Space-bounded Turing Machines Research Articles

Generalizations of Checking Stack Automata: Characterizations and Hierarchies

We examine different generalizations of checking stack automata by allowing multiple input heads and multiple stacks, and characterize their computing power in terms of two-way multi-head finite automata and space-bounded Turing machines. For various models, we obtain hierarchies in terms of their computing power. Our characterizations and hierarchies expand or tighten some previously known results. We also discuss some decidability questions and the space/time complexity of the models.

Reversibility of computations in graph-walking automata

Graph-walking automata (GWA) are finite-state devices that traverse graphs given as an input by following their edges; they have been studied both as a theoretical notion and as a model of pathfinding in robotics. If a graph is regarded as the set of memory configurations of a certain abstract machine, then various families of devices can be described as GWA: such are two-way finite automata, their multi-head and multi-tape variants, tree-walking automata and their extension with pebbles, picture-walking automata, space-bounded Turing machines, etc. This paper defines a transformation of an arbitrary deterministic GWA to a reversible GWA. This is done with a linear blow-up in the number of states, where the constant factor depends on the degree of the graphs being traversed. The construction directly applies to all basic models representable as GWA, and, in particular, subsumes numerous existing results for making individual models halt on every input.

Deterministic One-Way Turing Machines with Sublinear Space

Deterministic one-way Turing machines with sublinear space bounds are systematically studied. We distinguish among the notions of strong, weak, and restricted space bounds. The latter is motivated by the study of P automata. The space available on the work tape depends on the number of input symbols read so far, instead of the entire input. The class of functions space constructible by such machines is investigated, and it is shown that every function f that is space constructible by a deterministic two-way Turing machine, is space constructible by a strongly f space-bounded deterministic one-way Turing machine as well. We prove that the restricted mode coincides with the strong mode for space constructible functions. The known infinite, dense, and strict hierarchy of strong space complexity classes is derived also for the weak mode by Kolmogorov complexity arguments. Finally, closure properties under AFL operations, Boolean operations and reversal are shown.

Parallel computation using active self-assembly

We study the computational complexity of the recently proposed nubots model of molecular-scale self-assembly. The model generalises asynchronous cellular automata to have non-local movement where large assemblies of molecules can be moved around, analogous to millions of molecular motors in animal muscle effecting the rapid movement of macroscale arms and legs. We show that nubots is capable of simulating Boolean circuits of polylogarithmic depth and polynomial size, in only polylogarithmic expected time. In computational complexity terms, we show that any problem from the complexity class NC is solved in polylogarithmic expected time on nubots that use a polynomial amount of workspace. Along the way, we give fast parallel algorithms for a number of problems including line growth, sorting, Boolean matrix multiplication and space-bounded Turing machine simulation, all using a constant number of nubot states (monomer types). Circuit depth is a well-studied notion of parallel time, and our result implies that nubots is a highly parallel model of computation in a formal sense. Asynchronous cellular automata are not capable of such parallelism, and our result shows that adding a movement primitive to such a model, to get the nubots model, drastically increases parallel processing abilities.

Reversibility in space-bounded computation

In this paper, we investigate how computing power of a space-bounded Turing machine (TM) is affected by reversibility and determinism. We show an irreversible deterministic TM (IDTM), and a reversible non-deterministic TM (RNTM) can be simulated by a reversible and deterministic TM (RDTM) that uses exactly the same numbers of storage tape symbols and storage tape squares. Thus, an RDTM has relatively high capability in spite of the constraints of reversibility and determinism. Here, we also discuss a space-bounded symmetric TM.

Negative Interactions in Irreversible Self-assembly

This paper explores the use of negative (i.e., repulsive) interactions in the abstract Tile Assembly Model defined by Winfree. Winfree in his Ph.D. thesis postulated negative interactions to be physically plausible, and Reif, Sahu, and Yin studied them in the context of reversible attachment operations. We investigate the power of negative interactions with irreversible attachments, and we achieve two main results. Our first result is an impossibility theorem: after t steps of assembly, Ω(t) tiles will be forever bound to an assembly, unable to detach. Thus negative glue strengths do not afford unlimited power to reuse tiles. Our second result is a positive one: we construct a set of tiles that can simulate an s-space-bounded, t-time-bounded Turing machine, while ensuring that no intermediate assembly grows larger than O(s), rather than O(s⋅t) as required by the standard Turing machine simulation with tiles. In addition to the space-bounded Turing machine simulation, we show another example application of negative glues: reducing the number of tile types required to assemble “thin” (n×o(logn/loglogn)) rectangles.

Machine semantics

Using simple systems with a notion of discrete deterministic evolution over time, we study discrete causality via tools from theoretical computer science and logic. We consider the set of all representations (i.e. partial descriptions) of such systems, from algebraic, domain-theoretic, and categorical viewpoints. The order theory introduced, is based in the notion of comparing high-level and low-level descriptions of the same system. This is shown to give a Complete Partial Order where the down-closure of each element is a locale. This partial order has a very close connection to a categorical construction known as the ‘particle-style’ trace, via analogues of domain-theoretic equations. Thus, the trace may be thought of as the computation of suprema in this partial order. As a sample application, we show how to construct algebraic models of space-bounded Turing machines in these terms, and derive compositionality from the abstract properties of the trace.

