proc. of the 15ù International Cotloquium on Automata, Languagés Td ltg*ITming'Lecrufe Notes in compuær Science vol. 317, Springer (Berlin, 1988)' 188-200

THEcoMPLExITYoFMATn.txTn.ANsPoslTloNoNoNE-T^PEOFF-LTNE TUn.tNG MACHINES 1VITH OUTPUT TAPE

(Extcnded Abstnctl

by

Martln Dletzfetbingcr r'r'

Lehrsluhl lnformattl' llUnlverslttt DortmundPostfach S0 0S 00

D - +600 DorÈmund S0, F'R'G'

and

Wolfgang M¡acc r

pept.-of Mathemltlcl' Statlstlcs' and

ComPuter Sclencellniverslty of Illlnol¡ aÈ Chlcago

Box {348Chicago, tL 60680' U'S'A'

a Wrlttcn under prrtlrl suPPort by NSF-grrnt DCR-ES0| 217

Thl¡ çorL l¡ br¡ed on ¡ prrt'of thc flr¡t ru3hor'¡ Ph'D'-the¡lr

rt tbe llnlver¡lty of llllnol¡ rt cblcrgo, chlcr3o. Illlnol¡, ll's'A'¡a¡

ß

I{

o

ÂlsUrst.

A serles of existing lower bound results for one-tape Turing machlnes(TM'sl ls extended to the strongest such model for the computatlon

of functions: one-tape off-llne TM's with a write-only output tape'("Off-line' means: havlng a tÌvo-way input tape')

The following optlmal lower bound ls shown:

Computlng the transpose of Boolean f xf -matrices takes

A(fS'/21=-ç¡¡,15/41 stlps on such TM's. ln = 92 ls the length of the

tnput.)

-3-

Rl lntroductlon.

Durlng the last few years one has developed lower bound argumcntr

for a sequence of restricted Turing machlnes (TM's) of lncreasing

power. Techniques have been devised that allow onc to prove optlmal

superllnear lower bounds on the computation time for several concrete

computatlonal problems on one-ttpe TM's çithout lnput tape [2J, on

one-tape TM's wlth a one-way lnput tape ('on-line one-taPe TM'¡'l[{], and flnally on one-tape TM's wlth a two-way lnput tape ('off-llneone-tape TM's"¡ thts ls the standard model for the deflnltlon ofBpace-complextty clas_sesl. For thls model onc has proved an optlmal

lower bound of O(n3/2'/ llog n¡l/2¡ for the matrlx transposltlonfunctlon [5], and a barely auperlinear lower bound of O(n'log n/loglogn)for a related declsion problem [6J.

In thts psper we consider the next more powerful type of restrlctedTM'¡ (for whlch the preceding lower bound arguments do not sufflce):

off-llne one-tape TM's with an additlonal onc-way outPut tape. where¡e

the addition of the output tape obvlously makes no dtfference for solvlng

declsion problems, it was alre.ady noted tn [5] that these machlnes can do

metrlx tr"rrrpo"itll" t" Of"S/41 ¡tePs (as opposed to O(n3l2 /llo1 n¡l/2¡

cteps for the prevlously consldered verslon çithout outPut tape, where the

output has to tPpetr on the worl tapel'

Thlg ctronger model lc ¡lso of ¡ome lnterest from e technlcel polnt

of vler, because lt exhlblts a ferture that l¡ char¡cterl¡tlc for TM'¡

rlth cevcr¡l work taPet (whlch rrc to f¡r lntrrctrblc for lower

bound ¡rgumcntsl: thc extcn¡lve usc of thc rort tepc lr an lntermedlate

rtorage devlce. Thl¡ fcaturc plryed only ¡ mlnor rolc ln the analyrle

of mrtrfu tranaporltlon on one-tlpc off-llne TM'¡ tlthout outputtrpc, beceucc one could eaally ¡hoç that any usc of the work tepc

Ir ln lntcr¡nedl¡ts ¡tor¡gc devlce l¡ tnefflclent for ùbl¡ modal. (Oncc

ronc blÈ¡ h¡vc been çrlÈten on thc worl t¡PG, thcy crn bc movcd

lrtar only bt tlmc-conrumlng ¡tGGPs of thc rorl trpc berd: äurlng

i':'

-l-

each sweep only 3 log n btts ctn be moved' wherc n ls the length

of the lnput. The number of blts that can be moved ls È log n rather

thanconstantsincethetnputtapecanbeusedasaunarycounter'thus can store log n bits')r

In this paper, we prove an optlmal lower bound of O(n5/4) for the

transposition of Boolean matrices on one-tape off-line TM's wlth

outputtape.ThelowerboundargumentemploysKolmogorovcomplexltyto enable us to analyze the posslble flow of lnformatlon durtng

the transpositton of a sultably chosen matrlx on such ¡ machlne'

ThlsanalystsdiffersfrompreviouslowerboundargumentswlthKolmogorov complexity by its emphasls on the tlme-dlmension of

the computatlon: lt ls not enough to watch whlch lnformatlon ryreaches a certaln lnterval on the work tape, rather lt ls essentlal

to note which informatlon may be present ln such an lnterval at

specific time points. tn partlcular, the argument explolts the fact

that ln certain situations the same lnformatlon may have to be brought

lntothes8metapeareaseveraltimes(becauseefterlswasflrstbroughtthere,ithadtobeoverwrlttentomakesPaceforsomeother lnformatlon).

