Theoretical Computer Science

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1 Theoretical Coputer Science 411 (2010) Contents lists available at ScienceDirect Theoretical Coputer Science journal hoepage: Theory of one-tape linear-tie Turing achines Kohtaro Tadai a,1, Tooyui Yaaai b,, Jac C.H. Lin b a ERATO Quantu Coputation and Inforation Project, Japan Science and Technology Corporation, Toyo, , Japan b School of Inforation Technology and Engineering, University of Ottawa, Ottawa, Ontario, Canada, K1N 6N5 a r t i c l e i n f o a b s t r a c t Article history: Received 16 June 2008 Received in revised for 24 April 2009 Accepted 31 August 2009 Counicated by O. Watanabe Keywords: One-tape Turing achine Crossing sequence Finite state autoaton Regular language One-way function Low set Advice Many-one reducibility A theory of one-tape two-way one-head off-line linear-tie Turing achines is essentially different fro its polynoial-tie counterpart since these achines are closely related to finite state autoata. This paper discusses structural-coplexity issues of one-tape Turing achines of various types (deterinistic, nondeterinistic, reversible, alternating, probabilistic, counting, and quantu Turing achines) that halt in linear tie, where the running tie of a achine is defined as the length of any longest coputation path. We explore structural properties of one-tape linear-tie Turing achines and clarify how the achines resources affect their coputational patterns and power Elsevier B.V. All rights reserved. 1. Prologue Coputer science has revolved around the study of coputation incorporated with the analysis and developent of fast and efficient algoriths. The notion of a Turing achine, proposed by Turing [40,41] and independently by Post [34] in the id 1930s, is now regarded as a atheatical odel of any existing coputers. This achine odel has long been a foundation of extensive studies in coputational coplexity theory. Early research unearthed the significance of various restrictions on the resources of achines: for instance, the nuber of wor tapes, the nuber of heads, execution tie bounds, eory space bounds, and achine types in use. This paper ais at the better understanding of how various resource restrictions directly affect the patterns and the power of coputations. The nuber of wor tapes and also achine types of tie-bounded Turing achines significantly alter their coputational power. For instance, two-tape Turing achines are shown to be ore powerful than any one-tape Turing achines [12,35]. Even using the odel of ultiple-tape Turing achines, Paul, Pippenger, Szeeredi, and Trotter [32] proved in the early 1980s that linear-tie nondeterinistic Turing achines are ore powerful than their deterinistic counterparts. An earlier version appeared in the Proceedings of the 30th SOFSEM Conference on Current Trends in Theory and Practice of Coputer Science, Lecture Notes in Coput. Sci., vol. 2932, pp , Springer-Verlag, January 24 30, This wor was in part supported by the Natural Sciences and Engineering Council of Canada. Corresponding address: School of Coputer Science and Engineering, University of Aizu, 90 Kai-Iawase, Tsuruga, Ii-achi, Aizu-Waaatsu, Fuushia , Japan. Tel.: ; fax: E-ail address: tooyuiyaaai@gail.co (T. Yaaai). 1 Present address: 21st Century COE Progra: Research on Security and Reliability in Electronic Society, Chuo University, Kasuga, Bunyo-u, Toyo , Japan. This wor was partly done while he was visiting the University of Ottawa between Noveber 1 and Deceber 1 in /$ see front atter 2009 Elsevier B.V. All rights reserved. doi: /j.tcs

2 K. Tadai et al. / Theoretical Coputer Science 411 (2010) Of particular interest in this paper is the odel of one-tape (or single-tape) two-way one-head off-line linear-tie Turing achines, apart fro well-studied polynoial-tie achines. Not surprisingly, this rather siple odel proves a close tie to finite state autoata. Despite its siplicity, such a odel still offers coplex structures. As a result, a theory of one-tape (one-head) linear-tie coplexity draws a picture quite different fro ultiple-tape odels as well as polynoial-tie odels. It is thus possible for us to prove, for instance, the collapses and separations of nuerous one-tape linear-tie coplexity classes without any unproven assuption, such as the existence of one-way functions. Hennie [18] ade the first ajor contribution to the theory of one-tape linear-tie Turing achines in the id 1960s. He deonstrated that no one-tape linear-tie deterinistic Turing achine can be ore powerful than deterinistic finite state autoata. To prove his result, Hennie described the behaviors of a Turing achine in ters of the sequential changes of the achine s internal states at the tie when the tape head crosses a boundary of two adjacent tape cells. Such a sequence of state changes is nown as a crossing sequence generated at this boundary. Using this technical tool, he argued that (i) any one-tape linear-tie deterinistic Turing achine has short crossing sequences at every boundary and (ii) if any crossing sequence of the achine is short, then this achine recognizes only a regular language. Using the non-regularity easure of Dwor and Stoceyer [13], the second clai asserts that any language accepted by a achine with short crossing sequences has constantly-bounded non-regularity. Extending Hennie s arguent, Kobayashi [25] later showed that any language recognized by one-tape o(n log n)-tie deterinistic Turing achines should be regular as well. This tie bound o(n log n) is actually optial since certain one-tape O(n log n)-tie deterinistic Turing achines can recognize non-regular languages. Unlie polynoial-tie coputation, one-tape linear-tie nondeterinistic coputation is sensitive to the definition of the achine s running tie. Such sensitivity is also observed in average-case coplexity theory [43]. By taing his wea definition that defines the running tie of a nondeterinistic Turing achine to be the length of a shortest accepting path, Michel [30] deonstrated that one-tape nondeterinistic Turing achines running in linear tie (in the sense of his wea definition) solve even NP-coplete probles. Clearly, his wea definition gives an enorous power to one-tape nondeterinistic achines and therefore it does not see to offer any interesting features of tie-bounded nondeterinis. On the contrary, the strong definition (in Michel s ter) requires the running tie to be the length of any longest (both accepting and rejecting) coputation path. This strong definition provides us with a reasonable basis to study the effect of linear tie-bounded coputations. We therefore adopt his strong definition of running tie and, throughout this paper, all