[Paper Review] On Loop Formulas with Variables

Joohyung Lee, Yunsong Meng|arXiv (Cornell University)|Jul 15, 2023

Logic, Reasoning, and Knowledge16 references22 citations

TL;DR

The paper generalizes loop formulas to handle variables in stable model semantics, extends to disjunctive programs and arbitrary first-order sentences, and shows how extended programs with quantifiers can be analyzed using first-order theorem provers. It also establishes conditions under which query answering reduces to first-order entailment and discusses relationships between Ferraris et al.'s stable models and first-order loop formulas.

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.

Motivation & Objective

• Relate first-order loop formulas to the new stable model semantics that avoids grounding.
• Extend first-order loop formulas to disjunctive programs and arbitrary first-order sentences.
• Introduce extended programs with explicit quantifiers and analyze their semantics as a subclass of the Ferraris et al. stable model language.
• Identify syntactic conditions under which query answering reduces to first-order entailment.
• Demonstrate applicability of first-order theorem provers for reasoning about non-Herbrand stable models.

Proposed method

• Define SM[F] as the second-order stable model operator and explain its relation to grounding-free semantics.
• Generalize first-order loop formulas to disjunctive programs and arbitrary sentences; extend the FES/FLF construction accordingly.
• Introduce NFES and NES formalisms to connect loop formulas with first-order representations of extended programs.
• Develop dependency-graph-based and unbounded-loop notions to handle loops in first-order and second-order settings.
• Prove equivalences between SM[F] and sets of first-order loop formulas under various formalisms (grounded vs. non-grounded).
• Provide conditions (finite universe, normal form, finite complete sets of loops) under which SM[F] can be reduced to first-order entailment.

Experimental results

Research questions

• RQ1 How do first-order loop formulas relate to the new non-grounding stable-model semantics?
• RQ2 Can loop formulas be extended to disjunctive programs and to arbitrary first-order sentences?
• RQ3 Under what syntactic conditions can query answering for extended programs be reduced to first-order entailment?
• RQ4 What is the relationship between non-Herbrand stable models and extended programs with explicit quantifiers?
• RQ5 When can SM[F] be characterized by first-order loop formulas, possibly with finite universes or complete loop sets?

Key findings

• Loop formulas with variables generalize grounding-free loop formulas to retain variables and enable probabilistic reasoning about unnamed objects.
• First-order loop formulas extend to disjunctive programs and to arbitrary first-order sentences, preserving a link to stable-model semantics.
• Extended programs with explicit quantifiers define a semantics as a subclass of Ferraris et al.’s stable models, enabling succinct loop formulas.
• Certain syntactic conditions allow query answering to be reduced to first-order entailment, enabling the use of first-order theorem provers for non-Herbrand stable models.
• Propositions establish equivalences between SM[F] and sets of first-order loop formulas under normal/form constraints and finite universe assumptions, with extensions to unbounded loops and extended loops.
• The work connects non-ground logic programs with first-order reasoning, enabling reasoning about unnamed objects without grounding.