Moreover,theKolmogorovcomplexityLemmata(Lemma2andS)employ e new trick (from [1]), whtch allows us to Prove opttmal

lower bounds for matrlx tranposltlon even ln the case where the

matrlx entrles are slngle blts. (The technlque of [s] could only

handle the case wlth entrles of bltlength > log n,)

The followlng notlons and deflnltlons Ere used in thls paper' The

deflnttton of Turlng machlnes that we use ls atandard (cec e'g' t3])'A L-tape TM lc ¡ TM wtth k (read/wrlte) wort tepec. The work

tape alphebet l¡ assumed to be {0,1,8}. (lf larger worl tape alphabets

r Thl¡ fcature of one-tape TM'c wlth trro-Ì"ay lnput tape can be

used to shoç that such machlnes can slmulate f(nl-tlme-boundcd

L-tape TM's tn O(f(n)2llog n) steps'

-5-

Í rrere u¡ed, the lower bound ln thl¡ pePer would change by thcconstrnt fector t/log(lfl l.l The functlon MATRIX TRANSPOSTTION

l¡ lnduccd by t,he operetlon of trensposlng ¡ m¡trlx: glven ¡n lnputxe (Q,lln, ¡ = 12, regerd x as the representatlon of a Booleen matrlxAe{O,l}ftl in row-major order, and output the transPose Af ln row-mafor order (or, equlvalently, A ln column-maior orderl. That meanEr

tf the lnput lE x = blb2...bn wlth br,{0,1}, then the output ls

I'br6lb:r(21...b¡(nl, ìvhere Èhe permutatlon t of (1,2,...,n} ls deflned

by r((t-t).1+J) = (¡-l)'i+1. for I 3 ¡.¡ < t. (A variatlon of thlg functlonwas used ln [S,ó] for ceparatlng tto-taPe TM's from one-tape off-llneTM's wtthout output tape¡ before that, lt had occured tn [7] ac

an examplc of a permutation that is hard to reallze on devlces simllarto Turing machlnes.) .

.

The Kotmogoroy complexlty of a binary qtring ls deflned as follow¡.Let an effectlve codlng of all determ lnistlc Turing machlnes (wlthany number of tapesl as blnary strings bc given and assumc thatno code ts a preflx of any other codc. Denote the code of ¡ TM Mby lir{1. Then the Kolmogorov comglexlty of x, [0,1]r (w.r.t. thtc flxedcodingl lc

Klxl:. m¡ntlrMl ul lu¡{0,1}t, M on lnput u prtntr x}.

A rtrlng x.[O,l]' tc called lncompre¡slble lf K(xl > lxl. (A trtvtalcountlng ¡rgumenÈ ¡howa Èhat for rll n there lc ¡n x¡{0,1}n çlthKtx)>n.lrl.l

The peper lr orgenlzed l¡ follow¡: ln 32' çc ctlte thc thcorern ud¡lctch thc proof of thc upPcr bound; tn 393-5 rc lndlc¡te thc mllnrtcpr of thc proof of thc loçer bound¡ h 06 çc dlrcu¡¡ relrtedopen problcnr. (Thc proof of thc Kolmogorov complexlty Ler¡m¡trll 3lrcn l¡ thc Appendt:.!

-ó-

$2 Matn Result.

Theore¡r- The tlme complexlty of MATRIX TRANSPOSITTON on one-Èape

off-line Turing machlnes with a one-way output tape lsg¡ j5/2¡ = g¡nSze¡. (Here n = 12 ls the length of the lnput, whtch lsa Boolean 9x I -matrix given ln row-maJor order'l

R"m*L The lower bound still holds lf the output tape ls two-way,

and even lf the TM can prlnt several tlmes to each cell on the outputtape, ac long as the same symbol is prlnted each tlme. The changes

needed ln the proof are contained ¡n [1] and will be glven in the fullpaper.

Sketch of the oroof of the uDDer bound (Thts was already noted

,tn [5]): Let the lnput x.(0,1]I-."pr"."nt the Boolean f xl-matrlx

A = (ai¡)1<i,l< ¡, l.e. x = all...al Í42,1... a2n ..,tall...ar!.

Spllt A lnto submatric-es A¡, l. t(.Í 9l/2, where A¡ consis,t^s ofthe columns (k-l) ' ú/2 * l,].. ,k' Il/2 of A. For k = 1,2,"',Íllz compute

and output the transpose e[ of A¡ as follows: flrst, write A¡ lnro¡-wise order on the work-tape (this takes one sweep over the

lnput, hence Oll2l steps); then, output A¡ column by column (thlc

takes gl/2 "*""ps over the representatlon of A¡ on the work tape'

whtch conslsts o¡ i3/2 blts, hence Ol9:¿! stepsl. Altogether, A icprinted column by column, and 91f /2¡ - g¡n5/4) steps are made.

