# NEARL LINEAR TIME uri Gurevic Electrical Engineering and Computer Science Univ ersit of Mic higan Ann Arbor MI Saharon Shelah Mathematics Hebrew Univ ersit Jerusalem Israel Mathematics Rutgers Univ PDF document - DocSlides

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In this connection w ein tro duce a class NL T of functions computable in ne arly line ar time log 1 on random access computers NL is ery robust and do es not dep end on the particular c hoice of random access computers Kolmogoro v mac hines Sc h on ID: 45489

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## Presentations text content in NEARL LINEAR TIME uri Gurevic Electrical Engineering and Computer Science Univ ersit of Mic higan Ann Arbor MI Saharon Shelah Mathematics Hebrew Univ ersit Jerusalem Israel Mathematics Rutgers Univ

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NEARL LINEAR TIME uri Gurevic Electrical Engineering and Computer Science Univ ersit of Mic higan, Ann Arbor, MI 48109-2122 Saharon Shelah Mathematics, Hebrew Univ ersit Jerusalem 91904, Israel Mathematics, Rutgers Univ ersit New Brunswic k, NJ 08903 Jan uary 1989 Abstract The notion of linear-time computabilit yis v ery sensitiv to mac hine mo del. In this connection, w ein tro duce a class NL T of functions computable in ne arly line ar time (log (1) on random access computers. NL is ery robust and do es not dep end on the particular c hoice of random access computers. Kolmogoro v mac hines, Sc h onhage mac hines, random access uring mac hines, etc. also compute exactly NL functions in nearly linear time. It is not kno wn whether usual ultitap e uring mac hines are able to compute all NL functions in nearly linear time. do not b eliev they are and do not consider them necessarily appropriate for this relativ ely lo w complexit ylev el. It turns out, ho ev er, that nondeterministic T uring mac hines accept exactly the languages in the nondeterministic v ersion of NL T. W giv e also a mac hine-indep enden t de nition of NL T and a natural problem complete for NL T. Springer LNCS 363, 1989, 108{118. P artially supp orted b y an NSF gran t and a gran t from Binational US-Israel Science oundation. substan tial p ortion of the ork w as done during aw eek in all 1985 when both authors visited Rutgers Univ ersit y; during the last stage of the ork, the rst author as with Stanford Univ ersit y and IBM Almaden Researc hCen ter (on a sabbatical lea e from the Univ ersit of Mic higan).

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In troduction What is Linear Time? In other ords, what is the correct notion of linear time com- putabilit y? The answ er to this question is not clear at all. The notion of Linear Time seems to be badly dep enden on computational mo del. It is p ossible that there is no univ ersal notion of Linear Time and di eren ersions of Linear Time are appropriate to di eren applications. On the other hand, olynomial Time is ery robust. Ev en uring mac hines (w alking painfully from to on their tap es to ac hiev e what their luc kier com- p etitors can do in one step) are adequate for P olynomial Time. An analysis of argumen ts in fa or of the robustness of P olynomial Time turns up a m uc h more mo dest but still v ery robust extension of the apparen tly ill-de ned notion of Linear Time. pro that mac hine mo dels giv the same notion of p olynomial time com- putabilit one often hec ks that an )-time-b ounded mac hine of one kind can be sim ulated b some mac hine of the other kind in time where the erhead is b ounded p olynomial of or ev en (W restrict atten tion to computations with .) It is often the case that the erhead is b ounded p olynomial of the logarithm of ); let us call suc sim ulations eÆcient noticeable exception is the sim ulation of random access mac hines b uring mac hines. Call function ne arly line ar if it is b ounded the pro duct of and some p olynomial of log (F unctions b ounded p olynomials of log are often called olylo functions of n. Th us, nearly linear function is the pro duct of and p olylog function of .) If t omac hine mo dels eÆcien tly sim ulate eac h other (i.e. ev ery mac hine of one kind is eÆcien tly sim ulated mac hine of the other kind) then they giv the same notion of nearly linear time computabilit (as ell as the same notion of nearly square time computabilit , etc. ). uring mac hine mo dels with 2, 3, etc. (linear) tap es form one cluster of mac hine mo dels that eÆcien tly sim ulate eac h other. Sc hnorr [12 ]in tro duced and studied the class QL (for Quasilinear Time) of functions computable on suc uring mac hines in nearly linear time and the class NQL of languages accepted b y nondeterministic m ultitap e T uring mac hines in nearly linear time. He sho ed in particular that SA is complete for NQL with resp ect to QL reductions; see also [4] in this connection. e iden ti ed an apparen tly di eren t cluster of mac hine mo dels that eÆcien tly sim ulate eac h other. Cho osing random access computers (RA Cs) of Angluin and V alian t [3] as our basic mo del in the cluster, w ein tro duce a class NL T of functions computable on RA Cs in nearly linear time. (A language is NL T if its c haracteristic function is NL T. ) Among other mo dels in the cluster are random access uring mac hines, Kolmogoro vmac hines, storage mo di cation mac hinesofSc h onhage, and T uring mac hines with the tap e in the form of a tree. All mo dels in the cluster giv the same notion of nearly linear time computabilit Th us NL Tisv ery robust. In particular, the de nition of NL T do es not dep end on whether RA Cs can ultiply (or ev en add) in one step. Section is dev oted to the robustness of NL T. (The class NL has been announced in [6].) Whether NL is the \robust closure" of Linear Time, b eliev that it is useful appro ximation to and an extension of Linear Time. QL, on the other hand, ma not con tain some functions computable in linear time on an y | whatev er mo dest | mac hine mo del in the cluster of RA Cs. or, if (as one ma y exp ect) NL T prop erly con tains QL, then padding inputs of an y non-QL function in NL Tgiv es a non-QL function computable

