Parallelizing SMT solving: Lazy decomposition and conciliation

Abstract Satisfiability Modulo Theories (SMT) is the satisfiability problem for first-order formulae with respect to background theories. SMT extends the propositional satisfiability by introducing various underlying theories. To improve the efficiency of SMT solving, many efforts have been made on low-level algorithms but they generally cannot leverage the capability of parallel hardware. We propose a high-level and flexible framework, namely lazy decomposition and conciliation (LDC), to parallelize solving for quantifier-free SMT problems. Overall, a SMT problem is firstly decomposed into subproblems, then local reasoning inside each subproblem is conciliated with the global reasoning over the shared symbols across subproblems in parallel. LDC can be built on any existing solver without tuning its internal implementation, and is flexible as it is applicable to various underlying theories. We instantiate LDC in the theory of equality with uninterpreted functions, and implement a parallel solver PZ 3 based on Z 3. Experiment results on the QF_UF benchmarks from SMT-LIB as well as random problems show the potential of LDC, as (1) PZ 3 generally outperforms Z 3 in 4 out of 8 problem subcategories under various core configurations; (2) PZ 3 usually achieves super-linear speed-up over Z 3 on problems with sparse structures, which makes it possible to choose an appropriate solver from Z 3 and PZ 3 in advance according to the structure of input problem; (3) compared to PCVC 4, a state-of-the-art portfolio-based parallel SMT solver, PZ 3 achieves speed-up on a larger portion of problems and has better overall speed-up ratio.