal.

We remark that the previous proposition implies that for all n P N pLnqKKis a principal behaviour and Ln is a principal set. Obviously the previousproof is much more simple then Proposition 15prop.15, but the preliminaryresult that we use to prove Proposition 15prop.15, once generalized, permitus to prove that ListpNatq is principal. For the moment we do not investigate


about the principality of ListpAq, when A is principal. We remark that toprove the principality of ListpNatq we use some structural properties of theset Nat.

2.1.4 Records

We give an informal presentation of records and record types, for a moreformal thorough presentation see [5].

• A record type is a sequence of fields, such that each field contains alabel li and a type Ai

T “ă l1 : A1, ..., ln : An ą.

In dependent record types the type Ai may depend on the precedentlabels l1, ..., li´1.

• A record R is a sequence of fields that contain a label li and an objectai

R “ă l1 “ a1, ..., ln “ an ą.

A record R is of type T when for each field in T there is a correspondingfield in R with the same label, such that the object in the field of R is of thetype in the field of T , i.e., for i “ 1, ..., n ai : Ai.

Given a record R of type T , the usual operations defined on it are thefollowing:

• the dot operation, that allows us to recover the object contained is afield.

For all i P t1, ..., nu, we have R.i “ ai.

• Add a field to R. For instance, when we add a field in the last positionof R we obtain the record R1 “ă l1 “ a1, ..., ln “ an, ln`1 “ an`1 ą.

• Eliminate a field in R. For instance, when we eliminate the field inthe last position of R, we obtain the record R0 “ă l1 “ a1..., ln´1 “

an´1 ą.

There is a notion of subtyping on types: given two types A,A1, A is asubtype of A1, noted A Ď A1 when for all a : A, we have that a : A1 and forall a “ b : A, a “ b : A1.


Given two record types T and T 1, T is a subtype of T 1 when for each fieldl : A1 in T 1 there is a corresponding field l : A in T such that A Ď A1.For instance the record type ă l1 : N, l2 : ListpNq ą is a subtype of ă l :ListpNq ą.

Example 16. The record ă l1 “ 2, l2 “ 1.nil ą whose fields contain theobject 2 : N and 1.nil : ListpNq is of type ă l1 : N, l2 : ListpNq ą.

Now we recall an example of record type in Ludics [13]. We show then thatthe presentation proposed in [13] holds principality, but not separability.

Example 17. ([13])Let R be the following record

R “ă l1 “ p3, 4q, l2 “ green, l3 “ circle ą.

R is a term of type

T “ă l1 : Coord, l2 : Colour, l3 : Shape ą.

We represent the record R by the following design

Dăp3,4q,g,cą “

H

$ 2.6.1

2.6 $

H

$ 2.9.1

2.9 $

$ 2

H

$ 3.8.1

3.8 $

$ 3

H

$ 8.0.1

8.0 $

$ 8ăą$

p´,ăą, t2uq p´,ăą, t3uq p´,ăą, t8uq

p`, 2, t6, 9uq

p´, 2.6, t1uq

p`, 2.6.1,Hq

p´, 2.9, t1uq

p`, 2.9.1,Hq

p`, 3, t8uq

p´, 3.8, t1uq

p`, 3.8.1,Hq

p`, 8, t0uq

p´, 8.0, t1uq

p`, 8.0.1,Hq

where the three fields of the record are represented by means of the addresses2, 3, 8.

Let us suppose that the types Coord, Colour, Shape are represented by meansof the behaviours Coord,Colour,Shape. Then the record type T is repre-sented by the behaviour Ò Coord & Ò Colour & Ò Shape. Remark that thedesign Dăp3,4q,g,cą is material in it.

Let us consider the record type T 1 “ă l2 : Colour, l3 : Shape ą. SinceT is a subtype of T 1, then R is a record of type T 1.


Themistery of incarnation (Section 1.1.2.2The Mistery of the Incarnationsubsubsection.1.1.2.2)gives us two ways to reach the material designs of Ò Colour& Ò Shape: asan intersection or as a cartesian product. This property gives us a very nat-ural notion of subtyping.

The representation of a record of type T 1 is the disjoint union of the represen-tation of a record of type ă l2 : Colour ą and a record of type ă l3 : Shape ą.Let us consider the record R1 “ă l2 “ green, l3 “ circle ą of type T 1.We represent R1 by the design

Dăg,cą “

H

$ 3.8.1

3.8 $

$ 3

H

$ 8.0.1

8.0 $

$ 8ăą$

p´,ăą, t3uq p´,ăą, t8uq

p`, 3, t8uq

p´, 3.8, t1uq

p`, 3.8.1,Hq

p`, 8, t0uq

p´, 8.0, t1uq

p`, 8.0.1,Hq

The record ă l2 “ green ą of type ă l2 : Colour ą is represented by

Dăgą “

H

$ 3.8.1

3.8 $

$ 3ăą$

p´,ăą, t3uq

p`, 3, t8uq

p´, 3.8, t1uq

p`, 3.8.1,Hq

The record ă l3 “ circle of type ă l3 : Shape ą is represented by

Dăcą “

H

$ 8.0.1

8.0 $

$ 8ăą$

p´,ăą, t8uq

p`, 8, t0uq

p´, 8.0, t1uq

p`, 8.0.1,Hq

Dăg,cą is material in Ò ColourKK and Ò ShapeKK. Since Dăg,cą is theunion of Dăgą and Dăcą, then Dăg,cą belongs to Ò ColourKK and Ò ShapeKK.This corresponds to the fact that ă l2 : Colour, l3 : Shape ą is a subtype ofthe types ă l2 : Colour ą and ă l3 : Shape ą.


The representation of records proposed in [13] holds principality, i.e., whenwe represent the types A1, ..., An by means of principal behaviours, thenthe behaviour that represents the record type ă l1 : A1, ..., ln : An ą is aprincipal behaviour.

Lemma 30. Let the types A1, ..., An be represented by means of the principalpositive behaviours A1, ...,An. Then the behaviour Ò A1&...& Ò An thatrepresents the record type ă l1 : A1, ..., ln : An ą is a principal behaviour.

Proof. From Proposition 7prop.7 principality is closed w.r.t. Ò and &, thenÒ A1&...& Ò An is principal.

On the other side, this presentation does not hold separability.We remark (Example 11example.11) that separability is not closed w.r.t. theconnective &. Therefore the fact that the types A1, ..., An are representedby means of the separable behaviours A1, ...,An does not imply that thebehaviour Ò A1&...& Ò An is separable.

We can define the usual operations on records. In [13] records are repre-sented by designs on a negative base ăą$. To define functions on recordswe need them to have a positive base. We consider them based on α.0 $and add a shift to obtain designs on a positive base $ α.Let Recn be the set of records with n fields. The elements of Recn are recordsof the form R “ă l1 “ a,..., ln “ an ą. For all i P t1, ..., nu let us representthe term ai by means of a design Ai, based on $ α.0.i. We represent R bythe following design DR.

DR “

A1

$ α.0.1 . . .An$ α.n

α.0 $$ α

p`, α, t0uq

p´, α.0, t1uq

A1

. . . p´, α.0tnuq

An

We denote by Rn the set of such designs, i.e., Rn “ tDR |R P Recnu.

• We represent the identity function on Rec by the design IdRn (Defini-tion 4example.4).For all i P t1, ..., nu let Ai be the set of designs Ai that represent theterm in the ith field of a record of Recn, that is

Ai “ tAi | DR P Recn such that ai is the term in the ith field of Ru.


IdRn “

IdA1

α.0.1 $ σ.0.1

$ σ.0.1, α.0 . . .

IdAnα.0.n $ σ.0.n

$ σ.0.n, α.0

σ.0 $ α.0

$ α.0, σ

α $ σ

p´, α, t0uq

p`, σ, t0uq

p´, σ.0, t1uq

p`, α.0, t1uq

IdA1

. . . p´, σ.0, tnuq

p`, α.0, tnuq

IdAn

• We can then define easily the operation dot.For all i P t1, ..., nu, R.i “ ai is represented by the subset Fi of IdRn

defined as follows

Fi “

IdAiα.0.i $ σ.0.i

$ σ.0.i, α.0

σ.0 $ α.0

$ α.0, σ

α $ σp´, α, t0uq

p`, σ, t0uq

p´, σ.0, tiuq

p`, α.0, tiuq

IdAi

• Let us represent the term an`1 by a design An`1, based on $ α.0.n` 1.We can represent the function that takes as argument a record R P

Recn and adds (the field that contains) an`1 at the end of R, by thefollowing design Addn`1. This design is constructed by IdRn adding anew branch that allows us to add the new field.


Addn`1 “

IdA1

α.0.1 $ σ.0.1

$ σ.0.1, α.0 . . .

IdAnα.0.n $ σ.0.n

$ σ.0.n, α.0An`1

$ σ.0.n` 1, α.0

σ.0 $ α.0

$ α.0, σ

α $ σ

p´, α, t0uq

p`, σ, t0uq

p´, σ.0, t1uq

p`, α.0, t1uq

IdA1

. . . p´, σ.0, tnuq

p`, α.0, tnuq

IdAn

p´, σ.0, tn` 1uq

An`1

• We can represent the functions that eliminates the last field of a recordof Recn by the following design En. This design constructed by IdRn

eliminating the chronicles that correspond to the field in position n.

En “

IdA1

α.0.1 $ σ.0.1

$ σ.0.1, α.0 . . .

IdAn´1

α.0.n´ 1 $ σ.0.n´ 1

$ σ.0.n´ 1, α.0

σ.0 $ α.0

$ α.0, σ

α $ σ

p´, α, t0uq

p`, σ, t0uq

p´, σ.0, t1uq

p`, α.0, t1uq

IdA1

. . . p´, σ.0, tn´ 1uq

p`, α.0, tn´ 1uq

IdAn´1

2.2 Constructions and Dependency

Now that we have introduced the basic types we consider the constructionsfunctions and pairs. The types introduced until now hold principality. Un-

2.2. CONSTRUCTIONS AND DEPENDENCY 77

fortunately we will see that the representation of the type Ñ has not thisproperty. But thanks to the property of principality and separability forbasic types we will be able to define very simply functions on them.

2.2.1 Functions

Given two types A and B, AÑ B is the set of functions from A to B. Howcan we interpret it in Ludics? Given two behaviours A and B, Girard defines[13] the sequent of behaviours A $ B as follows.

Definition 29. Given two positive behaviours A,B, of disjoint bases $ ξand $ σ.

A $ B “ ttA,Fu |A P A,F P BKuK.

Remark 16. We remark from [13] that A $ B is a behaviour of base ξ $ σand

A $ B “ tD | @A P A, JD,AK P Bu.

The designs of A $ B are such that their interaction with any element of Agives an element of B. Since computation corresponds to interaction, thenA $ B looks like a good candidate to represent AÑ B.

Since we represent the canonical terms a of type A by designs A P |A|z´free,then we consider a superset of A $ B, whose designs D are such that forall A P |A|z´free, we have JD,AK P B. We will see later how this choicepermits us to construct a design that represents the graph of a function in asimple way.

Definition 30. Given two positive behaviours A,B of disjoint bases $ ξand $ σ, we define AÑ B as

AÑ B “ tD | @A P |A|z´free, JD,AK P Bu.

Lemma 31. Given two positive behaviours A and B, AÑ B is a behaviour.

Proof. We prove that AÑ B “ pAÑ BqKK.From Remark 1rem.1 AÑ B Ď pAÑ BqKK.Now we prove that pAÑ BqKK Ď AÑ B.

1. We first prove that ttA,Gu |A P |A|z´free,G P BKu Ď pAÑ BqK.Let A P |A|z´free and G P BK, for all D P A Ñ B JD,A,GK “JJD,AK,GK “ tzu, i.e., tA,Gu P pAÑ BqK.Then pAÑ BqKK Ď ttA,Gu |A P |A|z´free,G P BKuK.


2. Now we prove that ttA,Gu |A P |A|z´free,G P BKuK Ď AÑ B.Let F P ttA,Gu |A P |A|z´free,G P BKuK, i.e., F is such that forall A P |A|z´free, G P BK JF,A,GK “ tzu. Since JJF,AK,GK “JF,A,GK, then JF,AK P BKK. By hypothesis B is a behaviour, thenBKK “ B and JF,AK P B. Therefore F P AÑ B.

From (1) pA Ñ BqKK Ď ttA,Gu |A P |A|z´free,G P BKuK and from p2qttA,Gu |A P |A|z´free,G P BKuK Ď AÑ B. Thus pAÑ BqKK Ď AÑ B.

2.2.1.1 Principality and AÑ B

Unfortunately principality is not closed under the construction Ñ.Given two principal behavioursA andB, the behaviourAÑ B is not alwaysprincipal, as shown in the following example.

Example 18. Let A “ tAuKK, B “ tBuKK where

A “

α.0.2.0 $ α.0.1.0.0

$ α.0.2, α.0.1.0.0

α.0.1.0 $ α.0.2

$ α.0.1, α.0.2

α.0 $$ α A “ p`, α, t0uq

p´, α.0, t1, 2uq

p`, α.0.1, t0uq

p´, α.0.1.0, t0uq

p`, α.0.2, t0uq

B “

β.1.0.0 $

$ β.1.0

β.1 $

β.2.0.0 $

$ β.2.0

β.2 $

$ β B “ p`, β, t1, 2uq

p´, β.1, t0uq

p`, β.1.0, t0uq

p´, β.2, t0uq

p`, β.2.0, t0uq

A and B are principal.

E “

α.0.1.0.0 $ β.1.0

$ β.1.0, α.0.1.0

β.1 $ α.0.1.0

β.2.0.0 $

$ β.2.0

β.2 $

$ α.0.1.0, β

α.0.1 $ β

z$ α.0.2.0

α.0.2 $

$ α.0, β

α $ β


E “ p´, α, t0uq

p`, α.0, t1, 2uq

p´, α.0.1, t0uq p´, α.0.2, t0uq

zp`, β, t1, 2uq

p´, β.1, t0uq

p`, α.0.1.0, t0uq

p´, β.2, t0uq

p`, β.2.0, t0uq

|A|z´free “ tAu and JE,AK P B therefore E P AÑ B.Let us consider the path

pz “ p´, α, t0uqp`, α.0, t1, 2uqp´, α.0.1, t0uqp`, β, t1, 2uqp´, β.2, t0uqp`, β.2.0, t0uqp´, β.1, t0uqp`, α.0.1.0, t0uqp´, α.0.2, t0uqz

pz P VAÑB, but there exists no action κ` ‰z such that pκ` P VA$B. Indeedthe only positive action κ` ‰z in E that is not already in p is p`, β.1.0, t0uq,but pp`, β.1.0, t0uq is not a path, because the justifier κ of κ` does not belongto xpy.From Proposition 6prop.6 follows that AÑ B is not principal.

