Symbolic Execution of Constrained Horn Clauses

Abstract: Constrained Horn clauses (CHCs) are a fragment of first-order logic widely used as an intermediate language for verification. We study constrained resolution as a calculus for reasoning about the satisfiability of sets of CHCs. Constrained resolution forms a simple alternative to model checking algorithms such as CEGAR and IC3 and is here shown to exhibit complementary features. We show that, when CHCs encode programs, forward symbolic execution corresponds to positive unit hyper-resolution, and backward symbolic execution to SLD resolution, which are special cases of constrained resolution. We formalize forward and backward reasoning for linear and non-linear CHCs and transition systems, prove refutational completeness under fair strategies, and develop subsumption criteria for pruning redundant derivations. We moreover show that backward resolution with subsumption generalizes $k$-induction to non-linear CHCs, and that forward constrained resolution is related to the problem of constructing proofs in incorrectness logic. Lastly, we provide an experimental evaluation in the Eldarica CHC solver on benchmarks from CHC-COMP.