Turing Machines with Sublogarithmic Space

126 Pages·1994·2.84 MB·English

by  Andrzej Szepietowski (eds.)

AndrSzzeejp ietowski
TuriMnagc hinweist h
SublogaritShpmaicce
Spri-nVrgaelegr
BerliHne idelbNeerwgY ork
LondonP ariTso kyo
HongK ongB arcelona
Budapest

LectNuortieenC s o mpuStceire nce 843
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AdvisoBroya rdW:.B raueDr. G rieJs. S toer
SeriEedsi tors
GerhaGrodo s JurHiasr tmanis
UniversKiaUrilts ruhe CornelUln iversity
Postfa6c98h 0 DepartmoenfCt o mputeSrc ience
Vincen-zPriessnitz1- StraBe 4130U psoHna ll
D-7613K1a rlsruGheer,m any IthacNaY, 1 4853U,S A
Author
AndrzSezje pietowski
MathematiIcnaslt ituGtdea,n sUkn iversity
Ul.W itaS twosz5a7 ,8 0-95G2d ansPko,l and
UntiMla rch1 995:
FB 17I-nformaUtniikv,e rsGiHU Piatd erborn
Warburgers1t0r0aD,B- e3 309P5a derboGrne,r many
CR SubjeCclta ssific(1a 9t9i)1:o F.n1 .1F,. 41.
ISBN3 -540-583S5p5r-i6n ger-VBeerrllaiHgne idelbNeerwgY ork
ISBN0 -387-583S5p5r-i6n ger-VNeerwlY aogr kB erliHne idelberg
CIPd ataap plifeodr
Thiwso rki ss ubjetcoct o pyriAglhlrt i.g hatrser e servwehde,t htehrew holoer p art
oft he matiesrc ioanlc erned, spetchiref iigchaotlftsl r ya nslarteiporni,n trien-gu,s e
ofi llustr,ar teicointsabtriooand,c asrteipnrgo,d ucotnmi iocnr ofiolrim nsa nyo ther
way,a nds toraigned atbaa nksD.u plicatoifto hnip su blicaotrip oanr ttsh ereiosf
permitotnelduy n detrh ep rovisiooftn hseG ermaCno pyriLgahwto fS eptemb9e,r
1965i,ni tcsu rrevnetr siaonnd,p ermissfioourns em usta lwaybseo btained from
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Preface
Them aino bjecotfis n vestigoafts ipoancc eo mplextihteyo arrye T urinmga chines
witbohu ndeds pac(et hneu mbeorf c elolnst het apues eddu rincogm putatainodn )
languagaecsc eptbeyds uchm achineAsl.t housgohm ei mportant prsotbillelm s
remaionp enm,u chw orkh asbe en donea,n dm anye xcitirnegs ulhtasv eb een
obtainMeadn.y o ft hesree sulhtosw,e vehra,v ebe en proveudn detrh ea ssumption
thatth ea mounotfa vailasbplaeci esa tl ealsotg aritwhimtirhce spetcott h el engotfh
thei nput.
Int hiboso k hIa vper esenttheekd e yr esuolnts pacceo mplexbiuttcy o,n centrated
onp robleWmhsa:t h appenwshe nw e drotph ea ssumptoifoa ntl easlto garithmic
spacaen dw hadt ol anguaagcecset pabwliet shu blogaristphamclieoc o lki ke?
Them anuscrfioprtt h iboso k wasw rittaettn h eT echniUcnailv ersoifGt dya nsk
andG dansUkn iversfirtoym1 99t1o 1 99P2a.r tosf t hem anuscrwieprteu seda s
notefso rl ectugrievse antt hesuen iversiIwt ainettos .tha nk thep articiipnat nhtess e
coursfeosr t heicrr iticTihsemf .i naclo rrecttioot nhsem anuscrwieprte m ade
durinmgy staayt t heU niversoiftP ya derboirn1n 99u4n dearn A lexandveorn
HumboldRte searFcehl lowship.