Of course, there are many varlatlons of thls method that all lead

to the destred effect, ln partlcular lf the output tape la two-way.In the lower bound proof lt ls shown that anv method for computlng

Aù ln c . tílz stepl, for ¡ sufftcleçtly rmall c > 0, mult act on

a conctant frectlon of the lnput blt¡ ln a way slmllar ec the method

luat deccrlbed. (The head movement¡ muct ¡lloç for copylng thercblts to a certaln worktape ares' end later uslng the lnforrnatlontn thls area to prtnt them on the output tapc')

-7-

ln thlc and thc followlng two ¡cctlonc wc outllnc the of of hcoro

Fixaone-tspeoff.lineTMMthatcomputesMATRlxTRANsPostTtoN.Choose f large enough (how large' can be seen- from the proofs

of the Kolmogoro" "ã-plexlty Lemmata), and fir an lncompressible

srrrng x,t',lln ,;;";"";': ;''. (^rru-" for slmplicltv that 1l/2¡220',¡1'¡

conslder the computatlon of M on x as lnpg-t, conslstlng of, rry,

T steps. \f,fe want to show that T > C a5/2' for some ftxed C (c'g"

Ç,2-20¡. The lnput x'blbz...bn represents A = (e¡¡)l<t¡-<-¡-t{0'illtl'

where "il = b(¡-tl'l+l'The outpt''i' '

= b'¡t(tlbrtZl"'bnf"¡-í"pt"s"nts At'

DeffniËon. (Printlng tlmes' prlnting phases')

Forllm<n, let

t^.(m) := the timestep at whlch the copy of b- is prlnted to thc

¡(m)-th outPut cell'

(Then tor(r(l)) < tor(n(2)) < "' ' tpt(¡(n));noten'n-l')

spltt t1,2,...,TÌ lnto l3/2 dlsloint lntervals Pt (the prlntlng pharerl'

co that each P, contcl rs exactly l/2 o¡ the'gd"sj ùlme¡ tn'(nl'

lnform¡llr, ìrc t¡lt of 1 aa the 'g!or' of prlntlng p-harc.Pr' Tr"

bltr bm whoce prlntlng tlme belon-gs to-fr.lnh-erlt the color:

lf t^-(m)¡P., botù copler of b- (the m-th lnput blt ¡nd the r(nl-th

outËlt blt)'¡re ¡¡ld to have color 1'

Flr¡t, we observe ¡ trlvlal fact: lf T ls to be < C ' 1S/2 ' thcn M

c¡nnot prlnt too ¡loçly' Morc preclsely'wc can rasunc r'l'o'g'

th¡ù ¡ fl2¡2of thc P, conalat of <t etepe' (Otherrl¡c' M nrkc¡

, ¡3t2 ñ; ' I ' f /2 ¡2 rtepr' rnd rc arc donc'l ltc choo¡c r ¡et

Gl s {1,2,...,t3/21of color¡ 1 auch that P., haa length <l for dl

1rG¡, and lGll ' ¡3f2 ¡2' Furthcr' re lct81 :. {nltp¡(rnleP, for ¡omc t'o¡l t!rr3 re1 tlt¡ rlth color ln G1l'

wc rrn focu¡ ;;t;";" iri" L tíc followl (end rcgard thc othcr

blt¡ t¡ 'u¡colored'l' .îtc ll¡t comc elmplc ob¡crratlon¡'

.fI

-8-

!-eEqqa 1.. Let 1eG¡. Then(") on the outputiape, the bits of color Y occupy an interval of

1l/2 ".¡1",

correspondlng to a part of a column of A;

(b) on the input tape, all bits of color t have distance >f from

one another;

(c) durlng P' the work tape head visiÈs <I cells'

(Proof omitted.)

.lñIe now turn to the aree on the work tape that the work tape head

scans during Pr. lntultlvely, at the beginning "f Pt-llå".11"" must

contain all the'information necessary to prinù the f rzz þ¡¿s of color

f. (By-Lemma I (b),(c), at most one blt of color 1 on the lnput tape

can be inspected during Pr. The incompressibility of x entails thatthe other bits of the lnpui, lnspected durlng P.l will not contaln

any information that can be used to weaken this requirement') For

technical reasonsr we need these areas to be either ldentlcal or

dtsJoint for different y. This can be achieved as follows, reducing

the number of useful (colored) bits only by a constant fraction'

Deftnltlon. ('Work tape lntervals.)spt" the work tape lnto blocks of I cells each. For 1eG1, let v,be an interval consistlng of two adlacent blocks such that durlng

V- (posslble by Lemma I fcl).o"t to the left end to the rlght

a set G2 s G1 wlthfor 1,y'eG2 the lntervels lV, and

ol. Let

BZ r. {raeB1 I tprfm)e P, for some 1eG2}, the set of bltr wlth color

ln-G2, "o¿io"rii on thL¡e colorE and bttc from here on.

