A Completion–Based Method for Adding Equality to Free Variable Semantic Tableaux

For both classical and nonclassical first–order logic equality is a crucial feature to increase expressivity of the object language. It is, therefore, of great importance to have a method at hand that allows tableau–based theorem provers to handle equality in an efficient way. In the first approaches to adding equality to semantic tableaux [8, 10, 11] additional tableau rules were defined that allow the application of equalities occurring on a branch to other formulas. In [4] a completion–based method has been described; it uses, however, as the approaches in [8, 10, 11], the ground version of tableaux. Recent methods, described in [5] and [2, 3], are based on the much more efficient free variable tableau system [5]. In [3] it has been shown that one of the reasons for the inefficiency of previous approaches is that they all mix up the application of equalities and of classical tableau rules. In contrast, the method presented in [3] for closing a tableau branch, consists of two separate tableau expansion stages. In the first stage the tableau is expanded using the classical tableau rules; in the second stage, from a branch B the set of equalities