one-tape tie-bounded Turing achines are assued to accoodate this strong definition. By expanding Kobayashi s result, we prove that one-tape o(n log n)-tie nondeterinistic Turing achines recognize only regular languages. The odel of alternating Turing achines of Chandra, Kozen, and Stoceyer [6] naturally expand the odel of nondeterinistic achines. The nuber of alternations of such an alternating Turing achine sees to enhance the coputational power of the achine; however, our strong definition of running tie aes it possible for us to prove that a constant nuber of alternations do not give any additional coputational power to one-tape linear-tie alternating Turing achines; naely, such achines recognize only regular languages. Apart fro nondeterinis, probabilistic Turing achines with fair coin tosses of Gill [16], can present distinctive features. Any language recognized by a certain one-head one-way probabilistic finite autoaton with unbounded-error probability is nown as a stochastic language [35]. By eploying a crossing sequence arguent, we can show that any language recognized by one-tape linear-tie probabilistic Turing achines with unbounded-error probability is just stochastic. This collapse result again proves a close relationship between one-tape linear-tie Turing achines and finite state autoata. The odel of Turing achines, nonetheless, presents distinguishing loos when we discuss functions rather than languages. Beyond the fraewor of foral language theory, Turing achines are capable of coputing (partial ultivalued) functions by siply odifying their tape contents and producing output strings (which are soeties viewed as nubers). Such functions also serve as any-one reductions between two languages. To explore the structure of language classes, we introduce various types of any-one one-tape linear-tie reductions. Nondeterinistic anyone reducibility, for instance, plays an iportant role in showing the aforeentioned collapse of alternating linear-tie coplexity classes. Naturally, we can view any-one reducibility as oracle echanis of the siplest for. In ters of such oracle coputation, we can easily prove the existence of an oracle that separates the one-tape linear-tie nondeterinistic coplexity class fro its deterinistic counterpart. The existence of a one-way function is a ey to the building of secure cryptosystes. Intuitively, a one-way function is a function that is easy to copute but hard to invert. Restricted to one-tape linear-tie deterinistic coputation, we can show that no one-way function exists. The nuber of accepting coputation paths of a tie-bounded nondeterinistic Turing achine has been a crucial player in coputational coplexity theory. With the notion of counting Turing achines, Valiant [42] initiated a systeatic study in the late 1970s on the structural properties of counting such nubers. Counting Turing achines have been since then used to study the coplexity of counting on nuerous issues in coputer science. The functions coputed by these achines are called counting functions and coplexity classes of languages defined in ters of such counting functions are generally referred to as counting classes. We show that counting functions coputable by one-tape linear-tie counting Turing achines are ore powerful than deterinistically coputable functions. By contrast, we also prove that certain counting classes induced fro one-tape linear-tie counting Turing achines collapse to the faily of regular languages.

3 24 K. Tadai et al. / Theoretical Coputer Science 411 (2010) The latest variant of the Turing achine odel is a quantu Turing achine, which is seen as an extension of a probabilistic Turing achine. While a probabilistic Turing achine is based on classical physics, a quantu Turing achine is based on quantu physics. The notion of such achinery was introduced by Deutsch [9] and later reforulated by Bernstein and Vazirani [5]. Of all the nown types of quantu Turing achines, we study only the following two achine types: bounded-error quantu Turing achines [5] and nondeterinistic quantu Turing achines [1]. We give a characterization of one-tape linear-tie nondeterinistic quantu Turing achines in ters of counting Turing achines. We also discuss suppleental echanis called advice to enhance the coputational power of Turing achines. Karp and Lipton [24] foralized the notion of advice, which eans additional inforation supplied to underlying coputation besides an original input. We adapt their notion in our setting of one-tape Turing achines as well as finite state autoata. We can deonstrate the existence of context-free languages that cannot be recognized by any one-tape linear-tie deterinistic Turing achines with advice. 2. Fundaental odels of coputation This paper uses a standard definition of a Turing achine (see, e.g., [11,19,20]) as a coputational odel. Of special interest are one-tape one-head Turing achines of various achine types. Here, we give brief descriptions of fundaental notions and notation associated with our coputational odel. Let Z, Q, R be the sets of all integers, of all rational nubers, of all real nubers, respectively. In particular, let R 0 be {r R r 0}. Moreover, let N denote the set of all natural nubers (i.e., nonnegative integers) and set N + = N {0}. For any two integers n, with n, an integer interval [n, ] Z eans the set {n, n + 1, n + 2,..., }. We assue that all logariths are to the base two. Throughout this paper, we use the notation Σ (Σ 1, Σ 2, etc.) to denote an arbitrary nonepty finite alphabet. A string over alphabet Σ is a finite sequence of eleents fro Σ and Σ denotes the collection of all finite strings over Σ. Note that the epty string over any alphabet is always denoted λ. Let Σ + = Σ {λ}. For any string x in Σ, x denotes the length of x (i.e., the nuber of sybols in x). A language (or siply a set ) over alphabet Σ is a subset of Σ, and a coplexity class is a collection of certain languages. The copleent of A is the difference Σ A, and it is often denoted A if Σ is clear fro the context. For any coplexity class C, the copleent of C, denoted co-c, is the collection of all languages whose copleents belong to C. We often use ulti-valued partial functions as well as single-valued total functions. For any ulti-valued partial function f apping fro a set D to another set E, do(f ) denotes the doain of f, naely, do(f ) = {x D f (x) is defined} and, for each x do(f ), f (x) is a subset of E. Whenever f is single-valued, we write f (x) = y instead of y f (x) by identifying the set {y} with y itself. Notice that total functions are also partial functions. The characteristic function χ A of a language A over Σ is defined as, for any string x in Σ, χ A (x) = 1 if x A