There ¡re Eeveral waya for M to get the lnformatlon about t'hc

bit¡ of color 1 lnÈo v.r, before Pr. The mo¡t natural one glves rlreto the following deflnltlon.

Deflnltlon. (VtstbllltY.)i"t W bc any lnterval on the work topc, and b-' Î<m<n, any lnput

blt. \¡Ve say that brn ls-vlslble-from \f,t at steo t lf at rtep t the

lnput tape head ."äh" b- and the work tape head Ecans a cell ln lt'

-t- I

For b- I btt of color T, we know that M prlnts b- to cell ¡(ml.from vr-, ùhat l¡, whlle the work ttpe head ts tn

.vr..lntultlvely,It seems reasonable for M to makc such bits bm vlslble at least

from \il, at some step before -t'r(m),

to allow for a'dlrect" transferof b,,, fiom the lnput tape'to W, end from there to the outputt"p".-tQtherwlse, these bits would have to be moved by the work

tape head oyer >f cells, and thls t¡kes too much tlme.l By e KolmoSorov

complexlty argument we show ln the followlng lemme that ln factIt costs M too much tlme to devlate too far from thls obvlous strategy.

R{ Thc case of manv 'underlnformed' lnterval¡'

Lglqnt-a 2. (The 'underlnformed' lnterval.)i"t U, l,n,x be as above, I large enough. Assume that \f,r l¡ an lntervalof length 4 f on the work tap-e.,-and that V consists of the 2l cellstn the center of lt¡. Let r 2 il/2¡2, and assume that there are > rblts bm that are prlnted "from V' (1.e. at a tlme step when the work

t"p" nåìa ls ln V) but are never vlslble from rrV before being prlnted.

Then the work tape head spends > r'f /116' log n) steps ln tñ,|.

Proof: See Appendlx.

Thls lemma allows us to show th¡t lf M ts to make a ç' f /2 steps,

then lt cannot be the case that too many 'colored' blte b. rrenever vlslble from the assoctated lork tape lnterv¡l Wy before

betng prlnted from Yr.

Ñamely, ascume that there l¡ ¡ ret G3 É G2, wlthlG3l ' lczl /2 - t?lzttt, ¡nd r ¡ct 83 s 82, wlth

trjt . ttit t+. t2ts2, ¡uch that for ãech i,c¡ there ere cx¡ctry tl/2¡2tnãtc¿¡ m.B3 ¡o thet b- har color 1 end b- lr nevcr vl¡tble from lV,

.

beforc tnr(ml. Thcn for c¡ch 1e G3.thera ¡rG

I rï ,' l{1'rG3l Wï'Wî'lt ' ,lf2 /2

bttr bn thàt ¡att¡fy the hypotherl¡ of Lemme 2 rrtth_W ' It' Y '-Vr,nenelfell btt¡ wtth r color 1' euch thet Wr' . Wï. From Lcmm¡ 2

tt follow¡ that M cpendr t tï ' llll6' log irl stepc çtth thc worl

-to-

tape head ln Wf . By summlng up these boundc for ¡ family of 1tG3

that form a set of,"p,esentatlves for the equlvalence relatlon over

G3 deflned bf 1ñ1, = Wy' r we see Èhat M makes

> lG3l ' l9l/2/21 ' 9/(ló'log n).= '3l(512'log

n)

steps altogether, whlch^i:^t C ' 15/2' for I large enough' Thus

lf M ls to make . C ' ¡5l2 t¡"Os, the above assumptlon cannot be

true. This case ls consldered ln the following section'

QS Tlre case of manv 'overburdeled" lntervals'

Wehaveseenintheprecedlngsectlonthatw.l.o.g.wecanassumethat many blts b- are vlstble from the associated lnterval before

they are prlnted. irfo." preclsely, we will assume that there ls a set-¿;

;C2, wlth lG4l = i3/2¡15, and a set

84 c 82, with lB4l = l}2l/4=g2lgZ such that for each 1tG4

there ^r" !l/2/2 indices meB4 so that b- has color l and b- ls

vlsible from \U'- at some "t"i before tor(m)' rv\le focus on the'''ù)áîirr,

i" "|

and 'uncoror.. alr rheither bits (and prtnrrng phases).

pgþitlon. ('VlstbilttY timee"')Ftt Y.C4 "nd

me 84, where bm has color 1' we let

t"¡r(m) := Èhe largest t < t'r(ml such that b- ts vlstble

from lüt at, sÈeP t.

