Specifying Latin Square Problems in Propositional Logic

This chapter discusses how to specify various Latin squares so that their existence can be eeciently decided by computer programs. The computer programs considered here are so-called general-purpose model generation programs (or simply model generators) that are used to solve constraint satisfaction problems in AI, to prove theorems in nite domains, or to produce counterexamples to false conjectures. For instance, any example of nite structures in Larry Wos's book 16] can be easily solved using these model generators. In the recent years, model generators have been used to solve the existence problem of Latin squares with speciied properties. Numerous previously open cases of Latin squares were rst solved by these model generators. These Latin square problems are attacked along the two lines: (a) develop eecient model generation programs; (b) provide eecient speciications of the same problem. This chapter will focus on the latter as we realize through our experience that this is a crucial issue to the success of solving these problems. In this chapter, we show how to specify various Latin squares and their related structures, such as holey Latin squares, Steiner pentagon systems, and perfect Mendelsohn designs, in propositional logic. We examine various performance enhancing techniques used in the study, such as adding extra constraints, eliminating isomorphic models and the cyclic group construction. We provide the justiication of these techniques, either theoretical or experimental, whenever possible. Some remaining open Latin square problems are mentioned along the presentation. Most experimental results provided in the chapter are obtained by SATO 17, 19] on a SUN Sparc 2 workstation. SATO (Satissability Testing Optimized) is a model generator based on the Davis-Putnam algorithm for propositional clauses. One of the major motivations to develop SATO was to attack open Latin square problems. To solve a Latin square problem or any nite model generation problem, SATO encodes the problem into a set of propositional clauses such that the satissability of the clauses ensures a solution to the problem. During the past three years, SATO solved over two hundred previously open cases of Latin square problems. In the rst