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in linear time on mac hines. EÆcien sim ulations of Section surviv if deterministic mo dels are replaced cor- resp onding nondeterministic mo dels (and computing functions is replaced accepting languages). The nondeterministic ersion NNL of class NL is ev en more robust than NL T: sho in Section that NNL NQL. The coincidence of NNL and NQL sheds some ligh on the diÆcult of the problem whether NL prop erly includes QL. It implies that NQL con tains ev ery NL T language. Moreo er, if NL Tcon tains some non-QL function, then NQL con tains some non-QL language without loss of generalit some NTM accepts in linear time. This is uc wider gap bet een deterministic and nondeterministic v ersions of T uring time that is presen tly kno wn [10]. In order to stress the mac hine indep enden tc haracter of NL T, w epro vide a calculus of total NL op erations on binary strings in Section 4. Finally , what reductions are appropriate for NL T decision problems? A natural c hoice is to use QL reductions. Problems that are QL hard for NL are imp osible to solv on uring mac hines in nearly linear time unless QL=NL T. In Section 5, w e exhibit a decision problem QL complete for NL T. The Robustness of NL 2.1 Random Access Computers A random access computer [3] is an abstract mac hine with a sequence of memory lo cations. The size and the um ber of lo cations dep end on the input size, so that RA can be seen as sequence of nite mac hines. Eac lo cation con tains binary string of length log where is the size of input and dep ends on the giv en mac hine only There are exactly lo cations. RA is con trolled program (that do es not dep end on n) consisting of nite sequence of instructions. The time-complexit of computation is the n um b er of instructions executed. RA Cs do all the usual (for random access mac hines) op erations: store, load, etc. The exact instruction set is immaterial for our purp oses. The instruction set used in [3] is ne. Those instructions do not use registers. or some purp oses, it ma be con enien to use registers. Call a RA frugal if the visited lo cations alw ys form an initial segmen t. In particular, a frugal RA with a nearly linear time b ound uses only nearly linear man lo cations. Lemma Every RA an eÆciently simulate by frugal RA C. Pro of Whenev er the sim ulated RA uses new lo cation the sim ulating RA uses the rst un used lo cation where it stors b oth and the con ten of be able to nd ) promptly (whenev er needs lo cation ), uses the balanced tree searc h and insertion algorithm [3, 7], namely , the lo cations )withw eigh ts form a balanced tree. In accordance with [9], call RA is write-onc memory machine (or simply non- er asing )ifan y bit of (an y lo cation in) the memory ma ybec hanged from 0 to 1 but nev er from 1 to 0. Lemma Every frugal RA an b e eÆciently simulate d by an appr opriate frugal non- er asing RA

