The article retraces major events and milestones in the mutual influences between mathematical logic and computer science since the 1950s.

Mathematical Logic in Computer Science

Publisher Summary This chapter focuses on algorithmic procedures, generalized turning algorithms, and elementary recursion theory. The theory of recursive or computable functions has been generalized in many ways: (1) to provide techniques for use in other parts of mathematics, particularly logic, set theory, and classical recursive function theory itself, (2) to gain better understanding of advanced recursion theory, for example, degree theory and hierarchy theory, and (3) to gain better understanding of the nature of computation. The generalizations resulting from (1) and (2) have been very successful but have not helped with (3) very much because they usually involve generalizing the notion of finite and the use of infinitely long computations. In classical recursion theory, the natural numbers are computed over, and they not only form the data objects but also can be used for auxiliary arithmetic operations, such as counting. The second section of the paper is a lightning comprehensive survey of how the theorems of elementary recursion theory lift to arbitrary structures when the notion of recursive is taken to mean computable by a gTa.

Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory

A general form of parallel procedure is defined for computation over abstract relational structures. These procedures are compared over both total and partial structures with Friedman's effective definitional schemes [12]. Various kinds of serial procedure are considered, also conditions under which parallel procedures are equivalent to serial (e.g. over total structures). Comparisons are made with the classical theory of partial recursive functionals with partial functions as arguments. The effects of different assumptions about the times taken by processors are explored, also the effect of asynchronous operation.

Computation Over Abstract Structures: Serial and Parallel Procedures and Friedman's Effective Definitional Schemes

Over many familiar datatypes the notion of “computable” coincides with the notion of “flowchartable”. It is also known that flowcharts are not a universal programming formalism over arbitrary datatypes, in the sense that there are datatypes over which not all computable functions are flowchartable. In this paper we consider various extensions and restrictions of the basic formalism of flowcharts, and then for every such formalism, we characterize the datatypes over which the computable functions are exactly the functions programmable in this formalism. We say that a function is computable over a datatype if it is effective relative to the primitive operations and relations of the datatype.

Necessary and sufficient conditions for the universality of programming formalisms

LED, the Logic of Effective Definitions, is an extension of first order predicate calculus used for making assertions about programs. Programs are modeled as effective definitional schemes (following Friedman). Logical properties of LED and its relations to classical logics and other programming logics are surveyed.

A. J. Kfoury

Papers

Translatability of schemas over restricted interpretations