Space-bounded simulation of multitape turing machines

A new proof of a theorem of Hopcroft, Paul, and Valiant is presented: Every deterministic multitape Turing machine of time complexityT(n) can be simulated by a deterministic Turing machine of space complexityT(n)/logT(n). The proof includes an overlap argument.Supported by National Science Foundation Grant No. MCS-78-04343.Supported by National Science Foundation Grant No. MCS-77-19754 and the Fannie and John Hertz Foundation.     

Records of turing machines

Suppose a Turing machine is equipped with an extra tape. At each step of a computation being performed, it prints symbol read move symbol symbol printed on a square of the extra tape. It then moves the extra tape one square to the left. This procedure yields arecord of the computation.If is a finite alphabet, let be the set of triples (a, b, c) wherea ,c , andb {–1, 0, 1}. We will characterize those sequences of symbols of that are records of computations of Turing machines.This research was supported in part by AFOSR Grant GP-68-1402.     

Three-way tape-bounded two-dimensional turing machines

In this paper, we introduce a three-way tape-bounded two-dimensional Turing machine (which can be considered as an extension of the on-line tape-bounded one-dimensional Turing machine to two dimensions), and investigate hierarchical and closure properties of the classes of sets accepted by several types of such machines.     

Simulations among multidimensional turing machines

For all d ? 1 and all e >d, every deterministic multihead e-dimensional Turing machine of time complexity T(n) can be simulated on-line by a deterministic multihead d-dimensional Turing machine in time O(T(n)^1+1?d?1?(logT(n))⁰⁽¹⁾). This simulation almost achieves the known lower bound $\text{Ω(T(n)}{}^{\mathrm{1+1?}}\text{d?1?e)}$ on the time required. The simulation is interpreted in terms of dynamic embeddings among arrays with local access.     

Stateless turing machines and fixed points

Thes _n ^m function defined in the iteration theorem is shown to be quite subelementary through the use of stateless Turing machines. This means that recursion theorems can be used subrecursively with no change in computational complexity.     

On the transition graphs of turing machines

As for pushdown automata, we consider labelled Turing machines with ε-rules. With any Turing machine M and with a rational set C of configurations, we associate the restriction to C of the ϵ-closure of the transition set of M. We get the same family of graphs by using the labelled word rewriting systems. We show that this family is the set of graphs obtained from the binary tree by applying an inverse mapping into F followed by a rational restriction, where F is any family of recursively enumerable languages containing the rational closure of all linear languages. We show also that this family is obtained from the rational graphs by inverse rational mappings. Finally we show that this family is also the set of graphs recognized by (unlabelled) Turing machines with labelled final states, and even if we restrict to deterministic Turing machines.     

The complexity of on-line simulations between multidimensional turing machines and random access machines

To study different implementations of arrays, we present four results on the time complexities of on-line simulations between multidimensional Turing machines and random access machines (RAMs). First, everyd-dimensional Turing machine of time complexityt can be simulated by a log-cost RAM running inO(t(logt)^1–(1/d)(log logt)^1/d) time. Second, everyd-dimensional Turing machine of time complexityt can be simulated by a unit-cost RAM running inO(t/(logt)^1/d) time, provided that the input length iso(t/(logt)^1/d). Third, there is a log-cost RAMR of time complexityO(n), wheren is the input length, such that, for anyd-dimensional Turing machineM that simulatesR on-line,M requires (n ^{1 + (1/d)})/(logn(log logn)^{1 + (1/d)})) time. Fourth, every unit-cost RAM of time complexityt can be simulated by ad-dimensional Turing machine inO(t ²(logt)^1/2) time ifd = 2, and inO(t ²) time ifd 3. This result uses the weight-balanced trees of Nievergelt and Reingold.This paper was prepared while M. C. Loui was visiting the National Science Foundation in Washington, DC, and the Institute for Advanced Computer Studies, University of Maryland, College Park, MD. The views, opinions, and conclusions in this paper are those of the authors and should not be construed as an official position of the National Science Foundation, Department of Defense, U.S. Air Force, or any other U.S. government agency. The research of M. C. Loui was supported by the National Science Foundation under Grant CCR-8922008.     

Unbounded hardware is equivalent to deterministic turing machines

The purpose of this paper is to investigate models of computation from a realistic viewpoint. We introduce the concept of physical computation as opposed to functional computation, and by referring to the laws of physics we study the basic criteria which a model of computation must meet in order to be realistic. With this formal apparatus, we define a very general, realistic model of planar, digital circuits, which allows for full parallelism. This model is especially well suited to describe systolic architectures. Actually, the assumption of planarity serves only practical purposes, and can be removed without altering our main results. We compare the complexity classes in this parallel model with those associated with sequential models such as the Deterministic Turing Machine (DTM) model. Our main result is that both models are space and time equivalent in the polynomial class. In particular, any circuit can be simulated in polynomial time on a DTM. One consequence is that unbounded hardware does not make NP-hardness tractable. We also address the issue of area-time tradeoffs and show that the area of a circuit can always be bounded by a polynomial function of the sequential time.     

Some observations concerning alternating turing machines using small space

We make some observations concerning alternating Turing machines operating in small space. For example, we show that alternating Turing machines using o(log n) space are more powerful than nondeterministic Turing machines using the same space-bound. In fact, we show that there is a language over a unary alphabet that can be accepted by an on-line alternating Turing machine in log n space, but not by any off-line nondeterministic Turing machine in o(log n) space. We also investigate the weak vs. strong space bounds and on-line vs. off-line machines at these low tape bounds.     

Uniform simulations of nondeterministic real time multitape turing machines

A new and uniform technique is developed for the simulation of nondeterministic multitape Turing machines by simpler machines. For many types of restricted nondeterministic Turing machines it can now be proved that both linear time is no more powerful than real time, and multitape machines are no more powerful than machines with two tapes, one of which is a simple and normalized comparison tape. This is an improvement over all previously known simulations in terms of weaker machines. As an application we obtain that, for all such machines, the class of languages accepted in real time by multitape machines is principal and has a simple trio generator. Moreover, multitape machines with different types of tapes are hierarchically related, contrasting with the case of one-tape machines, and some important families of languages are characterized in a new way.     

Simulating one-reversal multicounter machines by partially blind multihead finite automata

This work is concerned with simulating nondeterministic one-reversal multicounter automata (NCMs) by nondeterministic partially blind multihead finite automata (NFAs). We show that any one-reversal NCM with k$k$ counters can be simulated by a partially blind NFA with k$k$ blind heads. This provides a nearly complete categorization of the computational power of partially blind automata, showing that the power of a (k+1)$(k+1)$-NFA lies between that of a k$k$-NCM and a (k+1)$(k+1)$-NCM.     

Closure properties of three-way and four-way tape-bounded two-dimensional turing machines

This paper investigates closure properties (under row and column catenations, row and column closures, row and column cyclic closures, and projection) of the classes of sets accepted by four-way and three-way tape-bounded two-dimensional Turing machines whose input tapes are not restricted to square ones.     

An improved simulation of space and reversal bounded deterministic turing machines by width and depth bounded uniform circuits

We present an improved simulation of space and reversal bounded Turing machines by width and depth bounded uniform circuits. (All resource bounds hold simultaneously.) An S(n) space, R(n) reversal bounded deterministic k-tape Turing machine can be simulated by a uniform circuit of O(R(n)log²S(n)) depth and O(S(n)^k) width. Our proof is cleaner, and has slightly better resource bounds than the original proof due to Pippenger (1979). The improvement is resource bounds comes primarily from the use of a shared-memory machine instead of an oblivious Turing machine, and the concept of a ‘special situation’.