To motlvate the followlng deftnlÈtonsr wG cketch the ldee behlnd

thc re¡t of the proof. lve a¡k et whet frequency thc tlmc lteprtr¡"(ml, me 84, occur' Clcarly' lf tlie ¡vere8e dlgtancc between two

con¡ccutlve ¡uch rtepc ls > 32 ' C ' f/2' wc erc donc: M make¡

;ïïs7ï;."0r. Thu¡ we may es ¡ume tht on the ¡verage the vtatblllty

tlmes are clocer together, ll!:nce that thc followlng sltuatlon occurE

frequentlr, -*, (>'c1 . ú/2 for a btg const¡nt c'l bltE bm- ell h¡ve

thclr vlslbtllty timeg tr¡r(ml wlthln a tlme lnterv¡l of length l'

-tl-

From thc deflnltlon¡ lt followr lmncdl¡tcly thrt rll thc¡a blt¡ h¡vc

dtfferent color¡ end thaù the rrorl tlpc rretE rscoclqted wtth the¡e

blts are tdenttcal or ad¡acent. Taklng lnto ¡ccount the other ll/2 nblts of e¡ch of these colors, wG c¡n rhow (Lemma fl thrt for ¡ome

work tape lnterval the following ls true: there ls a tlme step t5

and there are many colors 1 wlth Wï = W, so that for altogether> C' ' I btts b- we have

tr¡.(m) < t5 tor(m).

Here C' >> 4. log3, so that C' , f¡r exceeds the 'capaclty' of \t.Thls means that on the one hand all these blts could have been

stored ln til before t5, and that at some step after t5 they are need¿d

ln W, but tt ls lmpos-sible that at t5 all the required lnformetlonls contalned tn w. By a Kolmogorov complextty argument (Lemma 5l

It can be shown that ln thls sttuation M must spend a lot of tlme

carrylng the mlsslng lnformatlon into \f,r'

Definitlon.

BE4,= {me 84 I t"r"lm) < tvts(m') for ¿ 3 ' 1l/2 ¡g

blts bm. of the samc color 1 as b-l

(the bttc rvlth 'early' vlalblllty ttmer w.r.t. thc other blt¡ of thc

Eame color)'

B! '. tnrD¡lt tr¡r(m,l < tr¡r(ml for ¿ 1l/2¡1blt3 bm, of the same color 1er b- rnd

tr¡.lml < tr¡"(m') for > fl/8/8 bltr b,,,, of the Eame color 1ar b-)

(thc blt¡ tlth 'lete' vl¡lblllty tlnerl'

cleerly, trft . lull . ll+llt' t2ttzl-

P¡rtltlon {1,2,...,T} tn¿o l3/2 ' 2-16 dl¡folnt lntcrv¡t¡

il:i;'.";i'ii':;:6:.; that e¡ch P! contatn¡ cxactlv t28 ' tttztilertepr rr¡.(ml wlth m¡E!. fn" P! ire cellcd thc vl¡lbllltv oha¡cr.

-12-

.We may assume w.l.o.g. thaÈ there ls ¡ set D s {1,2,..., F/2 ' 2-16},

lDl = ,3/2 . 2'17 so that for Ee D the vlslbtllt¡r.phase Pi conslsts

of < f /2 steps. (Otherwlse M makes > C IS/2 steps, and we are

done.) The followlng ls lmmedlate from the deftnltlons'

Leru¡r-a-t(d If m / m'm,m',g!, and t"¡.(m), tr¡"(m')ePi for some õeD' then

b-,b-' have dtfferent colors'

(b) If tr¡"(m), t"¡r(m')eP! for somc 6eD, m-,m"B!'and 1'1'are the

""f"ii of U-,i-', rãspectlvely, then W, and \Ãltr are elther

adjacent' r ldentlcal.

(Proof omitted.)

For each te D, we choose 54 ' il/2 blts bm wlth me B! and t"¡r(m)e P!

so that atl the blts chosen have the same Wr. (Thls ls posslble

by Lemma 3(b).) \tre then call this interval Wi,lts central 2l cells

Vi. The subset .i t¡ ".""i"ritt of all the btts chosen ls called B!'

ctearly, lBll = lntl / 2= t2tzsø.

D-eflnitlon.ior ô.D, let t5 be the ftrst step of Pi , and let

BE ,= {m.n! I for some m',8!, tr¡"(m')e P!

and b-,bm. have the same color 1)

(the set of 'carly" blt¡ whose color 'occurE' tn Pi)'

Lemm¡ f.t"l u|u l¡ never vlslble fro¡a \[li after t5,, and b- ls prtntcd frorn v!

efter t5, for ¡ll öeD and b-e B¡.

(b) lBSl ' 32' I , for ¡ll teD'

(c) m occurs tn ¡t most û/818 of thc Bg,, for all meB!.

(Proof o¡nltted¡ u¡e Lemma 1 (b) and thc deflnltlons.l

-t3-

Lemma S. (The 'overburdened' lntenrl.lLet M.l.n,x be ¡s ebove, I lerge enough, ¡nd let W be tn lntervrlof ll cellc on the work tape, V the 2l cells ln the center of lV.Let to < t1 be ttme stcpe. Let r > 16' l, and assume that there ere>r blts b- that are prlnted 'from V' durlng ( to, tl] but are nevervisible from lV after to before being prlnted from V. Then the worltape head spends > r' l/18'log n) steps in W durlng the lnterval(ro,rll.

