# Constraint Solving for Finite Model Finding in SMT Solvers

## TL;DR

This paper enhances SMT solvers by integrating finite model finding techniques, enabling them to produce counter-models for quantified formulas, thus broadening their applicability in automated reasoning tasks.

## Contribution

It introduces a novel solver for cardinality constraints and on-demand instantiation methods, improving SMT solvers' ability to find finite models with quantified formulas.

## Key findings

- Competitive performance with state-of-the-art SMT solvers
- Enables counter-model generation for quantified formulas
- Orthogonal to existing automated theorem proving approaches

## Abstract

SMT solvers have been used successfully as reasoning engines for automated verification and other applications based on automated reasoning. Current techniques for dealing with quantified formulas in SMT are generally incomplete, forcing SMT solvers to report "unknown" when they fail to prove the unsatisfiability of a formula with quantifiers. This inability to return counter-models limits their usefulness in applications that produce queries involving quantified formulas. In this paper, we reduce these limitations by integrating finite model finding techniques based on constraint solving into the architecture used by modern SMT solvers. This approach is made possible by a novel solver for cardinality constraints, as well as techniques for on-demand instantiation of quantified formulas. Experiments show that our approach is competitive with the state of the art in SMT, and orthogonal to approaches in automated theorem proving.