Showing papers on "Turing machine published in 1986"

Algebraic Approaches to Program Semantics

Abstract: In the 1930s, mathematical logicians studied the notion of "effective comput ability" using such notions as recursive functions, A-calculus, and Turing machines. The 1940s saw the construction of the first electronic computers, and the next 20 years saw the evolution of higher-level programming languages in which programs could be written in a convenient fashion independent (thanks to compilers and interpreters) of the architecture of any specific machine. The development of such languages led in turn to the general analysis of questions of syntax, structuring strings of symbols which could count as legal programs, and semantics, determining the "meaning" of a program, for example, as the function it computes in transforming input data to output results. An important approach to semantics, pioneered by Floyd, Hoare, and Wirth, is called assertion semantics: given a specification of which assertions (preconditions) on input data should guarantee that the results satisfy desired assertions (postconditions) on output data, one seeks a logical proof that the program satisfies its specification. An alternative approach, pioneered by Scott and Strachey, is called denotational semantics: it offers algebraic techniques for characterizing the denotation of (i. e., the function computed by) a program-the properties of the program can then be checked by direct comparison of the denotation with the specification. This book is an introduction to denotational semantics. More specifically, we introduce the reader to two approaches to denotational semantics: the order semantics of Scott and Strachey and our own partially additive semantics."

Digital optical computing with symbolic substitution.

Abstract: Symbolic substitution logic is based on optical pattern transformations This space-invariant mechanism is shown to be capable of supporting space-variant operations An optical implementation is proposed It is based on splitting an image, shifting the split images, superimposing the results, regenerating the superimposed image with an optical logic array, splitting the regenerated image, shifting the resulting images, and superimposing the shifted images Experimental results are presented Examples demonstrate how symbolic substitution logic can be used to implement Boolean logic, binary arithmetic, cellular logic, and Turing machines

Parallel Computation with Threshold Functions

Abstract: We study two classes of unbounded fan-in parallel computation, the standard one, based on unbounded fan-in ANDs and ORs, and a new class based on unbounded fan-in threshold functions. The latter is motivated by a connectionist model of the brain used in Artificial Intelligence. We are interested in the resources of time and address complexity. Intuitively, the address complexity of a parallel machine is the number of bits needed to describe an individual piece of hardware. We demonstrate that (for WRAMs and uniform unbounded fan-in circuits) parallel time and address complexity is simultaneously equivalent to alternations and time on an alternating Turing machine (the former to within a constant multiple, and the latter a polynomial). In particular, for constant parallel time, the latter equivalence holds to within a constant multiple. Thus, for example, polynomial-processor, constant-time WRAMs recognize exactly the languages in the logarithmic time hierarchy, and polynomial-word-size, constant-time WRAMs recognize exactly the languages in the polynomial time hierarchy. As a corollary, we provide improved simulations of deterministic Turing machines by constant-time shared-memory machines. Furthermore, in the threshold model, the same results hold if we replace the alternating Turing machine with the analogous threshold Turing machine, and replace the resource of alternations with the corresponding resource of thresholds. Threshold parallel computers are much more powerful than the standard models (for example, with only polynomially many processors, they can compute the parity function and sort in constant time, and multiply two integers in O(log*n) time), and appear less amenable to known lower-bound proof techniques.

On play by means of computing machines

Abstract: This paper examines the 'bounded rationality" inherent in play by means of computing machines. The main example is the finitely repeated prisoners' dilemma game which is discussed under different models. The game is played by Turing machines with restricted number of internal states using unlimited time and space. The observations in this paper strongly suggest that the cooperative outcome of the game can be approximated in equilibrium. Thus, the cooperative play can be approximated even if the machines memorize the entire history of the game and are capable of counting the number of stages.

Fast recognition of pushdown automaton and context-free languages

Abstract: We prove: 1) every language accepted by two-way nondeterministic pushdown automaton can be recognized on RAM in O(n3/log n) time; 2) every language accepted by two-way loop-free pushdown automaton can be recognized in O(n3/log2n) time; 3) every context-free language can be recognized on-line in O(n3/log2n) time. We improve the results of [1,7,4].

