A theoretical analysis of reasoning by symmetry in first-order logic (extended abstract)

Many classes of reasoning problems display a large amount of symmetry. In mathematical and common-sense reasoning, such symmetries are often used to reduce the diiculty of reasoning. In this paper we show how symmetries can be used in automated reasoning both to reduce or avoid case analysis, and to reduce the scope of existential quantiication. We show further that the computational complexity of the symmetry detection problem is equivalent (within a polynomial factor) to that of the graph iso-morphism problem. This result is signiicant because the complexity of the graph isomor-phism problem is a hard open problem (it is believed to lie between P and NP). Further, for graphs with bounded degree, graph isomor-phism is known to be polynomial decidable. We then show how PP olya's theorem can be used to count the number of interpretations of a logical theory which are distinct under a given set of symmetries. This provides a method for predicting the eeect of the symmetries on the ease of reasoning with the theory. We consider several commonly occurring types of symmetries and show how they aaect the size of the space of distinct interpretations. Finally, we verify experimentally that in one well studied problems class, pigeonhole problems, the use of symmetries succeeds in reducing the complexity of determining unsatissability from exponential (using resolution or a tableau based method) to polynomial.