Extending SMTCoq, a Certified Checker for SMT (Extended Abstract)

This extended abstract reports on current progress of SMTCoq, a communication tool between the Coq proof assistant and external SAT and SMT solvers. Based on a checker for generic first-order certificates implemented and proved correct in Coq, SMTCoq offers facilities both to check external SAT and SMT answers and to improve Coq's automation using such solvers, in a safe way. Currently supporting the SAT solver zChaff, and the SMT solver veriT for the combination of the theories of congruence closure and linear integer arithmetic, SMTCoq is meant to be extendable with a reasonable amount of effort: we present work in progress to support the SMT solver CVC4 and the theory of bit vectors.Comment: In Proceedings HaTT 2016, arXiv:1606.0542

Partial Quantifier Elimination By Certificate Clauses

We study partial quantifier elimination (PQE) for propositional CNF formulas. In contrast to full quantifier elimination, in PQE, one can limit the set of clauses taken out of the scope of quantifiers to a small subset of target clauses. The appeal of PQE is twofold. First, PQE can be dramatically simpler than full quantifier elimination. Second, it provides a language for performing incremental computations. Many verification problems (e.g. equivalence checking and model checking) are inherently incremental and so can be solved in terms of PQE. Our approach is based on deriving clauses depending only on unquantified variables that make the target clauses $\mathit{redundant}$. Proving redundancy of a target clause is done by construction of a ``certificate'' clause implying the former. We describe a PQE algorithm called $\mathit{START}$ that employs the approach above. We apply $\mathit{START}$ to generating properties of a design implementation that are not implied by specification. The existence of an $\mathit{unwanted}$ property means that this implementation is buggy. Our experiments with HWMCC-13 benchmarks suggest that $\mathit{START}$ can be used for generating properties of real-life designs

Circuit complexity, proof complexity, and polynomial identity testing

We introduce a new algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits (VNP is not equal to VP). As a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof complexity. More specifically, we introduce certain propositional axioms satisfied by any Boolean circuit computing PIT. We use these PIT axioms to shed light on AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. We show that either: a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size circuits of depth d - a notoriously open question for d at least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. Using the algebraic structure of our proof system, we propose a novel way to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity

DepQBF 6.0: A Search-Based QBF Solver Beyond Traditional QCDCL

We present the latest major release version 6.0 of the quantified Boolean formula (QBF) solver DepQBF, which is based on QCDCL. QCDCL is an extension of the conflict-driven clause learning (CDCL) paradigm implemented in state of the art propositional satisfiability (SAT) solvers. The Q-resolution calculus (QRES) is a QBF proof system which underlies QCDCL. QCDCL solvers can produce QRES proofs of QBFs in prenex conjunctive normal form (PCNF) as a byproduct of the solving process. In contrast to traditional QCDCL based on QRES, DepQBF 6.0 implements a variant of QCDCL which is based on a generalization of QRES. This generalization is due to a set of additional axioms and leaves the original Q-resolution rules unchanged. The generalization of QRES enables QCDCL to potentially produce exponentially shorter proofs than the traditional variant. We present an overview of the features implemented in DepQBF and report on experimental results which demonstrate the effectiveness of generalized QRES in QCDCL.Comment: 12 pages + appendix; to appear in the proceedings of CADE-26, LNCS, Springer, 201