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Pro of When writes in to lo cation for the -th time, creates (on new blo c of lo cations) binary tree with lea es equidistan from the ro ot. If and when writes in to for the (2 )-th time, where writes (the same con ten t) in to the -th leaf of mak es sure that all ancestors of the rst lea es, except for the ro ot, are mark ed. Marking mak es it easy to nd the last leaf that has been written in to (and th us facilitates easy reads) as w ell as the rst leaf that has not b een written in to y et. When all lea es of are written in to, the address of the next relev an tbloc k of lo cations is written in to the ro ot of 2.2 Generalized uring Mac hines It is not diÆcult to c hec k that RA Cs can b e eÆcien tly sim ulated b y random access T uring mac hines. or the sak e of de niteness, let us form ulate a sp eci c v ersion of random access uring mac hines. De nition nR TM is a T uring machine with thr eline ar tap es, c al le d the main tap e the address tap e and the auxiliary tap e such that the he ad of the main tap (the main head ) is always in the c el l whose numb er is the c ontents of the addr ess tap e. n instruction for an TM has the form p; ; ; q; ; ; ; ; and me ans the fol lowing: If the ontr ol state is and the symb ols in the observe el ls on the thr tap es ar binary digits ; ; esp ctively, then print binary digits ; ; in the esp ctive el ls, move the addr ess he ad to the left (r esp. to the right) if (r esp. +1 ), move the auxiliary-tap he ad with esp ct to and go to ontr ol state TM is frugal if at any time the length of the addr ess tap is 1 + log(1 + and the length of the auxiliary tap is (1 + log(1 + TM is non-erasing if, for every instruction, Lemma Every frugal RA an eÆciently simulate by frugal TM Mor e- over, if is non-er asing then so is Pro of Supp ose that uses -bit lo cations. Then uses the -th blo c of cells of the main tap e to store the con ten ts of the -th lo cation of The auxiliary tap e is used to p erform arithmetical op erations, to mak esurethat the address head do es not fall o the address tap e, etc. De nition BTM is uring machine with jumping and bise cting abilities. It has one tap and one he ad. In addition to usual uring instructions, pr gr am for BTM an use the fol lowing instructions. Goto( Move the he ad to the el l with the rst, i.e., leftmost curr enc eof if ado es not cur on the tap then do nothing. Bisect( If symb ol app ars on the tap to the left of the he ad and the distanc etwe en the osition of the he ad and the osition of the rst curr enc of is even then move the he ad to the midd le el l otherwise do nothing.

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Lemma rugal TMs an eÆciently simulate by BTMs. Pro of De ne BTMs with sev eral heads and call them GTMs. The Bisect command has the follo wing form for GTMs: Bisect( ;h ). It sends the activ e head to cell n um ber where i; j are the p ositions of ;h resp ectiv ely (pro vided that i; j ha the same parit y; otherwise the command has no e ect). GTM can be sim ulated some BTM with a constan to erhead. The desired BTM uses additional tap e sym b ols to mark the lo cations of the heads of the giv en GTM. No wlet b e the giv en frugal R TM. One head (the right guar ) of the desired GTM o ccupies the p osition of the righ tmost used cell of Another head (the left guar sta ys in the leftmost cell of the tap e. mimics the address tap e of on a sp ecial trac of its only tap e. Using the guards and some auxiliary heads, is able to sim ulate one step of in (log steps where is the curren p osition of the righ t guard. 2.3 Storage Mo di cation Mac hines There are brands of storage mo di cation mac hines in the literature: Kolmogoro (or Kolmogoro v-Usp ensky) mac hines [8] and Sc h onhage mac hines [13]. A "philosophical" discussion on storage mo di cation mac hines v ersus uring mac hines can b e found in [5]. Lemma BTMs an eÆciently simulate by Kolmo gor ov machines. Pro of Easy Lemma Kolmo gor ov machines an eÆciently simulate by Sch onhage machines. Pro of Easy Lemma Sch ongage machines an eÆciently simulate by RA Cs. Pro of In [13], Sc h onhage pro es that his mac hines can be real-time sim ulated ery restricted RAMs (RAM1 mo del in his terminology). The same pro of sho ws that Sc h onhage mac hines can be eÆcien tly sim ulated b ery restricted frugal RA Cs. 2.4 Robustness Theorem The lemmas of this section imply the follo wing theorem. Theorem RA Cs, frugal non-er asing RA Cs, TMs, frugal noner asing TMs, BTMs, Kolmo gorov machines and Sch onhage machines al l eÆciently simulate ach other and ther efor ompute exactly NL functions in ne arly line ar time.

