Showing papers on "Computability published in 1993"

The asynchronous computability theorem for t-resilient tasks

Abstract: We give necessary and sufficient combinatorial conditions characterizing the computational tasks that can be solved by N asynchronous processes, up to t of which can fail by halting The range of possible input and output values for an asynchronous task can be associated with a high-dimensional geometric structure called a simplicial complex Our main theorem characterizes computability y in terms of the topological properties of this complex Most notably, a given task is computable only if it can be associated with a complex that is simply connected with trivial homology groups In other words, the complex has “no holes!” Applications of this characterization include the first impossibility results for several long-standing open problems in distributed computing, such as the “renaming” problem of Attiya et al, the “k-set agreement” problem of Chaudhuri, and a generalization of the approximate agreement problem

Computability by Probabilistic Machines

Abstract: This chapter contains sections titled: Introduction Probabilistic Machines Vs. Deterministic Machines A More General Class of Machines Appendix Bibliography

Continuous refinement equations and subdivision

Abstract: This paper is concerned with the study of a general class of functional equations covering as special cases the relation which defines theup-function as well as equations which arise in multiresolution analysis for wavelet construction. We discuss various basic properties of solutions to these functional equations such as regularity, polynomial containment within the space spanned by their integer shifts and their computability by subdivision algorithms.

Computability Properties of Low-dimensional Dynamical Systems

Abstract: It has been known for a short time that a class of recurrent neural networks has universal computational abilities. These networks can be viewed as iterated piecewise-linear maps in a high-dimensional space. In this paper, we show that similar systems in dimension two are also capable of universal computations. On the contrary, it is necessary to resort to more complex systems (e.g., iterated piecewise-monotone maps) in order to retain this capability in dimension one.

Computational Tractability and Conceptual Coherence: Why Do Computer Scientists Believe that P≠NP?

Abstract: According to Church’s thesis, we can identify the intuitive concept of effective computability with such well-defined mathematical concepts as Turing computability and partial recursiveness. The almost universal acceptance of Church’s thesis among logicians and computer scientists is puzzling from some epistemological perspectives, since no formal proof is possible of a thesis that involves an informal concept such as effectiveness. Elliott Mendelson has recently argued, however, that equivalencies between intuitive notions and precise notions need not always be considered unprovable theses, and that Church’s thesis should be accepted as true. I want to discuss a thesis that is nearly as important in current research in computer science as Church’s thesis. I call the newer thesis the tractability thesis, since it identifies the intuitive class of computationally tractable problems with a precise class of problems whose solutions can be computed in polynomial time. After briefly reviewing the theory of intractability, I compare the grounds for accepting the tractability thesis with the grounds for accepting Church's thesis. Intimately connected with the tractability thesis is the mathematical conjecture, whose meaning I shall shortly explain, that P≠NP. Unlike Church's thesis, this conjecture is precise enough to be capable of mathematical proof, but most computer scientists believe it even though no proof has been found. As we shall see below, understanding the grounds for acceptance of the conjecture that P≠NP has implications for general questions in the philosophy of mathematics and science, especially concerning the epistemological importance of explanatory and conceptual coherence.

A declarative laboratory approach for discrete structures, logic, and computability

Abstract: 19 Overview Many students find it hard to grasp and retai n the ideas presented in courses covering discret e structures, logic, and computability. These subjects provide a foundation for required upper division courses in computer science . Therefore a major effort must be made to improve the learnin g environment for students studying these ideas a t the lower division level . Many of us succeeded academically in spite o f the way we were taught . But how many people hav e not succeeded because of the way material was presented to them? Since people learn in different ways, it makes sense to present students with a va riety of learning experiences . We have created a laboratory component for a year long sophomore course in discrete structures , logic, and computability for students majoring i n computer science or computer engineering . The labs consist of experiments in declarative programming environments . The experiments are designed to reinforce the learning of material o n a daily basis, just like the regular homework assignments . In other words, the lab experiment s are short in duration and relevant to the material covered by each lecture . Short programming labs that correspond t o each lecture should be useful learning tools fo r many traditional courses . The instant feedbac k that students get from wrong assumptions can give them incentive to try something new to experiment and see what happens . The lab component can also encourage the use of laboratory partners, interaction of students, team presenta -

On the equivalence of two-way pushdown automata and counter machines over bounded languages

Abstract: It is known that two-way pushdown automata are more powerful than two-way counter machines. The result is also true for the case when the pushdown store and counter are reversal-bounded. In contrast, we show that two-way reversal-bounded push-down automata over bounded languages (i.e., subsets of for some distinct symbols a1,…, ak) are equivalent to two-way reversal-bounded counter machines. We also show that, unlike the unbounded input case, two-way reversal-bounded pushdown automata over bounded languages have decidable emptiness, equivalence and containment problems.

