Showing papers on "Computability published in 1973"

Metamathematical investigation of intuitionistic arithmetic and analysis

Abstract: Intuitionistic formal systems.- Models and computability.- Realizability and functional interpretations.- Normalization theorems for systems of natural deduction.- Applications of Kripke models.- Iterated inductive definitions, trees and ordinals.- Erratum.

D-SCRIPT: a computational theory of descriptions

Abstract: This paper describes D-Script, a language for representing knowledge in artificial intelligence (AI) programs. D-Script contains a powerful formalism for descriptions, which permits the representation of statements that are problematical for other systems. Particular attention is paid to problems of opaque contexts, time contexts, and knowledge about knowledge. The design of a deductive system for this language is also considered.

The Computability of Group Constructions, Part I

Abstract: Publisher Summary This chapter discusses the computability of group constructions. The chapter also discusses relative Grzegorczyk hierarchy and free products with amalgamation. A free product with amalgamation is a useful construction when dealing with decision problems in groups because intuitively the normal form theorem yields decision procedures for such products modulo the decision procedures for the groups and the amalgamated subgroups. The chapter discusses strong Britton extensions.

Hauptsatz for Intuitionistic Simple Type Theory

Abstract: Publisher Summary This chapter discusses cut-free rules for classical and intuitionistic simple type theory with and without extensionality. The proof uses an extension of the method of computability previously applied to the intuitionistic theory of iterated inductive definitions and the theory of species. Function constants denote functions whose arguments and values are individuals. Free and bound occurrences of a variable in a term or formula are defined as usual. Terms and formulae which only differ in the naming of their bound variables are identified. Free and bound occurrences of a variable in a deduction are defined as in MARTIN-LOF. Deductions that only differ in the naming of their bound variables are identified.

I – mathematical basis

Abstract: This chapter contains the basic mathematical concepts and terminology used in the theory of computability. The presentation is brief but selfcontained. Some of the material may be a review for readers who are knowledgeable in set theory; however enough terminology is peculiar to computability theory so that every reader should at least skim through this chapter. Examples of new material may include partial functions, madic number notation, predicates, and induction applied in nonnumeric contexts. Readers who plan to cover only the informal aspects of computability theory (Chapters I and II) may wish to skip the material which relates to predicates and logical formulas. In general, the reader is encouraged to try all the exercises.

The computability of group constructions II

Abstract: Finitely presented groups having word, problem solvable by functions in the relativized Grzegorczyk hierarchy, {En(A)| n ε N, A ⊂ N (N the natural numbers)} are studied. Basically the class E3 consists of the elementary functions of Kalmar and En+1 is obtained from En by unbounded recursion. The relativization En(A) is obtained by adjoining the characteristic function of A to the class En. It is shown that the Higman construction embedding, a finitely generated group with a recursively enumerable set of relations into a finitely presented group, preserves the computational level of the word problem with respect to the relativized Grzegorczyk hierarchy. As a corollary it is shown that for every n ≥ 4 and A ⊂ N recursively enumerable there exists a finitely presented group with word problem solvable at level En(A) but not En-1(A). In particular, there exist finitely presented groups with word problem solvable at level En but not En-1 for n ≥ 4, answering a question of Cannonito.

Toward Abstract Numerical Analysis

Abstract: This paper deals with a technique for proving that certain problems of numerical analysis are numerically unsolvable. So that only methods which are natural for dealing with analytic problems may be presented, notions from recursive function theory have been avoided. Instead, the number of necessary function evaluations is taken as the measure of computational complexity. The role of topological concepts in the study of computability is examined. Last, a topological result is used to prove that a simple initial-value problem is numerically unsolvable.

Development of guidelines for the definition of the relavant information content in data classes

Abstract: The problem of experiment design is defined as an information system consisting of information source, measurement unit, environmental disturbances, data handling and storage, and the mathematical analysis and usage of data. Based on today's concept of effective computability, general guidelines for the definition of the relevant information content in data classes are derived. The lack of a universally applicable information theory and corresponding mathematical or system structure is restricting the solvable problem classes to a small set. It is expected that a new relativity theory of information, generally described by a universal algebra of relations will lead to new mathematical models and system structures capable of modeling any well defined practical problem isomorphic to an equivalence relation at any corresponding level of abstractness.