Finding countermodels is an effective way of disproving false conjectures. In first-order predicate logic, model finding is an undecidable problem. But if a finite model exists, it can be found by exhaustive search. The finite model generation problem in the first-order logic can also be translated to the satisfiability problem in the propositional logic. But a direct translation may not be very efficient. This paper discusses how to take the symmetries into account so as to make the resulting problem easier. A static method for adding constraints is presented, which can be thought of as an approximation of the least number heuristic (LNH). Also described is a dynamic method, which asks a model searcher like SEM to generate a set of partial models, and then gives each partial model to a propositional prover. The two methods are analyzed, and compared with each other.

Abstract. The finite model generation problem in the first-order logic is a generalization of the propositional satisfiability (SAT) problem. An essential algorithm for solving the problem is backtracking search. In this paper, we show how to improve such a search procedure by lemma learning. For efficiency reasons, we represent the lemmas by propositional formulas and use a SAT solver to perform the necessary reasoning. We have extended the first-order model generator SEM, combining it with the SAT solver SATO. Experimental results show that the search time may be reduced significantly on many problems. 1