A Three-Valued Semantics for Typed Logic Programming

TL;DR

This paper introduces a three-valued semantics for typed logic programming, incorporating true, false, and a new 'wrong' value to represent runtime type errors, along with a sound type system.

Contribution

It proposes a novel three-valued semantics for logic programming and a corresponding type system that is proven to be semantically sound.

Findings

Semantics includes a 'wrong' value for runtime type errors

Type system is proven sound with respect to the semantics

Program semantics are independent of type semantics

Abstract

Types in logic programming have focused on conservative approximations of program semantics by regular types, on one hand, and on type systems based on a prescriptive semantics defined for typed programs, on the other. In this paper, we define a new semantics for logic programming, where programs evaluate to true, false, and to a new semantic value called wrong, corresponding to a run-time type error. We then have a type language with a separated semantics of types. Finally, we define a type system for logic programming and prove that it is semantically sound with respect to a semantic relation between programs and types where, if a program has a type, then its semantics is not wrong. Our work follows Milner's approach for typed functional languages where the semantics of programs is independent from the semantic of types, and the type system is proved to be sound with respect to a relation between both semantics.