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Nondeterministic Nearly Linear Time Recall that NNL (resp. NQL) is the class of languages accepted nondeterministic RA Cs (resp. nondeterministic m ultitap e T uring mac hines) in nearly linear time. Theorem NQL NNL T. Pro of It is ob vious that NNL includes NQL. The pro of that ev ery NNL language is NQL builds on the fact that ultitap e T uring mac hines can sort in nearly linear time [12]. Fix an TM that accepts in nearly linear time ). If is computation of of some length de ne the trace of as the sequence t; q ;a ;I ;b ;J ;c ): where ;a ;I ;b ;J ;c are resp ectiv ely the state, the con guration of the address tap e, the p osition of the head of the address tap e, the con guration of the auxiliary tap e, the p osition of the head of the auxiliary tap e, and the observ ed haracter on the main tap e in the momen Giv en string of length on sp ecial input tap e, the desired nondeterministic ultitap e uring mac hine guesses the trace t; q ;a ;I ;b ;J ;c of presumably accepting computation of It guesses the 6-tuples one one and hec ks that the rst tuple is correct, and ev ery +1-st tuple is consisten with the -th one, and ) is accepting. Ho ev er, do es not c hec k whether c haracters are correct; this ma require going to o far in to the history of the computation. In order to c hec k the correctness of c haracters +1 sorts the tuples rst b y the third and then the rst comp onen ts. Then it reads the sorted list from the left to righ t. It uses head on the input tap e to hec the correctnes of the rst tuple in the blo c of tuples with the same con ten of the address tap e. No supp ose that t; q ; a; I ; b; J ;c ), ;q ;a;I ;b ;J ;c ) are in the same blo c kand kno ws already that is correct. The rst of the tuples has enough information for to decide whether and what writes on the main tap e at momen If do es not write at momen then else is exactly the haracter that writes on the main tap e at momen Calculus for NL In this section, the term op er ation is used to denote total functions from binary strings to binary strings. The lo er case letters u; v ; w ; x; y ; z (with or without subscripts) denote binary strings. De nition Initial replacemen op er ations ar as fol lows. u;y If is an initial se gment of then eplac it with else do nothing. u;y If app ars in (as a substring) then r eplac e the rst, i.e., leftmost o curr enc of in with else do nothing. u;v;y;z If and app ar in and their rst o curr enc es do not overlap then r eplac the rst o curr enc es of and with and esp ctively. It is assume d that al l four ar ameter strings have the same length.

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u;v;w;y;z If has a form :::ux vx 00 w ::: wher e the shown o curr enc es of and ar leftmost and 00 then eplac the shown o curr enc es of and with and esp ctively; else do nothing. It is assume that al l ve ar ameter strings have the same length. De nition If is an op er ation, then +1 )) and The op er ation is al le the iteration of De nition iterated replacemen is the iter ation of the omp osition of initial e- plac ement op er ation. Lemma Every iter ate eplac ement is NL T. Pro of Sk etc Supp ose an to compute the iteration of comp osition of initial replacemen t op erations. The initial replacemen t op eration in ha e parameter strings ( or u; v or u; v ; w ) that need to be found and ma;y be replaced; let be the collection of all suc parameter strings. The idea of the desired algorithm is to main tain tree suc that the curren string is the string of lea es and ev ery no de of the tree eeps the list of -strings that occur in the p ortion of b elo (trees tend to gro do wn ard in computer science). It is easy to see that ev ery replacemen and the resulting up date of the tree can be p erformed in p olylog time. De nition Initial extension and con traction op er ations ar as fol lows. u;v Simultane ously eplac every with and every with Her and ar di er ent strings of the same length. u;v If u;v for some then else dd tail of ln many opies of to Her is the length of and is the length of the binary notation for Delete the (maximal) tail of 's in Let b e the closure of initial op erations and iterated replacemen ts under comp osition. Theorem is exactly the set of NL op er ations. Pro of It is easy to see that ev ery initial op eration is NL T and that NL T is closed under comp osition. It remain to apply Lemma 8 to sho that ev ery op eration is NL T. osho w that ev ery NL T op eration is in , x a BTM (see Section 2) that computes in nearly linear time and consider the computation of on an input of length Let b e the set of states of b e the set of tap e sym bols of and ). Assign di eren binary strings of xed length to elemen ts of if an elemen is assigned string :::b co de with string )= 11 :::b of length 2+2 If is string :::a er co de with :::c ). (The arti cial form of letter co des serv es the follo wing purp ose: There are no "illegal" sneaking o ccurrences of letter co des in an ).) If is the initial state of co des the pair ( p; 0) and co des p; 1), then

