Showing papers on "Computability published in 1974"

Automata, Languages, and Machines

Abstract: The 80 revised papers presented together with two keynote contributions and four invited papers were carefully reviewed and sele... The study of formal languages and automata has proved to be a source of much interest and discussion amongst mathematicians in recent times. This book, written by Professor Ian Chiswell, attempts to provide a comprehensive textbook for undergraduate and postgraduate mathematicians with an interest i...

Computability and Logic

Abstract: Part I. Computability Theory: 1. Enumerability 2. Diagonalization 3. Turing computability 4. Uncomputability 5. Abacus computability 6. Recursive functions 7. Recursive sets and relations 8. Equivalent definitions of computability Part II. Basic Metalogic: 9. A precis of first-order logic: syntax 10. A precis of first-order logic: semantics 11. The undecidability of first-order logic 12. Models 13. The existence of models 14. Proofs and completeness 15. Arithmetization 16. Representability of recursive functions 17. Indefinability, undecidability, incompleteness 18. The unprovability of consistency Part III. Further Topics: 19. Normal forms 20. The Craig interpolation theorem 21. Monadic and dyadic logic 22. Second-order logic 23. Arithmetical definability 24. Decidability of arithmetic without multiplication 25. Non-standard models 26. Ramsey's theorem 27. Modal logic and provability.

Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers)

Abstract: COMPUTABILITY AND ITS RELATION TO THE GENERAL PURPOSE ANALOG COMPUTER (SOME CONNECTIONS BETWEEN LOGIC, DIFFERENTIAL EQUATIONS AND ANALOG COMPUTERS) BY MARIAN BOYKAN POUR-EL(l) ABSTRACT. Our aim is to study computability from the viewpoint of the analog computer. We present a mathematical definition of an analog generable function of a real variable. This definition is formulated in terms of a simultaneous set of nonlinear differential equations possessing a \"domain of generation.\" (The latter concept is explained in the text.) Our definition includes functions generated by existing general-purpose analog computers. Using it we prove two theorems which provide a characterization of analog generable functions in terms of solutions of algebraic differential polynomials. The characterization has two consequences. First we show that there are entire functions which are computable (in the sense of recursive analysis) but which cannot be generated by any analog computer in any interval—e.g. l/r(x) and 2^-i (x°/n^')). Second we note that the class of analog generable functions is very large: it includes special functions which arise as solutions to algebraic differential polynomials. Although not all computable functions are analog generable, a kind of converse holds. For entire functions,/(x) = X\"o b,x', the theorem takes the following form. If f(x) is analog generable on some closed, bounded interval then there is a finite number of bk such that, on every closed bounded interval, f(x) is computable relative to these bk. A somewhat similar theorem holds if/is not entire. Although the results are stated and proved for functions of a real variable, they hold with minor modifications for functions of a complex variable. Our aim is to study computability from the viewpoint of the analog computer. We present a mathematical definition of an analog generable function of a real variable. This definition is formulated in terms of a simultaneous set of nonlinear differential equations possessing a \"domain of generation.\" (The latter concept is explained in the text.) Our definition includes functions generated by existing general-purpose analog computers. Using it we prove two theorems which provide a characterization of analog generable functions in terms of solutions of algebraic differential polynomials. The characterization has two consequences. First we show that there are entire functions which are computable (in the sense of recursive analysis) but which cannot be generated by any analog computer in any interval—e.g. l/r(x) and 2^-i (x°/n^')). Second we note that the class of analog generable functions is very large: it includes special functions which arise as solutions to algebraic differential polynomials. Although not all computable functions are analog generable, a kind of converse holds. For entire functions,/(x) = X\"o b,x', the theorem takes the following form. If f(x) is analog generable on some closed, bounded interval then there is a finite number of bk such that, on every closed bounded interval, f(x) is computable relative to these bk. A somewhat similar theorem holds if/is not entire. Although the results are stated and proved for functions of a real variable, they hold with minor modifications for functions of a complex variable. Introduction. This work represents a chapter in the development of a mathematical theory of the analog computer. As stated in the abstract, the definition of analog generable function which we present is expressed in terms of a simultaneous set of nonlinear differential equations. We will see that it includes functions generated by existing general purpose analog computers—i.e., the electronic analog computer and the mechanical differential analyzer. Our definition with its \"domain of generation\" appears to differ considerably from the approach taken in [17]. We have found our approach necessary for reasons stated in footnotes 4 and 12. Received by the editors April 12, 1971 and, in revised form, June 20, 1972. AMS (MOS) subject classifications (1970). Primary 02F50, 02F99; Secondary 26A42, 33A15, 34A10.

Notes on Pre-Set Pushdown Automata

Abstract: Motivated by practical implementation-methods for recursive program-schemata we will define and study presetting techniques for push-down automata. The main results will characterize the languages of preset pda's in terms of types of iterated substitution languages. In particular when conditions of "locally finiteness" and of "finite returning" are imposed we get a feasible machine-model for a class of developmental languages. The accepted family extends to the smallest AFL enclosing it when we drop the condition of locally finiteness. At the same time this family will be the smallest such full AFL. If all conditions are removed, present pda's exactly represent the family of iterated regular substitution languages, a sub-family of the indexed languages. Deterministic preset pda's are also studied, and the language-family they define is shown to be closed under complementation, generalizing a classical result.

On the conversion of programs to decision tables: method and objectives

Abstract: The problems of converting programs to decision tables are investigated. Objectives of these conversions are mainly program debugging and optimization in practice. Extensions to the theory of computation and computability are suggested.

Computability on Continuous Higher Types and its Role in the Semantics of Programming Languages

Abstract: This paper is about mathematical problems in programming language semantics and their influence on recursive function theory. In the process if constructing computable Scott models of the lambda calculus we examine the concepts of deterministic and non-deterministic effective operators of all finite types and continuous deterministic and non-deterministic partial computable operators on continuous inputs of all finite types. These are new recursion theoretic concepts which are appropriate to semantics and were inspired in part by Scott''s work on continuity.

On centain decompositions of Gödel numberings

Abstract: We define certain decompositions of the functions that describe Godel numberings of the partial recursive functions (Section 2). These decompositions reflect the way in which concrete Godel numberings may be obtained from the known computability formalisms. We show that such decompositions exist for all partial recursive functions (Section 3). It turns out that there is an intimate connection between these decompositions and Blum's step counting functions which yields a suggestive interpretation of Blum's notion (Section 4). In terms of these decompositions we, finally, give an exact formulation for a basic problem in the theory of computability formalisms, which, on an intuitive level, would read as follows: Find conditions on the expressive power of one step in a given computability formalism such that all partial recursive functions can be represented within that formalism. We derive two theorems which may be regarded as a first step in a thorough study of this problem.