Decomposable theories

We present in this paper a general algorithm for solving first-order formulas in particular theories called decomposable theories. First of all, using special quantifiers, we give a formal characterization of decomposable theories and show some of their properties. Then, we present a general algorithm for solving first-order formulas in any decomposable theory T . The algorithm is given in the form of five rewriting rules. It transforms a first-order formula φ, which can possibly contain free variables, into a conjunction φ of solved formulas easily transformable into a Boolean combination of existentially quantified conjunctions of atomic formulas. In particular, if φ has no free variables then φ is either the formula true or ¬true . The correctness of our algorithm proves the completeness of the decomposable theories. Finally, we show that the theory T of finite or infinite trees is a decomposable theory and give some benchmarks realized by an implementation of our algorithm, solving formulas on two-partner games in T with more than 160 nested alternated quantifiers.