The Complexity of Model Checking Higher-Order Fixpoint Logic

Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed \lambda-calculus and the modal \lambda-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal \lambda-calculus. This paper provides complexity results for its model checking problem. In particular we consider those fragments of HFL built by using only types of bounded order k and arity m. We establish k-fold exponential time completeness for model checking each such fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singly-exponential in the size of the formula and k-fold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence, we obtain an exponential time upper bound on the expression complexity of each such fragment. The lower bound is established by a reduction from the word problem for alternating (k-1)-fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are already hard with respect to k-fold exponential time, we obtain, as a corollary, k-fold exponential time completeness for the data complexity of our fragments of HFL, provided m exceeds 3. This also yields a hierarchy result in expressive power.

Characterizations of context-sensitive languages and other language classes in terms of symport/antiport P systems

We give “syntactic’’ characterizations of context-sensitive languages (CSLs) in terms of some restricted models of symport/antiport P systems. These are the first such characterizations of CSLs in terms of P systems. In particular, we show the following for any language L over a binary alphabet: (1) Let m be any integer ≥ 1 . Then L is a CSL if and only if it can be accepted by a restricted symport/antiport P system with m membranes and multiple number of symbols (objects). Moreover, holding the number of membranes at m, there is an infinite hierarchy in computational power (within the class of binary CSLs) with respect to the number of symbols. (2) Let s be any integer ≥ 14 . Then L is a CSL if and only if it can be accepted by a restricted symport/antiport P system with s symbols and multiple number of membranes. Moreover, holding the number of symbols at s, there is an infinite hierarchy in computational power with respect to the number of membranes. (Similar results hold for languages over an alphabet of k ≥ 2 symbols.) Thus (1) and (2) say that in order for the restricted symport/antiport P systems to accept all binary CSLs, at least one parameter (either the number of symbols or the number of membranes) must grow. These are the first results of their kind in the P systems area. They contrast a known result that (unrestricted) symport/antiport P systems with s ≥ 2 symbols and m ≥ 1 membranes accept (or generate) exactly the recursively enumerable sets of numbers even for s + m = 6 . We also note that previous characterizations of formal languages in the membrane computing literature are mostly for the Parikh images of languages. Variations of our model yield characterizations of regular languages, languages accepted by one-way log n space-bounded Turing machines, and recursively enumerable languages.

Continuous-time symmetric Hopfield nets are computationally universal.

We establish a fundamental result in the theory of computation by continuous-time dynamical systems by showing that systems corresponding to so-called continuous-time symmetric Hopfield nets are capable of general computation. As is well known, such networks have very constrained Lyapunov-function controlled dynamics. Nevertheless, we show that they are universal and efficient computational devices, in the sense that any convergent synchronous fully parallel computation by a recurrent network of n discrete-time binary neurons, with in general asymmetric coupling weights, can be simulated by a symmetric continuous-time Hopfield net containing only 18n + 7 units employing the saturated-linear activation function. Moreover, if the asymmetric network has maximum integer weight size w(max) and converges in discrete time t*, then the corresponding Hopfield net can be designed to operate in continuous time Theta(t*/epsilon) for any epsilon > 0 such that w(max)2(12n) </= epsilon2(1/epsilon). In terms of standard discrete computation models, our result implies that any polynomially space-bounded Turing machine can be simulated by a family of polynomial-size continuous-time symmetric Hopfield nets.

Separations by Random Oracles and “Almost” Classes for Generalized Reducibilities

Let ≤r and ≤sbe two binary relations on 2ℕ which are meant as reducibilities. Let both relations be closed under finite variation (of their set arguments) and consider the uniform distribution on 2ℕ, which is obtained by choosing elements of 2ℕ by independent tosses of a fair coin.Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤r X, differs from the lower ≤s-cone of X. By c osure under finite variation, the Kolmogorov 0-1 aw yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles.Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2ℕ which can be defined by a countable set of total continuous functionals on 2ℕ in the same way as the usual resource-bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises a natura resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations.

ON THE POWER OF ONE-WAY GLOBALLY DETERMINISTIC SYNCHRONIZED ALTERNATING TURING MACHINES AND MULTIHEAD AUTOMATA

The alternating model augmented by a special simple form of communication among parallel processes—the so-called synchronized alternating (SA) model, provides (besides others) nice characterizations of the space complexity classes defined by nondeterministic Turing machines. The model investigated in this paper — globally deterministic synchronized alternating (GDSA) model—is obtained by a feasible restriction of nondeterminism in SA. It is known that it characterizes the deterministic counterparts of the nondeterministic space classes characterized by the SA model. In the paper we resume in the investigation of GDSA solving the open questions about the computational power of the one-way GDSA models. It is known that in the case of space-bounded Turing machine and multihead automata, the one-way SA models are equivalent to their two-way counterparts. We show that the same holds for GDSA models. The results contribute to the knowledge about the model and imply new characterizations of the deterministic space complexity classes.

