Subsumption Demodulation in First-Order Theorem Proving

Abstract

Motivated by applications of first-order theorem proving to software analysis, we introduce a new inference rule, called subsumption demodulation, to improve support for reasoning with conditional equalities in superposition-based theorem proving. We show that subsumption demodulation is a simplification rule that does not require radical changes to the underlying superposition calculus. We implemented subsumption demodulation in the theorem prover Vampire, by extending Vampire with a new clause index and adapting its multi-literal matching component. Our experiments, using the TPTP and SMT-LIB repositories, show that subsumption demodulation in Vampire can solve many new problems that could so far not be solved by state-of-the-art reasoners.

SCL(EQ): SCL for First-Order Logic with Equality

“…The proof of Lemma 25 is based on the fact that conflict resolution eventually produces a clause smaller then the original conflict clause with respect to ≺ Γ * . All simplifications, e.g., contextual rewriting, as defined in [2,32,34,35,36,20], are therefore compatible with Lemma 25 and may be applied to the newly learned clause as long as they respect the induced trail ordering. In detail, let Γ be the trail before the application of rule Backtrack.…”

SCL(EQ): SCL for First-Order Logic with Equality

“…The proof of Lemma 25 is based on the fact that conflict resolution eventually produces a clause smaller then the original conflict clause with respect to ≺ Γ * . All simplifications, e.g., contextual rewriting, as defined in [2,20,33,[35][36][37], are therefore compatible with Lemma 25 and may be applied to the newly learned clause as long as they respect the induced trail ordering. In detail, let Γ be the trail before the application of rule Backtrack.…”

An Efficient Subsumption Test Pipeline for BS(LRA) Clauses

“…In order to be efficient, modern theorem provers need to decide multiple thousand subsumption checks per second. In the pure first-order case, this is possible because of indexing and filtering techniques that quickly decide most subsumption checks [24,25,[27][28][29][30]33,39,40,[45][46][47][48][49][52][53][54]56,59,61,62].…”

“…In order to reduce the number of clauses out of a set of clauses to be considered for pairwise subsumption checking, the best known practice in firstorder theorem proving is to use (imperfect) indexing data structures as a means for pre-filtering and research concerning appropriate techniques is plentiful, see [24,25,[27][28][29][30]33,39,40,43,[45][46][47][48][49][52][53][54]56,59,61] for an evaluation of these techniques. Here we concentrate on the efficiency of a subsumption check between two clauses and therefore do not take indexing techniques into account.…”