Truth versus information in logic programming

TL;DR

This paper unifies different three-valued semantics for logic programming using Belnap's four-valued logic, providing new insights into program analysis, debugging, and the relationship between specifications and executions.

Contribution

It introduces a four-valued logic framework that unifies existing three-valued semantics, enhancing understanding of logic programming semantics and related applications.

Findings

Unifies three-valued semantics using Belnap's four-valued logic.

Reveals the duality between truth and information orderings.

Provides new perspectives on program analysis and debugging.

Abstract

The semantics of logic programs was originally described in terms of two-valued logic. Soon, however, it was realised that three-valued logic had some natural advantages, as it provides distinct values not only for truth and falsehood, but also for "undefined". The three-valued semantics proposed by Fitting and by Kunen are closely related to what is computed by a logic program, the third truth value being associated with non-termination. A different three-valued semantics, proposed by Naish, shared much with those of Fitting and Kunen but incorporated allowances for programmer intent, the third truth value being associated with underspecification. Naish used an (apparently) novel "arrow" operator to relate the intended meaning of left and right sides of predicate definitions. In this paper we suggest that the additional truth values of Fitting/Kunen and Naish are best viewed as duals. We use Belnap's four-valued logic, also used elsewhere by Fitting, to unify the two three-valued approaches. The truth values are arranged in a bilattice which supports the classical ordering on truth values as well as the "information ordering". We note that the "arrow" operator of Naish (and our four-valued extension) is essentially the information ordering, whereas the classical arrow denotes the truth ordering. This allows us to shed new light on many aspects of logic programming, including program analysis, type and mode systems, declarative debugging and the relationships between specifications and programs, and successive executions states of a program. This paper is to appear in Theory and Practice of Logic Programming (TPLP).