Nondeterministic Turing Machines Research Articles

Reversible simulation of space-bounded computations

We give an alternative proof of Bennett's simulation of deterministic Turing machines by reversible ones (machines whose configuration graph has out-degree and in-degree one) with a quadratic loss of space. More importantly, our proof extends this result to nondeterministic Turing machines.

On the Computational Power of Neural Nets

This paper deals with finite size networks which consist of interconnections of synchronously evolving processors. Each processor updates its state by applying a "sigmoidal" function to a linear combination of the previous states of all units. We prove that one may simulate all Turing machines by such nets. In particular, one can simulate any multi-stack Turing machine in real time, and there is a net made up of 886 processors which computes a universal partial-recursive function. Products (high order nets) are not required, contrary to what had been stated in the literature. Non-deterministic Turing machines can be simulated by non-deterministic rational nets, also in real time. The simulation result has many consequences regarding the decidability, or more generally the complexity, of questions about recursive nets.

First-order linear logic without modalities is NEXPTIME-hard

The decision problem is studied for the nonmodal or multiplicative-additive fragment of first-order linear logic. This fragment is shown to be NEXPTIME-hard. The hardness proof combines Shapiro's logic programming simulation of nondeterministic Turing machines with the standard proof of the PSPACE-hardness of quantified boolean formula validity, utilizing some of the surprisingly powerful and expressive machinery of linear logic.

New lower bounds for element distinctness on a one-tape Turing machine

It is shown that the Element Distinctness Problem ( n numbers of k log n bits each, k ⩾ 2) on a one-tape Turing machine takes time proportional to almost the square of the size of the input. The proof holds for both deterministic and nondeterministic Turing machines. This proof improves the best known lower bound of Ω( n 2 log n ) for deterministic Turing machines and of Ω(nlog 2n) (B. Kalyanasundaram and G. Schnitger (1992)) for nondeterministic Turing machines to Ω(n 2log n) . The lower bound is generalized to the n-Element Distinctness problem; on inputs of size N = nm, with 1 ⩽ n < N log N , it is shown to take time Ω(max{ Nn, Nm}). The proof makes use of Kolmogorov Complexity.

Simple characterizations of P(#P) and complete problems

In this paper, P(#P) and PF(#P) are characterized in terms of a largely different computation structure, where P(#P) (resp., PF(#P)) is the class of sets (resp., functions) that are polynomial-time Turing reducible to #P functions. Let MidP be the class of functions that give the medians in the outputs of metric Turing machines, where a metric Turing machine is a polynomial-time bounded nondeterministic Turing machine such that each branch writes a binary number on an output tape. Then it is shown that every function in PF(#P) is polynomial-time one-Turing reducible to a function in MidP and MidP ⊆ PF (#P); that is, PF(#P) = PF(MidP[1]). Furthermore, it is shown that for all sets L, L is in P(#P) if and only if there is a function F ∈ MidP, such that for every string x, x ∈ L, iff F(x) is odd. Thus the problem of computing medians in the outputs of metric Turing machines captures the computational complexity of P(#P) and PF(#P). As applications of the results, several natural polynomial-time many-one complete problems for P(#P) are shown, for example, given an undirected graph with integer edge weights, checking that the parity of the middle cost among all the simple circuits is complete for P(#P).

On 1-inkdot alternating Turing machines with small space

This paper investigates the accepting powers of nondeterministic and alternating 1-inkdot Turing machines using small space. Let NTM (ATM, UTM) denote a nondeterministic Turing machine (alternating Turing machine, alternating Turing machine with only universal states). For each Xϵ{N, A, U}, let STRONG-XSPACE[ L( n)] (STRONG-XSPACE ∗[ L( n)]) denote the class of languages accepted by strongly L( n) space-bounded XTMs (1-inkdot XTMs), and let WEAK-XSPACE[ L( n)] (WEAK-XSPACE ∗[ L( n)]) denote the class of languages accepted by weakly L( n) space-bounded XTMs (1-inkdot XTMs). We show that 1. (1) STRONG-ASPACE∗[log log n ] - WEAK-ASPACE[o(log n )]≠∅, 2. (2) STRONG-USPACE∗[log log n ] - WEAK-USPACE[o(log n )]≠∅, 3. (3) STRONG-ASPACE∗[log log n ] - WEAK-NSPACE∗[o(log n )]≠∅, and 4. (4) STRONG-ASPACE∗[log log n ] - WEAK-USPACE∗[o(log n )]≠∅.