Proof:By e Kolmogoroy complexlty argument, see Appendlx.

One last compllcatlon arises when re try to put together the lowerbounds for the tlme that the wort tape head spends ln v¡rlour U¡¡durlng various ttme lntervals. Lower bounds for dlfferent W! can beadded up, r lnce they are dlsjoint. But there may be many 6e D wlththe same ttri. The dynamlcs of such an 'overburdened' lnterval maybe qulÈe complex, because the set of blts that'belong'to thls lntervelwhose vtstblllty time ls over but thrt h¿ve not been prlnted yetchanger constantly. The following tcchnlcal lemm¡, whose proofls based on Lemma 4, resolves thls problem. It shows th¡t {1,2,...,T}can be cpllt lnto sufflctently many dl¡lolnt tlme lntervels to whlchLer¡m¡ 5 can be applled.

Lenma 6.Lct Do ß D,.¡nd let lt be cuch thet Sf . \[t¡¡ for ¡ll 6eDo. Let V betha 2l cell¡'ln the center of ìV. Then there ls e qeN and there rrettmestepc t .lå ' li ' ... ' tä wlth tf ,...,tä . (t¡l teDol such th¡tfor thc rctr Bf,...,ll deftned ty

E! :. t.. rlJ,

86l b,o lr prlntcd from V tn ttl, t!-1)

¡nd l¡ not vl¡lble from lY ln ft!, t'r{mtl}

-ll-

holds:{a) lBll > 4! for I < s < q, and

(b) ,1,

ttlt > sz'lDol ' Ir/2.

(Proof omitted¡ use Lemma 4')

ByapplylngLemmaSinthlssltuation,wegettheresultweneed.

Cetollar;.lf M,f ,n,x are as ln Lemma 5, and Do and \il are as ln Lemma 6'

then M spends> 4 ' lDol ' i3/2 ¡6t n

steps with the work taPe head ln \t'(Proof omitted.)

Uslng the corollary, we can finlsh the proof of the'Theorem. Let

t¡D be arblrrary. Let Do ,= {d'e Dlrñli. = \ilå }, and apPly the corollary

to conclude that the wòrk tape head spends

>lDol '4'13/2./logn

steps ln lüi. Summtng up these lower bounds for e famlly of 6 e D that

form a system of class representatlves for the equlvalence relatlon

on D deftned by \ñ/ô = wå' ' we conclude that M makes

> lDl . + . t3/2 / bg n = 13 I r,l3' log nl

steps eltogether, and thls ls > C ' iS/2 ¡oi. f large enough. Thls

ftnlshc¡ thc Proof of the Theorem'

-ls-

06 Opcn oroblemg.

It seem¡ that thc methods prerently ¡v¡llablc do noÈ ¡ufflcc forproving cuperl lnear lower bounds on the comPutttlon tlme for concreteproblems from P on TM's with two (or more) work tapes. Thereforeone has started to develop lower bound arguments for e sequenceof more restrlcted types of TM's çhere various abilittes of two-tapc TM's are 'phased ln' in e stepwlse process. ln thls cequenceof models that approximate the Power of trvo-tape TM's from belowwc hove, wlth the model consldered ln thts paPer, reached the 'endof the rope' as far as already existlng models are concerned.

A TM wlth two work tapes ls substanttally more powerful slncelÈ can move lnformatlon repeatedly forth and back between thetwo work tapes. This suggest that one should constder thc followlngnew hlerachy of restrlcted two-tapc TM'¡: ot thc k-th level of thlchlerarchy consider TM's with two tsPes where at any tlme polntoni of the two tapes ls a two-way read-only tspe, the other onels a read/wrlte tape, and the two tapes may switch thelr roles up

to k ttmes. (One may also allow that k Srolvs with n.) For k . I sucha TM can perform matrlx transpositlon as fast ¡s the modcl conslderedtn this paper, for k . 2 lt can do lt ¡ubstanttally faster, ¡ad fork ' O(log nl one can lmplement on thls model thc standard O(n'log nl-algortthm for matrlx transposltion on trro-tapc TM'r (vle the 'butterfly-graph').

Already for k . 2 the known argument¡ do not ¡ufflce for provlnglower bounds for TM'¡ from tht¡ hierarchy. In partlcular, lt wouldbe de¡lr¡ble to ¡how that th: Potcr of these m¡chlnc¡ rtrlctlylncrease¡ çtth t (¡t lea¡t for I < log.n).

-ló-

Bgleggn""s.

[t] Uietzfelbinger, M.: Lower bounds on computatlon tlme for varlous

models ln computational complexlty theory'Ph.D.-Thesis, university of Illinois at Chicago, 1987'

[2] Hennte,F.C.: One-tape off-line Turing machine computatlon.lnf. and Control 8: 553-578' l9ó5.

[3] Hopcroft,J.E., and ullman, J.D.: lntroduction to Automata Theory,Languages, and Computation. Reading (Mass.), Addlson-lVesley,l979.

