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SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs
"... This paper introduces a propositional encoding for recursive path orders (RPO), in connection with dependency pairs. Hence, we capture in a uniform setting all common instances of RPO, i.e., lexicographic path orders (LPO), multiset path orders (MPO), and lexicographic path orders with status (LPO ..."
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This paper introduces a propositional encoding for recursive path orders (RPO), in connection with dependency pairs. Hence, we capture in a uniform setting all common instances of RPO, i.e., lexicographic path orders (LPO), multiset path orders (MPO), and lexicographic path orders with status (LPOS). This facilitates the application of SAT solvers for termination analysis of term rewrite systems (TRSs). We address four main interrelated issues and show how to encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (A) the lexicographic comparison w.r.t. a permutation of the arguments; (B) the multiset extension of a base order; (C) the combined search for a path order together with an argument filter to orient a set of inequalities; and (D) how the choice of the argument filter influences the set of inequalities that have to be oriented (socalled usable rules). We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power.
Author manuscript, published in "PSATTT'11: International Workshop on ProofSearch in Axiomatic Theories and Type Theories (2011)" Verifying SAT and SMT in Coq for a fully automated decision procedure
, 2011
"... Abstract Enjoying the power of SAT and SMT solvers in the Coq proof assistant without compromising soundness requires more than a yes/no answer from them. SAT and SMT solvers should also return a proof witness that can be checked by an external tool. We propose a fully certified checker for such wit ..."
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Abstract Enjoying the power of SAT and SMT solvers in the Coq proof assistant without compromising soundness requires more than a yes/no answer from them. SAT and SMT solvers should also return a proof witness that can be checked by an external tool. We propose a fully certified checker for such witnesses written in Coq. It can currently check witnesses from the SAT solvers ZChaff and MiniSat and from the SMT solver VeriT. Experiments highlight the efficiency of this checker. On top of it, new reflexive Coq tactics have been built that can decide a subset of Coq’s logic by calling external provers and carefully checking their answers. 1
DPLL(T) as a goaldirected proofsearch mechanism
"... in its version DPLL(T) for Satisfiability Modulo Theory (SMT), can be interpreted as a proofsearch mechanism for the incremental construction of a proof tree in sequent calculus. For this we use a sequent calculus with polarities and focusing and show how its metalogical control features allow for ..."
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in its version DPLL(T) for Satisfiability Modulo Theory (SMT), can be interpreted as a proofsearch mechanism for the incremental construction of a proof tree in sequent calculus. For this we use a sequent calculus with polarities and focusing and show how its metalogical control features allow for a precise simulation of runs of DPLL(T). This simulation also accounts for backjumping and learning steps. Finally we describe Psyche, an extensible prototype for this sequent calculus, that includes the implementation of a proofsearch component based on this simulation of DPLL(T). 1
Proofsearch, Satisfiability Modulo Theories
"... banner above paper title A bisimulation between DPLL(T) and a proofsearch strategy for the focused sequent calculus ..."
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banner above paper title A bisimulation between DPLL(T) and a proofsearch strategy for the focused sequent calculus
A Modular Integration of SAT/SMT Solvers to Coq through Proof Witnesses ⋆
"... Abstract We present a way to enjoy the power of SAT and SMT provers in Coq without compromising soundness. This requires these provers to return not only a yes/no answer, but also a proof witness that can be independently rechecked. We present such a checker, written and fully certified in Coq. It i ..."
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Abstract We present a way to enjoy the power of SAT and SMT provers in Coq without compromising soundness. This requires these provers to return not only a yes/no answer, but also a proof witness that can be independently rechecked. We present such a checker, written and fully certified in Coq. It is conceived in a modular way, in order to tame the proofs ’ complexity and to be extendable. It can currently check witnesses from the SAT solver ZChaff and from the SMT solver veriT. Experiments highlight the efficiency of this checker. On top of it, new reflexive Coq tactics have been built that can decide a subset of Coq’s logic by calling external provers and carefully checking their answers. 1
DOI: 10.1145/2503887.2503892 A Bisimulation between DPLL(T) and a ProofSearch Strategy for the Focused Sequent Calculus
, 2013
"... We describe how the DavisPutnamLogemannLoveland procedure DPLL is bisimilar to the goaldirected proofsearch mechanism described by a standard but carefully chosen sequent calculus. We thus relate a procedure described as a transition system on states to the gradual completion of incomplete proo ..."
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We describe how the DavisPutnamLogemannLoveland procedure DPLL is bisimilar to the goaldirected proofsearch mechanism described by a standard but carefully chosen sequent calculus. We thus relate a procedure described as a transition system on states to the gradual completion of incomplete prooftrees. For this we use a focused sequent calculus for polarised classical logic, for which we allow analytic cuts. The focusing mechanisms, together with an appropriate management of polarities, then allows the bisimulation to hold: The class of sequent calculus proofs that are the images of the DPLL runs finishing on UNSAT, is identified with a simple criterion involving polarities. We actually provide those results for a version DPLL(T) of the procedure that is parameterised by a background theory T for which we can decide whether conjunctions of literals are consistent. This procedure is used for Satisfiability Modulo Theories (SMT) generalising propositional SAT. For this, we extend the standard focused sequent calculus for propositional logic in the same way DPLL(T) extends DPLL: with the ability to call the decision procedure for T. DPLL(T) is implemented as a plugin for PSYCHE, a proofsearch engine for this sequent calculus, to provide a sequentcalculus based SMTsolver. *Categories and Subject Descriptors F.4.1 [Mathematical Logic]: Mechanical theorem proving