Universal Traversal Sequences

In this article we discuss a purely combinatorial problem, the construction of short universal traversal sequences, and its relationship to questions about logspace computation. We state the problem formally, show how it arises naturally in complexity theory, and review some of the known partial results. A basic introduction to complexity theory can be found in [6]. The P vs. NP problem is recognized by the mathematical world as the central open question in the theory of computation. Less widely known outside of computer science is the fact that the analogous question for space-bounded computation was resolved long ago': Savitch [8] shows that any language accepted by a nondeterministic Turing machine that uses space O(s(n)) is also accepted by a deterministic Turing machine that uses space O((s(n))2). Hence deterministic polynomial space is equivalent (in terms of language-recognition power) to nondeterministic polynomial space: PSPACE = NPSPACE. However, one question about the relationship of nondeterministic spacebounded computation and its deterministic counterpart remains open: Is the quadratic blow-up in space complexity exhibited by Savitch's construction necessary? This question turns out to be most interesting for computations that use very little space. Let L be the class of languages accepted by deterministic Turing machines using only O(log n) space, and NL be the class of languages accepted by nondeterministic Turing machines using only O(log n) space. It could be the case that Savitch's theorem is optimal at this low end of the space-complexity spectrum. On the other hand, it could be that any language recognizable by a nondeterministic Turing machine is also recognizable by a deterministic Turing machine with the same space complexity; if that's true, then NL is exactly equal to L. The truth might also lie somewhere between these two extremes.

Simulation of three-dimensional one-marker automata by five-way Turing machines

This paper investigates the simulation of three-dimensional one-marker automata by five-way Turing machines. We show that the necessary and sufficient space for fiveway three-dimensional deterministic Turing machines to simulate three-dimensional deterministic one-marker automata (three-dimensional nondeterministic one-marker automata) is 2 Θ( lm log( lm)) (2 Θ(l 2m 2) and the necessary and sufficient space for five-way three-dimensional nondeterministic Turing machines to simulate three-dimensional deterministic one-marker automata (three-dimensional nondeterministic one-marker automata) is lm log( lm)( l 2 m 2), where l( m) is the number of rows (columns) on each plane of three-dimensional rectangular input tapes.

A note on multi-inkdot nondeterministic Turing machines with small space

A note on multi-inkdot nondeterministic Turing machines with small space

A speed-up theorem without tape compression

We shall show that, for each nondeterministic single-tape Turing machine M of time complexity T( n)⩾ n 2 and each K⩾1, there exists and equivalent K times faster nondeterministic Turing machine M′ writing only zeroes and ones on its tape. In other words, we can obtain the linear speed-up while preserving the binary worktape alphabet. Therefore, nondeterministic single-tape Turing machines do not require tape compression for speeding up.