Principality is a very strong property that is not stable under ” Ñ ”.

In the previous example we show that there is a path pz that is visitable inA Ñ B and there exist no proper, positive action κ` such that pκ` is stillvisitable in AÑ B, i.e., p can not be extended. From Proposition 6prop.6this implies that AÑ B is not principal.Let us consider two principal behaviours A and B, what can we say aboutthe visitable paths pz that can not be extended in AÑ B?Let F P |A Ñ B| and pz be a visitable path of F such that p can not beextended.Let κ´0 be the last action of p. If κ´0 is an action of B, then there exists anegative action κ´ that extends κ´0 in BK. Since B is principal, then we canextend p with κ´ (contradiction). Therefore the last action of p belongs toa design of AK.This means that p “ qrsz, where q, s only contain actions of AK and eitherr is empty or it only contains actions of B. Let r be empty, then the onlyaction of B in pz is z. Since B is principal, then we can extend p (contra-diction). Then r is not empty.Therefore p starts in AK, then it passes in B and continues in AK to z. This


means that pz starts reading a part of the argument of the function, thenlooks at the output and finally finishes to read the argument. Therefore thepaths pz that can not be extended, represent functions that give the outputbefore having finished to read the argument. We wonder if a "new princi-pality" that accepts these functions could be stable under Ñ.

New "principality":A behaviour A is "principal" when @D P |A|

• either DE P |A|z´free such that D P tEuz

• or there exists a path pz P PtDu X VA of the following form

pz “ w1κ`1 w2κ

`2 w3κ

´1 w4κ

´2 z

where κ`1 justifies κ´1 and κ`2 justifies κ´2 .

Conjecture: Let A and B be "principal" behaviours, then A Ñ B is aprincipal behaviour.

In next section we generalize the construction A Ñ B to propose a rep-resentation of dependent functions. Since principality is not stable under” Ñ ”, then we will have the same problem with its generalization. There-fore we will not investigate about principality in next section.

2.2.1.2 Encoding

We represent the type of natural numbers and lists by positive behaviourson an atomic base. This feature really simplifies the definition of functionsof them. Let A and B be based on $ α and $ β, then the behaviour AÑ Bhas got base α $ β and to have a positive atomic base would simplify therepresentation of higher order types as for instance pA Ñ Bq Ñ C. Wedecide then to encode the elements of A Ñ B to define a behaviour on apositive, atomic base that represents the type AÑ B.

Definition 31. Given a design D with base α $ β, where α “ γ.0.0.0 andβ “ γ.0.1, we define the design

Dcod “

Dcod

$ γ wherecod

$ γ “

γ.0.0.0 $ γ.0.1

$ γ.0.0, γ.0.1

γ.0 $

$ γ .


Dcod “ p`, γ, t0uq

p´, γ.0, t0, 1uq

p`, γ.0.0, t0uq

D

Since we choose arbitrarily the base of designs we consider negative basesα $ β where α and β have a common prefix.

Definition 32. Given a behaviour A we define the behaviour Acod as

Acod “ tAcod |A P AuKK.

From now we always suppose that a behaviour with base α $ β can beencoded to obtain a behaviour on an atomic positive base $ γ.

2.2.2 Dependent Functions

In this section we generalize to the type of dependent functions the mod-elling given for the type Ñ. Then we define some examples of functions onnatural numbers and lists.

Let us give a type A and a family of types pBxqx:A, suppose that theyare respectively represented by means of the behaviour A and the family ofbehaviours pBxqxP|A|z´free . We represent the type of dependent functionsfrom A to pBxqx:A, noted pΠx : AqBx, by the set of designs (based on α $ β)pΠx P |A|z´freeqBx defined below.

pΠx P |A|z´freeqBx :“ tD | @A P |A|z´free, JD,AK P BAu

When there is no dependency on A, pΠx P |A|z´freeqB is equal to AÑ B.

Lemma 32. The set pΠx P |A|z´freeqBx is a behavior, i.e.,

pΠx P |A|z´freeqBx “ ppΠx P |A|z´freeqBxqKK.

Proof. • From Remark 1rem.1 pΠx P |A|z´freeqBx Ď ppΠx P |A|z´freeqBxqKK.

• Now we prove that ppΠx P |A|z´freeqBxqKK Ď pΠx P |A|z´freeqBx.

1. We first prove that ppΠx P |A|z´freeqBxqKK Ď

ttA,Gu |A P |A|z´free,G P BKAuK.

Let A P |A|z´free and G P BKA . For all D P pΠx P |A|z´freeqBx

we have JD,A,GK “ JJD,AK,GK “ tzu. This means thattA,Gu P ppΠx P |A|z´freeqBxq

K.Thus ppΠx P |A|z´freeqBxq

KK Ď ttA,Gu |A P |A|z´free,G P BKAu

K.


2. Now we prove that ttA,Gu |A P |A|z´free,G P BKAuK Ď pΠx P

|A|z´freeqBx.Let F P ttA,Gu |A P |A|z´free,G P BKAu

K, i.e., F is such thatfor all A P |A|z´free, G P BKA we have that JF,A,GK “ tzu.Since JJF,AK,GK “ JF,A,GK, then JF,AK P BKKA . By hypothesisBA is a behaviour, then BKKA “ BA and JF,AK P BA. ThereforeF P pΠx P |A|z´freeqBx.

• From (1) we have ppΠx P |A|z´freeqBxqKK Ď ttA,Gu |A P |A|z´free,G P

BKAuK and from (2) ttA,Gu |A P |A|z´free,G P BKAu

K Ď pΠx P

|A|z´freeqBx. Therefore ppΠx P |A|z´freeqBxqKK Ď pΠx P |A|z´freeqBx.

We represent the type pΠx : AqBx by the behaviour pΠx P |A|z´freeqBx.Let us represent the family of terms pbxqx:A by the family of designs tBX P

|BX|z´free |X P |A|z´freeu. The function pλxqbx : pΠx : AqBx is suchthat for all x : A we have bx : Bx. We want to represent it with a designD P |pΠx P |A|z´freeqBx|z´free such that for all X P |A|z´free, we haveJD,XK “ BX P BX. Can we construct such a design?

When A is a separable behaviour (Definition 28defi.28) we can separatebetween them the designs of |A|z´free thanks to a set of visitable pathsppAqAP|A|z´free . These visitable paths give us a very simple way to define adesign that represents the graph of a function, i.e., the kind of design thatwe are asking for.

The idea is to define a design such that for all A P |A|z´free it containsthe chronicle xpAy (where pA characterizes A) followed by the design BA

that corresponds to the term ba.We define then the set of paths Dλ “ txpAyBA |A P |A|z´freeu. To provethat it is a design the hypothesis on the paths pA and the chronicles xpAy arefundamental.

Lemma 33. Let A be a separable behaviour characterized by the visitablepaths ppAqAP|A|z´free, let pBAqAP|A|z´free be a family of designs. Then the

set Dλ “Ť

AP|A|z´freexpAyBA is a design.

Proof. We prove that all the chronicles of Dλ are pairwise coherent.Let c and c1 be two distinct chronicles of Dλ.

• Either there exist A P |A|z´free, and two distinct chronicles q, q1 ofBA such that c “ xpAyq and c1 “ xpAyq1. In this case, since q and q1 arechronicles of the same design, then they are coherent. Therefore c andc1 are coherent.


• Or there exist A,A1 P |A|z´free such that A ‰ A1, c is a prefix of xpAy

and c1 is a prefix of xpA1y. Since A is separable, then xpAy and xpA1y arecoherent. Therefore c and c1 are coherent.

Therefore the chronicles of Dλ are pairwise coherent.

Since we represent canonical terms by z-free, material designs we have toprove that Dλ P |pΠx P |A|z´freeqBx|z´free. The hypothesis on the pathsxpAy are fundamental to prove this property.

Lemma 34. Given a separable behaviour A, a family of separable behaviourstBA |A P |A|z´freeu and a family of designs tBA P |BA|z´free |A P |A|z´freeuthe design Dλ belongs to |pΠx P |A|z´freeqBx|z´free.

Proof. By definition for all A P |A|z´free JDλ,AK “ BA and BA belongs toBA. Therefore Dλ P pΠx P |A|z´freeqBx.We prove by contradiction that Dλ P |pΠx P |A|z´freeqBx|.Let Dλ R |pΠx P |A|z´freeqBx|, then there exists a design E such thatE Ĺ Dλ and E P pΠx P |A|z´freeqBx. This means that there exists achronicle c such that c P Dλ and c R E. By definition of Dλ there existsA0 P |A|z´free and a chronicle c1 of BA0 such that c is a prefix of xpA0

yc1.Since E Ĺ Dλ, Dλ “

Ť

AP|A|z´freexpAyBA and for all A1 ‰ A0 P |A|z´free

the path xpA1y does not belong to A0, then the normalization of E and A0 isnot a design of BA0 .Therefore E R pΠx P |A|z´freeqBx (contradiction).

From Example 10example.10 there exist behaviours that are not separable.Since separability gives us a simple way to define functions, then we decideto consider only separable behaviours.

From Propositions 12prop.12, 16prop.16, 20prop.20 we have that NatKK,LKKn and pListpNatqqKK are separable behaviours. Therefore we can de-fine the design Dλ in the cases A “ pNatqKK, A “ pLnqKK and A “

pListpNatqqKK.

Notationin the following we denote ListpNatq by ListpNatq.

2.2.2.1 Examples of functions

In this section we propose some examples of functions on Nat and ListpNatq.We define the successor, the predecessor, the function that adds a term onthe head of a list, the function that eliminates a term from a list and thefunction that gives the length of a list.


Example 19. We construct the design Ds P pΠx : NatKKqNatKK such thatfor all x P Nat, Bx “ x+1, i.e., the design that represents the successorfunction.

From Lemma 12prop.12, for all x P Nat, we have that px is the maximalchronicle of x:p0 “ p`, α,Hq, p1 “ p`, α, t0uqp´, α.0, t1uqp`, α.1,Hq, etc.

We define then

Ds “Ť

xPNat px x+1

where px is the maximal chronicle of x and x+1 is based on $ β.

Ds “

H

$ β.1

$ β

H

$ β.2

$ β

...$ α.1.0, β

α.1 $ β

$ α.0, β

α $ βDs “p´, α,Hq

1

p´, α, t0uq

p`, α.0, t1uq

p´, α.1,Hq

2

p´, α.1, t0uq

...

Remark 17. The structure of the designs of Nat is very simple, indeed thedesigns of Nat have only one maximal chronicle. These chronicles are thepaths that "separate" the designs of Nat. This means that for any termpλxqbx : pΠx : NqBx, the design Dλ that represents it is

Dλ “Ť

xPNat px Bx

where px is the maximal chronicle of x

We remark that the design defined in the Example 19example.19 is not theonly one such that its interaction with any x P Nat gives the design x+1 PNat.Given pλxqbx : pΠx : AqBpxq there is a class of terms that are extensionallyequal to it, i.e., they compute the same function but in different ways. Wedefine a corresponding notion in Ludics.

Definition 33. Let D,D1 P |pΠx P |A|z´freeqBx|z´free, we say that D andD1 are extensionally equal in pΠx P |A|z´freeqBx, noted D “eq D

1, when forall A P |A|z´free, we have JD,AK “ JD1,AK “ BA P BA.

Example 20. We define a design S that is extensionally equal to the de-sign that represents the successor function (Example 19example.19).S tests if n “ 0 (in this case the result is directly 1), and if n ą 0 makes 2steps (to say that the result is at least 2) and copies the rest of actions of n,


n´ 1 steps, with Idtn-1u ( Definition 4example.4) to finally have n+1.

S “

H

$ β.1

$ β

Idtn-1u

σ.1 $ β.2

$ σ.0, β.2

$ σ.0, β

σ $ β S “ p´, σ,Hq

1

p´, σ, t0uq

p`, β, t0uq

p´, β.0, t1uq

p`, β.1, t0uq

p´, β.1.0, t1uq

p`, σ.0, t1uq

Idtn´1u

For all n P Nat, we have JS,nK “ n+1 P Nat. Moreover S P |pΠx :NatKKqNatKK|z´free. Therefore S “eq Ds (the Example 19example.19).

Example 21. Let B0 “ 0 and for all x P Nat, Bx`1 “ x.The predecessor (total) function may be represented by the following designDp P pΠx P NatKKqNatKK

Dp “Ť

xPNat px Bx

where px is the maximal chronicle of x

By definition JDp,0K “ 0 and for all x ‰ 0, we have JDp,xK “ x´ 1.

Dp “

H

$ β

H

$ β

H

$ β.1

$ β

...$ α.2.1, β

α.2 $ β

$ α.1.0, β

α.1 $ β

$ α.0, β

α $ β


Dp “p´, α,Hq

0

p´, α, t0uq

p`, α.0, t1uq

p´, α.1,Hq

0

p´, α.1, t0uq

p`, α.1.0, t1uq

p´, α.2,Hq p´, α.2, t0uq

1...

We define a design P1 that is extensionally equal to Dp.

P1 “

H

$ β

Idtx´1u

α.1 $ β

$ α.0, β

α $ βP1 “p´, α,Hq

0

p´, α, t0uq

p`, α.0, t1uq

Idtx´1u

p00 “ p´, α,Hqp`, β,Hq belongs to P1, then JP1,0K “ 0.Since p´, α, t0uqp`, α.0, t1uqIdtx´1u P P1, then @x ‰ 0, JP1,xK “ x´ 1.

We define a design that represents the construction function on lists of nat-ural numbers, i.e., the function that adds a term on the head of a list ofnatural numbers.

Example 22. We consider lists of natural numbers, then for all l : ListpNqthere is design Dξ

l P ListpNatq that represents l. We choose the path pl tobe the path that covers Dξ

l following the order of the elements of the list (seeLemma 15lemme.15). For instance:pnil “ p`, ξ,Hq,pD0 “ p`, ξ, t0, 1uqp´, ξ.0, t1uqp`, ξ.0.1,Hqp´, ξ.1, t1uqp`, ξ.1,Hq,pD0.0 “ p`, ξ, t0, 1uqp´, ξ.0, t1uqp`, ξ.0.1,Hqp´, ξ.1, t1uqp`, ξ.1, t0, 1uqp´, ξ.1.0, t1uq, p`, ξ.1.0.1,Hqp´, ξ.1.1, t1uqp`, ξ.2,Hq, etc.

The function which adds b1 in head position is represented by the designConsb1.

Consb1 “Ť

Dξl PListpNatqplD

σb1.l.

where the notations Dξl and Dσ

b1.l specify that these designs are respectivelybased on $ ξ and $ σ. We have to consider them with distinct bases, other-wise Consb1 would be based on ξ $ ξ that is not a base.


