To Ackermann-ize or Not to Ackermann-ize? On Efficiently Handling Uninterpreted Function Symbols in $\mathit{SMT}(\mathcal{EUF} \cup \mathcal{T})$

Abstract

Satisfiability Modulo Theories $(\mathit{SMT}(\mathcal{T}))$ is the problem of deciding the satisfiability of a formula with respect to a given background theory ${\mathcal{T}}$. When ${\mathcal{T}}$ is the combination of two simpler theories ${{\mathcal{T}}_1}$ and ${{\mathcal{T}}_2} ({\mathit{SMT}({{\mathcal{T}}_1\cup{\mathcal{T}}_2})})$, a standard and general approach is to handle the integration of ${{\mathcal{T}}_1}$ and ${{\mathcal{T}}_2}$ by performing some form of search on the equalities between the shared variables.

A frequent and very relevant sub-case of ${\mathit{SMT}({{\mathcal{T}}_1\cup{\mathcal{T}}_2})}$ is when ${{\mathcal{T}}_1}$ is the theory of Equality and Uninterpreted Functions $({\mathcal{EUF}})$. 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 ${\mathit{SMT}}({{\mathcal{T}}_2})$ 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 ${\mathit{SMT}}$ tool, of applying Ackermann’s expansion to all or part of the function symbols.

A thorough experimental analysis, including the benchmarks of the SMT’05 competition, shows that the proposed technique is extremely effective in improving the overall performance of the ${\mathit{SMT}}$ tool.

This work has been partly supported by ISAAC, an European sponsored project, contract no. AST3-CT-2003-501848, by ORCHID, a project sponsored by Provincia Autonoma di Trento, and by a grant from Intel Corporation.