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(1) ;b (0) ;a (0) ;c (1) is the co de for the the initial con guration of where the blanks are ignored. Let the co de of the blank tap e sym bol of and be nearly linear upp er b ound on b oth the run-time of and the space used There is comp osition of sev eral copies of suc that the length of the string )=( )) exceeds lm or exp ositional purp oses, it is con enien t to assume that has a nite tap e of length Then instan taneous descriptions (IDs) of ha the form :::a where exactly one of the sym bols b elongs to and the others b elong to The co de )of an ID will be called the binary instantane ous description (BID) of In particular, is the initial BID of or ev ery instruction of there is a comp osition of initial replacemen t op erations suc that for ev ery BID of )= [If I is applicable at then the next BID, else ]. In the case of uring instruction, the desired is comp osition of R1 op erations. In the case of Goto, is the comp osition of R2 op erations. And in the case of Bisect, is comp osition of R3 op erations. Let be the comp osition of all Then for ev ery BID of )is the next BID of (if is halting BID then )= ). It is easy to see that )= )) is the nal BID of Let b e the comp osition of with a comp osition of R1 op eration that "erases" the state, so that )is )) with a tail of 's. Then (0) ;c (1) Complete Problem start with generalization of b o olean form ulas. The boolean constan ts \true" and \false" are iden ti ed with and resp ectiv ely simple (b o olean) ariable is an expression of the form where isabinary string. omplex (b o olean) v ariable is an expression of the form where is string of b o olean constan ts and simple boolean ariable. (A simple ariable is also complex ariable.) An quation is an expression of the form where is complex ariable and is b o olean com bination of complex v ariables. gener alize ole an formula is a sequence of equations. An envir onment is function that ev aluates (i.e. assigns b o olean constan ts to) some simple ariables. An en vironmen identi es complex ariable if ev ery simple ariable in b elongs to the domain of or example, if =01 and )=0 and )=1 then iden ti es as the simple ariable 0101 An en vironmen evaluates complex v ariable if it iden ti es and ev aluates the resulting simple v ariable. Supp ose that iden ti es as some simple v ariable and supp ose that ev aluates ev ery v ariable

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in a b o olean com bination and therefore ev aluates to some b o olean v alue ;then the equation alters to a new en vironmen suc h that dom )= dom [f , and )= for ev ery in dom f and )= (the original en vironmen could be de ned or unde ned at .) A generalized b o olean form ula ;:::;E is true if there are en vironmen ts ;:::;e suc that is the empt en vironmen t, alters to +1 ,forev ery i and ev aluates to \true". Theorem The pr oblem of evaluating gener alize db ole an formulas is c omplete for NL under QL ductions. Pro of It is clear that the problem is NL T. will pro that the problem is hard for NL T. Giv en a language in NL T, x a frugal non-erasing R TM (see Section 2) that accepts the language. The desired generalized b o olean form ula describ es the computation of on some input of length Let range o er the steps in the computation and er p ossible con ten ts of the address tap e. The diÆcult is ho to describ e the main tap e of at all times; cannot ha separate b o olean ariable for ev ery pair t; a ). get around similar diÆcult in his pro of of NQL completeness of SA T, Sc hnorr used the fact that T uring mac hines can b e eÆcien tly sim ulated b y oblivious T uring mac hines [11 ]. Ho ev er no obliviousness result is kno wn for random access uring mac hine. Our generalized b o olean form ula will use nearly linear man y simple b o olean v ariables whic split in to sev eral groups. or example, is group of ab out log simple b o olean ariables where is bound on the length of the computation and is the um ber of con trol states; the in tended meaning of is the con trol state at momen The in tended meaning of other groups of simple v ariables is as follo ws: The con ten ts of the address tap e at momen t;i The -th digit in The p osition of the head of address tap e at momen The con ten ts of the auxiliary tap e at momen t;j The -th digit of The p osition of the head of auxiliary tap e at momen The con ten ts of the cell um ber of the main tap e. (Notice the lac of the time parameter.) ac ept A sp ecial b o olean v ariable to signal the acceptance. or eac , the desired system of equations con tains a subsystem of p olylog( )man equations eac of length p olylog( ). If then equations precede equations. equations re ect the initial conditions of they