Dσb1

$ σ

Dσb1.0$ σ

...ξ.1.0.1 $ ξ.1.1, σ

$ ξ.1.0, ξ.1.1, σ

ξ.1 $ σ

$ ξ.1, σ

Dσb1.1$ σ

...$ ξ.1.1, σ

ξ.1 $ σ

$ ξ.1, σ

...$ ξ.0.1.1.0, ξ.1, σ

ξ.0.1.1 $ ξ.1, σ

$ ξ.0.1.0, ξ.1, σ

ξ.0.1 $ ξ.1, σ

$ ξ.0, ξ.1, σ

ξ $ σ

p´, ξ,Hq

Dσb1

p´, ξ, t0, 1uq

p`, ξ.0, t1uq

p´, ξ.0.1,Hq

p`, ξ.1, t1uq

p´, ξ.1,Hq

Dσb1.0

p´, ξ.1, t0, 1uq

...

p´, ξ.0.1, t0uq

p`, ξ.0.1.0, t1uq

p´, ξ.0.1.1,Hq

p`, ξ.1, t1uq

p´, ξ.1,Hq

Dσb1.1

p´, ξ.1, t1uq

...

p´, ξ.0.1.1, t0uq

...

We define a design C extensionnally equal to Consb1 , where b1 P N is repre-sented by the design B1 on the base $ σ.0.1 and l P ListpNq. The idea is toput b1 on the address σ.0.1 (the first element of a list based on σ), copy thefirst element a of l on the address σ.1.0.1 (the second element of a list basedon σ) and b, i.e., the rest of l, on σ.2. In this way we obtain a list based onσ whose first element is b1 and then l, i.e., Dσ

b1.l.


C “

Dσb1

$ σ

B1

$ σ.0.1

σ.0 $

IdtAuξ.0.1 $ σ.1.0.1

$ σ.1.0.1, ξ.0

σ.1.0 $ ξ.0

IdtDbuξ.1 $ σ.2

$ σ.2, ξ.1

σ.1.1 $ ξ.1

$ σ.1, ξ.0, ξ.1

σ.1 $ ξ.0, ξ.1

$ ξ.0, ξ.1, σ

ξ $ σ

C “ p´, ξ,Hq

Dσb

p´, ξ, t0, 1uq

p`, σ, t0, 1uq

p´, σ.0, t1uq

B1

p´, σ.1, t1uq

p`, σ.1, t0, 1uq

p´, σ.1.0, t1uq

p`, ξ.0, t1uq

IdtAu

p´, σ.1.1, t1uq

p`, ξ.1, t1uq

IdtDbu

Since Nil “ p`, ξ,Hq and p´, ξ,HqDσb1 P C, then JC,NilK “ Dσ

b1.For all a.b P ListpNatq JC,Dξ

a.bK “ Dσb1.a.b.

Remark 18. As for Nat in the Remark 17rem.17, the representation of aterm pλxqbx : pΠx : ListpNqqBx has the form

Ť

Dξl PListpNatqpl BDξl

.

where pl covers Dξl following the order of the elements of l P ListpNq and

the family of designs tBDξl|Dξ

l P ListpNatqu represents the family of termspbxqx:ListpNq.

We define the representation of the function that eliminates the second ele-ment of a list of natural numbers.

Example 23. Let l2 be the list l without the element in the position 2 (Ifl has got at most one element, then l2 “ l) and l1 the list l2 without its firstelement (if there is one).

• If l “ nil or l “ a.nil, then l2 “ l and l1 “ nil.


• If l “ a.a2.b.nil, then l2 “ a.b.nil and l1 “ b.nil.

We define then the design El as

El “Ť

DlPListpNatq pl Dσl2.

We define a design F extensionally equal to El

F “

H

$ σ

IdtAuξ.0.1 $ σ.0.1

$ σ.0.1, ξ.0

σ.0 $ ξ.0

IdtDl1uξ.2 $ σ.1

$ ξ.1.0, ξ.1.1, σ.1

ξ.1 $ σ.1

$ ξ.1, σ.1

σ.1 $ ξ.1

$ ξ.0, ξ.1, σ

ξ $ σ

F “ p´, ξ,Hq

p`, σ,Hq

p´ξ, t0, 1uq

p`, σ, t0, 1uq

p´, σ.0, t1uq

p`, ξ.0, t1uq

IdtAu

p´, σ.1, t1uq

p`, ξ.1, t1uq

p´, ξ.1, t0, 1uq

p`, ξ.1.1, t1uq

IdtDl1u

If l “ nil, then JF,DξnilK is Dσ

nil “ p`, σ,Hq.Otherwise the idea is to copy the first element of l2 on the address σ.0.1 andthe rest of l2 on the address σ.1. In this way we obtain the representation ofl2 on the base $ σ.For all Dξ

l P ListpNKKq, JDξ

l ,FK gives as result Dσl2.

We define a design that may represent the function that associates its lengthto a list.

Example 24. We define a design Length such that for all l : ListpNqof length n : N represented by means of Dl P ListpNatq, we have thatJLength,DlK “ n.


Let l : ListpNq be represented by means of Dl, and pl be the path that coversDl following the order of the elements of l. We define then Length as follows

Length “Ť

lPListnpNq pln

Length “

H

$ σ

H

$ σ.1

$ σ

...$ ξ.1.0, ξ.1.1, σ

ξ.1 $ σ

$ ξ.1, σ

...ξ.0.1.0, ξ.1, σ

ξ.0.1 $ ξ.1, σ

$ ξ.0, ξ.1, σ

ξ $ σ

p´, ξ,Hq

p`, σ,Hq

p´, ξ, t0, 1uq

p`, ξ.0, t1uq

p´, ξ.0.1,Hq

p`, ξ.1, t1uq

p´, ξ.1,Hq

n

p´, ξ.1, t0, 1uq

...

p´, ξ.0.1, t0uq

...

For all n P N, for all l P ListnpNq represented by means of Dl P ListpNatq,the path pl covers Dl and pln P Length. Therefore for all Dl P ListpNatq wehave JLength,DlK “ n.

By definition, for all Dl P ListpNatq there exists n P N such that Dl containsa chronicle whose last action is p`, ξ.n,Hq. This chronicle tells us the lengthof the list l.

We can then define another design G that tells us the length of a list, i.e.,@Dl P ListpNatq, if n is the length of l, then JG,DlK “ n.


G “

0$ σ

1$ ξ.0, σ

2$ ξ.0, σ

...$ ξ.2.1, ξ.0, σ

ξ.2 $ ξ.0, σ

$ ξ.1.1, ξ.0, σ

ξ.1 $ ξ.0, σ

$ ξ.0, ξ.1, σ

ξ $ σ

p´, ξ,Hq

0

p´, ξ, t0, 1uq

p`, ξ.1, t1uq

p´, ξ.1,Hq

1

p´, ξ.1, t1uq

p`, ξ.1.1, t1uq

p´, ξ.2,Hq

2

p´, ξ.2, t1uq

...

2.2.3 Pairs

In this section we propose a representation for pairs. Principality is notstable w.r.t. this representation, but when we consider pairs on principal,separable behaviours, then we can easily define functions on them.

Let A be a type and pBxqx:A a family of types respectively represented bymeans of the behaviour A (based on $ ξ.0.1) and the family of behaviourspBxqxP|A|z´free (based on $ ξ.1.1 that we shorten as $ ξ.1). Let a : A andb : Ba, we represent the pair pa, bq by the design CA,B defined as follows

CA,B “

A$ ξ.0.1

ξ.0 $

B$ ξ.1

ξ.1 $

$ ξCA,B “ p`, ξ, t0, 1uq

p´, ξ.0, t1uq

A

p´, ξ.1, t1uq

B

EΣ :“ tCA,B |A P |A|z´free, B P |BA|z´freeu

We denote the type of pairs of this form by pΣx : AqBx.


Remark that CA,B may be seen as a tensor product d defined in [13].The elements of EΣ are p`, ξ, t0, 1uqpp´, ξ.0, t1uqAY p´, ξ.1, t1uqBq, for allA P |A|z´free and B P |BA|z´free, that we denote as wA d w1B wherew “ p`, ξ, t0uqp´, ξ.0, t1uq and w1 “ p`, ξ, t1uqp´, ξ.1, t1uq.

We remark thatEΣ is not principal, even ifA is principal and pBAqAP|A|z´freeis a family of principal behaviours. In the following example we define abehaviour A and a family of behaviours pBAqAP|A|z´free such that EΣ Ĺ

|pEΣqKK|z´free. In particular in |pEΣq

KK|z´free there are also some designsthat do not belong to EΣ and represent pairs pa, bq such that a : A and thereexists a1 : A, a ‰ a1 such that b : Ba1 .

Example 25. Let A “ tA1,A2,A3,A4uKK as in the Example 10example.10

that is

A1 “

σ.1.1.0 $

$ σ.1.1

σ.1.2.0 $

$ σ.1.2

σ.1 $$ σ

p`, σ, t1uq

p´, σ.1, t1uq

p`, σ.1.1, t0uq

p´, σ.1, t2uq

p`, σ.1.2, t0uq

A2 “

σ.1.1.0 $

$ σ.1.1

σ.1.2.3 $

$ σ.1.2

σ.1 $$ σ

p`, σ, t1uq

p´, σ.1, t1uq

p`, σ.1.1, t0uq

p´, σ.1, t2uq

p`, σ.1.2, t3uq

A3 “

σ.1.1.1 $

$ σ.1.1

σ.1.2.0 $

$ σ.1.2

σ.1 $$ σ

p`, σ, t1uq

p´, σ.1, t1uq

p`, σ.1.1, t1uq

p´, σ.1, t2uq

p`, σ.1.2, t0uq

A4 “

σ.1.1.1 $

$ σ.1.1

σ.1.2.3 $

$ σ.1.2

σ.1 $$ σ

p`, σ, t1uq

p´, σ.1, t1uq

p`, σ.1.1, t1uq

p´, σ.1, t2uq

p`, σ.1.2, t3uq

|A|z´free “ tA1,A2,A3,A4u.

Let BA1 “ tB1uKK, BA2 “ tB2u

KK, BA3 “ tB3uKK, BA4 “ tB4u

KK where


B1 “

β.1 $

$ β p`, β, t1uq, B2 “

β.2 $

$ β p`, β, t2uq,

B3 “

β.3 $

$ β p`, β, t3uq, B4 “

β.4 $

$ β p`, β, t4uq

By definition for i “ 1, 2, 3 |BAi |z´free “ tBiu.

We have then EΣ “ tCAi,Bi | i P t1, 2, 3, 4uu, for instance

CA1,B1 “

A1$ σ

ξ.0 $

B1

$ β

ξ.1 $

$ ξp`, ξ, t0, 1uq

p´, ξ.0, t1uq

A1

p´, ξ.1, t1uq

B1

where σ “ ξ.0.1 and β “ ξ.1.

The following designs E1 and E2 belong to pEΣqK.

E1 “p´, ξ, t0, 1uq

p`, ξ.0, t1uq

p´, σ, t1uq

p`, σ.1, t1uq

p´, σ.1.1, t0uq

p`, ξ.1, t1uq

p´, β, t1uq

z

p´, β, t2uq

z

p´, β, t3uq

z

p´, β, t4uq

z

p´, σ.1.1, t1uq

p`, ξ.1, t1uq

p´, β, t1uq

z

p´, β, t2uq

z

p´, β, t3uq

z

p´, β, t4uq

z


E2 “p´, ξ, t0, 1uq

p`, ξ.0, t1uq

p´, σ, t1uq

p`, σ.1, t2uq

p´, σ.1.2, t0uq

p`, ξ.1, t1uq

p´, β, t1uq

z

p´, β, t2uq

z

p´, β, t3uq

z

p´, β, t4uq

z

p´, σ.1.2, t3uq

p`, ξ.1, t1uq

p´, β, t1uq

z

p´, β, t2uq

z

p´, β, t3uq

z

p´, β, t4uq

z

Let us consider the design CA1,B2 that represents the pair pa1, b2q whereb2 : Ba2.

The path p “ p`, ξ, t0, 1uqp´, ξ.0, t1uqp`, σ, t1uqp´, σ.1, t1uqp`, σ.1.1, t0uqp´, ξ.1, t1uqp`, β, t2uq and the path q “ p`, ξ, t0, 1uqp´, ξ.0, t1uqp`, σ, t1uqp´, σ.1, t2uqp`, σ.1.2, t0uqp´, ξ.1, t1uqp`, β, t2uq belong to CA1,B2 and theirdual rp,rq are respectively chronicles of E1 and E2.

Therefore p and q are visitable paths of EΣ. Since p and q are coherentbetween them and xxtp, quyy “ CA1,B2 then CA1,B2 belongs to |EKKΣ |z´free.Therefore in |EKKΣ |z´free we do not find just the representation of pairspa, bq where a : A and b : Ba.Thus EΣ Ĺ |E

KKΣ |z´free.

From Example 25example.25 follows that principality is not stable w.r.t. ourrepresentation of pairs.The behaviour A that we consider in this example isnot a separable behaviour (Example 10example.10).For the moment we do not know if separability is stable w.r.t. the con-struction of pairs, but we decide to request the properties of separabilityand principality for A and for all A P |A|z´free BA to define functionson pairs generalizing the construction Dλ given in Section 2.2.2DependentFunctionssubsection.2.2.2.


2.2.3.1 Funtions on Pairs

Definition 34. Let A and @A P |A|z´free BA be separable, principal be-haviours. Let tpAuAP|A|z´free be the set of visitable paths that distinguishthe designs of |A|z´free (resp. tpBuBPBA

for |BA|z´free) following the Def-inition 28defi.28. Let D be a type and f a function from pΣx : AqBx to it,that is f : pΣx : AqBx Ñ D, for all a : A, b : Ba we have fppa, bqq : D.Let us represent the family of terms pfppa, bqqqa:A,b:Ba by a family of designspFA,BqAP|A|z´free,BP|BA|z´free

.

We represent then f by the design E

E “Ť

AP|A|z´free,BP|BA|z´freep´, ξ, t0, 1uqp`, ξ.0, t1uqxpAyp`, ξ.1, t1uqxpBy

FA,B.

Proposition 22. The set E of chronicles defined above is a design, i.e., allits chronicles are pairwise coherent.