I woulldi ktoe expremsys p rofound grtaotP irtoufdeesA s.oWr. M ostowsfkoir
hiesn courageamnedan dtv ice.
Io wea specidaelb tto a na nonymoruesf erfeoehr i so rh erh elpfcuolm menttos ,
Mrs.K . Mostowsaknad J .S kurczynsfkoirt heicra reful reatdhieon rgi goifn al
versioofnt hem anuscriapntdt, o L .C harlkfoo rh elpimneg w itht ypesettthien g
manuscriinTp EtX .
PaderbomM,a y 1994 AndrzeSjz epietowski

Contents
1
Inrtoduction
1 ... . . . ... . . .... . . . . .... . .. . . . .. . . .. . . . . . . . . .. .
BasiNco tions. . . . . . . . . . . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 .
2.1T uriMnacgh in.e s. . . . . . . . .7 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
2.C2o nfigaunrCdao tmipotunit oa.n . . . . . . 9. . . . . . . . . . . . . . . . . .
2.I3n terniaglu rCao.tn if.o n. . . . . . .1 1. . . . . . . . . . . . . . . . . . . . . . . .
.
2.A4l terTnuartMiinancggh in.e s. . . . . . 1.2 . . . . . . . . . . . . . . . . . . .
.
2.5S paCcoem ple.x i.t y. . . . . . . .1 3. . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . . .1 5. . . . . . .
LanguageAsc ceptabwliet hL ogarithmSipca ce
3 3.1T wo ExamLpalneggseuA saco cfe pwtiatbhl. e
LogtahrmiSipca c.e . . . . . . . . .1 5. . . . . . . . . . . . . . . . . . . . . . . . . .
.
3.P2e bAbluet oma.t a. . . . . . . . .1 6. . . . . . . . . . . . . . . . . . . . . . . . . . .
.
3.3N SPAClEongComplLeatnegg eu. a . 18
( ) . . . . . . . . . . . . . . . . . .
Exampleosf L anguageAsc ceptabwliet hS ublogarithmic
4
. . . . . . . . . . . . . .2 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Space
.
4.1L anguAacgceesp wtiatob(hll eoS gpcnae) . . . . .2 1. . . . . . . . . . .
.
4.T2al lLya nggeuAsac ecpbtlawei tShul bogtahrmSiipccae . . 2.4 . .
. .2 .7 . .
LowerB oundsf orA cceptiNnogn -regulLaarn guages
5 .
5.1L owBeoru nfdroTs w o-yTw uariMnagc hin.e s. . .2 8. . . . . . . . .
.
5.2L owBeoru nfdroO sn e-TwuaryMi nacgh in.e s . . . 3.3 . . . . . . . .
.
5.3L owBeoru nfdrots h Mei ddMlodeoe fS paCcoem ple.x .i. t 3y.4
SpaceC onstructiFubnlcet ions. . . . . . . 3.7 . . . . . . . . . . . . . . . . . . .
6 .
6.1D efinaintBdias oiPncrs o per.t i.e s. . . . .3 7. . . . . . . . . . . . . . . . . .
6.2F ulSlpaycC eo nrsutctFiubnlcet. i o.n s. . . . 39. . . . . . . . . . . . . . . .
6.3N ondetiesrtmiicnaSlplayCc oeFn usltlryFu ucntcitbiloen 4s2
HaltinPgr opertayn dC losuruen derC omplemetn . . .4 7. . . . .
7
7.1H altPirnogp oeTfru tryiM nacgh iwnietLsho g iatrhomri c
GreatSepcrae . . . . . . . . . . .4 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
7.C2l osure undeorfS tCroDomenptgle ermmeinnti stic
ComplCelxasiste.ys . . . . . . . . 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . .
.
7.3C losuunrdCeeo rm pleomfSe tnrtNo onngd eterSmpiancies tic
ComplCelxsasietasyb ovei tLhogm.a r. . . . . . . . . . . . . . . . . . . 52
.