Some formal properties of stratified feature grammars

In this paper, we investigate the formal expressive properties of Stratified Feature Grammar (SFG), a new logic-based linguistic framework motivated by relational grammar and metagraph grammar, as well as by Kasper-Rounds logic. The driving force behind SFG is the generalization of the conceptfeature from an unanalyzable atomic one to a sequence of so-called R-signs. The linguistic interpretation of thesestratified features is that each R-sign in such a sequence denotes a primitive grammatical relation such as subject or direct object in different syntactic “strata”. This generalization permits the specification of a rigorous feature-structure-based formalism for natural-language grammars based on the view that syntax is “multistratal” and “relational”. The introduction of stratified features leads to several other innovations, two of which might have utility in other frameworks. One is the idea of imposing a partial order on features. The other is the concept of(data) justification: essentially, this is a stipulation that for an S-graph to be well-formed with respect to some grammarG it must, in addition to satisfying the rules ofG, have each of its “core” data (each feature occurrence, each node-label occurrence and each instance of so-called structure-sharing) justified in a formally precise manner by some rule of Justification ensures that. Justification ensures that satisfying S-graphs do not have more structure than absolutely necessary and so makes it appealing to a notion of “minimal model” otiose. Justification plays a key role in a number of our proofs. The formal results presented here include the following. First, it is proved that in the unrestricted SFG framework, every type 0 (r.e.) language is generated by some SFG. Then, we restrict the framework to so-calledbounded SFG with two linguistically motivated principles: Lexical Anchoring and Boundedness. Anchoring requires, in essence, that each core datum of an S-graph be justified by aword in its yield. Boundedness insures that S-graph features are short. Although these restrictions together put the class of bounded SFG languages well within the class of recursive languages, we go on to demonstrate that the recognition problem for bounded SFG is NP-hard. Further, we establish that a bounded SFG language need not be semi-linear. On the matter of upper bounds, we show that every bounded SFL can be recognized by a nondeterministic Turing machine inn logn space and polynomial time and that the recognition problem for bounded SFL's is NP-complete.

A note on three-dimensional alternating Turing machines with space smaller than log m

This paper deals with the accepting powers of five-way three-dimensional alternating machines and six-way three-dimensional alternating machines whose input tapes are restricted to cubic ones. We first show that for space smaller than log m, five-way three-dimensional alternating Turing machines are less powerful than six-way three-dimensional alternating Turing machines. We then show that the set of all the three-dimensional connected tapes can be accepted by a six-way three-dimensional alternating finite automaton, but not accepted by any o(log m) space-bounded five-way three-dimensional alternating Turing machine. Finally, we show that there exists a language accepted by a five-way three-dimensional alternating finite automaton, but not accepted by any o(log m) space-bounded six-way three-dimensional nondeterministic Turing machine.

${\text{ASPACE}}(o(\log \log n))$ is Regular

One of the common results of resource bounded Turing machines is the $\log \log n$ lower bound for the space usage of deterministic and nondeterministic Turing machines that accept nonregular languages. In this paper this result is extended to alternating Turing machines: It is proved that if $f(n) = o(\log \log n)$, then $f(n)$-space-bounded (off-line) alternating Turing machines can accept only regular sets. The problem has been open for a decade.

A relationship between nondeterministic Turing machines and 1-inkdot turing machines with small space

A relationship between nondeterministic Turing machines and 1-inkdot turing machines with small space

On intractability of the classUP

The classUP [V] is the class of sets accepted by polynomial-time nondeterministic Turing machines which have at most one accepting path for every input. The complexity of this class closely relates to that of computing inverses ofone-way functions, where a one-way function is a one-to-one, length-increasing, and polynomial-time computable function whose inverse cannot be computed within polynomial time. It is known [GS], [K] that there exists a one-way function if and only ifP ≠UP. In this paper the intractability of sets inUP is investigated in terms of polynomial-time reducibility to a sparse set. It is shown thatUP has a set that is ≤mP-reducible to no sparse set ifP ≠UP. We interpret this structural property in the relation with approximation algorithms: it is shown that ifP ≠UP, thenUP has a set with no 1-APT approximation and, furthermore,UP has a set that is not ≤mP-reducible to any set with a 1-APT approximation. The implication of this result in the study of one-way functions is also discussed. In order to prove the main theorem, we introduce a variation of “tree-pruning” methods.

