More Computation Power for a Denotational Semantics for First Order Logic

Abstract

This paper starts from a denotational semantics for first order logic proposed by Apt. He interprets first order logic as a programming language within the declarative paradigm, but different from the usual approach of logic programming. For his proposal Apt obtains soundness with respect to the standard interpretation of first order logic. Here we consider the expressivity (or:computation power) of the system proposed, i.e., the programs for which no error messages are produced.

To gain computational realism a decrease in expressivity compared to first order logic is inevitable. In [1] the expressivity was compared with logic programming with both positive and negative results. Here we consider three ways of improving the situation: replacing the original interpretation of conjunction-as-sequential-composition by a symmetric interpretation of conjunction; extending the framework with the option of recursive definitions by Clark completions; replacing the interpretation of disjunction-as-choice by an interpretation which is in the style of ‘backtracking’: only produce an error if all options lead to error.

For each improvement we obtain a soundness result. The three improvements are developed independently in a way that makes sure that also their combination preserves soundness.

The combination gives a considerable improvement of the expressivity of the denotational semantics that Apt proposed, both in the comparison with first order logic and in the comparison with logic programming.