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10 set to the initial state of initialize all ;i and ;j to zero, set ariables with resp ect to the initial con guration. The +1 equations re ect the up dates p erformed at time They set eac of +1 ;a +1 ;I ;I +1 ;b +1 ;J ;J +1 equal to an expression of the form ;a t;I ;b t;J ;c for appropriate b o olean functions And is set to: ;a t;I ;b t;J for an appropriate function The nal equation in our system (in addition to all equations in all ) sets the v ariable ac ept to 1 if the nal state of is accepting. Remark During the presen tation of this pap er in IBM Almaden Cen ter on Ba Area Theory Da No em ber 11, 1988, Moshe ardi noted the sp ecial haracter of the generalized b o olean form ulas, constructed in the pro of of Theorem 4: if an en vironmen is altered to an en vironmen and then 1. In other ords, en vironmen ts are revised only up ard; this giv es a or of least xed p oin ts to our generalized boolean form ulas. Moshe ask ed ab out natural NL problem explicitly in terms of least xed p oin ts. The though is attractiv e; the details need to b e ork ed out. Remark The reductions of Theorem are not only QL but also logspace. There is ho ev er problem with logspace QL reductions: there is no reason to b eliev that they are closed under comp osition.

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11 References [1] G. M. Adelson-V elsky So viet Math. 3 (1962), 1259-1263. [2] A. V. Aho, J. E. Hop croft and J. D. Ullman. The Design and A nalysis of Computer lgorithms. Addison-W esley Reading, Mass. 1974. [3] Dana Angluin and Les alian t. ast Pr ob abilistic lgorithm for Hamiltonian Cir- cuits and Matchings. J. of Computer and System Sciences 18 (1979), 155{193. [4] Stephen A. Co ok. Short Pr op ositional F ormulas R epr esent Nondeterministic Com- putations. IPL 26 (1987/88), 269{270. [5] uri Gurevic h. Kolmo gor ov Machines and elate Issues: The Column on gic in Computer Scienc e. Bulletin of Europ ean Asso c. for Theor. Comp. Science 35, June 1988, 71{82. [6] uri Gurevic h and Saharon Shelah. unctions Computable in Ne arly Line ar Time. AMS Abstracts 7:4 (1986), p. 236. [7] Donald E. Kn uth. The rt of Computer Pr gr amming: olume Sorting and Se ar ching. Addison-W esley Reading, Mass. 1973. [8] A. N. Kolmogoro v and V. A. Usp ensky On the De nition of an A lgorithm. Usp ekhi Mat. Nauk 13:4 (1958), 3{28 (Russian) or AMS T ranslations, ser. 2, v ol. 21 (1963), 217{245. [9] Sandy Irani, Moni Naor, Ronitt Rubinfeld. On the Time and Sp ac Complexity of Computation Using Write-Onc e Memory. Man uscript, Computer Science Division, UC Berk eley No v. 1988. [10] W. J. P aul, N. Pipp enger, E. Szemeredi and W. T. T rotter, On determinism versus non-determinism and elate pr oblems. Pro c. 24th IEEE Symp osium on ounda- tion of Computer Science, No em ber 1983, ucson, Arizona, 429{438. [11] N. Pippinger and M. J. Fisc her. elations among Complexity Me asur es. J. CM 26:2 (1979), 361{381. [12] Claus Sc hnorr. Satis ability is Quasiline ar Complete in NQL. Journal of CM 25:1 (1978), 136{145. [13] A. Sc h onhage. Stor age Mo di c ation Machines. SIAM J. Computing 9:3, August 1980, 490{508.