Coinductive Techniques for Checking Satisfiability of Generalized Nested Conditions

TL;DR

This paper introduces coinductive tableau-based methods for checking satisfiability of nested conditions in a categorical setting, extending previous graph condition results and enabling finite model generation in specific categories.

Contribution

It generalizes satisfiability checking to a categorical framework using coinductive techniques and introduces witnesses for detecting infinite models.

Findings

Provides a semi-decision procedure for satisfiability and model generation.

Introduces a notion of witnesses for infinite model detection.

Ensures soundness and completeness using coinductive proof methods.

Abstract

We study nested conditions, a generalization of first-order logic to a categorical setting, and provide a tableau-based (semi-decision) procedure for checking (un)satisfiability and finite model generation. This generalizes earlier results on graph conditions. Furthermore we introduce a notion of witnesses, allowing the detection of infinite models in some cases. To ensure completeness, paths in a tableau must be fair, where fairness requires that all parts of a condition are processed eventually. Since the correctness arguments are non-trivial, we rely on coinductive proof methods and up-to techniques that structure the arguments. We distinguish between two types of categories: categories where all sections are isomorphisms, allowing for a simpler tableau calculus that includes finite model generation; in categories where this requirement does not hold, model generation does not work, but we still obtain a sound and complete calculus.