Proof. Let c ‰ c1 be two chronicles of E, then

• If there exist A P |A|z´free, B P |BA|z´free and two chronicles w,w1 ofFA,B such that c is a prefix of p´, ξ, t0, 1uqp`, ξ.0, t1uqxpAyp`, ξ.1, t1uqxpBywand c1 is a prefix of p´, ξ, t0, 1uqp`, ξ.0, t1uqxpAyp`, ξ.1, t1uqxpByw1. Sincew and w1 are coherent, then c and c1 are coherent.Otherwise

• Either there exist A ‰ A1 P |A|z´free such that c is a prefix ofp´, ξ, t0, 1uqp`, ξ.0, t1uqxpAy and c1 is a prefix of p´, ξ, t0, 1uqp`, ξ.0, t1uqxpA1y.Since A is a separable behaviour, then xpAy and xpA1y are coherent.Therefore c and c1 are coherent.

• Or there existsB ‰ B1 P |BA|z´free such that c is a prefix of p´, ξ, t0, 1uqp`, ξ.0, t1uqxpAyp`, ξ.1, t1uqxpBy and c1 is a prefix of p´, ξ, t0, 1uqp`, ξ.0, t1uqxpAyp`, ξ.1, t1uqxpB1y. Since BA is a separable behaviour, then xpBy

xpB1y are coherent. Therefore c and c1 are coherent.

Remark 19. @A P |A|z´free, B P |BA|z´free we have that JE,CA,BK “FA,B.

Example 26. We define (following the construction given above) the designE such that for all x P N, for all pair c “ px, x+1q, we have JE, cK “ Fx,x+1.

E “Ť

xPNatp`, ξ, t0, 1uqp´, ξ.0, t1uq pxp`, ξ.1, t1uq px+1 Fx,x+1

where for all x P Nat, px is the maximal chronicle of x.


E “

F0,0

$ γ

F0,1

$ γ

...$ ξ.1.1.0, γ

ξ.1.1 $ γ

$ ξ.1.0, γ

ξ.1 $ γ

$ ξ.1, γ

...$ ξ.1, γ

...$ ξ.0.1.1.0, ξ.1, γ

ξ.0.1.1 $ ξ.1, γ

$ ξ.0.1.0, ξ.1, γ

ξ.0.1 $ ξ.1, γ

$ ξ.0, ξ.1, γ

ξ $ γ

E “ p´, ξ, t0, 1uq

p`, ξ.0, t1uq

p´, ξ.0.1,Hq p´, ξ.0.1, t0uq

p`, ξ.1, t1uq

p´, ξ.1,Hq

F0,0

p´, ξ.1, t0uq

p`, ξ.1.0, t1uq

p´, ξ.1.1,Hq

F0,1

p´, ξ.1.1, t0uq

...

p`, ξ.0.1.0, t1uq

p´, ξ.0.1.1,Hq

...p´, ξ.0.1.1, t0uq

...

We define now the representation of the projections on the first and thesecond component.

Example 27. We represent the projections on the first and the second com-ponent with the following designs

E1 “Ť


A.

E2 “Ť


B.

For all CA,B P EΣ, JE1,CA,BK “ A and JE2,CA,BK “ B.

We define two designs π1, π2 such that for all CA,B P EΣ, we have that


Jπ1,CA,BK “ A and Jπ2,CA,BK “ B, i.e., π1 and π2 are respectively exten-sionally equal to E1 and E2.

π1 “

IdtAuξ.0.1 $ α

$ ξ.0, ξ.1, α

ξ $ α , π2 “

IdtBuξ.1 $ α

$ ξ.0, ξ.1, α

ξ $ α .

π1 “ p´, ξ, t0, 1uq

p`, ξ.0, t1uq

IdtAu

π2 “ p´, ξ, t0, 1uq

p`, ξ.1, t1uq

IdtBu

Since p´, ξ, t0, 1uqp`, ξ.0, t1uqIdtAu P π1, p´, ξ, t0, 1uqp`, ξ.1, t1uqIdtBu Pπ2 and p`, ξ, t0, 1uqp´, ξ.0, t1uqA, p`, ξ, t0, 1uqp´, ξ.1, t1uqB P CA,B, thenJCA,B, π1K “ A and JCA,B, π2K “ B.

We define a design that may represent the function sum as follows.

Example 28. We represent a pair of natural numbers pn,mq by

Cn,m “

n$ ξ.0.1

ξ.0 $

m$ ξ.1

ξ.1 $

$ ξCn,m “ p`, ξ, t0, 1uq

p´, ξ.0, t1uq

n

p´, ξ.1, t1uq

m

The function sum is then represented by Ds

Ds “Ť

n,mPNatp´, ξ, t0, 1uqp`, ξ.0, t1uqpnp`, ξ.1, t1uqqm n`m

where pn is the maximal chronicle of n and qm is the maximal chronicleof m, n and m are respectively based on $ ξ.0.1 and $ ξ.1, while n`m isbased on $ α.

Ds “

0` 0$ α

0` 1$ α

...$ ξ.1.1.0, α

ξ.1.1 $ α

$ ξ.1.0, α

ξ.1 $ α

$ ξ.1, α

...$ ξ.1, α

...$ ξ.0.1.1.0, ξ.1, α

ξ.0.1.1 $ ξ.1, α

$ ξ.0.1.0, ξ.1, α

ξ.0.1 $ ξ.1, α

$ ξ.0, ξ.1, α

ξ $ α


Ds “ p´, ξ, t0, 1uq

p`, ξ.0, t1uq

p´, ξ.0.1,Hq p´, ξ.0.1, t0uq

p`, ξ.1, t1uq

p´, ξ.1,Hq

0` 0

p´, ξ.1, t0uq

p`, ξ.1.0, t1uq

p´, ξ.1.1,Hq

0` 1

p´, ξ.1.1, t0uq

...

p`, ξ.0.1.0, t1uq

p´, ξ.0.1.1,Hq

...p´, ξ.0.1.1, t0uq

...

We define the design S` extensionally equal to Ds.

S` “

G0

$ ξ.0, ξ.1, α

ξ $ α , where Gi “

F0

$ ξ.1, α.i

Gi`1

$ ξ.0.1.i.0, ξ.1, α.i` 1

$ ξ.0.1.i.0, ξ.1, α.i

ξ.0.1.i $ α.i, ξ.1 and

Fi “

H

$ α.i

Fi`1

$ ξ.1.i.0, α.i` 1

$ ξ.1.i.0, α.i

ξ.1.i $ α.i @i P N.

S` “ p´, ξ, t0, 1uq

p`, ξ.0, t1uq

S0

Si “ p´, ξ.0.1.i,Hq

p`, ξ.1, t1uq

F0

p´, ξ.0.1.i, t0uq

p`, α.i, t0uq

p´, α.i.0, t1uq

p`, ξ.0.1.i.0, t1uq

Si`1


Fi “ p´, ξ.1.i,Hq

p`, α.i,Hq

p´, ξ.1.i, t0uq

p`, α.i, t0uq

p´, α.i.0, t1uq

p`, ξ.1.i.0, t1uq

Fi`1

For all Cn,m we have JS`,Cn,mK “ n `̀̀m. The intuition behind S` is thefollowing: read n and stock step by step n on α and then do the same withm. In particular the design Gi reads n, while Fi reads m.

2.2.4 Dependent Record Types

In Section 2.1.4Recordssubsection.2.1.4 we recalled the presentation of recordsproposed in [13] and proved that this presentation preserves principality butnot separability. The records considered in this presentation do not have anydependency. Now we use the constructions defined in the previous sections(Nat, Ln, ListpNatq) to discuss an example of dependent record type in Lu-dics.In this section we propose a representation of the dependent record typeă l1 : N, l2 : Listl1pNq ą. We remark that the presentation of records of [13]does not represent dependency. Then we propose another representationsuch that we do not loose dependency.

Example 29. Let us consider Girard’s presentation of records [13].The record ă l1 “ n, l2 “ l ą of type L “ă l1 : N, l2 : Listl1pNq ą may berepresented by the design

n$ σ

Dξln

$ ξ

ăą$p´,ăą, tσuq

n

p´,ăą, tξuq

Dξln

where n P Nat and DξlnP Ln.

This design is material in the behaviour G “Ò NatKK& Ò ListpNatqKK.But |G|z´free does not contain only the designs that represent a record oftype L. It also contains the designs that represent a record whose first fieldcontains the object n : N and the second one a list of length m ‰ n of naturalnumbers. This means that with this representation we lose the dependencyof the record type.


How can we represent the record type L without loosing the expression ofdependency?

1. We first consider pŤ

nPNpÒ tnuKK& Ò LKKn q.

But we have the same problem as above: we can not distinguish betweenthe records of the form ă l1 “ n, l2 “ l ą where l : ListnpNq and theothers where l : ListmpNq and m ‰ n.

2. We consider then the same union but delocalized,Ţ

nPNpÒ tnuKK& Ò

LKKn q, i.e., we distinguish the designs between them on the first action..

For all i P N, for all ln P ListipNq

i$ σ.i

Dln

$ ξ.i

ăą$p´,ăą, tσ.iuq

i

p´,ăą, tξ.iuq

Dξ.iln

With this delocalization we can distinguish the records as we wanted,but p

Ţ

nPNpÒ tnuKK& Ò LKKn qKK “ T, i.e., the behaviour whose ele-

ments are all the designs based on ăą$ and of course it is too large.

3. Another possibility is the tensor d as for the representation of pairs(Section 2.2.3Pairssubsection.2.2.3). We consider the set of pairs whosefirst component is a natural number n, represented by n, and the secondone a list of length n, represented by Dln.

F “ tCn,Dln|n P Nat,Dln P Lnu.

The designs of F are of the following form

Cn,Dln“

n$ ξ.0.1

ξ.0 $

Dln

$ ξ.1

ξ.1 $

$ ξp`, ξ, t0, 1uq

p´, ξ.0, t1uq

n

p´, ξ.1, t1uq

Dln

Remark that Cn,Dlnmay be seen as a tensor product d defined in [13].

Let w “ p`, ξ, t0uqp´, ξ.0, t1uq and w1 “ p`, ξ, t1uqp´, ξ.1, t1uq, thenCn,Dln

“ pwnq d pw1Dlnq.

With this representation we can distinguish the records as we wanted,


i.e., for all n ‰ m Cn,DlmR FKK.

Let us justify this remark by defining a design G such that G P FK andfor all n ‰ m G M Cn,Dlm

.

For all n P N, for all Dln P Ln based on $ ξ.1, let c1n be the chronicleof Dln that represents the length of the list ln:

• c10 “ p`, ξ.1,Hq

• for all i ‰ 0, c1i “ p`, ξ.1, t0, 1uqp´, ξ.1.1, t1uq...p`, ξ.i` 1,Hq.

The idea to construct G is to visit the first component n of the pair,and then visit the chronicle c1n that is common to all the lists of lengthn. For all n P Nat let pn be its only maximal chronicle.We define G as follows

G “ tp´, ξ, t0, 1uqp`, ξ.0, t1uqpnp`, ξ.1, t1uqc1nz |n P Natu.

G “

z$

ξ.1 $

$ ξ.1

z$

ξ.2 $

ξ.1 $

$ ξ.1

...$ ξ.0.1.1.0, ξ.1

ξ.0.1.1 $ ξ.1

$ ξ.0.1.0, ξ.1

ξ.0.1 $ ξ.1

$ ξ.0, ξ.1

ξ $


p´, ξ, t0, 1uq

p`, ξ.0, t1uq

p´, ξ.0.1,Hq

p`, ξ.1, t1uq

p´, ξ.1,Hq

z

p´, ξ.0.1, t0uq

p`, ξ.0.1.0, t1uq

p´, ξ.0.1.1,Hq

p`, ξ.1, t1uq

p´, ξ.1, t1uq

p`, ξ.1.1, t1uq

p´, ξ.2,Hq

z

p´, ξ.0.1.1, t0uq

...

By definition G contains the dual of a path of each Cn,Dln, therefore

G P FK. For all n ‰ m, let n ă m, we have that G M Cn,Dlmbecause the normalizations fails on the action p´, ξ.n` 1,Hq. ThusCn,Dlm

R FKK.This means that we can distinguish the pairs pn, lnq from pn, lmq.

But with this representation we can not talk about subtyping as before.

To summarize:

• With the & representation we have a very natural representation ofsubtyping and we can represent record types with an infinite number offields, but we are not able to represent dependency.

• With the pairs representation (d) we can represent dependency, butwe lose the representation of subtyping and we can not represent recordtypes with a not a priori limited number of fields.

We propose to mix up these representations.For all n P N we consider the designs of the followign form


Fn,ăn,lną

n$ ξ.0.1

ξ.0 $

n$ ξ.1.0

Dln

$ ξ.1

ξ.1 $

$ ξp`, ξ, t0, 1uq

p´, ξ.0, t1uq

n

p´, ξ.1, t1uq

n

p´, ξ.1, t1uq

Dln

Fn,ăn,lną “ pwnq d pp`, ξ, t1uqpÒ n& Ò Dlnqq.

These designs represent a sort of pair whose first component contains theparameter n and the second one contains the record ă l1 “ n, l2 “ ln ąwhere ln : Listl1pNq.

We consider then G “ tFn,ăn,lną |n P N,Dln P Lnu.With GKK we can represent records with an infinite numbers of fields.Moreover we can represent the dependency, i.e., for all n ‰ m the designsFn,ăm,lną and Fn,ăn,lmą do not belong to GKK.Let us justify this remark by defining two designs E,D P GK such that for alln ‰ m we have E M Fn,ăm,lmą and D M Fn,ăn,lmą.

For all i P Nat, based on $ σ, let ci be the maximal chronicle of i. Weconsider JFaxσ$ξ.0.1, ciK whose result is the delocalization of i on the base$ ξ.0.1. We denote the maximal chronicle of this design by ciξ.0.1. We do thesame with the base $ ξ.1.0 and denote this chronicle by ciξ.1.0.

To construct E, for all i P N we follow the chronicle ciξ.0.1 that covers theparameter i based on $ ξ.0.1, then the chronicle ciξ.1.0 that covers the firstfield of the record, based on $ ξ.1.0.

E “ tp´, ξ, t0, 1uqp`, ξ.0, t1uqciξ.0.1p`ξ.1, t0uqciξ.1.0z | i P Nu

z$

ξ.1 $

$ ξ.1

z$

ξ.1.0.1 $

ξ.1.0 $

$ ξ.1

...$ ξ.0.1.1.0, ξ.1

ξ.0.1.1 $ ξ.1

$ ξ.0.1.0, ξ.1

ξ.0.1 $ ξ.1

$ ξ.0, ξ.1

ξ $


p´, ξ, t0, 1uq

p`, ξ.0, t1uq

p´, ξ.0.1,Hq

p`, ξ.1, t0uq

p´, ξ.1.0,Hq

z

p´, ξ.0.1, t0uq

p`, ξ.0.1.0, t1uq

p´, ξ.0.1.1,Hq

p`, ξ.1, t0uq

p´, ξ.1.0, t0uq

p`, ξ.1.0.0, t1uq

p´, ξ.1.0.1,Hq

z

p´, ξ.0.1, 1, t0uq

...

