On Loop Formulas with Variables

TL;DR

This paper extends the concept of loop formulas with variables to more general logic programs and first-order sentences, enabling more succinct representations and reasoning in nonmonotonic logic without grounding.

Contribution

It generalizes loop formulas to disjunctive and first-order sentences, extends syntax with explicit quantifiers, and links stable model semantics to first-order logic entailment.

Findings

Generalized loop formulas for disjunctive programs and first-order sentences.

Extended syntax allows explicit quantifiers in logic programs.

Reduced query answering to first-order entailment under certain conditions.

Abstract

Recently Ferraris, Lee and Lifschitz proposed a new definition of stable models that does not refer to grounding, which applies to the syntax of arbitrary first-order sentences. We show its relation to the idea of loop formulas with variables by Chen, Lin, Wang and Zhang, and generalize their loop formulas to disjunctive programs and to arbitrary first-order sentences. We also extend the syntax of logic programs to allow explicit quantifiers, and define its semantics as a subclass of the new language of stable models by Ferraris et al. Such programs inherit from the general language the ability to handle nonmonotonic reasoning under the stable model semantics even in the absence of the unique name and the domain closure assumptions, while yielding more succinct loop formulas than the general language due to the restricted syntax. We also show certain syntactic conditions under which query answering for an extended program can be reduced to entailment checking in first-order logic, providing a way to apply first-order theorem provers to reasoning about non-Herbrand stable models.