Formulas as Programs

TL;DR

This paper presents a novel computational interpretation of first-order logic where formulas are viewed as programs, offering a practical approach to programming inspired by logic programming and implemented in ALMA-0.

Contribution

It introduces a new interpretation of first-order logic as programs, distinct from formulas-as-types, and demonstrates its application in a practical programming language ALMA-0.

Findings

Provides a constructive interpretation of satisfiability in logic

Shows how formulas can be directly executed as programs

Supports reasoning about correctness of non-recursive ALMA-0 programs

Abstract

We provide here a computational interpretation of first-order logic based on a constructive interpretation of satisfiability w.r.t. a fixed but arbitrary interpretation. In this approach the formulas themselves are programs. This contrasts with the so-called formulas as types approach in which the proofs of the formulas are typed terms that can be taken as programs. This view of computing is inspired by logic programming and constraint logic programming but differs from them in a number of crucial aspects. Formulas as programs is argued to yield a realistic approach to programming that has been realized in the implemented programming language ALMA-0 (Apt et al.) that combines the advantages of imperative and logic programming. The work here reported can also be used to reason about the correctness of non-recursive ALMA-0 programs that do not include destructive assignment.