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