For each Fn,n,ln P G there exists a path p of Fn,ln such that its dual rp is achronicle of E. Therefore E P GK.

For all n ‰ m the normalization between Fn,m,lm and E fails on the ac-tion p´, ξ.0.1.n,Hq. Therefore Fn,m,lm R G

KK.

We define now a design D that belongs to GK and such that for all n ‰ mwe have Fn,n,lm M D. From which follows that Fn,n,lm R GKK.

We define F as the design G defined above. For all n P N, we have thatG K Cn,ln. Since Cn,ln Ă Fn,ăn,lną then G K Fn,ăn,lną, i.e., G P GK.For all n ‰ m, let n ă m, the normalization between G and Fn,n,lm fails onthe actions p´, ξ.n` 1,Hq. Therefore G M Fn,ăn,lmą, i.e., Fn,ăn,lmą R GKK.

With this representation we still have a sort of subtyping:

• For all n P Nat, since Cn,ln Ă Fn,ăn,lną, then Fn,ăn,lną P FKK. This

corresponds to the fact that the type of pairs pn,ă n, ln ąq is a subtypeof the type of pairs pn, lnq.

• Let us consider the set A of designs An,n that represent the pair pn, nqwhere the second component is based on $ ξ.1.0. For all n P N, we

2.3. SUMMARY AND COMMENTS 105

have An,n Ď Fn,ăn,lną. Therefore Fn,ăn,lną P AKK. This corresponds

to the fact that the type of pairs pn,ă n, ln ąq is a subtype of the typeof pairs pn, nq.

2.3 Summary and Comments

In this chapter we propose a representation for basic types and constructionsin Ludics. The representation of basic types holds the property of principal-ity. This is no more true with constructions. Anyway restricting ourself toseparable behaviour we find a very natural representation of functions andpairs.

As we recall in Chapter 1Ludics: Original Setting and New Developmentschapter.1,Terui proposes a representation of data in Computational Ludics [26]. Herepresents data by negative c-designs, while we choose a positive polarity.Quoting [26]: "We have chosen data to be negative c-designs, even thoughthey are positive "in spirit", as their main ingredients are positive actions(the negative action Ò is just used for adjusting polarity). The reason isthat a c-design may have in general multiple variables, for which negativec-designs can be substituted. Hence our choice allows for natural definitionsof multi-arity partial functions".We choose a positive polarity because data are usually positive, moreoverthe fields of a record are represented with a positive base [13] and one of ourgoal was to represent records with dependent types. On the other side wecould always change polarity of the objects we just defined adding a shift asfirst action (as Ò in [26]).

Surely Ludics has got interesting properties, but there is also a limitation:Ludics is linear. Such a restriction can be very constrictive, in particularwhen we want to represent dependent functions pΠx : AqBpxq. So far wepropose a representation of pΠx : AqBpxq in Ludics only when A is repre-sented by a behaviour with some further properties (a principal, separablebehaviour, Section 2.2.1Functionssubsection.2.2.1).Choosing to work in Ludics we choose a linear world instead that non-linearone. But there are also bright sides. For instance the characterization of theincarnation thanks to visitable paths [11] allows us to characterize the z´free

incarnation of a behaviour, i.e., the representation of canonical terms. So farthere is no such a characterization in the non-linear extensions of Ludics.

Chapter 3

Martin-Löf’s Type Theory andLudics

In this chapter we recall the basic types and the constructions of Martin-Löf’s Type Theory [20] and propose a representation for their rules in Ludics.For such a purpose we extend the results of Chapter 2Types and terms inLudicschapter.2.Having seen the applications of Ludics and Type Theory in Linguistics, thena representation of Martin-Löf’s Type Theory in Ludics would then be a firststep to fill the gap between these domains.

3.1 Martin- Löf’s Type Theory

Types have been studied from the initial motivation to improve the para-doxical structure of sets, and were found to be much closer to the notion ofcomputation, thanks to the Curry-Howard isomorphism. This latter is a one-to-one correspondence between logical systems and type systems such thatpropositions are mapped to types and proofs of a proposition are mapped toterms of the corresponding type. The Curry-Howard isomorphism (proposi-tions as types, proofs as programs) is the base to understand Martin Löf’sType Theory [19], [20]. A proposition is interpreted as a type whose termsrepresent the proofs of the proposition. A false proposition is then repre-sented by the empty type, while a true one by a non-empty type. A termof type A is a program whose execution gives a canonical term of type A,that corresponds to a cut-free proof. The original aim of Martin-Löf wasto propose a foundation for constructive mathematics. He defined a con-structive type theory where no external notion on types can be assumed.He starts from the very beginning defining the kind of judgements he wantsto use and their meaning. Together with the notion of judgement thereare two fundamental notions: the notion of type and the notion of canon-ical term of a certain type. The notion of canonical term is related with

107

108 CHAPTER 3. MARTIN-LÖF’S TYPE THEORY AND LUDICS

the notion of "computation". Given a type A the canonical terms of typeA are the terms constructed using the introduction rules for A (Section3.1.3Rulessubsection.3.1.3) while the non canonical terms are such that their"computation" gives a canonical term of type A. As we said above the firststep is the definition of the judgements we want to use. There are two kindsof judgements: categorical judgements and hypothetical judgements. Thefirst ones do not depend on any assumption, while the latter ones are madeunder assumptions.

We recall categorical and hypothetical judgements, then we define the rulesof formation, introduction, elimination and equality that describe types. InSection 3.2Martin-Löf’s Type Theory in Ludicssection.3.2 we recall the rulesfor basic types N, ListpNq, the constructions pΠx : AqBpxq, pΣx : AqBpxqand intensional equality.

3.1.1 Categorical Judgements

There are four forms of categorical judgement:

• $ A type may be read as A is a type.

• $ A “ B may be read as A and B are equal types.

• $ a : A may be read as a is a term of type A.

• $ a “ b : A may be read as a and b are equal terms of type A.

What do these judgements mean? What does it allow for?

• $ A type.To be able to say that A is a type we need to prescribe how a canonicalterm of A is formed, and when two canonical terms of type A are equal.We need then some rules that define the construction of canonical termsof type A, i.e., the introduction rules of A.

Example 30. Let N denote the set of natural numbers. To assert the judge-ment $ N type we have to say how a canonical term of type N is formedand how two equal canonical terms of type N are formed.

A canonical term of type N is either 0 or the successor of a term of typeN. This is expressed by the following rules.

$ 0 : N,$ a : N

$ sucpaq : N where sucpaq denotes the successor of a

The canonical term 0 is only equal to itself. If two canonical terms of Nare equal then they have equal successors.

3.1. MARTIN- LÖF’S TYPE THEORY 109

$ 0 “ 0 : N$ a “ b : N

$ sucpaq “ sucpbq : N.

• $ a : A

A term of type A is a "program" that if executed gives a canonicalterm of type A. Obviously if a is already a canonical term, then theprogram gives directly the term a.

• $ a “ b : A

Two terms of a type are equal if they are two "programs" such thattheir computation gives the same result, i.e., the same canonical term.

• $ A “ B

Two types A and B are equal if every term of type A is a term of typeB, every term of type B is a term of A, equal terms of type A are equalterms of type B and equal terms of type B are equal terms of type A.

3.1.2 Hypothetical Judgements

Categorical judgements are not enough to represent mathematical reasoning.We consider then judgements made under assumptions.

Let A be a type, then we have the following hypothetical judgements underone assumption, i.e., the assumption is of the form x : A.

• x : A $ Bpxq type .Bpxq is a type, assumed that x : A, i.e., Bpxq is a family of types overthe type A.

• x : A $ Bpxq “ Dpxq.Bpxq and Dpxq are equal types, assumed that x : A.

• x : A $ bpxq : Bpxq.bpxq is a term of type Bpxq, assumed that x : A.

• x : A $ bpxq “ dpxq : Bpxq.bpxq and dpxq are equal terms, assumed that x : A.

For every judgement we have the relative substitution rules, for instance forx : A $ Bpxq type we have

$ a : A x : A $ bpxq : Bpxq

$ bpaq : Bpaq

$ a “ c : A x : A $ bpxq : Bpxq

$ bpaq “ bpcq : Bpaq

We call the assumption x : A the context of the judgement and generalizethis notion with the following definition.


Definition 35. A context is a list (possibly empty) x1 : A1, ..., xn :Anpx1, ..., xn´1q where for all i P t1, ..., nu Aipx1, ..., xi´1q is a type underthe assumptions x1 : A1, ..., xi´1 : Ai´1px1, ..., xi´2q.

x1 : A1, ..., xi´1 : Ai´1px1, ..., xi´2q $ Aipx1, ..., xi´1q type

$ x1 : A1, ..., xi : Aipx1, ..., xi´1q ctx

where xi ‰ x1, ..., xi´1.

We often shorten a context by Γ in the following. We have then the judge-ment $ Γ ctx that must be read as Γ is a context.

We can rewrite the categorical judgements defined above with a context fol-lowing the notations of [24], but since there is no dependence on the context,then their meaning does not change.

3.1.3 Rules

Together with each type are given some rules to describe it. These rules areneeded to define new types, introduce their canonical terms, explain how tocompute functions on them.

For each type there are four kinds of rules:

• TYPE FORMATION, i.e., how to form a new type (eventually usingother types already defined)

• INTRODUCTION, i.e, how to form canonical terms of a type and whatare two equal canonical terms of a type.

• ELIMINATION, i.e., how to define operations on a type.

• EQUALITY, i.e., how to compute the functions defined thanks to theelimination rule on the canonical terms of a type.

When we define a new type A, there is a formation rule that states howto form it. Then we define the canonical terms of type A thanks to theintroduction rules. This is a very important step because it completely de-termines the type A, since the other rules will depend on it. Since a termof type A is a program such that its computation is a canonical term, thenthe elimination rules are determined. Indeed we can define functions on Aknowing how they work on canonical terms. The equality rules show thenhow to compute functions defined by the elimination rule.

In Section 3.2Martin-Löf’s Type Theory in Ludicssection.3.2 we recall therules for the basic types N, ListpNq, the constructions pΠx : AqBpxq, pΣx :AqBpxq and intensional equality.

3.1. MARTIN- LÖF’S TYPE THEORY 111

3.1.4 Universes

To strengthen the language Martin-Löf adds transfinite type to the construc-tions, i.e, he introduces universes.A universe U is a type whose elements are types. From Girard’s paradox[14] there is no type of all types, i.e., U can not be a term of itself. To avoidthis paradox a hierarchy on universes is introduced.

U0 : U1 : U2 : . . .

where for all i Ui is a term of Ui`1 and the terms of Ui are also terms ofUi`1, i.e, if A : Ui then A : Ui`1. When we say that A is a type, it meansthat A is a term of some universe Ui.Using universes we can define family of types, i.e., functions on a type A,F : AÑ U whose codomain is a universe.

Ludics models (a variant) of second-order multiplicative additive LinearLogic, then it would be interesting to investigate in which extent we couldrepresent universes in Ludics.For the moment we do not treat universes in our proposition to representconstructions of Martin-Löf’s type theory.

3.1.5 Comments

In this section we shortly introduce the Intuitionistic type Theory (ITT)of Martin-Löf [19], [20]. Type Theory is very connected with computationtheory [4] and recently inspired interesting developments connected to otherfields as homotopy theory [24]. We are particularly interested in the con-struction of dependent types of ITT, to make a first step in the representationof records with dependent types in Ludics. ITT is non linear, differently fromLudics, then we would like to represent a linear version of the constructionsΠ and Σ. One motivation to work on linear dependent types is that it is adomain quite unexplored where recently there have been interesting develop-ments [17], [27]. ITT is based on the Curry-Howard isomophism: proofs aremapped into programs and the execution of these programs correpond to theapplication of the cut-elimination procedure to the proofs. The correspond-ing of the cut-elimination procedure in Ludics is the notion of normalization(Chapter 1Ludics: Original Setting and New Developmentschapter.1). Thisnotion, also called interaction, is central in Ludics, since the objects of Lu-dics, called designs, are characterized from how they interact with otherdesigns. This means that ITT and Ludics are both based on the notion ofcomputation. This is another motivation to investigate a representation ofthe types of ITT in Ludics.


3.2 Martin-Löf’s Type Theory in Ludics

In Chapter 2Types and terms in Ludicschapter.2 we represented the objectsof type theory in Ludics. Now we focus on Martin-Löf’s Type Theory andpropose a representation of some basic types and constructions. In particularwe interpret their rules: formation, introduction, elimination and equality,where our interpretation of the conclusion follows from the interpretation ofthe hypotheses.In Sections 3.2.2The Rules of Nsubsection.3.2.2, 3.2.3The Rules of ListpNqsubsection.3.2.3we represent the basic types N and ListpNq, while in Sections 3.2.4The Rulesof Πsubsection.3.2.4, 3.2.5The Rules of Σsubsection.3.2.5 we consider theconstructions pΠx : AqBpxq and pΣx : AqBpxq. Then in Section 3.2.6Equalitysubsection.3.2.6we propose a representation for intensional equality.

To interpret the rules of Martin-Löf’s type theory we first have to interpretthe judgments.

3.2.1 Context and Judgements

In this section we propose a representation for hypothetical judgements.

In Chapter 2Types and terms in Ludicschapter.2 we propose a represen-tation of types and terms. A type A is represented by a behaviour A basedon $ α. The canonical terms of type A are represented by the designs of|A|z´free and the non-canonical terms by nets such that their normalizationis a design of |A|z´free.The equality between terms a “ b : A is represented by the equality of thenormalizations of the representation of a and the representation of b.

Now we want to represent the judgements of Martin-Löf’s Type Theory,then we start with the judgement $ Γ ctx that must be read as "Γ is acontext" and then consider the judgements of the form Γ $ A type.

To represent the judgement $ Γ ctx we first have to represent the vari-ables of the context Γ.

For each variable x we consider an address x‹. Let the variables of Γ bex1, ..., xn, then we shorten the addresses x‹1, ...,x‹n by Γ‹ (∆‹ for the vari-ables of the context ∆).

Once that we have represented the variables of Γ, we can represent thejudgement $ Γ ctx.

The judgement $ x1 : A1, ..., xn : Anpx1, ..., xn´1q ctx means that

3.2. MARTIN-LÖF’S TYPE THEORY IN LUDICS 113

• A1 is a type.

• if x1 : A1, then A2px1q is a type.

• if x1 : A1, ..., xn´1 : An´1px1, ..., xn´2q, then Anpx1, ..., xn´1q is a type.

