# On the Expressivity and Applicability of Model Representation Formalisms

## TL;DR

This paper investigates the limitations and capabilities of various model representation formalisms in automated reasoning, establishing a finite model property for certain classes and proposing an approximation method to handle infinite models.

## Contribution

It proves a finite model property for MSLH clause sets and introduces a novel approximation called reflexive relation splitting for satisfiability checking.

## Key findings

- MSLH clause sets cannot represent inherently infinite models.
- Translation to tree automata extends the finite model property to other formalisms.
- Reflexive relation splitting enables automatic satisfiability detection for certain infinite models.

## Abstract

A number of first-order calculi employ an explicit model representation formalism for automated reasoning and for detecting satisfiability. Many of these formalisms can represent infinite Herbrand models. The first-order fragment of monadic, shallow, linear, Horn (MSLH) clauses, is such a formalism used in the approximation refinement calculus. Our first result is a finite model property for MSLH clause sets. Therefore, MSLH clause sets cannot represent models of clause sets with inherently infinite models. Through a translation to tree automata, we further show that this limitation also applies to the linear fragments of implicit generalizations, which is the formalism used in the model-evolution calculus, to atoms with disequality constraints, the formalisms used in the non-redundant clause learning calculus (NRCL), and to atoms with membership constraints, a formalism used for example in decision procedures for algebraic data types. Although these formalisms cannot represent models of clause sets with inherently infinite models, through an additional approximation step they can. This is our second main result. For clause sets including the definition of an equivalence relation with the help of an additional, novel approximation, called reflexive relation splitting, the approximation refinement calculus can automatically show satisfiability through the MSLH clause set formalism.