Founded Semantics and Constraint Semantics of Logic Rules

TL;DR

This paper introduces founded and constraint semantics for logic rules, providing a unified, intuitive, and computationally efficient framework that handles unrestricted negation and quantification in logic programming.

Contribution

It proposes new semantics that unify and simplify prior approaches, supporting unrestricted negation and quantification with explicit assumptions and linear-time computation.

Findings

Supports unrestricted negation and quantification

Unifies core of prior semantics

Computes in linear time

Abstract

Logic rules and inference are fundamental in computer science and have been studied extensively. However, prior semantics of logic languages can have subtle implications and can disagree significantly, on even very simple programs, including in attempting to solve the well-known Russell's paradox. These semantics are often non-intuitive and hard-to-understand when unrestricted negation is used in recursion. This paper describes a simple new semantics for logic rules, founded semantics, and its straightforward extension to another simple new semantics, constraint semantics, that unify the core of different prior semantics. The new semantics support unrestricted negation, as well as unrestricted existential and universal quantifications. They are uniquely expressive and intuitive by allowing assumptions about the predicates, rules, and reasoning to be specified explicitly, as simple and precise binary choices. They are completely declarative and relate cleanly to prior semantics. In addition, founded semantics can be computed in linear time in the size of the ground program.