7.4C losuunrdCeeo rm plefmroBe onutn Ldaendg uAacgceesp table
byN ondeterTmuriinMniagsc thii.cn es . 53
. . . . . . . .. . . . . . . . . . .
VIIIC ontents
7.5N oncluonsduCerorem pleomfLea nnatgg uAec cepbtya ble
WeaklSypc aeB ounTduerdiM nacgh in.e s.. . . .5 5. . . . . . . . . . . .
8 Stronvge rsuWse ak Mode ofS paceC omplexit.y . . .6 1. . . . . . .
8.1W eaakn Sdt ornMgo doefS p aCcoem plferoxF iutlSylp cyae
ConrsutctFiubnlcet ions . . . . . .6 1. . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2W eaakn Sdt rCoonmgp lCelxasistaeybs oL voeg ari.t h.m 6 .2 . .
83. Weaakn Sdt rCoonmgp lCelxasistbeyes lL oowgi atrh.m . 6.3 . .
8.4S trMoondgoe fS paCcoem plfeoxRrie tcyio gzniMnagc hi.n e6s6
8.5W eaakn Mdi dMdoldeoe Sfsp aCcoem plaebxoiLvtoeyrg aith6m6
Padding.. . . .. . . .. .. .. ... ... ... ... ..6. 7 .. .. .. .. .. .. . . . . .
9 . . . . .
9.1P addaibnogLv oerg ait.h m. . .. .. .. .. .. 6.7. .. .. .. .. . . . . . . .
.
9.2P addbienlgLo owrg ait.h m. . . . . . . 7.0 . . . . . . . . . . . . . . . . . . . . . .
Determisntiivce rsuNso ndeterminiTusrtiincg M achines 77
10
10.D1e termvienriNssoumns d etearbmoiLvnoeig samr ithm 77
10.D2e termvienriNssoumnes dt ermbienliLosowmg ar.i t..h m.. 7 8.
SpaceH ierarch.y . . . . . . . . . .8 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.11D igaonali.z a.t i.o n. . . . . . . 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112 . SpHaicererc ahbye lLoowrg iat.h m . . . . . . . . . . . . . . . . . . . . . . . 82
Closuruen derC oncatenat.io n. . . . . . .8 5. . . . . . . . . . . . . . . . . . . . . .
12
AlternatHiinegr arch.y . . . . . . . . 8.9 . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
13.C1o llaopfts hiAenl gt teirHnneigar aracbhoLyvo egi atrh.m .9 0.
13.N2o ncposliolnftag h Ael terHniaetriabnrgec lLhooywg arit9h2m
IndependeCnotm plemen.t . . . . . . . .9 5. . . . . . . . . . . . . . . . . . . . . . . .
14
Other ModoeflTu sr ingM achine.s . . . . . .9 9. . . . . . . . . . . . . . . . . .
15
15.T1w o-dimleT nusriiMonancgah i.n e.s . . . . 1.0 0. . . . . . . . . . . . . . . .
1.52I nkTduortiM nacgh in.e s. . . . . . . 1.0 4. . . . . . . . . . . . . . . . . . . . . . .
15.13- peTbubrliMenac gh in.e s. . . . . . . 1.0 7. . . . . . . . . . . . . . . . . . . . . .
15.D4e moTnu riMnacgh in.e s. . . . . . . 1.0 9. . . . . . . . . . . . . . . . . . . . . . .
Referenc.es . . . . . . . . . . .1 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SubjecItn dex 13
1
SymbolI ndex 151
Introduction
1.
Thimso nogirsaan pi hn vestoiftg haceto imoptnui toapnoawloe fTr u rimnag­
chiwnietsshu blogasrpiaWtcehec m.oi ncs tihdTe·u errmi ancgh miondeie nlt ro­
ducbeySd t eaHranrst,m aannLdie sw(,i9 1s6 w5i)ta th w o-rweaayd,-i onnpluyt
tapaena ds epatrwaot-erw eaayd,- wworrikt et ape. tTahpecee lnlusm ber of
useodnt hweo rtka pcea,ls lpeadic oseu ,mr e asuorfce o mptuitocanoamlp lexity.