We represent this judgement by the designs X1 P |A1|z´free, ...,Xn P |AnpX1, ...,Xn´1q|z´free and the behavioursA1,A2pX1q, ...,AnpX1, ...,Xn´1q.

Let the judgement $ A type be represented by means of a behaviour Abased on $ α. We first consider judgements that do not depend on a con-text, as for instance Γ $ A type. Let us consider the behaviour A, butbased on $ α,Γ‹. Since Ludics is affine, then we can enlarge the base of thebehaviour A adding the adresses of Γ‹ without changing the behaviour. Werepresent the judgement Γ $ A type by such a behaviour.In the case of judgements made under assumptions as Γ, x : A $ Bpxq typewe represent it by a family of behaviours tBpXq |X P |A|z´freeu based on$ β,x‹,Γ‹, where the address x‹ explicits the dependency.

Let us fix a term a : A and represent it by means of a design A P |A|z´freebased on $ α. Let x‹ “ γ.0.0 and α “ γ.0.0.0, that is α “ x‹.0. Then werepresent the judgement Γ $ Bpaq type by means of the behaviour obtainedfrom BpAq replacing each design BX based on $ β,x‹,Γ‹ by its normaliza-tion with the design Òα A. The result of such normalization is a design basedon $ β,Γ‹, that we denote by BA. We make an abuse of notation denotingthe behaviour obtained by BpAq.

BX

$ β,x‹,Γ‹, Òα A “

A$ α

x‹ $ p´,x‹, t0uq

A

JBX, Òα AK “BA

$ β,Γ‹.

Let the term bpxq : Bpxq be represented by means of a design BX P BpXq.We represent then the judgement Γ $ bpaq : Bpaq by the design BA P BpAqdefined above, based on $ β,Γ‹.


Type Theory Ludics$ Γ ctx X1 P |A1|z´free, ...,Xn P |AnpX1, ...,Xn´1q|z´free

Γ, x : A $ Bpxq type X P |A|z´free, BpXq

based on $ β,x‹,Γ‹

Γ, x : A $ bpxq : Bpxq R such that JRK “ BX P |BpXq|z´free

Γ, x : A $ bpxq “ cpxq : Bpxq R1,R2 such that JR1K “ JR2K P |BpXq|z´free

Γ, x : A $ Bpxq “ Cpxq BpXq “ CpXq

Γ $ bpaq : Bpaq BA “ JBX, Òα AK based on $ β,Γ‹

Γ $ Bpaq type BpAq obtained replacing each BX with BA

Now that we have a representation for the judgements we can consider therules for the basic types and the constructions.

We start with the type of natural numbers N.

3.2.2 The Rules of N

In this section we propose a representation for the rules of formation, intro-duction, elimination and equality of the type Nat.

• N-formation

Γ $ N type

This rule says that N is a type.

In Section 2.1.1Natural Numberssubsection.2.1.1 we have defined theset Nat whose designs have base $ σ. The definition of Nat followsthe inductive definition of natural numbers. Moreover Nat is principal,then |NatKK|z´free “ Nat. We decide then to represent the canonicalterms of type N by Nat and the type N by NatKK. As we explainedabove the judgements of the N-rules have a context Γ, then we con-sider the same elements of Nat but based on $ σ,Γ‹ and make anabuse of notation calling it Nat. Γ $ N type is then represented bythe behaviour NatKK.

• N-introduction

Γ $ 0 : NΓ $ a : N

Γ $ sucpaq : N


This rules tell us that a canonical term of N is either 0 or the successorof another term of N.

0 “H

$ σ,Γ‹ p`, σ,Hq a` 1 “

a$ σ.1,Γ‹

σ.0 $ Γ‹

$ σ,Γ‹p`, σ, t0uq

p´, σ.0, t1uq

a

The definition of Nat explains the N-introduction rules.

The design 0 with base $ σ,Γ‹ represents the term 0 : N and thefact that 0 P Nat represents the judgement Γ $ 0 : N.Let a : N be represented by means of the design a P Nat. We representthe term sucpaq by the design a` 1 and the judgement Γ $ sucpaq : Nby the fact that a` 1 P Nat.

• N-elimination

Γ $ c : N ∆ $ d : Cp0q ∆, x : N, y : Cpxq $ epx, yq : Cpsucpxqq

Γ,∆ $ Rpc, d, px, yqepx, yqq : Cpcq

The notation px, yqepx, yq means that epx, yq depends on x, y.

The N-elimination defines functions by induction on N. The com-putation of R is explained by the equality rules defined later. First weconsider the canonical term computed by c: either it is 0 or it is thesuccessor of some a : N. In the first case R gives as result d : Cp0q,in the latter case it executes epa,Rpaqq and obtain a canonical term oftype Cpsucpaqq. This process continues until when we reach the term 0.

Representation of the hypotheses:Let us represent the family of types tCpxq |x : Nu by the family of be-haviours tCpxq |x P Natu. The judgement Γ $ c : N is represented bya net C such that its normalization is a design of Nat. ∆ $ d : Cp0q isrepresented by a net such that its normalization is a design D P Cp0q.The judgement ∆, x : N, y : Cpxq $ epx, yq : Cpsucpxqq is representedby a family of designs based on $ γ,x‹,y‹,∆‹ tEx,y P Cpx` 1q |x PNat,y P |Cpxq|z´freeu.

Representation of the conclusion:Since Ludics is affine, then we can only represent linear induction. Inthat purpose we suppose to have already computed the terms ep0, dq, ep1, ep0, dqq,


etc respectively represented by means of the designs E0,D,E1,E0,D, etc.

We define then a design Rec that represents the graph of the functionRp_, d, px, yqepx, yqq in the following way.

Rec “Ť

iPNat qiFi

where @i ě 0, qi is the maximal chronicle of i, F0 “ D and Fi`1 “

Ei,Fi P Cpi` 1q.

Lemma 35. Rec is a design, i.e., its chronicles are pairwise coherent.

Proof. Let c ‰ c1 be two chronicles of Rec. By Lemma 7lemme.7, forall i P Nat we have that qi is a chronicle.

– Either there exist i P Nat and two chronicles ci, c1i P Fi such thatc “ qici and c1 “ qic

1i. Since ci and c1i belong to the same design

then they are coherent. Therefore c and c1 are coherent.– Or there exists i ‰ j P Nat, and two chronicles ci P Fi, cj P Fi such

that c is a prefix of qici and c1 is a prefix of qjcj . By definition ofNat follows that qi and qj start differ on a negative action on thesame address. Therefore c and c1 are coherent.

Rec “

D$ γ,∆‹

E0,D

$ γ,∆‹

E1,E0,D

$ γ,∆‹

...$ σ.2.0, γ,∆‹

σ.2 $ γ,∆‹

$ σ.1.0, γ,∆‹

σ.1 $ γ,∆‹

$ σ.0, γ,∆‹

σ $ γ,∆‹

p´, σ,Hq

D

p´, σ, t0uq

p`, σ.0, t1uq

p`, σ.1,Hq p`, σ.1, t0uq

E0,D p`, σ.1.0, t1uq

p´, σ.2,Hq

E1,E0,D

p´, σ.2.0, t1uq

...


Since Rec contains q0D “ p´, σ,HqD, then JRec,0K “ D P Cp0q.Moreover for all a` 1 P Nat we have thatRec contains qa`1Ea,Fa whereqa`1 is the maximal chronicle of a` 1 P Nat. Therefore JRec,a` 1K “Ea,Fa P Cpa` 1q.

The judgement Γ $ Rpc, d, px, yqepx, yqq : Cpcq is then representedby the net tRec,Cu.

We remark that Ex,y is based on $ γ,x‹,y‹,∆‹, while E0,D is basedon $ γ,∆‹. Indeed E0,D is obtained by the normalization of Ex,y withÒx‹ 0 and Òy‹ D.

• N-equality

Γ $ d : Cp0q Γ, x : N, y : Cpxq $ epx, yq : Cpsucpxqq

Γ $ Rp0, d, px, yqepx, yqq “ d : Cp0q

Γ $ a : N ∆ $ d : Cp0q Γ, x : N, y : Cpxqq $ epx, yq : Cpsucpxqq

Γ $ Rpsucpaq, d, px, yqepx, yqq “ epa,Rpa, d, px, yqepx, yqqq : Cpsucpaqq

The equality rules define the computation of the function R on thecanonical terms of N.

From above JRec,0K “ D P Cp0q and for all a` 1 P Nat, we haveJRec,a` 1K “ Ea,Fa that is Ea,JRec,aK P Cpa` 1q. These equalitiesdefine the computation of Rec. We recover exactly the computationof R on the canonical terms of type N defined by the N-equality rules,i.e., Rp0, d, px, yqepx, yqq “ d : Cp0q and Rpsucpaq, d, px, yqepx, yqq “epa,Rpa, d, px, yqepx, yqqq : Cpsucpaqq.

Summary


N Type Theory Ludics

FORM Γ $ N type NatKK based on $ σ,Γ‹

INTRO Γ $ 0 : N 0 P Nat

Γ $ a : N a P Nat

Γ $ sucpaq : N a` 1 P Nat

ELIM Γ $ d : Cp0q D P Cp0q

Γ, x : N, y : Cpxq $ epx, yq : Cpsucpxqq Ex,Y P Cpx` 1q

Γ $ Rpc, d, px, yqepx, yqq : Cpcq tRec,Cu

EQ Rp0, d, px, yqepx, yqq “ d : Cp0q, JRec,0K “ D P Cp0q,

Γ $ Rpsucpaq, d, px, yqepx, yqq “ JRec,a` 1K “epa,Rpa, d, px, yqepx, yqqq : Cpsucpaqq Ea,JRec,aK P Cpa` 1q

We remark that Nat does not contain other designs apart the interpretationof the canonical terms of N.

In Section 2.1.3Listssubsection.2.1.3 we represent lists of terms of type Aby a set of designs ListpAq. When A “ N we prove that this set is principaland the behaviour generated by it is separable. Moreover we define functionsof it. For we moment, when A ‰ N we are not able to prove its principalityand represent functions on it. For this reason we only consider here theparticular case of lists of natural numbers, i.e., ListpNq.

3.2.3 The Rules of ListpNq

In this section we propose a representation for the rules of the type of listsof natural numbers ListpNq.

• ListpNq-formation

Γ $ N type

Γ $ ListpNq type

This rule says that the lists of elements of type N, written ListpNq, isa type.

In Section 2.1.3Listssubsection.2.1.3 we define the set ListpNatq “ListpNatq based on $ ξ, that represents the lists of terms of type N. Asfor Nat in the previous section we consider the set ListpNatq based on$ ξ,Γ‹ and we make an abuse of notation calling it ListpNatq. We rep-resent the judgement Γ $ ListpNq type by the behaviour ListpNatqKK.


• ListpNq-introduction

Γ $ nil : ListpNqΓ $ a : N ∆ $ b : ListpNq

Γ,∆ $ pa.bq : ListpNq

where pa.bq denotes the list obtained adding the term a on the head ofthe list b.

A canonical term of ListpAq is either the empty list nil or pa.bq where ais a term of A and b is a term of ListpAq. For instance if A “ t1, 2, 3u,then p3.p1.nilqq is a canonical term of ListpAq.

Nil “H

$ ξ p`, ξ,Hq, Da.b “

A$ ξ.0.1,Γ‹

ξ.0,Γ‹ $

Db

$ ξ.1,∆‹

ξ.1 $ ∆‹

$ ξ,Γ‹,∆‹ @n ą 0.

Da.b “ p`, ξ, t0, 1uq

p´, ξ.0, t1uq

A

p´, ξ.1, t1uq

Db

The definition of ListpNatq explains the ListpNq introduction rules.

The design Nil represents the term nil : ListpNq and the fact thatNil P ListpNatq represents the judgement Γ $ nil : ListpNq.

Let the terms a : N and b : ListpNq be represented by means of thedesigns A based on $ ξ.0.1,Γ‹ and Db based on $ ξ.1,∆‹.The judgements Γ $ a : N and ∆ $ b : ListpNq are represented bythe fact that A P Nat and Db P ListpNatq. The term pa.bq : ListpNq isrepresented by the design Da.b and Γ $ pa.bq : ListpNq is representedby the fact that Da.b P ListpNatq.

• ListpNq-elimination

Γ $ c : ListpNq ∆ $ d : Cpnilq

∆, x : N, y : ListpNq, z : Cpyq $ epx, y, zq : Cppx.yqq

Γ,∆ $ listrecpc, d, px, y, zqepx, y, zqq : Cpcq


The ListpNq-elimination is the definition of functions by induction onListpNq. The explanation of the computation of listrec also explainsthe equality rules. The computation of the term c gives a canonicalterm of type ListpNq. This term is either nil or pa.bq for some a : Aand b : ListpNq. In the first case listrec gives the term d : Cpnilq,while in the latter it executes epa, b, listrecpb, d, px, y, zqepx, y, zqqq andobtain a canonical term of type Cppa.bqq.

Representation of the hypotheses:Let us represent the family of types tCpxq |x : ListpNqu by the familyof behaviours tCpXq |X P ListpNatqu. We represent Γ $ c : ListpNq,by a net C such that JCK P ListpNatq. The judgement Γ $ d : Cp0qis represented by a design D P CpNilq. The judgement ∆, x : N, y :ListpNq, z : Cpyq $ epx, y, zq : Cppx.yqq is represented by the family ofdesigns tEX,Y,Z P CpX.Yq |X P Nat,Y P ListpNatq,Z P |CpYq|z´freeubased on $ γ,x‹,y‹, z‹,∆‹.

Representation of the conclusion:As we explained for Nat we can only represent linear induction inLudics. We define a design Listrec that represents the graph of thefunction listrecp_, d, px, y, zqepx, y, zqq.

From Proposition 20prop.20, there exist a set of visitable paths pplqlPListpNqsuch that we can distinguish the designs of ListpNatq. For all Dl P

ListpNatq pl is the path that covers Dl following the order of the ele-ments of l.Let Gnil “ D, Ga.b “ EA,Db,JListrec,DbK .

We define Listrec as

Listrec “Ť

lPListpNq plGl.

Lemma 36. Listrec is a design.

Proof. From Lemma 20prop.20, for all Dl ‰ Dl1 P ListpNatq, we havethat pl and pl1 are two distinct coherent chronicles. Therefore Listrecis a set of pairwise coherent chronicles, i.e., it is a design.