A deterministic optimal control theory for discrete event systems

Abstract: In certain discrete event applications it may be desirable to find a particular controller, within the set of acceptable controllers, which extremises some quantitative performance measure In this paper we propose a theory of optimal control to meet such design requirements for deterministic systems The discrete event system (DES) is modelled by a regular language Event and cost functions are defined which induce costs on controlled system behaviour The event costs associated with the system behaviour can only be reduced, in general, by increasing the control costs Thus it is nontrivial to find the optimal amount of control to use and consequently the formulation captures the fundamental tradeoff motivating classical optimal control Results on the existence and computability of minimally restrictive optimal controllers are presented Such controllers are also polynomially computable under certain assumptions It is shown that this framework subsumes the prior graph-theoretic formulation >

Real Number Computability and Domain Theory

Abstract: We propose a possible implementation, using lazy functional programming, of the exact computation on real numbers Using domain theory we can analyze this kind of computation and give a definition of computability for the functions on the real number This definition turns out to be equivalent to other definitions given in the literature using different methods

Effective Real Dynamics

Abstract: The study of computability in analysis has a long history, going back to the papers of Lacombe [6] in the 1950’s. There has been much work on the connection between recursive function theory and computable analysis. One key result which we will use here is a theorem of Nerode’s from [7] that the existence of a recursive continuous function mapping a real a to a real b implies that b is truth-table reducible to a. Another connection which we will use is given in the papers of Soare [11, 12] on recursion theory and Dedekind cuts, where the effectively closed real intervals are characterized. One important aspect of computable analysis is the search for effective versions of classical theorems. This is in the spirit of the well-known Nerode program for applying recursion theory to mathematics. As an example, if K is a closed subset of the real line ℜ, then the distance function δK’ defined by δK(x) = min{|x − y|: y ∈ K} is continuous. Thus the question arises whether an effectively closed set K has an effectively continuous (i.e., recursive) distance function. (In general, the answer is no.) Closed sets play an important role in the study of analysis. For example, the set of zeros of a continuous function F is a closed set, as is the set of fixed points of F. We are particularly interested in the role of effectively closed, or Π0 1 classes. Π1 0 classes are important in the applications of recursion theory and have been studied extensively. (See [4] for a survey of results.)

Real Computation with Cellular Automata

Abstract: Two definitions about computability of real-valued functions by cellular automata are proposed, each requiring exact computation (unlike Turing-based computability). Recursive functions, some polynomials, and even logistic and chaotic maps are shown to be exactly computable even under discrete space, time and states of the computer model, on representations more general than standard signed expansions. A number of consequences of these definitions are presented that point to computational primitives different from classical continuous objects based on addition and multiplication. Several open questions pertaining characterization of real-valued functions computable by cellular automata are briefly discussed, notably the encoding/representation problem and the halting criterion.

Polynomial time computable stable models

Abstract: We study the relations between the expressive power of non-monotonic formalisms and polynomial-time computability in the framework of stable models semantics. While the problem of deciding whether a logic program has a total stable model isNP-complete, we introduce a polynomial-time algorithm that generates such a model for several important classes of programs, that are discussed in this paper. In the general case, the algorithm generates a (not necessarily total)p-stable model of the input program.

Computability in Unitary Representations of Compact Groups

Abstract: This paper deals with the computability of unitary representations of compact groups. An algorithm is given to effectively decompose a unitary representation into its irreducible parts. Difficulties in finding the effective procedure are caused by the absolute lack of a priori information about the irreducible representations and the obligation of making decisions from inexact data. Several lemmas on group representations (classical, i.e. computability not mentioned) have been proved in order to design the algorithm which overcomes these difficulties.

A Step Size Control for Lohner's Enclosure Algorithm for Ordinary Differential Equations with Initial Conditions

Abstract: Publisher Summary Lohner's enclosure algorithm for ordinary differential equations with initial conditions is supplemented by an automatic control of the step size In this chapter, a control has been developed mainly in view of the computability of the upper and lower bounds of an enclosure in a close neighborhood of a pole in a restricted three-body problem Applications to other problems are investigated in the chapter For any system of equations, the practical determination of the value(s) of a true solution rests on the execution of a suitable algorithm Rounding errors are then unavoidable, and there are additional procedural errors if the algorithm is chosen as a truncation of an infinite sequence of arithmetic operations If there are numerical errors of these kinds, an algorithm delivers only an approximation of the value(s) of the unknown true solution Applications of Lohner's enclosure algorithm are particularly important in the case of perturbation-sensitive neighborhoods in phase spaces Controls of the kind presented in the chapter are instrumental for the practical computability of enclosures

The planning of tracking control via search

Abstract: The issue of computability which is associated with the design of systems combining feedforward and feedback controls is addressed. Hybrid systems of this kind are shown to be suitable for the separation of the control problem into two parts, i.e., planned operation and compensation for unmodeled dynamics, which permits better tracking of assignments than any feedback scheme without prediction. It is shown that the synthesis of feedforward commands relies on the solution of (often ill-posed) inverse problems, and that there exists a need for methods which provide approximate solutions in such cases. A definition of approximate inversion is constructed and a simple algorithm for performing this operation is described. Results are shown for SISO and nonlinear MIMO cases, and avenues for future research are suggested. >

What is a Universal Higher-Order Programming Language?