Twod efinoifst piacoconems p lheaxvbieet euyns eidnl iterLaeLttn(u ) rb eea.
fnuctoinot nhn ea tunruamlb Ae Trusr.im nacgh iissna eti od sbter oLn(ng) l y
s:p-abcoeunindfoe c do mptuitooanna niyn pouflte nng,ut shme osr teh aLn( n)
spaIctie ss. a itdob ew eakLl(yns )p a-cobeundieffdro e vearcyc eipntpeud t
ofl engatht leasactc eopncteoi mnptgui tooanrt rferoea lterTnuraitnign g
n, (
machiunsenesosm orteh aLn(n s)p aDcSeP.A CE[LN(SnP)A]C,E []Lo,(r n )
)
ASPACE[Ld(enn)ot]th ceesl asosfl anggeuasac cebpytd eedt ermnionni­stic,
dletmeirniosrt iacl,t Le(rnsn)pa atbcioenu-gn Tdureidmn agc hirneessp,e ctively.
These acrlceasa slelse dc osmppalccelexas i,sdt eeyst ermnionnidsettiecr,m inis­
tiocra, l terrneastpiencgt,i vely.
A grenautm boefpr r oblems scpoancccoeem rpnlisentxgrii eltmlya oipne n.
'1\Ten odkton omwu cahb otuhtre e lahtiibpoe ntswdeeetne rmainnndio snt­ic
determsipnacicosemt pilcce lxai.sBt suyert se cetnhtehlrayev,b e e esno mvee ry
excidteivnegl oipntm hesent tuosdfs y p abcoeu-ncdoemdp utIantt ihboionsos k.
wer evtihekewe rye saublotsuspt a ccoem plexciotnyc ebonunttt rh saept ee­
dfpirco ipeeasrn td ptreocohfnr ieqqrueueidisnt h set uodfly a nggeuasac cepted
witshuibnl ogasrpiatich.esme,pi. ac bc oeu nbdyea fd nu ctLi(onns) u ch that
E(n=) o logonrl iimn��f�0 .
( ) =
Turmiancgh wiinsetushb logasrpiadtciheffsm ehirac r fprlotymh owshiech ha ve
lorgiathomrgi rce astpeaIrcnt e h.bi osow kes hadlils cmuasnsy exoaftm hspi les
faacntd i ltlotu es tporouairwn ept r esbeenlstoo wmo eft hebmr iefly.
Classoefls a nggeuasac cewpitteshdui bgnla oriUsupnadicecep ehneda voinl y
JL.
l;mhaec hmiondeea lntsdh meo doefs paccoem plMeaxnimyto yd.eo lfTs u ring
machwiintbehuo ndsepda acreee q iuvalteoen atc h iotfth hseep rab coeu nd
fnuctiiseoq nu taol gorre atthealrnoa gritHhemn.ct ehre,e suclotnsc erning
lorgiathomrgi rce atearr iesns,p o amsceee n msoed,ie nled pendTehniitsns. o t
1>chaseef ro s ublogasrpiatHchoemw.ie ctv heerar,re ae l msood ienld ependent
1sruelitnts hc eas oefs ublogasrpiaFtcoheerm.x i acm tphlfeenu ,c tlioolgnon gi s
1�lhoew er frboa ocucnedpn toinn-gr leagnuggleuafsraor m andyie ffremnotd els
ofT urimnagc hainnde moofsd peascc oem plexity.
Manrye stuhlattsf rohl oolgda roirgtr hemasitpcea arcrn eeo vta lfrios dul bog­
::!.
:a.ristphamFcioecer. x ampilTeh:ea lntaetrhiinegr afrroS cPhAyC E[L(n)]
()