Listrec is the following design


D$ γ

G0.nil$ γ

...ξ.1.0.1 $ ξ.1.1, γ

$ ξ.1.0, ξ.1.1, γ

ξ.1 $ γ

$ ξ.1, γ

G1.nil$ γ

...$ ξ.1.1, γ

ξ.1 $ γ

$ ξ.1, γ

...$ ξ.0.1.1.0, ξ.1, γ

ξ.0.1.1 $ ξ.1, γ

$ ξ.0.1.0, ξ.1, γ

ξ.0.1 $ ξ.1, γ

$ ξ.0, ξ.1, γ

ξ $ γ

p´, ξ,Hq

D

p´, ξ, t0, 1uq

p`, ξ.0, t1uq

p´, ξ.0.1,Hq

p`, ξ.1, t1uq

p´, ξ.1,Hq

G0.nil

p´, ξ.1, t0, 1uq

...

p´, ξ.0.1, t0uq

p`, ξ.0.1.0, t1uq

p´, ξ.0.1.1,Hq

p`, ξ.1, t1uq

p´, ξ.1,Hq

G1.nil

p´, ξ.1, t1uq

...

p´, ξ.0.1.1, t0uq

...

We remark that EX,Y,Z is based on$ γ,x‹,y‹, z‹,∆‹, while EA,Db,JListrec,DbK

is based on $ γ,Γ‹. Indeed EA,Db,JListrec,DbK is the result of the normal-ization of EX,Y,Z with Òx‹ A, Òy‹ Db, Òz‹ JListrec,DbK.

The judgement Γ $ listrecpc, d, px, y, zqepx, y, zqq : Cpcq is then repre-sented by the net tListrec,Cu.

For all Dl P ListpNatq, we have that the path pl covers Dl and plGl P

Listrec. Therefore JListrec,DlK “ Gl. This means that JListrec,NilK “D and JListrec,Da.bK “ EA,Db,JListrec,DbK .

• ListpNq-equality

Γ $ d : Cpnilq Γ, x : A, y : ListpNq, z : Cpyq $ epx, y, zq : Cppx.yqq

Γ $ listrecpnil, d, px, y, zqepx, y, zqq “ d : Cpnilq


Γ $ a : A Γ $ b : ListpNq $ d : Cpnilq

Γ, x : A, y : ListpNq, z : Cpyq $ epx, y, zq : Cppx.yqq

Γ $ listrecppa.bq, d, px, y, zqepx, y, zqq “ epa, b, listrecpb, d, px, y, zqepx, y, zqqq : Cppa.bqq

The ListpNq-equality explains the computation of listrec on the canon-ical terms of ListpNq.

From above we have JListrec,NilK “ D P CpNilq and JListrec,Da.bK “EA,Db,JListrec,DbK P CpDa.bq.

These equalities represent the computation of Listrec and we recoverexactly the computation of listrec.

The judgement Γ $ listrecpnil, d, px, y, zqepx, y, zqq “ d : Cp0q is rep-resented by JListrec,NilK, while Γ $ listrecppa.bq, d, px, y, zqepx, y, zqq “epa, b, listrecpb, d, px, y, zqepx, y, zqqq : Cpa.bq is represented byJListrec,Da.bK “ EA,Db,JListrec,DbK P CpDa.bq.


Summary

List(N) Type Theory LudicsFORM Γ $ ListpNq type ListpNatqKK

based on $ ξ,Γ‹

INTRO Γ $ nil : ListpNq, Nil P ListpNatq

Γ $ a : A A P |A|z´free

Γ $ b : ListpNq Db P ListpNatq

Γ $ pa.bq : ListpNq Da.b

ELIM Γ $ c : ListpNq C such that

JCK P ListpNatq

Γ $ d : Cpnilq D P CpNilq

x : A, y : Cpxq, z : Cpx, yq $ epx, y, zq : Cpx.yq EX,Y,Z P CpDx.yq

Γ $ listrecpc, d, px, y, zqepx, y, zqq : Cpcq tListrec,Cu

EQ Γ $ listrecpnil, d, px, y, zqepx, y, zqq “ d : Cpnilq JListrec,NilK “ D,

Γ $ listrecppa.bq, d, px, y, zqepx, y, zqq “ JListrec,Da.bK “

epa, b, listrecpb, d, px, y, zqepx, y, zqqq : Cpa.bq EA,Db,JListrec,DbK

We remark that ListpNatq does not contain other designs apart the inter-pretation of the canonical terms of ListpNq.

Now that we have represented the basic types, we consider the construc-tion pΠx : AqBpxq of dependent functions and pΣx : AqBpxq. We remarkthat in Ludics we represent with the same object the graph of a functionand its computation.

3.2.4 The Rules of Π

In this section we propose a representation for the rules of pΠx : AqBpxq.

Notation: in the following α “ γ.0.0.0, β “ γ.0.1 and x‹ “ γ.0.0, thatis α “ x‹.0.

• Π-formation

Γ $ A type Γ, x : A $ Bpxq type

Γ $ pΠx : AqBpxq type

This rule states that the product of a family of types is a type.


In Section 2.2.2Dependent Functionssubsection.2.2.2 we define a be-haviour pΠx P |A|z´freeqBx that allows us to represents the type ofdependent functions. This behaviour (once encoded) has got base $ γ.We enlarge its base with Γ‹ and still denote it by pΠx P |A|z´freeqBx.We represent the judgement Γ $ pΠx : AqBpxq type by the behaviourpΠx P |A|z´freeqBx.

• Π-introduction

Γ, x : A $ bpxq : Bpxq

Γ $ pλxqbpxq : pΠx : AqBpxq

A canonical term of the product of a family of types is the abstractionpλxqbpxq such that for all x : A, we have bpxq : Bpxq.

Dπ “

D$ x‹, β,Γ‹

γ.0 $ Γ‹

$ γ,Γ‹p`, γ, t0uq

p´, γ.0, t0, 1uq

D

Dπ is an encoding of D. It is the result of two delocalisations, that weneed to switch from the base $ β,x‹,Γ‹ of D to the base $ γ,Γ‹.

Let the judgement Γ, x : A $ bpxq : Bpxq be represented by a designD of base $ x‹, β,Γ‹. The first action of D, p`,x‹, t0uq, gives us thebase α $ β,Γ‹. Let us represent the family of terms tbpxq |x : Au bythe family of designs tBA |A P |A|z´freeu. Then for all A P |A|z´freewe have JD,AK “ BA P BpAq.

We represent then the judgement Γ $ pλxqbpxq : pΠx : AqBpxq by thefact that the design Dπ defined above belongs to pΠx P |A|z´freeqBx.

• Π-elimination

Γ $ t : pΠx : AqBpxq ∆ $ a : A

Γ,∆ $ Appt, aq : Bpaq

Appt, aq must be read t applied to a. The computation of t gives acanonical term pλxqbpxq. By a : A and the substitution rule we obtainbpaq : Bpaq. Calculating bpaq we find a canonical term of type Bpaq.As for the other types introduced above, the computation of Ap isexplained by the equality rules.


Tπ “

T$ x‹, β,Γ‹

γ.0 $ Γ‹

$ γ,Γ‹p`, γ, t0uq

p´, γ.0, t0, 1uq

T

decA “

A$ α,∆‹

γ.0.0 $ ∆‹

IdtBAu

β $ τ,Γ‹

$ γ.0, τ,Γ‹,∆‹

γ $ τ,Γ‹,∆‹ decA “ p´, γ, t0uq

p`, γ.0, t0, 1uq

p´, γ.0.0, t0uq IdtBAu

A

The Π-elimination considers terms t : pΠx : AqBpxq possibly non-canonical.

The judgement Γ $ t : pΠx : AqBpxq is represented by a net R whosenormalization is a design Tπ that belongs to pΠx P |A|z´freeqBx.

The judgement ∆ $ a : A is represented by a net whose normal-ization is a design A P |A|z´free based on $ α,∆‹.

For all A P |A|z´free the design decA decodes the encoding of Tπand then copies the design BA associated to A.

The judgement Γ,∆ $ Appt, aq : Bpaq is represented by the nettR, decAu.

The application of t to a, Appt, aq : Bpaq is then represented by thenormalization of this net, i.e., JR, decAK. Since JRK “ Tπ, then by asso-ciativity JR, decAK “ JTπ, decAK. By construction JTπ, decAK “ JT,AKthat is BA P BpAq.

• Π-equality

Γ, x : A $ bpxq : Bpxq ∆ $ a : A

Γ,∆ $ Apppλxqbpxq, aq “ bpaq : Bpaq.

The Π-equality defines the computation of Ap on canonical termspλxqbpxq : pΠx : AqBpxq.

Let pλxqbpxq be represented by Dπ as explained for the Π-introduction.


Then Γ,∆ $ Apppλxqbpxq, aq “ bpaq : Bpaq is represented by JDπ, decAK “BA P BpAq.

Summary

Π Type Theory LudicsFORM Γ $ pΠx : AqBpxq type pΠx P |A|z´freeqBx

based on $ γ,Γ‹

INTRO Γ, x : A $ Bpxq type BpXq

Γ, x : A $ bpxq : Bpxq D based on α $ β,x‹,Γ‹

Γ $ pλxqbpxq : pΠx : AqBpxq Dπ based in $ γ,Γ‹

ELIM Γ $ t : pΠx : AqBpxq R s.t. JRK “ Tπ

∆ $ a : A A based on $ α,∆‹

Γ,∆ $ Appt, aq : Bpaq tR, decAu

Appt, aq JR, decAK

EQ Γ,∆ $ Apppλxqbpxq, aq “ bpaq : Bpaq JDπ, decAK “ BA P BpAq

3.2.5 The Rules of Σ

In this section we propose a representation for the rules of the type pΣx :AqBpxq.

• Σ-formation

Γ $ A type Γ, x : A $ Bpxq type

Γ $ pΣx : AqBpxq type

The formation rule states that the disjoint union of a family of typesis a type.

In Section 2.2.3Pairssubsection.2.2.3 we define a set EΣ, based on $ ξ,that represent the pairs pa, bq where a : A and b : Bpaq. As we ex-plained in Section 2.2.3Pairssubsection.2.2.3, we suppose A and for allA P |A|z´free BpAq to be separable.Let us consider EΣ but based on $ ξ,Γ‹,∆‹ and still denote it byEΣ. We represent the judgement Γ,∆ $ pΣx : AqBpxq type by thebehaviour EKKΣ based on $ ξ,Γ‹,∆‹.

• Σ-introduction

Γ $ a : A ∆ $ b : Bpaq

Γ,∆ $ pa, bq : pΣx : AqBpxq


A canonical term of type pΣx : AqBpxq is a pair pa, bq where the firstcomponent a is a term of type A and the second component b is a termof type Bpaq.

CA,B “

A$ ξ.0.1,Γ‹

ξ.0 $ Γ‹

B$ ξ.1,∆‹

ξ.1 $ ∆‹

$ ξ,Γ‹,∆‹ CA,B “ p`, ξ, t0, 1uq

p´, ξ.0, t1uq

A

p´, ξ.1, t1uq

B

Let the judgements Γ $ a : A and ∆ $ b : Bpaq be represented by netssuch that their normalizations are respectively a design A P |A|z´freebased on$ ξ.0.1,Γ‹ and a designB P |BpAq|z´free based on$ ξ.1,∆‹.

The term pa, bq is represented by the design CA,B and Γ,∆ $ pa, bq :pΣx : AqBpxq is then represented by the fact that CA,B P EΣ.

• Σ-elimination

Γ,∆ $ c : pΣx : AqBpxq Γ, x : A, y : Bpxq $ dpx, yq : Cppx, yqq

Γ,∆ $ Epc, px, yqdpx, yqq : Cpcq

The elimination rule defines the function E over elements of pΣx :AqBpxq. First E executes the term c and gives a canonical term pa, bqwhere a : A and b : Bpaq. Then it substitutes a and b respectivelyfor x, y in dpx, yq to obtain dpa, bq : Cppa, bqq. This also explains theΣ-equality defined later.

Let us represent the family of types tCppx, yqq | a P A, b P Bpaqu bya family of behaviours tCpCA,Bq |CA,B P EΣu.

Γ,∆ $ c : pΣx : AqBpxq is represented by a net C such that itsnormalization is a design CA,B.

C “

A$ σ.0.1

σ.0 $ ,

B$ σ.1

σ.1 $,

IdtAuσ.0.1 $ ξ.0.1,Γ‹

$ ξ.0.1, σ.0,Γ‹

ξ.0 $ σ.0,Γ‹

IdtBuσ.1 $ ξ.1,∆‹

$ ξ.1, σ.1,∆‹

ξ.1 $ σ.1,∆‹

$ ξ, σ.0, σ.1,Γ‹,∆‹


C “ p´, σ.0, t1uq

A

, p´, σ.1, t1uq

B

, p`, ξ, t0, 1uq

p´, ξ.0, t1uq

p`, σ.0, t1uq

IdtAu

p´, ξ.1, t1uq

p`, σ.1, t1uq

IdtBu

For all A P |A|z´free, B P BpAq, JCK “ CA,B based on $ ξ,Γ‹,∆‹.

The judgement Γ, x : A, y : Bpxq $ dpx, yq : Cppx, yqq is repre-sented by a family of designs tDX,Y P CpCX,Yq |X P |A|z´free,Y P

|BpXq|z´freeu based on $ δ,x‹,y‹,Γ‹.

We represent the judgement Γ,∆ $ Epc, , px, yqdpx, yqq : Cpcq by thenet tC,Eu (Definition 34defi.34), where E is based on ξ $ δ,x‹,y‹.JE,CK “ JE,CA,BK and by definition for all CA,B JE,CA,BK “ DA,B.

• Σ-equality

Γ $ a : A ∆ $ b : Bpaq Γ, x : A, y : Bpxqq $ dpx, yq : Cppx, yqq

Γ,∆ $ Eppa, bq, px, yqdpx, yqq “ dpa, bq : Cppa, bqq

The Σ-equality explains the computation of E on canonical terms.

Since we represent the pair pa, bq with the design CA,B, then the judge-ment Γ $ Eppa, bqq “ dpa, bq : Cpa, bq is represented by JCA,B,EK “DA,B.


Summary

Σ Type Theory LudicsFORM Γ,∆ $ pΣx : AqBpxqtype EKKΣ

based on $ γ,Γ‹,∆‹

INTRO Γ, x : A $ Bpxq type BpXq

Γ $ a : A A P |A|z´free

∆ $ b : Bpaq B P |BpAq|z´free

Γ,∆ $ pa, bq : pΣx : AqBpxq CA,B P EΣ

ELIM Γ,∆ $ c : pΣx : AqBpxq C such that JCK “ CA,B

Γ, x : A, y : Bpxq $ dpx, yq : Cppx, yqq tDX,Y P CpCX,Yq |X P |A|z´free,Y P |BpXq|z´freeu

Γ,∆ $ Epc, px, yqdpx, yqq : Cpcq tC,Eu

EQ Γ,∆ $ Eppa, bq, px, yqdpx, yqq “ JCA,B,EK “ DA,B

dpa, bq : Cpa, bq

Up to now we proposed a representation for basic types N, ListpNq andthe constructions pΠx : AqBpxq, pΣx : AqBpxq. Now we consider the typeequality and propose a representation for intensional equality.