[4] Maass,W.: Combinatorlal lower bound arguments for determlnlstlcand nondetermlnistlc Turlng machlnes, Trans. Amer. Math.Soc.2922 675'693, 1985.

[5] Maass,'!V'., and Schni.tger,G.: An opttmal lower bound for Turingmachines wlth one work tape and a two-way lnput tape. lnstructure in complexity Theory, ed. A.selman, pp. 249'264.Lecture Notes ln Comput,er Science 223. Berlin, Sprlnger-Verlag,1986.

[6] Maass, W., Schnitger,G., and Szemerèdl, E.: Two.tapes are beÈter

than one for off-ltne Turlng machlnes. ln Proc. l9th AçM Sympoa.

on Theory of Computlng, pp. 94-100' 1987'

[7] Paul,lÃI., and Stoss, H.-J.t Zur Komplexltãt von Sortlerproblemen'Act¡ Informatlca 3: 217-225' t971.

A.l

Appendlx: Proofs of the Kolmogorov complexlty Len¡nata

tlo bcain rith Lcr¡¡ 5. Thr proof of Le¡r¡ 2 sill b¡ r rli¡ht vrri¡tion of

th¡t of Lcrr¡ 5.

Proof of Lar¡¡ 5: (fni¡ i¡ ¡n ¡rt¡n¡ion of ¡¡ ¡r3u¡.nt ln [5¡.) L¡t

L(f) Uthrl¡ftro¡t(rí¡htrort) I c¡ll¡of I. (So, ll i¡th¡unionof L, t, I .) Choo¡r ¡ c¡ll boundrry c¡ to tho ri3ht ol ¡ c¡ll in L ro

¡¡ to ríni¡iz¡ th¡ nurbrr ol tir¡¡ in (tgrt¡l th¡ rorl trpr hotd crorr..

thir boundery fror l¡ft to ri3ht. rnd lrt th¡ nurbrr of cro¡¡in3r bo ICL .

sirilrrly, chooll ¡ c¡ll boundery c¡ to tho lott of ¡ crll ín t th¡t

¡ini¡iz¡r th¡ nurbor of ti¡rr in (tgrt¡l th¡ rorl ttpo hord cro¡¡¡¡ thí¡

boundrry lror ri3ht to l¡ft, rnd lrt th¡ nurb¡r of cro¡¡in3r br lcl .

A.2

Clcarly, the work tapc head spcnds ¿ ,'(/|CL+fCR) rtepr in L end R t¿tcn

togcthcr (by nininal ity) , and hcnce in t{ . Thu¡, it suf f i.cc¡ to ¡hon that

thc worh tapc head entcr¡ [c¡,c¡l (tttc ¡ntcrval bctscen c¡ ¡nd "n ) ¡t

lc¡¡t rl$'log n) tine¡ in (tg,t¡l . For this, rlc dcccribc ¡ ¡cthod for

producing thc input r ¿t output of ¡omc Turing rachine.

Supporc rrc are givcn:

(i) thc progran of ll ' codcd ¡¡ a bitrtring in sooc ¡tandard foni

(ii) thc content¡ of [c¡,c¡l at ti¡c tO i

(iii) the nunbcr / ¿nd thc positionr of all thrce tapc hcad¡ ¡t thc fir¡t

tioc in (tg,t1l at shich thc sork tapc hced vi¡it¡ [c¡,c¡l ;

(iv) thc porition of the input and output tape hc¡dr, and thc ¡t¡tc of ll rt

cech ti¡¡e the work tapc hcad crocsc¡ c¿ or c¡ tonard¡ Y ;

(v) thc bit¡ bE thaÈ arc vi¡ibtc fro¡ [c¡,c¡l during (tg,t¡l but ¡rr

not printcd fror [c¡,c¡l during (tg,t¡l bcforc bcing vi¡ítcd on thc

input tapc (thc¡c bit¡ ¡rc givcn.! ¡ rínglc rtring in thc ordor thry

¡rr ví¡itcd by thc input tepr hcad) i

(vi) tho bit¡ br of thr input thrt ¡ro ncithrr vi¡itcd by thr input trpr

hcrd during (tg,t1l ehil¡ th¡ vorh trpr hred i¡ in lctr,c¡l nor

printed fror [e¿,c¡l durin3 (tg,t¡l (thr¡o bit¡ ¡r¡ givon i¡ì onr

con¡¡cutivr rtrin3 in tl¡¡ ordrr thcy rppr¡r in. r );

(vii) tho codo f¡rl for ¡ Turíng ¡¡chin¡ th¡t rorl¡ e¡ folloer. Fír¡t,

¡i¡ul¡tc thc corput¡tion of ll on ingut r during tlt tirr poriodr

th¡t th¡ *oih t"p. hcad rpcnd¡ in [c¡,c¡l : St¡rtin3 rith rn orgty

ingut tepc, uring th¡ infon¡tion 3ivcn by (ii) .nd (iíi), rtrrt¡

I

:

I

A.3

¡iruleting ll ¡t thr fir¡t tincstcp in (tg,t¡l ¡t vhich the vorl tagc

hc¡d il in fc¡,c¡]. Hhcnevcr thc input tapc hcad vi¡it¡ ¡ cctt th¡t

h¡¡ no bit sritten to it ar yct, copy thc ncrt bit 3ivcn by tho rtrin3

dc¡eribcd in (v) to thi¡ cell, and continuo thc ¡irul¡tion. tfhcncvcr

It príntr e bi t br to thr ¡(¡)-tt¡ ccl I on thc output tapc,

i¡¡cdi¡tcly coPy thi¡ bit to thc corrcsponding r-th cell on thc input

tapr. lltrencvcr thc norl, tapc head lc¿vc¡ [c¡,c¡l , intcrrugt tho

¡i¡ulation ¿nd rcru¡c it sith thc rtcp tho uorl tepc hoad cnt¡r¡

[c¡,c¡l again, uring thc infornation givcn by (iv). The ¡irul¡tion i¡

finírhcd nhen thc uorh tepc hcad lcavc¡ fc¡,c¡l for thr lrrt tíar, or

uhcn H halt¡. Aftcr thit happcnr, fill in thc bit¡ ¡till ritrin3 on

the input tapc, urin3 th¡ ¡tring dc¡cribcd in (vi). Output the contcnt¡

of thc input tapc.

rt i¡ clc¡r that tho proccdurr jurt dcrcribcd outputr r ¡ So it ¡ro

r¡tireto thc nuabcr of bit¡ nocdcd to codc th¡ intorr¡tion dc¡cribrd in

(i)-(vii), in th¡ fon rcguirod by tho d¡finition of Kolro3oroy co¡plority, ro

obt¡in rn upp.r bound for f(r) . For th¡ differrnt prrtr of th¡ rtrin3. uo

got thr lollorín3 r¡irrt¡¡3

(i)¡ .Ë bit¡, for ¡o¡¡ con¡t¡nt .f i

(ii): s 1l'lo3 3 J E', bít¡;(i¡i): f0'lo3n bítr;

(iv¡ s J (¡lcl+rC¡)'(2'lor(n)+c") bit¡l(v), (vi) ¡

f !2-t bit¡

A.¡l

(rccall that, by thc hypotherir, I r bit¡ ¡rr printcd fror Y

bcfore thcy erc visibte fro¡ tl; ¡t lcart thcsc bit¡ ¡r¡ printcd

for¡ [c¡,c¡l duríng (tg,t¡l beforc bcing visitcd by thc input

tape head) i

(vii), "O

bit¡, for ¡onc constant tO'

Furthcrnore, 0(tog n) bitr arc necdcd to rcparaÈc thc rubrtrinS¡ th¡t

bclong to (i),...,(vii), whcn concatcnated to e ringlc rtring. (For crarnple,

eG can precede cach of thc¡e rubrtrings by itr lcn¡,th in binary, rith cach bit

doublcd.) llc gct

K(r) 3 c, + 8l + (fCL+fCR)'(2'tog(n)+c") + 12 - t * "O

* "l log n .

Since x i¡ incornprcs¡iblc, 12 S K(x) . lfc gct, for tn ro lerSc thet

log l¡ a ", , and for ¡onc eon¡t¡nt "; t

r-8/-c s (rcL+fcR)'3'los n

By esrunption,

tr¡n¡ fornat i on,

'lo3 n

t>-16l'

yicldr:

ñ

hcnc¡ r-81¿r/2 thir, togcthcr rith r trivi¡l

r'( 4 - S'"i'ro¡(n) lr )l (24'lo¡ n) s lcl+rcR

For lâ ro lrr3o thet 8'.i'lo3 n <'161 3 ¡ ur 3ot

tl(8'los n) s tCr+tc¡ )

¡¡ dc¡irod.

Â.5

Proof of LC¡OA 2z Thir proof i¡ cc¡cntially thc t¿É. .t thc prcviour

on., cxccptin3 that r|c ¡i¡rulate thc coargutation of ll f or ¡l l, tiac pcriodr

in [1,2,...,T1 thc sorh tapc hcad tpcndr in [c¡.c¡l , ¡nd th¡t in (ií) so

jurt nacd thc nu¡bcr of ccll¡ bctsccn c¡r c¡ - tho uorh tapc bcín3 initielly

blant. llc gct thc folloning c¡ti¡¡tc for K(r) 3

¡2 3 ¡(r) s

"' * 2'log n + (/C¡+iC¡)'(2'lo3 n * .r)

hcnc¡, for lâ lergc cnough'

r-c¡l

'lo3 n s (llCu+ICR)'3'los n

Ar bcforc' eG concludc th¡t

cl$'[os n) s ,cL+rCR

for tí lrr3c cnoughi henc¡ ll rpcndr

¿ l'tl(t'lo¡ n)

rtopr in ll

+ ¡2 rlc +0 I 'lo3 n