and χ A (x) = 0 otherwise. For any single-valued total function g fro N to N, O(g(n)) denotes the set of all single-valued total functions f such that f (n) c g(n) for all but finitely any nubers n in N, where c is a positive constant independent of n. Siilarly, o(g(n)) is the set of all functions f such that, for every positive constant c, f (n) < c g(n) for all but finitely any nubers n in N. Let us give the basic definition of one-tape (one-head) Turing achines. A one-tape two-way one-head off-line Turing achine (abbreviated 1TM) is a septuple M = (Q, Σ, Γ, δ, q 0, q acc, q rej ), where Q is a finite set of (internal) states, Σ is a nonepty finite input alphabet, Γ is a finite tape alphabet including Σ, q 0 in Q is an initial state, q acc and q rej in Q are an accepting state and a rejecting state, respectively, and δ is a transition function. In later sections, we will define different types of transition functions δ, which give rise to various types of 1TMs. A halting state is either q acc or q rej. Our 1TM is equipped only with one input/wor tape such that (i) the tape stretches infinitely to both ends, (ii) the tape is sectioned by cells, and (iii) all cells in the tape are indexed with integers. The tape head starts at the cell indexed 0 (called the start cell) and either oves to the right (R), oves to the left (L), or stays still (N). A configuration of a 1TM M, which represents a snapshot of a coputation, is a triplet of an internal state, a head position, and a tape content of M. The initial configuration of M on input x is the configuration in which M is in internal state q 0 with the head scanning the start cell and the string x is written in an input/wor tape, surrounded by the blan sybols, in such a way that the leftost sybol of x is in the start cell. A coputation of a 1TM M generally fors a tree (called a coputation tree) whose nodes are certain configurations of M. The root of such a coputation tree is an initial configuration, leaves are final configurations, and every non-root node is obtained fro its parent node by a single application of δ. Each path of a coputation tree, fro its root to a certain leaf is referred to as a coputation path. An accepting (a rejecting, a halting, resp.) coputation path is a path terinating in an accepting (a rejecting, a halting, resp.) configuration. We say that a TM halts on input x if every coputation path of M on the input x eventually reaches a certain halting state. Of particular iportance is the synchronous notion for 1TMs. A 1TM is said to be synchronous if all coputation paths terinate at the sae tie on each input; naely, all the coputation paths have the sae length. Throughout this paper, we use the ter running tie for a 1TM M taing input x, denoted Tie M (x), to ean the height of the coputation tree produced by the execution of M on the input x; in other words, the length of any longest coputation path (no atter what halting state the achine reaches) of M on x. We often use the notation T(n) to denote a tie-bounding function of a given 1TM that aps N to N. Furtherore, a linear function eans a function of the for cx + d for a certain constant c, d R 0. A 1TM M is said to run in linear tie if its running tie Tie M (x) on any input x is upper-bounded by f ( x ) for a certain linear function f.

4 K. Tadai et al. / Theoretical Coputer Science 411 (2010) Although our achine has only one input/wor tape, the tape can be split into a constant nuber of tracs. To describe such tracs, we use the following notation. For any pair of sybols a, b Σ, [ a b ] denotes the special tape sybol for which a is written in the upper trac and b is written in the lower trac of the sae cell. By extending this notion, for any strings x, y Σ with x = y, we write [ x y ] to denote the concatenation [ x 1 y1 ][ x 2 y2 ] [ xn ] yn if x = x 1x 2 x n and y = y 1 y 2 y n, where all x i s and y i s are in Σ. For the definition of language recognition, we need to ipose certain reasonable accepting criteria as well as rejecting criteria onto our 1TMs to define the set of accepted input strings. With such criteria, we say that a 1TM recognizes a language A if, for every string x, (i) if x A then M halts on the input x and satisfies the accepting criteria and (ii) if x A then M halts and satisfies the rejecting criteria. The non-regularity easure has played a ey role in autoata theory. For any pair x and y of strings and any integer n N, we say that x and y are n-dissiilar with respect to a given language L if there exists a string z such that (i) xz n and yz n and (ii) xz L yz L. For each n N, define N L (n) (the non-regularity easure of L at n) to be the axial cardinality of a set in which any distinct pair is n-dissiilar with respect to L [13]. It is iediate fro the Myhill Nerode theore [20] that a language L is regular if and only if N L (n) = O(1) [13]. This is further iproved by the results of Karp [23] and of Kaņeps and Freivalds [22] as follows: a language L is regular if and only if N L (n) n + 1 for all 2 but finitely any nubers n in N. We assue the reader s failiarity with the notion of finite (state) autoata (see, e.g., [19,20]). The class of all regular languages is denoted REG, where a language is called regular if it is recognized by a certain (one-head one-way) deterinistic finite autoaton. The languages recognized by (one-head one-way) nondeterinistic push-down autoata are called context-free and the notation CFL denotes the collection of all context-free languages. A rational (one-head) one-way generalized probabilistic finite autoaton (for short, rational 1GPFA) [37,39] is a quintuple N = (Q, Σ, π, {T(σ ) σ Σ}, η), where (i) Q is a finite set of states, (ii) Σ is a finite alphabet, (iii) π is a row vector of length Q having rational coponents, (iv) for each σ Σ, T(σ ) is an Q Q atrix whose eleents are rational nubers, and (v) η is a colun vector of Q rational entries. A word atrix T(x) of N on input string x Σ is defined as T(λ) = I for the epty string λ, where I is the identity atrix of order Q, and T(x 1... x ) = T(x 1 )... T(x ) for x 1,..., x Σ. For each x Σ, the acceptance function p N (x) is defined to be π T(x) η. A atrix T is called stochastic if every row of T sus up to exactly 1. A rational (one-head) one-way probabilistic finite autoaton (for short, rational 1PFA) [35] N is a rational 1GPFA (Q, Σ, π, {T(σ ) σ Σ}, η) such that (i) π is a stochastic row vector whose entries are all nonnegative, (ii) for each sybol σ Σ, T(σ ) is stochastic with nonnegative coponents, and (iii) η is a colun vector whose coponents are either 0 or 1. Fro this η, we define the set F of all final states of N as F = {a Q the ath entry of η is 1}. Moreover, since p N (x) equals the probability of N accepting x, p N (x) is called the acceptance probability of N on the input x. Let ε be any rational nuber. For each rational 1GPFA N, let L(N, ε) = { x Σ p N (x) > ε} and L = (N, ε) = { x Σ p N (x) = ε}, where ε is called a cut point of N. Let GSL rat and SL rat denote the collections of all sets L(N, ε) for certain rational 1GPFAs N and for certain rational 1PFAs, respectively, where ε is a certain rational nuber. Siilarly, GSL = rat and SL = rat are defined fro GSL rat and SL rat, respectively, by substituting L = (N, ε) for L(N, ε). Sets in SL rat are nown as stochastic languages [35]. Turaainen [39] deonstrated the equivalence of GSL rat and SL rat. With a siilar idea, we can show that GSL = = rat SL= rat. The proof of this clai is left to the avid reader. 