Abstract: Classic recursion theory asserts that all conventional programming languages are equally expressive because they can define all partial recursive functions over the natural numbers. This statement is misleading because programming languages support and enforce a more abstract view of data than bitstrings. In particular, most real programming languages support some form of higher-order data such as potentially infinite streams (input and output), lazy trees, and functions. In this paper, we develop a theory of higher-order computability suitable for comparing the expressiveness of sequential, deterministic programming languages. The theory is based on the construction of a new universal domain T and corresponding universal language KL. The domain T is universal for “sequential” domains; KL can define all the computable elements of T, including the elements corresponding to computable sequential functions. In addition, T preserves maximality of finite elements in embeddings, so the termination behavior of programs is preserved by embeddings in T.

Real-Valued, Continuous-Time Computers: A Model of Analog Computations, Part I

Abstract: We develop a system of recursive functions on the reals analogous to classical recursion theory on the natural numbers. This system turns out to include many sets and functions that are uncomputable in the traditional sense. These functions can be computed by an idealized computer that runs on continuous states in continous time; however, this computer turns out to be highly unphysical. Looking more closely, we find that we can stratify these functions according to how many idealizations or infinite limits they are away from physical computability. We conclude that, in a certain sense, finite-dimensional analog computation is more powerful than digital computation: however, physically realizable analog computation would seem to be equivalent. Thus the {\it Physical Church-Turing Thesis}, that no physical computer is more powerful than a Turing machine, is false in a perfect, classical world but probably ture in the world we live in.

The Computability of the Distribution of the Nearest Neighbour Distance on a Network

Abstract: In GIS field, road networks have been studied mainly from the view point of traffic flows or tranportation problems. This paper regards a road network as aspaceitself where geographic objects are located rather than as a component of a geographic space. Some objects can be located only along roads, especiallyin an urban area. Even in this case some researchers think that a plane can be used as a substitute for a network and it is an extra work to take account of network. This paper shows that some computations on a network are quite practical and easier rather than those on a plane. First the computation of the distribution about the nearest neighbour distance for given objects is shown.Then the computability on a stochastic model is examined.

Parameterized Recursion Theory - A Tool for the Systematic Classification of Specification Methods

Abstract: We examine four specification methods with increasing expressiveness. Parameterized recursion theory allows to characterize the power of parameterization in the methods, using a computational model based on Moschovakis’ search computability. The four specification methods can be characterized by four different notions of semicomputable parameterized abstract data type, which differ in the availability of the parameter algebra and of nondeterminism.

Communication and computability: The case of Alan Mathison Turing

Abstract: This essay provides a preliminary examination of the relationships which exist between the disciplines of communication and computer science It isolates the original principles which determined the development of computer science and suggests how these early formative principles have and may continue to can affect the study of communication Because of his seminal role in virtually all areas during the origins of computer science, the published works of British logician and mathematician Alan M Turing (1912–1954) are employed as the foundation of computer science Turing's conceptions of computability theory, computer architecture and software, artificial intelligence, and the origins of life are examined for their potential consequences for communication and rhetorical theory It is argued here that both the limitations and the potentialities of the standards and interests which guided the emergence of computer science constitute an important self‐reflective framework for future developments in the dis

A Construction of Typed Lambda Models Related to Feasible Computability

Abstract: In this paper we develop an approach to the notion of computable functionals in a very abstract setting not restricted to Turing or, say, polynomial computability. We assume to start from some basic class of domains and a basic class of functions defined on these domains. (An example may be natural numbers with poly time computable functions). Then we define what are “all“ corresponding functionals of higher types which add nothing new to these basic functions. We call such functionals computable or, more technically and adequately speaking, substitutable. (Similarly, in D.Scott's domains we say about continuous functionals as about far-reaching abstraction of computable ones.) Our results are applicable t o quite arbitrary (complexity) classes of functions, satisfying a very general Assumption.

Non-perturbative Computability vs. Integrability in Susy QFT’s

Abstract: The present paper contains a short and rather informal summary of some work done in collaboration with Cumrun Vafa about exact results valid for any two-dimensional N = 2 supersymmetric theory. Our approach is based on the so-called topological-anti-topological fusion. However, here the emphasis is on the emergence of structures typical of an integrable theory, not on the underlying topological theory. We discuss the deep analogies between the non-perturbative computations in a general N = 2 theory and the usual techniques for integrable theories. The relationships of N = 2 susy with the 2d Ising model is analyzed in detail.