Decreasing the bandwidth of a transition matrix

Abstract Adapting a method that Freivalds used in the context of bounded-error probabilistic computation, we prove that the languages recognized by log-space un bounded-error probabilistic Turing machines (PL) are log-space reducible to languages recognized by automata of the same type but restricted to use at most e log n bits of storage space, for arbitrarily small e s 0. Furthermore, we show that the banded-matrix inversion problem Band-Mat-Inv( n e ) is log-space complete for PL, for any e ϵ (0, 1]. This strengthens a result of Jung that Band-Mat-Inv( n ) is log-space complete for PL, and may lead to new space-efficient deterministic simulations of space-bounded probabilistic Turing machines.

On three-way two-dimensional multicounter automata

In this paper several problems concerning three-way two-dimensional multicounter automata are solved. It is shown that the class of languages accepted by L( m) counter-bounded multicounter automata is equal to the class of languages accepted by log L( m) space-bounded Turing machines for any natural function L( m). It is shown that for every r ⩾ 1 there exists an infinite hierarchy, with respect to the number of counters, of languages accepted by m r -bounded (deterministic or nondeterministic) multicounter automata. It is also shown that for any k ⩾ 2 there is an infinite hierarchy, with respect to the capacity of the counters, of languages accepted by deterministic k-counter automata. Finally it is shown that the class of languages accepted by three-way L( m)-bounded k-counter automata is not closed under row or column cyclic closure if L( m) = m, L( m) = m 2, or log L( m) = o(log m). Only square inputs are considered.

Problems complete for ⊕ L

⊕ L is the class of languages acceptable by logarithmic space bounded Turing machines that work nondeterministically and are equipped with parity-acceptance. Several natural problems are shown to be complete for ⊕ L under NC 1-reductions. A consequence is that ⊕ L is the ▪ 2-analogon of Cook's class DET, the class of problems NC 1-reducible to the computation of determinants over ▪.

Polynomial Space Counting Problems

The classes of functions $# \textit{PSPACE}$ and $\natural \textit{PSPACE}$ that are analogous to the class $# P$ are defined. Functions in $# \textit{PSPACE}$ count the number of accepting computations of a nondeterministic polynomial space bounded Turing machine, and functions in $\natural \textit{PSPACE}$ count the number of accepting computations of nondeterministic polynomial space bounded Turing machines that on each computation path make at most a polynomial number of nondeterministic choices. In contrast to what is known about $# P$, exact characterizations of both $# \textit{PSPACE}$ and $\natural \textit{PSPACE}$ are found. In particular, $# PSPACE = FSPACE$ (the class of functions computable in polynomial space) and $\natural PSPACE = FSPACE (poly)$ (the class of functions computable in polynomial space with output length bounded by a polynomial). Both $# \textit{PSPACE}$ and $\natural \textit{PSPACE}$ can be characterized by counting problems related to alternating Turing machines. Both $# \te...

Some remarks on the alternating hierarchy and closure under complement for sublogarithmic space

We prove that strongly L(n) space-bounded nondeterministic Turing machines are not closed under the so-called independent (from the starting configuration) complement

Speed-Up of Turing Machines with One Work Tape and a Two-Way Input Tape

In this paper we consider the next more powerful restricted type of Turing machine, which cannot be handled by the existing lower bound arguments: Turing machines with one work tape and a two-way input tape. We show that one can simulate a deterministic Turing machine of this type with time bound $O(n^3 )$ by $\Sigma _2 $-Turing machine of the same type with time bound $O(n^2 \cdot \log ^2 n)$. This implies the new separation result $\Sigma _2 {\operatorname{TIME}}_1 (n) \nsubseteq {\operatorname{DTIME}}_1 ({{n^{{3 / 2}} } / {\log ^6 n}})$. Further, we improve Kannan’s separation result ${\operatorname{NTIME}}(n) \nsubseteq {\operatorname{DTIME}}_1 (n^{1.104} )$ to ${\operatorname{NTIME}}(n) \nsubseteq {\operatorname{DTIME}}_1 (n^{1.22} )$. Finally we show that with Turing machines of the considered type that use $2k$ alternations, the achieved speed-up increases and for large k approximates $t^{{1 / 2}} (n)$. In view of the close relationship between spacebounded Turing machines and time-bounded Turing m...

On pebble automata

We look at two simple variations of space-bounded Turing machines (TM's): An off-line S(n)-space bounded TM where S(n) is below log n, which can use a pebble on the input tape, and a TM with two (or three) pebbles and no workspace. We show that in the former case, a pebble increases the recognition power of the device. The latter model(s) can accept large families of languages. For example, 2-pebble automata can accept languages accepted by deterministic checking stack automata, and 3-pebble automata can accept languages accepted by ‘stack-resetting’ two-way deterministic pushdown automata. Moreover, the pebble automata are halting (i.e., halt on all inputs).

On computing the determinant in small parallel time using a small number of processors

The determinant, characteristic polynomial and adjoint over an arbitrary commutative ring with unity can be computed by a circuit with size O(n 3.496) and depth O(log 2n). Furthermore, the circuits can be constructed uniformly (by a log space bounded Turing machine).