3. Deterinistic and reversible coputations Of all coputations, deterinistic coputation is one of the ost intuitive types of coputations. We begin this section with reviewing the ajor results of Hennie [18] and Kobayashi [25] on one-tape deterinistic Turing achines. A deterinistic 1TM, ebodying a sequential coputation, is forally defined by a transition function δ that aps (Q {q acc, q rej }) Γ to Q Γ {L, N, R}. Since the notation DLIN is widely used for the odel of ultiple-tape linear-tie Turing achines, we rather use the following new notations to ephasize our odel of one-tape Turing achines. The general notation 1-DTie(T(n)) denotes the collection of all languages recognized by deterinistic 1TMs running in T(n) tie. Given a set T of tie-bounding functions, 1-DTie(T ) stands for the union of 1-DTie(T(n)) s over all functions T in T. The one-tape deterinistic linear-tie coplexity class 1-DLIN is then defined to be 1-DTie(O(n)). Earlier, Hennie [18] proved that REG = 1-DLIN by eploying a so-called crossing sequence arguent. Elaborating Hennie s arguent, Kobayashi [25] substantially iproved Hennie s result by showing REG = 1-DTie(o(n log n)). This tie bound o(n log n) is optial because 1-DTie(O(n log n)) contains certain non-regular languages, e.g.,{a n b n n N} and {a 2n n N}. These facts establish the fundaental collapse and separation results concerning deterinistic 1TMs. Proposition 3.1 ([18,25]). REG = 1-DTie(o(n log n)) 1-DTie(O(n log n)). In the early 1970s, Bennett [4] initiated a study of reversible coputation. Reversible coputations have recently drawn wide attention fro physicists as well as coputer scientists in connection to quantu coputations. We adopt the following definition of a (deterinistic) reversible Turing achine given by Bernstein and Vazirani [5]. A (deterinistic) reversible 1TM is a deterinistic 1TM of which each configuration has at ost one predecessor configuration. We use the notation 1-revDTie(T(n)) to denote the collection of all languages recognized by T(n)-tie reversible 1TMs and define 1-revDTie(T ) to be T T 1-revDTie(T(n)). Finally, let 1-revDLIN = 1-revDTie(O(n)). Obviously, 1-revDLIN is a subset of 1-DLIN.

5 26 K. Tadai et al. / Theoretical Coputer Science 411 (2010) Kondacs and Watrous [26] deonstrated that any one-head one-way deterinistic finite autoaton can be siulated in linear tie by a certain one-head two-way deterinistic reversible finite autoaton. Since any one-head two-way deterinistic reversible finite autoaton is indeed a reversible 1TM, we obtain that REG 1-revDLIN. Proposition 3.1 thus concludes: Proposition 3.2. REG = 1-revDLIN = 1-revDTie(o(n log n)). The coputational power of a Turing achine can be enhanced by suppleental inforation given besides inputs. Karp and Lipton [24] introduced the notion of such extra inforation under the nae of advice, which is given depending only on the size of input. Da and Holzer [8] later considered finite autoata that tae the Karp Lipton type advice. To ae ost of the power of advice, we should tae a slightly different forulation for our odels. In this paper, for any coplexity class C defined in ters of Turing achines (including finite state autoata as special cases), the notation REG/n is used to represent the collection of all languages A for which there exist an alphabet Σ, a deterinistic finite autoaton M woring with another alphabet, and a total function 2 h fro N to Σ with h(n) = n (called an advice function) satisfying that, for every x Σ x, x A if and only if [ h( x ) ] L(M). For instance, the context-free language L eq = {0 n 1 n n N} belongs to REG/n. More generally, every language L, over the alphabet Σ, whose restriction L Σ n for each length n has cardinality bounded fro above by a certain constant, independent of n, belongs to REG/n, because the advice can encode a finite loo-up table for length n. This gives the obvious separation REG REG/n. On the contrary, REG/n cannot include CFL since, as we see below, the non-regular language Equal = {x {0, 1} # 0 (x) = # 1 (x)}, where # i (x) denotes the nuber of occurrences of the sybol i in x, is situated outside of REG/n. This result will be used in Section 7. Lea 3.3. The language Equal is not in REG/n. Hence, CFL REG/n. Proof. Let Σ = {0, 1}. Assuing that Equal REG/n, choose a deterinistic finite autoaton M = (Q, Σ, q 0, F) and an advice function h fro N to Σ such that, for every string x Σ x, x Equal if and only if [ h( x ) ] L(M). Tae n = Q. For each nuber [0, n] Z, y denotes any string of length n satisfying # 0 (y ) =. There exist two distinct indices, l [0, n] Z such that (i) y z, y l z l Equal for certain strings z, z l Σ n and (ii) M enters the sae internal state after reading [ y y w n ] as well as [ l w n ], where w n is the first n bits of h(2n). Notice that such y a pair (, l) indeed exists because n + 1 > Q. It follows fro these conditions that M also accepts the input [ z l h( y z l ) ]. Thus, # 0 (y z l ) = # 1 (y z l ), which iplies # 0 (z l ) = n. However, since # 0 (y l z l ) = # 1 (y l z l ), we obtain # 0 (y l ) =. This contradicts the definition of y l. Therefore, Equal is not in REG/n. The second clai CFL REG/n follows fro the fact that Equal CFL. Up to now, we have viewed Turing achines as language recognizers (or language acceptors); however, unlie deterinistic finite state autoata, Turing achines are fully capable of coputing partial functions. Since a 1TM M has only one input/wor tape, we need to designate the sae input tape as the output tape of the achine as well. To specify an outcoe of the achine, we adopt the following convention. When the achine eventually halts with its output tape consisting only of a single bloc of non-blan sybols, say s, surrounded by the blan sybols, in a way that the leftost sybol of s is written in the start cell, we consider s as the valid outcoe of the achine. For notational convenience, we introduce the function