To Ackermann-ize or not to Ackermann-ize? On Efficiently Handling Uninterpreted Function Symbols in SMT(EUF U T)

Satisfiability Modulo Theories (SMT(T)) is the problem of deciding the satisfiability of a formula with respect to a given background theory T. When T is the combination of two simpler theories T1 and T2 (SMT(T1 U T2)), a standard and general approach is to handle the integration of T1 and T2 by performing some form of search on the equalities between the shared variables. A frequent and very relevant sub-case of SMT(T1 U T2) is when T1 is the theory of Equality and Uninterpreted Functions (EUF). For this case, an alternative approach is to eliminate first all uninterpreted function symbols by means of Ackermann’s expansion, and then to solve the resulting SMT (T2) problem. In this paper we build on the empirical observation that there is no absolute winner between these two alternative approaches, and that the performance gaps between them are often dramatic, in either direction. We propose a simple technique for estimating a priori the costs and benefits, in terms of the size of the search space of an SMT tool, of applying Ackermann’s expansion to all or part of the function symbols. We have implemented a preprocessor which analyzes the input formula, decides autonomously which functions to expand, performs such expansions and gives the resulting formula as input to an SMT tool. A thorough experimental analysis, including the benchmarks of the SMT’05 competition, shows that our preprocessor performs the best choice(s) nearly always, and that the proposed technique is extremely effective in improving the overall performance of the SMT tool.