On Loop Formulas with Variables
This work provides theoretical foundations for non-Herbrand stable models in answer set programming, enabling more efficient reasoning with first-order theorem provers.
The paper connects Ferraris, Lee and Lifschitz's grounding-free stable model definition to loop formulas with variables, generalizing them to disjunctive programs and arbitrary first-order sentences while extending logic programs with explicit quantifiers. This yields more succinct loop formulas and enables nonmonotonic reasoning without unique name/domain closure assumptions, with certain conditions allowing query answering to be reduced to first-order logic entailment.
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.