class 1-FLIN in the following fashion. A total function fro Σ 1 to Σ 2 is in 1-FLIN if there exists a deterinistic 1TM M satisfying that, on any input x Σ 1, (i) M halts by entering the accepting state in tie linear in x and (ii) when M halts, M outputs f (x) as a valid outcoe. When partial functions are concerned, we conventionally regard the rejecting state as an invalid outcoe. We thus define 1-FLIN(partial) to be the collection of all partial functions f fro Σ 1 to Σ 2 such that, for every x Σ 1, (i) if x do(f ) then M enters an accepting state with outputting f (x) and (ii) if x do(f ) then M enters a rejecting state (and we ignore the tape content). Historically, autoata theory has also provided the achinery that can copute functions (see, e.g., [20] for a historical account). In coparison with 1-FLIN, we herein consider only so-called Mealy achines. A Mealy achine (Q, Σ, Γ, q 0, δ, ν) is a deterinistic finite autoaton (Q, Σ, Γ, q 0, δ), ignoring final states, together with a total function ν fro Q Σ to Γ such that, on input x = x 1 x 2 x n, it outputs ν(q 0, x 1 )ν(q 1, x 2 ) ν(q n 1, x n ), where (q 0, q 1,..., q n ) is the sequence of states in Q satisfying δ(q i 1, x i ) = q i for every i [1, n] Z. Note that a Mealy achine coputes only length-preserving functions, where a (total) function is called length-preserving if f (x) = x for any string x. Consider the length-preserving function f defined by f (x 1 x 2 x n ) = x n x 1 x n 1 for any x 1, x 2,..., x n {0, 1}. It is clear that no Mealy achine can copute f. We therefore obtain the following proposition. Proposition 3.4. There exists a length-preserving function in 1-FLIN that cannot be coputed by any Mealy achines. 2 As standard in coputational coplexity theory, we allow non-recursive advice functions in general.

6 K. Tadai et al. / Theoretical Coputer Science 411 (2010) Nondeterinistic coputation Nondeterinis has been widely studied in the literature since any probles arising naturally in coputer science have nondeterinistic traits. In a nondeterinistic coputation, a Turing achine has several choices to follow at each step. We expand the collapse result of deterinistic 1TMs in Section 3 into nondeterinistic 1TMs. We also discuss the ulti-valued partial functions coputed by one-tape nondeterinistic Turing achines and show how to siulate such functions in a certain deterinistic anner Nondeterinistic languages As a language recognizer, a nondeterinistic 1TM taes a transition function δ that aps (Q {q acc, q rej }) Γ to 2 Q Γ {L,N,R}, where 2 A denotes the power set of A. An execution of a nondeterinistic 1TM produces a coputation tree. We say that a nondeterinistic 1TM M accepts an input x exactly when there exists an accepting coputation path in the coputation tree of M on the input x. Siilar to the deterinistic case, let 1-NTie(T(n)) denote the collection of all languages recognized by T(n)-tie 3 nondeterinistic 1TMs and let 1-NTie(T ) be the union of all 1-NTie(T(n)) s for all T T. We define the one-tape nondeterinistic linear-tie class 1-NLIN to be 1-NTie(O(n)). We first expand Kobayashi s collapse result on 1-DTie(o(n log n)) into 1-NTie(o(n log n)). Theore 4.1. REG = 1-NTie(o(n log n)) 1-NTie(O(n log n)). The proof of Theore 4.1 consists of two technical leas: Leas 4.2 and 4.3. The first lea has Kobayashi s arguent in [25, Theore 3.3] as its core, and the second lea is due to Hennie [18, Theore 2]. For the description of the leas, we need to introduce the ey terinology. Let M be any type of 1TM, which is not necessarily nondeterinistic. Any boundary that separates two adjacent cells in M s tape is called an intercell boundary. The crossing sequence at intercell boundary b along coputation path s of M is the sequence of internal states of M at the tie when the tape head crosses b, first fro left to right, and then alternately in both directions. To visualize the head ove, let us assue that the head is scanning tape sybol σ at tape cell i in state p. An application of a transition (q, τ, R) δ(p, σ ) aes the achine write sybol τ into cell i, enter state q, and then ove the head to cell i + 1. The state in which the achine crosses the intercell boundary between cell i and cell i + 1 is q (not p). Siilarly, if we apply a transition (q, τ, L) δ(p, σ ), then q is the state in which the achine crosses the intercell boundary between cell i 1 and cell i. The right-boundary of x is the intercell boundary between the rightost sybol of x and its rightadjacent blan sybol. Siilarly, the left-boundary of x is defined as the intercell boundary between the leftost sybol of x and its left-adjacent sybol. Any intercell boundary between the right-boundary and the left-boundary of x (including both ends) is called a critical boundary of x. Lea 4.2 observes that Kobayashi s arguent extends to nondeterinistic 1TMs without depending on their acceptance criteria. For copleteness, the proof of Lea 4.2 is included in the Appendix. Lea 4.2. Assue that T(n) = o(n log n). For any T(n)-tie nondeterinistic 1TM M, there exists a constant c N such that, for each string x, any crossing sequence at any critical boundary in any (accepting or rejecting) coputation path of M on the input x has length at ost c. In essence, Hennie [18] proved that any deterinistic coputation with short crossing sequences has constantlybounded non-regularity. We generalize his result to the nondeterinistic case as in the following lea. Different fro the previous lea, Lea 4.3 relies on the acceptance criteria of nondeterinistic 1TMs. Nonetheless, Lea 4.3 does not refer to rejecting coputation paths. For readability, the proof of Lea 4.3 is also placed in the Appendix. Lea 4.3. Let L be any language and let M be any nondeterinistic 1TM that recognizes L. For each n N, let S n be the set of all crossing sequences at any critical-boundary along any accepting coputation path of M on any input of length n. Then, N L (n) 2 S n for all n N, where S n denotes the cardinality of S n. Since REG is closed under copleentation, so is 1-NTie(o(n log n)) by Theore 4.1. In contrast, a siple crossingsequence arguent proves that 1-NTie(O(n log n)) does not contain the set of all palindroes, Pal = {x {0, 1} x = x R }, where x R is the reverse of x. Since Pal 1-NTie(O(n log n)), 1-NTie(O(n log n)) is different fro co-1-ntie(o(n log n)). Corollary 4.4. The class 1-NTie(o(n log n)) is closed under copleentation, whereas 1-NTie(O(n log n)) is not closed under copleentation. 3 As stated in Section 2, this paper accoodates the strong definition of running tie; naely, the running tie of a achine M on input x is the height of the coputation tree produced by M on x, independent of the outcoe of the coputation.

