Showing papers on "Computational logic published in 1977"

A course in mathematical logic

Abstract: 1. Beginning Mathematical Logic. 2. First-Order Logic. 3. First-Order Logic (continued). 4. Boolean Algebras. 5. Model Theory. 6. Recursion Theory. 7. Logic - Limitative Results. 8. Recursion Theory (continued). 9. Intuitionistic First-Order Logic. 10. Axiomatic Set Theory. 11. Nonstandard Analysis. Bibliography.

An Introduction to First-Order Logic

Abstract: Publisher Summary This chapter discusses the formulas that are certain finite strings of symbols. The “first” in the phrase “first-order logic” is to distinguish this form of logic from stronger logics, such as second-order or weak second-order logic, where certain extralogical notions (set or natural number) are taken as given in advance. The chapter provides information of what can and what cannot be expressed in first-order logic. Most of the examples are taken from the wealth of notions in modern algebra with which most mathematicians have at least a nodding acquaintance. The chapter also discusses many-sorted first-order logic, ω-logic, weak second-order logic, Infinitary logic, Logic with new quantifiers, and abstract model theory.

Logic in linguistics

Abstract: Preface Symbols and notational conventions 1. Logic for linguists 2. Set theory 3. Inference and logical analysis of sentences 4. Propositional logic 5. Predicate logic 6. Deduction 7. Modal logic 8. Intensional logic and categorial grammar 9. Further extensions 10. Logic for linguists? References Answers to exercises Index.

Computability and completeness in logics of programs (Preliminary Report)

Abstract: Dynamic logic is a generalization of first order logic in which quantifiers of the form “for all k” are replaced by phrases of the form “after executing program α” This logic subsumes most existing first-order logics of programs that manipulate their environment, including Floyd's and Hoare's logics of partial correctness and Manna and Waldinger's logic of total correctness, yet is more closely related to classical first-order logic than any other proposed logic of programs We consider two issues: how hard is the validity problem for the formulae of dynamic logic, and how might one axiomatize dynamic logic? We give bounds on the validity problem for some special cases, including a P02-completeness result for the partial correctness theories of uninterpreted flowchart programs We also demonstrate the completeness of an axiomatization of dynamic logic relative to arithmetic

On the theory of programming logics

Abstract: A new logic for reasoning about programs is proposed here, and its metamathematics is investigated. No new primitive notions are needed for the logic beyond those used in elementary programming and mathematics, yet the combination of these notions is remarkably powerful. The logic includes a programming language, designed with Michael O'Donnell, for program verification. It forms the core of the PL/CV verifier at Cornell. This study belongs to the discipline of Algorithmic Logic as conceived by Engeler. The logic is related to Park's mu-calculus based on functions instead of relations. The monadic quantifier free subset is developed in the style of the system in J.W. deBakker's Recursive Procedures. But this logic is substantially different from either of these. It is intended to be a practical programming logic in the spirit of E. Dijkstra's calculus. This paper proves the completeness and decidability of the monadic (iterative) programming logic. It discusses the polyadic logic and the programming language briefly, and considers general models for the iterative programming logic. The polyadic logic is shown to be incomplete with respect to standard models, but complete with respect to general models.

The Influence of Boole’s Search for a Universal Method in Analysis on the Creation of His Logic

Abstract: This paper deals with the influence exerted by Boole’s own work on differential equations on his creation of algebraic logic. The main traits of Boole’s methodology of logic, and the particular algorithms which he used in his 1847 The Mathematical Analysis of Logic. ore first pointed out. An examination of the mathematical papeis which Boole wrote before the publication of the mentioned logical treatise shows that both the methodology leading to the production of his logic and the algorithms used in its development were repeatedly used by him in his earlier work in analysis.

Hardware Description Languages: Voices from the Tower of Babel*

Abstract: A hardware description language can be used to describe the logic gates, the sequential machines, and the functional modules, along with their interconnection and their control, in a digital system. In a general sense, Boolean equations, logic diagrams, programrning languages, and Petri nets are hardware description languages: they can be used to describe some aspect of hardware and they have definable syntax and semantics. Specifically, what is more commonly referred to as a hardware description language is a variation of a programming language tuned to the overall needs of describing hardware. This article will discuss the rationale for using such languages in the first place, identify the problems attendant upon their proliferation, and describe the measures being taken to achieve a solution.

A proof-checker for dynamic logic

Abstract: We consider the problem of getting a computer to follow reasoning conducted in dynamic logic. This is a recently developed logic of programs that subsumes most existing first-order logics of programs that manipulate their environment, including Floyd's and Hoare's logics of partial correctness and Manna and Waldinger's logic of total correctness. Dynamic logic is more closely related to classical first-order logic than any other proposed logic of programs. This simplifies the design of a proof-checker for dynamic logic. Work in progress on the implementation of such a program is reported on, and an example machine-checked proof is exhibited.

an introduction to multiple-valued logic

Abstract: Publisher Summary Computer scientists, computer engineers, applied mathematicians, and physicists are familiar with options in which there are no middle choices between true and false Statisticians are familiar with the soft logic of probability, and physicists are familiar with the logic of uncertainty The lack of such choices is inconvenient and critical when trying to determine whether the status of a computer system is go, wait, or no-go Multiple-valued logic is concerned with these intermediate choices The major drawback to overcomplicated flowcharts developed by computer programmers is the difficulty with which they are checked, corrected, or modified This situation suggests a structured design approach, where a structured flowchart or well designed program is built up to an adequate level of detail according to definite rules from a small, simple, and sufficient set of elemental blocks or primitives Developments in multiple-valued logic as related to computer science include a range of disciplines in which comparisons to multiple-valued logic and computer science are being made, such as neural science and ethology

Applied algorithmic logic

Abstract: This is a survey of the last year's work of the group of algorithmic logic. Our studies have concetrated on two (not disjoint) tasks: design of programming language LOGLAN 77. studies of computational complexity.

Basic Concepts of Computer Science and Logic

Abstract: Some clarifications and mathematical definitions which are concerned with the many fundamental concepts of computability theory based directly on the computer field, are presented. The sole aim of this paper is to attempt to bridge the gulf between the computer field on the one hand, and mathematics and logic on the other. The particular concepts are presented here for justification only: and therefore almost no theorems are included. (Fortunately there is no employer who could press me to prove the theorems!). The intention is to provide such mathematical definitions as would be very relevant to computers and to their languages, and even more, to their development and design.

Logic and Foundations

Abstract: Publisher Summary This chapter discusses the development of mathematical logic. Mathematical logic is the mathematical study of the main activities of mathematicians: constructing, defining, proving, and computing. There are two radically different approaches to this subject. According to the first of these, mathematical logic has the task of justifying the activities of mathematicians—without logic, mathematicians might slip into conceptual muddles or even downright contradictions. This first approach is sometimes loosely called the foundational approach, and it was very much to the fore until about 1930. In the 1930s, a number of powerful logical techniques were invented and a different view of logic became popular. According to this second way of thinking, which is sometimes called the technical approach, logic is simply a branch of mathematics on a par with any other—the only distinguishing feature of logic being that it talks about definability, meanings, etc. The foundational kind of research has made relatively little progress in the last 40 years, except among those mathematicians who reject parts of classical mathematics.