A programming language for the inductive sets, and applications

Abstract: We introduce a programming language IND that generalizes alternating Turing machines to arbitrary first-order structures. We show that IND programs (respectively, everywhere-halting IND programs, loop-free IND programs) accept precisely the inductively definable (respectively, hyperelementary, elementary) relations. We give several examples showing how the language provides a robust and computational approach to the theory of first-order inductive definability. We then show: (1) on all acceptable structures (in the sense of Moschovakis [Mo]), r.e. Dynamic Logic is more expressive than finite-test Dynamic Logic. This refines a separation result of Meyer and Parikh [MP]; (2) IND provides a natural query language for the set of fixpoint queries over a relational database, answering a question of Chandra and Harel [CH2].

On nontrivial separators for k-page graphs and simulations by nondeterministic one-tape Turing machines

Abstract: We show that the following statements are equivalent: 1. Statement 1. 3-pushdown graphs have sublinear separators. 2. Statement 1∗. k-page graphs have sublinear separators. 3. Statement 2. A one-tape nondeterministic Turing machine can simulate a two-tape machine in subquadratic time. None of the statements is known to be true or false at present. However, our proof of equivalence is quantitative-it relates exactly the separator size of the two kinds of graphs to the running time of the simulation in Statement 2. Using this equivalence we derive several graph-theoretic corollaries. There are known examples where upper bounds on graph properties imply upper bounds on computation time or space. There are other examples where lower bounds on graph properties are used to derive lower bounds on computation time in restricted settings. However, our results may constitute the first example where a graph problem is shown to be equivalent to a problem in computational complexity. In a companion paper we construct graphs and prove a lower bound or their separators. Using the equivalence we prove an almost linear lower bound for the size of separators for 3-pushdown graphs and an almost quadratic lower bound for simulating two-tape nondeterministic Turing machines by one-tape machines. Specifically, for an integers s let ls(n), the s-iterated logarithm function, be defined inductively: l°(n)=n, ls+1(n)=log2(ls(n)) for s⩾0. Then: 1. For every fixed s and all n, there is an n-vertex 3-pushdown graph whose smallest separator contains at least ω(n/ls(n)) vertices. 2. There is a language L recognizable in real time by a two-tape nondeterministic Turing machine, but every on-line one-tape nondeterministic Turing machine that recognizes L requires ω(n2/ls(n)) time for any positive integer.

An optimal lower bound for Turing machines with one work tape and a two-way input tape

Abstract: This paper contains the first concrete lower bound argument for Turing machines with one work tape and a two-way input tape (these Turing machines are often called "offline 1-tape Turing machines"). In particular we prove an optimal lower bound of Ω(n3/2/(log n)1/2) for transposing a matrix with elements of bit length ∘(logn) (where n is the length of the total input). This implies a lower bound of Ω(n3/2/(log n)1/2) for sorting on the considered type of Turing machine. We also get as corollaries the first nonlinear lower bound for the most difficult version of the two tapes — versus — one problem, and a separation of the considered type of Turing machine from that with an additional write-only output tape.

On time versus space III

Abstract: Paul and Reischuk devised space efficient simulations of logarithmic cost random access machines and multidimensional Turing machines. We simplify their general space reduction technique and extend it to other computational models, including pointer machines, which model computations on graphs and data structures. Every pointer machine of time complexityT(n) can be simulated by a pointer machine of space complexityO(T(n)/logT(n)).

A simplified lower bound for context-free-language recognition

Abstract: For on-line recognition of the words in an arbitrary linear context-free language, there are known tight bounds on the time required by a deterministic multitape Turing machine. In terms of word length n , the time need never be worse than some constant times n 2 , even if only one worktape is available; and there is a linear context-free language that requires at least time proportional to n 2 /log n , no matter how many worktapes are available. Using Kolmogorov's notion of descriptional complexity as a tool, we present a simple proof of the latter result.