7 28 K. Tadai et al. / Theoretical Coputer Science 411 (2010) Reducibility between two languages has played a central role in the theory of NP-copleteness as a easuring tool for the coplexity of languages. We can see the reducibility as a basis of relativization with oracles. For instance, Turing reducibility induces a typical adaptive oracle coputation whereas truth-table reducibility represents a nonadaptive (or parallel) oracle coputation. Siilarly, we introduce the following restricted reducibility into one-tape Turing achines. A language A over alphabet Σ 1 is said to be any-one 1-NLIN-reducible to another language B over alphabet Σ 2 (notationally, A 1-NLIN B) if there exist a linear function T and a nondeterinistic 1TM M such that, for every string x in Σ 1, (i) M on the input x halts within tie T( x ) with the tape consisting only of one bloc of non-blan sybols, say y p, on every coputation path p, provided that the leftost sybol of y p ust be written in the start cell, (ii) when M eventually halts, the tape head returns to the start cell along all coputation paths, and (iii) x A if and only if y p B for soe accepting coputation path p of M on the input x. For any fixed set B, we use the notation 1-NLIN B to denote the collection of all languages A that are any-one 1-NLIN-reducible to B. Furtherore, for any coplexity class C, the notation 1-NLIN C stands for the union of sets 1-NLIN B over all sets B in C. A straightforward siulation shows that 1-NLIN REG is included in 1-NLIN. More generally, we can show the following proposition. This result will be used in Section 5. Proposition 4.5. For any language C, 1-NLIN 1-NLINC 1-NLIN C. Proof. This proposition is essentially equivalent to the transitive property of the relation 1-NLIN. Let A, B, and C be three arbitrary languages and assue that A 1-NLIN B 1-NLIN C. Our goal is to show that A 1-NLIN C. Tae a nondeterinistic 1TM M that any-one 1-NLIN-reduces A to B and another nondeterinistic 1TM M that any-one 1-NLIN-reduces B to C. Now, consider the following 1TM N. On input x, siulate M on x, and if and when it halts with an adissible value on the tape, start M on that value as its input. This achine N is clearly nondeterinistic and its running tie is O(n) since so are the running ties of M and M. It is not difficult to chec that N reduces A to C. Siilar to the any-one 1-NLIN-reducibility, we can define the any-one 1-DLIN-reducibility and its corresponding relativized class 1-DLIN B for any set B. Although 1-DLIN = 1-NLIN, two reducibilities, any-one 1-NLIN-reducibility and any-one 1-DLIN-reducibility, are quite different in their power. As an exaple, we can construct a recursive set B that separates between 1-DLIN B and 1-NLINB. The construction of such a set B can be done by a standard diagonalization technique. Proposition 4.6. There exists a recursive set B such that 1-DLIN B 1-NLINB. Proof. For any set B {0, 1}, define L B = {0 n x {0, 1} n [x B]}. Obviously, L B belongs to 1-NLIN B for any set B. Baer, Gill, and Solovay [2] constructed a recursive set B such that L B cannot be polynoial-tie Turing reducible to B. In particular, L B is not any-one 1-DLIN-reducible to B; that is, L B 1-DLIN B Multi-valued partial functions Conventionally, a Turing achine that can output values is called a transducer. Nondeterinistic transducers can copute ulti-valued partial functions in general. Let us consider a nondeterinistic 1TM that outputs a certain string in Σ 2 (whose leftost sybol is in the start cell) along each coputation path by entering a certain halting state. Siilar to partial functions introduced in Section 3, we invalidate any rejecting coputation path and let M(x) denote the set of all valid outcoes of M on input x. In particular, when M on the input x enters the rejecting state along all coputation paths, M(x) becoes the epty set. A ulti-valued partial function f fro Σ 1 to Σ 2 is in 1-NLINMV if there exists a linear-tie nondeterinistic 1TM M such that f (x) = M(x) for any string x Σ 1. Let 1-NLINSV be the subset of 1-NLINMV, containing only single-valued partial functions. In contrast, 1-NLINMV t and 1-NLINSV t denote the collections of all total functions in 1-NLINMV and in 1-NLINSV, respectively. Clearly, 1-FLIN(partial) 1-NLINSV 1-NLINMV and 1-FLIN 1-NLINSV t 1-NLINMV t. Note that, for any function f 1-NLINMV, we can decide nondeterinistically whether x is in do(f ), and thus do(f ) belongs to the class 1-NLIN, which equals REG by Theore 4.1. The basic relationship between functions in 1-FLIN and languages in 1-DLIN is stated in Lea 4.7. A ulti-valued partial function f fro Σ 1 to Σ 2 is called length-preserving if, for every x Σ 1 and y Σ 2, y f (x) iplies y = x. For convenience, we write LPF to denote the collection of all length-preserving ulti-valued partial functions fro Σ 1 to Σ 2, where Σ 1 and Σ 2 are arbitrary nonepty finite alphabets. Moreover, for any ulti-valued partial function f LPF, let L[f ] = {[ x y ] y f (x)}. Lea 4.7. For any ulti-valued partial function f LPF, f 1-NLINMV if and only if L[f ] is in 1-DLIN. Proof. Let f be any length-preserving ulti-valued partial function. Assue that f is coputed by a linear-tie nondeterinistic 1TM M. Consider the achine N that behaves as follows: on input [ x y ], nondeterinistically copute z in f (x) fro x and chec if y = z. This achine N places L[f ] in 1-NLIN, which equals 1-DLIN. Conversely, assue that L[f ] is recognized by a linear-tie deterinistic 1TM N. We define another achine M as follows: on input x, guess y Σ n (by writing y in the second trac), run N on input [ x y ]. If N accepts, output y. Clearly, M coputes f and thus f is in 1-NLINMV.