3.2.6 Equality

Equality between elements of a type and between types have already beendefined. There are two other kinds of equality: definitional equality andpropositional equality.

The first one is a relation between linguistic expressions, it is different frompropositional equality between objects (“) and it is denoted by ”.Propositional equality is an equality between objects such that we are allowedto operate on it with logical operations. There are two forms of proposi-tional equality: intensional equality and extensional equality. Thesetwo forms of equality have been presented by Martin Löf in two distinctpapers [19], [20], and the relation between them has been investigated inseveral works ( see e.g. [25], [1]).Intensional equality makes a distinction between judgemental and proposi-tional equality, while extensional equality identifies them.

Two terms are intensionally equal when they are exactly the same, whiletwo terms extensionally equal can have distinct forms as for instance two


functions that have the same graph (two distinct programs that give thesame result). In particular if two terms are intensionally equal then they areextensionally equal, while the inverse implication does not hold.

Example 31. We define two functions f and g on N that are extensionallyequal but not intensionally.Let f “ Rpa, 0, px, yq0q and g “ Rpa, 0, px, yqRpx, 0, px, yq0qq. Since f and gare distinct, then they are not intensionally equal. On the other side for alla : N, f applied to a and g applied to a give the same result 0. This meansthat they are extensionally equal and both represent the constant function 0.

During the last years Homotopy Type Theory has been developed. Thisproject involves intensional type theory and intensional equality is central[24]. This work interprets equality from an homotopical point of view, repre-senting terms as points and equality between terms as paths between thesepoints. The path between points can be intuitively seen as a deformation ofa proof into another. This approach looks very interesting, anyway in thefollowing we will only consider the original theory introduced by Martin-Löf.

What is the meaning of equality in Ludics? We represent a type A witha behaviour A and its canonical terms with |A|z´free. Since if |A|z´free “|B|z´free. then |A|KKz´free “ |B|

KKz´free. Therefore we represent the equal-

ity between types A “ B by the equality between |A|z´free and |B|z´free.The equality between terms of a type, a “ b : A, means that a and byield the same canonical term of type A. We represent it by the equality ofthe normalizations of the nets Ra,Rb, that respectively represent a and b,JRaK “ JRbK P |A|z´free.

For what concerns propositional equality, for the moment we are only ableto propose a representation for intensional equality in Ludics.

3.2.6.1 Intensional Equality in Ludics

In this section we recall the rules for the type IdpA, a, bq and propose a rep-resentation for its rules of formation, introduction, elimination and equality.

Id-formation

Γ $ Atype Γ $ a : A Γ $ b : A

Γ $ IdpA, a, bqtype

Id-introduction

Γ $ a : A

Γ $ idpaq : IdpA, a, aq


Given a type A and two terms a, b of type A, we define the type IdpA, a, bq.Given a : A, idpaq is a canonical term of type IdpA, a, aq.

Lemma 37. Given a behaviour A based on $ α and two designs A,B P

|A|z´free, A “ B iff |tAuKK X tBuKK|z´free ‰ H.

Proof. For all A,B P |A|z´free, we have that |tAuKKXtBuKK| “ |tAuKK|X|tBuKK| “ tAuz X tBuz. Since A and B are both material in A, then theycan not be one included in the other. Therefore |tAuKKXtBuKK|z´free ‰ H(in particular it is equal to tAu) iff A “ B.

From the previous Lemma we have that |tAuKK X tBuKK|z´free is nonempty if and only if the designs A and B are equal. The type IdpA, a, bqis non empty if and only if a “ b. Moreover if A “ B, then |tAuKK XtBuKK|z´free “ tAu “ tBu. This means that if we represent the typeIdpA, a, bq by the behaviour tAuKK X tBuKK, then there would be only onecanonical term of type IdpA, a, bq exactly as we wish from Id-introduction.

The behaviour tAuKK X tBuKK looks then like a good candidate to rep-resent the type IdpA, a, bq.

Given a type A and a, b : A, we represent the type IdpA, a, bq by the be-haviour Ia,b “ tAuKK X tBuKK based on $ α, where the designs A andB respectively represent the terms a and b. To represent the judgementΓ $ IdpA, a, bqtype we consider Ia.b based on $ α,Γ‹.

From Lemma 37lemme.37, A “ B iff |Ia,b|z´free ‰ H. In particular|Ia,a|z´free “ tAu, then we represent the canonical term idpaq : IdpA, a, aqby the design A.

Id-elimination

Γ $ a : A Γ $ b : A Γ $ c : IdpA, a, bq

Γ, x : A, y : A, z : IdpA, x, yq $ Cpx, y, zqtype Γ, x : A $ dpxq : Cpx, x, idpxqq

Γ $ idpeelpc, dq : Cpa, b, cq

The elimination rule expresses the structural induction on the type IdpA, a, bq.This rule is a substitution rule for equal terms (in the type A). The termidpeelpc, dq evaluates c and if c has value idpaq then gives the value of dpaq.

Let us represent the family of types tCpx, y, zq |x : A, y : A, z : IdpA, x, yquand the family of terms tdpxq |x : Au by a family of behaviours tCXY,Z |X,Y P

|A|z´free,Z P |IX,Y|z´freeu and a family of designs tDX |X P |A|z´freeu.


We represent c : IdpA, a, bq by a net R such that JRK “ C P |Ia,b|z´free,i.e., if a “ b, then C “ A “ B.

The term idpeelpc, dq : Cpa, b, cq is then represented by a design Idpeel suchthat JIdpeel,RK P CA,B,C. To define such a design we suppose A separa-ble (Definition 28defi.28), i.e., such that we can distinguish the designs of|A|z´free with a set of visitable paths ppAqAP|A|z´free with some furtherproperties.

Definition 36. Let A be a separable behaviour, ppAqAP|A|z´free the setof visitable paths that characterize the designs of |A|z´free and tDA |A P

|A|z´freeu the set of designs that represent the terms pdpxqqx:A. We definethen Idpeel as follows:

Idpeel “ txpAyDA |A P |A|z´freeu.

Remark 20. Since A is a separable behaviour, then for all A ‰ A1 P|A|z´free the chronicles xpAy and xpA1y are distinct and coherent. There-fore all the chronicles of Idpeel are pairwise coherent, i.e., it is a design.

The judgement Γ $ idpeelpc, dq : Cpa, b, cq is then represented by the nettIdpeel,Ru, where Idpeel has been defined above (Definition 36defi.36). Bydefinition, for all A P |A|z´free, JIdpeel,AK “ DA. Since JRK “ C, then byassociativity JIdpeel,RK “ JIdpeel,CK. If a “ b then C “ A and JIdpeel,CK “DA.

When c is a canonical term, i.e., there exists a : A such that c “ idpaq,we obtain the representation of the Id-equality.

Id-equality

Γ $ a : A Γ, x : A, y : A, z : IdpA, x, yq $ Cpx, y, zqtypeΓ, x : A $ dpxq : Cpx, x, idpxqq

Γ $ idpeelpidpaq, dq “ dpaq : Cpa, a, idpaqq

The Id-equality defines the computation of idpeel on canonical terms.

Since |Ia.,b|z´free is either empty (if A ‰ B) or equal to A (if A “ B),then we represent the canonical term idpaq by the design A. The judgementΓ $ idpeelpidpaq, dq “ dpaq : Cpa, a, idpaqq is represented by JIdpeel,AK “DA P CA,A,A. As for the other constructions, the interpretation of the elimi-nation and the equality rule are linked by the associativity of normalization.


Summary

Id Type Theory LudicsFORM Γ $ IdpA, a, bqtype Ia,b “ AKK XBKK

based on $ α,Γ‹

INTRO

Γ $ a : A

Γ $ idpaq : IdpA, a, aq IdA P |A $

A|z´free

Γ $ idpaq : IdpA, a, aq A P |Ia,a|z´free

ELIM Γ, x : A, y : A, z : IdpA, x, yqq $ Cpx, y, zqtype tCX,Y,Zu

Γ $ c : IdpA, a, bq JRK “ C P

|Ia,b|z´free

x : A $ dpxq : Cpx, x, idpxqq tDX |X P |A|z´freeu

DX P CX,X,X

Γ $ idpeelpc, dq : Cpa, b, cq tIdpeel,Ru

EQ Γ $ idpeelpidpaq, dq “ dpaq : Cpa, a, idpaqq JIdpeel,AK “ DA

3.2.6.2 Extensional Equality

For the moment we are not able to present a representation of the extensionalequality type in Ludics. We can just talk about extensional equality betweenfunctions. The point that we found particularly difficult to translate fromthe Eq-rules to Ludics is the notion of proof of a “ b : A. Given a behaviourA, the design IdA represents equality between functions that have exactlythe same chronicles, but this is not enough to recover equality between func-tions that have the same graph without having the same structure, that isextensional equality.

Conclusion and Future Works

Our goal was to represent the notion of type in Ludics, with a particularinterest in dependent record types. This interest is motivated from the ap-plications of Ludics and dependent record types in Linguistics [10], [7].We proposed a representation for basic types and constructions in Ludics.In particular we represented natural numbers, lists of natural numbers,records, dependent functions, pairs and discussed dependent record types.For our representation, we introduced the notion of principal and separablebehaviour. Principality is well-suited for representing faithfully canonicalterms and separablity gives us a tool to separate and then manipulate thedesigns that represent canonical terms. For all basic types our representationholds principality and for all but records it holds separability. We showedthat principality is closed with respect to MALL connectives ‘,b, Ò, Ó,&,but this notion is not stable under dependent product. We decided then torestrain ourself to separable behaviours and this allows us to define in a verynatural way dependent functions.We focused then on Martin-Löf’s Type Theory and proposed a representa-tion for the rules of N, ListpNq, pΠx : AqBpxq, pΣx : AqBpxq and intensionalequality.There is still lot of work to carry on: investigate more generally principalityand separability, in particular try to find a notion less strict than princi-pality that could be stable under dependent product and pairs. Moreoverwe have to find a generalization of the presentation of record proposed inSection 2.2.4Dependent Record Typessubsection.2.2.4. Finally it would beinteresting to investigate dependent types in a non-linear extension of Ludics[26], [2].

135

Bibliography

[1] Thorsten Altenkirch. Extensional equality in intensional type theory. In14th Symposium on Logic in Computer Science, pages 412 – 420, 1999.

[2] Michele Basaldella and Claudia Faggian. Ludics with repetitions (Ex-ponentials, Interactive types and Completeness). Logical Methods inComputer Science, 7(2):1–85, 2011.

[3] Michele Basaldella and Kazushige Terui. On the Meaning of LogicalCompleteness. Logical Methods in Computer Science, 6(4):1–35, 2010.

[4] Jan M. Smith Bengt Nordström, Kent Petersson. Programming inMartin-Löf type theory. An introduction. Oxford University Press, 1990.

[5] Gustavo Betarte and Alvaro Tasistro. Extension of Martin-Löf’s TypeTheory with Record Types and Subtyping. In Twenty-Five Years ofConstructive Type Theory, G. Sambin and J. Smith, eds. Oxford LogicGuides, 36. Oxford University Press, 1998.

[6] Robin Cooper. Austinian truth, attitudes and type theory. Research onLanguage and Computation, 2003.

[7] Robin Cooper. Records and record types in semantic theory. J. Log.Comput., 15(2):99–112, 2005.

[8] Claudia Faggian. On the dynamics of Ludics: a study of interaction.PhD thesis, Université d’Aix-Marseille, April 2002.

[9] Claudia Faggian and Martin Hyland. Designs, Disputes and Strategies.In CSL, pages 442–457, 2002.

[10] Marie-Renée Fleury, Myriam Quatrini, and Samuel Tronçon. Dialoguesin ludics. In Logic and Grammar, pages 138–157, 2011.

[11] Christophe Fouqueré and Myriam Quatrini. Incarnation in Ludics andmaximal cliques of paths. Logical Methods in Computer Science, 9(4):1–33, 2013.

137

138 BIBLIOGRAPHY

[12] Christophe Fouqueré and Myriam Quatrini. Ludics characterizationof multiplicative-additive linear behaviours. Manuscript avalaible athttp://arxiv.org/abs/1403.3772, march 2014. 29 pages.

[13] Jean-Yves Girard. Locus Solum: From the Rules of Logic to the Logicof Rules. Mathematical Structures in Computer Science, 11(3):301–506,2001.

[14] J.Y. Girard. Interprétation fonctionnelle et élimination des coupures del’arithmétique d’ordre supérieur. PhD thesis, Université Paris VII, 1972.

[15] J. M. E. Hyland and C.-H. Luke Ong. On full abstraction for PCF:.Information and Computation, 163(2):285–408, 2000.

[16] P. Tailor J. Y. Girard, Y. Lafont. Proofs and types. Cambridge Univer-sity Press, 1989.

[17] Ugo Dal Lago and Marco Gaboardi. Linear dependent types and relativecompleteness. In Proceedings of the 26th Annual IEEE Symposium onLogic in Computer Science, LICS 2011, pages 133–142, 2011.

[18] Alain Lecomte and Myriam Quatrini. Ludics and its applications tonatural language semantics. In Logic, Language, Information and Com-putation, 16th International Workshop, WoLLIC 2009, pages 242–255,2009.

[19] Per Martin-Löf. An intuitionistic Theory of Types : predicative part.Logic Colloquium ’73, pages 73–118, 1975.

[20] Per Martin-Löf. Intuitionistic Type Theory. Bibliopolis, Napoli, 1984.

[21] Francois Maurel. Un cadre quantitatif pour la Ludique. PhD thesis,Université Paris 7, November 2004.

[22] Paul-André Melliès. Asynchronous games 4: A fully complete modelof propositional linear logic. In 20th IEEE Symposium on Logic inComputer Science (LICS 2005), Proceedings, pages 386–395, 2005.

[23] Hanno Nickau. Hereditarily sequential functionals. In In proceedings ofthe Simposium on Logical Foundations of Computer Science: Logic atSt. Petersbourg. LFCS,1994,, pages 253–264, 1994.

[24] The Univalent Foundations Program. Homotopy type theory: Univalentfoundations of mathematics. Technical report, Institute for AdvancedStudy, 2013.

[25] Thomas Streicher. Investigations in Intensional Type Theory. Habilita-tion thesis. 1993.

BIBLIOGRAPHY 139

[26] Kazushige Terui. Computational ludics. Theor. Comput. Sci.,412(20):2048–2071, 2011.

[27] Matthijs Vákár. Syntax and semantics of linear dependent types. CoRR,Manuscript avalaibe at abs/1405.0033, 2014.

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