EFFICIENT DETECTORS AND CONSTRUCTORS FOR SIMPLE LANGUAGES

In this paper, we investigate the complexity of computing the detector, constructor and lexicographic constructor functions for a given language. The following classes of languages will be considered: (1) context-free languages, (2) regular sets, (3) languages accepted by one-way nondeterministic auxiliary pushdown automata, (4) languages accepted by one-way nondeterministic logspace-bounded Turing machines, (5) two-way deterministic pushdown automaton languages, (6) languages accepted by uniform families of constant-depth polynomial-size Boolean circuits, and (7) languages accepted by multihead finite automata. We show that for the classes (1)–(4), efficient detectors, constructors and lexicographic constructors exist, whereas for (5)– (7) polynomial-time computable detectors, constructors and lexicographic constructors exist iff there are no sparse sets in NP−P (or equivalently, E=NE). Our results provide sharp boundaries between classes of languages which have efficient detectors, constructors and classes of languages for which efficient detectors and constructors do not appear to exist.

An NP-complete language accepted in linear time by a one-tape Turing machine

We consider one-tape nondeterministic Turing machines, i.e. with a unique worktape on which the input is written. It is known that there are nonregular languages accepted by such machines in time O( n log log n), and an NP-complete language accepted in time O( n log n). We exhibit nonregular languages accepted by such machines in time n + O(√ nlog n) or n + O(log 2 n), and an NP-complete language accepted in time n + O(√ nlog n, by using a tight padding method.

Nondeterministic Computations in Sublogarithmic Space and Space Constructibility

The open problem of nondeterministic space constructibility for sublogarithmic functions is resolved. We show that there are no unbounded monotone increasing nondeterministically space constructible functions with supn→∞s(n)/log(n)=0. Consequently, the space constructibility cannot be used to separate nondeterministic space from deterministic one, since functions like loglog(n), and \(\sqrt {\log (n)}\)are not space constructible by nondeterministic Turing machines.

The complexity of ranking simple languages

Ranking is the problem of computing for an input string its lexicographic index in a given (fixed) language. This paper concerns the complexity of ranking. We show that ranking languages accepted by 1-way unambiguous auxiliary pushdown automata operating in polynomial time is inNC (2). We also prove negative results about ranking for several classes of simple languages.C is rankable in deterministic polynomial time iffP=P #P , whereC is any of the following six classes of languages: (1) languages accepted by logtime-bounded nondeterministic Turing machines, (2) languages accepted by (uniform) families of unbounded fan-in circuits of constant depth and polynomial size, (3) languages accepted by 2-way deterministic pushdown automata, (4) languages accepted by multihead deterministic finite automata, (5) languages accepted by 1-way nondeterministic logspace-bounded Turing machines, and (6) finitely ambiguous linear context-free languages.

Domino Games and Complexity

Domino games which describe computations of alternating Turing machines in the same way as dominoes (tiling systems) encode computations of deterministic and nondeterministic Turing machines are considered. The domino games are two-person games in the course of which the players build up domino tilings of a square of prescribed size. Acceptance of an alternating Turing machine corresponds to a winning strategy for one player—the number of moves in the game is the number of alternations of the Turing machine. Let $\operatorname{ATIME}(T(n), m)$ denote the class of all sets that are accepted by some alternating Turing machine in time $T(n)$ with at most m alternations. It is shown that any problem in such a complexity class can be reduced to the strategy problem for some domino game. In particular domino games which are complete in the classes $\Sigma_{m}^{p}$, and $\Pi_{m}^{p}$, of the polynomial time hierarchy are found. This corresponds to the approach of Savelsbergh and van Emde Boas and of Lewis and Papadimitriou, who have shown that the theory of NP-completeness may also be founded on a finite domino problem instead of the satisfiability problem for propositional formulae. Similar generalizations are possible for domino connectability problems: Domino thread games where the players build up a thread of dominoes are introduced; the first player wins if he can ultimately connect two given points. It is shown that a certain class of such games captures deterministic time complexity. In particular, games are found whose strategy problems are P-complete. In the last section applications to the complexity of simple prefix classes in logical theories are briefly discussed.