8 K. Tadai et al. / Theoretical Coputer Science 411 (2010) Fig. 1. The tape design of the folding achine N where = 2. The original tape of M is partitioned into 4 blocs of size n 1 and each bloc is siulated by a trac in the folded tape of N. For instance, bloc 0 is siulated by trac 0, and bloc 1 is siulated by trac 1 in the reverse order. The following ajor collapse result extends the collapse 1-NLIN = REG shown in Section 4.1. Theore NLINSV LPF = 1-FLIN(partial) LPF and thus 1-NLINSV t LPF = 1-FLIN LPF. Theore 4.8 is a direct consequence of the following ey lea. We first introduce the notion of refineent. For any two ulti-valued partial functions f and g fro Σ 1 to Σ 2, we say that f is a refineent of g if, for any x Σ 1, (i) f (x) g(x) (set inclusion) and (ii) f (x) = Ø iplies g(x) = Ø. (See, e.g., Selan s paper [36] for this notion.) Lea 4.9. Every length-preserving 1-NLINMV function has a 1-FLIN(partial) refineent. The crucial part of the proof of Lea 4.9 is the construction of a folding achine fro a given nondeterinistic 1TM. A folding achine rewrites the contents of cells in its input area, where the input area eans the tape region in which given input sybols are initially written. For later use, we give a general description of a folding achine. Construction of a folding achine. Let M = (Q, Σ, Γ, δ, q, 0 q acc, q rej ) be any 1TM that always halts in linear tie. The folding achine N is constructed fro M as follows. Choose the inial positive integer such that Tie M (x) x for all inputs x of length at least 3. Notice that, since M s tape head oves in both directions on its tape, M ay use tape cells indexed between 2( x 1) and 2( x 1) 1. Choose four new internal states q 0, q 1, q 2, q 3 not in Q and introduce new internal states of the for [ i q ] for each nuber i [ 2, 2 1] Z and each internal state q Q. Let x be an arbitrary string written on the input tape of M. (1) The achine N starts in the new initial state q 0. If the input x is epty, then N iediately enters M s halting state without oving its head. Hereafter, we assue that x is a nonepty string of the for σ 1 σ 2 σ n, where each σ i is a sybol in Σ. Note that σ 1 is written in the start cell. (2) In this preprocessing phase, the achine N re-designs its input/wor tape, as shown in Fig. 1, by oving its head. In the original tape of M, the cells indexed between 2( x 1) and 2( x 1) 1 are partitioned into 4 blocs of x 1 cells. These blocs are indexed in order fro the leftost bloc to the rightost bloc using integers ranging fro 2 to 2 1. In particular, bloc 0 contains the string σ 1 σ 2 σ n 1 (without σ n ). We split the tape of N into 4 tracs, which are indexed fro the top to the botto using 2 to 2 1. Intuitively, we want to siulate bloc i of M s tape using trac i of N s folded tape. The achine N first places the special sybol c (left end-arer) in all tracs of odd indices and then enters the internal state q 1 by stepping right. The achine eeps oving its head rightward in the state q 1. When the head encounters the first blan sybol, if x 3 then N enters the state q 2 and steps bac; otherwise, N enters M s halting state. In a single step, N places another special sybol $ (right end-arer) in all tracs of even indices, shifts σ n in trac 0 to trac 1, enters the state q 3, and steps to the left. The head then returns to the start cell in state q 3. Notice that this phase can be done in a reversible fashion. (3) The achine N siulates M s ove by folding M s tape content into 4 tracs of the input area. While M stays within bloc i in state q, N siulates M s ove on trac i with internal state [ i q ]. If i is even, then N oves its head in the sae direction as M does. Otherwise, N oves the head in the opposite direction. In particular, at the tie when M s head leaves the last (first, resp.) cell of bloc 2j to its adjacent bloc by rewriting sybol σ and entering the state q, N instead enters state [ 2j + 1 q ] ([ 2j 1 q ], resp.), writes sybol σ in trac 2j, and oves its head to the right. On the contrary, at the tie when M s head leaves bloc 2j + 1, M oves the head siilarly but in the opposite direction. It is clear that N s head never visits outside of the input area. This siulation phase taes exactly the sae aount of tie as M s. Consider the set S of all (possible) crossing sequences of the folding achine N. For any two crossing sequences v, v S and any tape sybol σ, we write v σ v if v is a crossing sequence of the left-boundary of σ and v is a crossing sequence of the right-boundary of σ along a certain coputation path of N on input xσ y for certain strings x and y. Along any coputation path p of N on any nonepty input x, it is iportant to note that v 0 = (), the epty sequence, and v f = (q 1, q 2 ) are respectively the unique crossing sequences at the left-boundary and the right-boundary of x. We can translate this coputation path p on input x = σ 1 σ 2 σ n into its corresponding series of crossing sequences, v 0, v 1,..., v n, satisfying the following conditions: v n = v f and v i 1 σi v i for every index i [1, n] Z. Now, let us return to the proof of Lea 4.9.

