Lazy Model Expansion: Interleaving Grounding with Search

TL;DR

This paper introduces a lazy grounding approach for model expansion in rich logic languages, reducing the grounding bottleneck by interleaving grounding with search, and demonstrates its effectiveness in FO(ID).

Contribution

It presents a theoretical framework and algorithm for lazy model expansion, integrating justification-based reasoning with partial grounding in FO(.) logic.

Findings

Reduces grounding size during search.

Integrates lazy grounding with FO(ID) model generation.

Shows improved performance in experiments.

Abstract

Finding satisfying assignments for the variables involved in a set of constraints can be cast as a (bounded) model generation problem: search for (bounded) models of a theory in some logic. The state-of-the-art approach for bounded model generation for rich knowledge representation languages, like ASP, FO(.) and Zinc, is ground-and-solve: reduce the theory to a ground or propositional one and apply a search algorithm to the resulting theory. An important bottleneck is the blowup of the size of the theory caused by the reduction phase. Lazily grounding the theory during search is a way to overcome this bottleneck. We present a theoretical framework and an implementation in the context of the FO(.) knowledge representation language. Instead of grounding all parts of a theory, justifications are derived for some parts of it. Given a partial assignment for the grounded part of the theory and valid justifications for the formulas of the non-grounded part, the justifications provide a recipe to construct a complete assignment that satisfies the non-grounded part. When a justification for a particular formula becomes invalid during search, a new one is derived; if that fails, the formula is split in a part to be grounded and a part that can be justified. The theoretical framework captures existing approaches for tackling the grounding bottleneck such as lazy clause generation and grounding-on-the-fly, and presents a generalization of the 2-watched literal scheme. We present an algorithm for lazy model expansion and integrate it in a model generator for FO(ID), a language extending first-order logic with inductive definitions. The algorithm is implemented as part of the state-of-the-art FO(ID) Knowledge-Base System IDP. Experimental results illustrate the power and generality of the approach.