A first course in computability

Abstract: Mathematical prerequisites Turing machines solvability and unsolvability formal languages recursive functions complexity theory appendix - the Turing machine simulator.

Will Machines Ever Think

Abstract: Artificial Intelligence research has come under fire for failing to fulfill its promises. A growing number of AI researchers are reexamining the bases of AI research and are challenging the assumption that intelligent behavior can be fully explained as manipulation of symbols by algorithms. Three recent books -- Mind over Machine (H. Dreyfus and S. Dreyfus), Understanding Computers and Cognition (T. Winograd and F. Flores), and Brains, Behavior, and Robots (J. Albus) -- explore alternatives and open the door to new architectures that may be able to learn skills.

Bounded Arithmetic Formulas and Turing Machines of Constant Alternation

Abstract: Publisher Summary This chapter discusses the bounded arithmetic formulas and Turing machines of constant alternation. Alternating Turing machines were introduced into computational complexity theory by Chandra, Kozen, and Stockmeyer. The chapter describes the Turing machines and some of their properties. An instantaneous description (ID) of M consists of the finite strings on the tapes, the head positions (relative to the tape contents), and the current state.

Alternation with restrictions on looping

Abstract: Alternating Turing machines with restrictions preventing them from returning to a previous configuration model games with rules enforcing such a restriction, for instance, the Chinese version of Go. Such restrictions do not affect the time complexity of problems for alternating Turing machines but space S on a machine with the restriction is equivalent either to time or to space exponential in S on a normal alternating machine, depending on the precise nature of the restriction.

Reliability Analysis of Computer Systems Using Dataflow Graph Models

Abstract: We use dataflow graphs to represent the computational structure, analogous to Petri nets and Turing machines, and have developed a method for analyzing the reliability of computer systems modeled as dataflow graphs. Because of the hierarchical nature of dataflow graphs, systems can be analyzed at several levels of abstraction. Reliabilities of subgraphs can be calculated using either traditional techniques or dataflow approach presented here (recursively). The reliabilities of subgraphs can then be combined leading to decomposition-aggregation approach. The time needed for an actor to complete its operation is not included in our analysis of dataflow graphs. Incorporation of the time element compounds the problem and we have not studied it yet.

Are Recursion Theoretic Arguments Useful in Complexity Theory

Abstract: Publisher Summary Recursion theory is that area of mathematical logic where one studies the qualitative aspects of computability. In complexity theory, which is part of computer science, one studies in addition quantitative aspects of computations. A number of open problems about the structure of NP where one can prove that even under the assumption PI≠NP recursion theoretic arguments will not suffice. The chapter presents polynomial time approximation schemes for some strongly NP-complete problems that arise—for example, in robotics. The chapter presents a survey for a proof of optimal lower bounds for two tapes versus one on deterministic and nondeterministic Turing machines. Results that show a substantial superiority of nondeterminism over determinism resp. co-nondeterminism over nondeterminism for one-tape Turing machines are given in the chapter. The chapter presents the proof of the desired result as the construction of a winning strategy for a two-person game.

The Mathematics Needed for Self-Organisation

Abstract: True self-organisation requires a new type of mathematics; the initial stages of its development are described and it is shown to be isomorphic to an earlier algebraic construction.

Characterizations of PUNC and precomputation

Abstract: Much complexity-theoretic work on parallelism has focused on the class NC, which is defined in terms of logspace-uniform circuits. Yet P-uniform circuit complexity is in some ways a more natural setting for studying feasible parallelism. In this paper, P-uniform NC (PUNC) is characterized in terms of space-bounded AuxPDA's and alternating Turing Machines with bounded access to the input. We also present a general-purpose parallel computer for PUNC; this characterization leads to an easy proof that NC = PUNC iff all tally languages in P are in NC. The characterizations of PUNC lead to natural methods for modelling precomputation. We show that for many classes of interest, there is a single “universal” table which can be used in place of any table of similar size and complexity, while for certain other classes, no such “universal” table exists.

Uniform simulations of nondeterministic real time multitape turing machines

Abstract: 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.