9 30 K. Tadai et al. / Theoretical Coputer Science 411 (2010) Proof of Lea 4.9. Let f be any length-preserving ulti-valued partial function in 1-NLINMV. There exists a linear-tie nondeterinistic 1TM M = (Q, Σ, Γ, δ, q 0, q acc, q rej ) that coputes f. Consider the folding achine N constructed fro M. Matching the output convention of 1TMs, we need to odify this folding achine to produce the outcoes of M. After N eventually halts, we further ove the tape head leftward. When we reach the left end-arer, we ove bac the head by changing the current tape sybol to the sybol written in the area of trac 0 and trac 1 where the original input sybols of N is written. When we reach the right end-arer, we step right to the first blan sybol by entering a halting state (either q acc or q rej ) of M. Evidently, this odified nondeterinistic 1TM produces the outcoes of M and also enters exactly the sae halting states of M. This odified achine is hereafter referred to as N for our convenience. Let CS be the set of all crossing sequences of N. Assue that all eleents in CS are enuerated so that we can always find the inial eleent in any subset of CS. For any two eleents v, v CS and any sybol σ with v σ v, Syb(v, σ, v ) denotes the output sybol written in the cell where σ is initially written. This sybol Syb(v, σ, v ) can be easily deduced fro (v, v, σ ) by tracing the tape head oves crossing the cell that initially contains the sybol σ. Finally, we want to construct a refineent g of f. This desired partial function g is defined by a deterinistic 1TM M that behaves as follows. Let n 3 and let x = σ 1 σ 2 σ n be an arbitrary input of length n. Set v 0 = () and v f = (q 1, q 2 ) as before. (1) In this phase, all internal states except v f are subsets of CS. Let S 1 = {v 0 } be the initial state of M. Let i [1, n] Z and assue that M currently scans the input sybol σ i in internal state S i. We define two ey sets V i = {v S i v CS [v σi v ]} and S i+1 = {v CS v V i [v σi v ]}. Intuitively, S i+1 captures all possible nondeterinistic oves fro S i. Notice that V i S i. When S i+1 is epty, M enters a new rejecting state. Provided that S i+1 is non-epty, M changes the tape sybol σ i to [ σ i V i ] and enters S i+1 as an internal state by stepping to the right. Unless x do(f ), after scanning σ n, M enters the internal state S n+1. By the property of the original folding achine, we ust have S n+1 = {v f }. For later convenience, let v n = v f and V n+1 = S n+1. When the tape head scans the first blan sybol, M then enters the internal state v n by stepping to the left. (2) In the beginning of this second phase, M is in the state v n, scanning the rightost tape sybol [ σn Vn ] in the input area. Notice that v n V n+1. For any index i [1, n] Z, let us assue that M scans the sybol [ σ i V i ] in the state v i, where v i V i+1 S i+1. Since M passes the first phase and enters the second phase, V i cannot be epty. Since v i S i+1, the set W i = {v V i v σi v i } is not epty, either. Choose the inial eleent, say v i 1, in W i. This crossing sequence v i 1 obviously satisfies that v i 1 σi v i. Now, M changes the sybol [ σ i V i ] to Syb(v i 1, σ i, v i ) and oves its tape head to the left by entering v i 1 as an internal state. After scanning [ σ 1 V 1 ], M enters the internal state v 0 because V 1 = S 1 = {v 0 }. Note that the resulting series (v 0, v 1, v 2,..., v n ) specifies a certain accepting coputation path of N and the output tape of M contains the outcoe produced along this particular coputation path. When the tape head reaches the blan sybol, M finally enters a new accepting state. This copletes the description of M. The above deterinistic 1TM M clearly produces, for each input x, at ost one output string fro the set f (x). Note that, if x do(f ), all coputation paths are rejecting paths, and thus M never reaches any accepting state. It is therefore obvious that the partial function g coputed by M is a refineent of f. Another application of Lea 4.9 is the non-existence of one-way functions in 1-FLIN. To describe the notion of one-way function in our single-tape linear-tie odel, we need to expand our trac notation [ x y ] to the case where x and y differ. To eep our notation siple, we also use the sae notation [ x y ] to express [ x# ] if x + = y and 1 and y express [ x y# ] if x = y + and 1, where # is a distinct blan sybol. A total function f is called one-way if (i) ( ( )) f 1-FLIN and (ii) there is no function g 1-FLIN such that f g [ f (x) 1 x ] = f (x) for all inputs x. When f is lengthpreserving, the equality f ( g Proposition There is no one-way function in 1-FLIN. ( [ f (x) 1 x ] )) = f (x) can be replaced by f (g(f (x))) = f (x). Proof. Assue that a one-way function f apping Σ 1 to Σ 2 exists in 1-FLIN. Let f 1 denote a ulti-valued partial function defined as follows. For each string of the for [ otherwise, we let f 1 ([ y 1 ([ n ], if y n, then we define f 1 y 1 n ] ) = {x# y n x = n, f (x) = y}; ) y 1 n ] = {x x = n, f (x) = y}. Note that f 1 is length-preserving and belongs to 1-NLINMV. Lea 4.9 ensures the existence of a 1-FLIN(partial) function g that is a refineent of f 1. Consider the following 1TM M: y on input [ 1 n ], chec if [ y 1 n ] do(g). If not, M outputs any fixed string of length n (e.g., 0n ). Otherwise, M coputes ( ) y g [ 1 n ] and outputs a string obtained fro it by deleting the sybol #. Since do(g) is in 1-DLIN, M can be deterinistic. Clearly, M inverts f ; that is, f be one-way. ( M ( [ f (x) 1 x ] )) = f (x) for all inputs x. This contradicts the one-wayness of f. Therefore, f cannot The third application concerns the advised class REG/n. Siilar to this class, we define 1-DLIN/lin as the collection of all languages A such that there are a linear-tie deterinistic 1TM M, an advice function h, and a constant c 1 for which (i)