Model Evolution with Equality Modulo Built-in Theories

Baumgartner, Peter; Tinelli, Cesare

doi:10.1007/978-3-642-22438-6_9

Cited by 15 publications

(17 citation statements)

References 13 publications

(28 reference statements)

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“…When classes of theories are generically integrated to automated reasoning with the use of constraints, as for the Model Evolution Calculus [3], these are usually described as first-order formulae over a particular theory's signature (as it is the case in [1,16] for LIA). Our abstract data-structures for constraints could be viewed as the semantic counter-part of such a syntactic representation, whose atomic construction steps are costless but which may incur expensive satisfiability checks by the background reasoner.…”

Section: Related Work and Further Workmentioning

confidence: 99%

Axiomatic Constraint Systems for Proof Search Modulo Theories

Rouhling

Farooque

Graham-Lengrand

et al. 2015

Frontiers of Combining Systems

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Goal-directed proof search in first-order logic uses metavariables to delay the choice of witnesses; substitutions for such variables are produced when closing proof-tree branches, using first-order unification or a theory-specific background reasoner. This paper investigates a generalisation of such mechanisms whereby theory-specific constraints are produced instead of substitutions. In order to design modular proofsearch procedures over such mechanisms, we provide a sequent calculus with meta-variables, which manipulates such constraints abstractly. Proving soundness and completeness of the calculus leads to an axiomatisation that identifies the conditions under which abstract constraints can be generated and propagated in the same way unifiers usually are. We then extract from our abstract framework a component interface and a specification for concrete implementations of background reasoners.

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Section: Related Work and Further Workmentioning

confidence: 99%

Axiomatic Constraint Systems for Proof Search Modulo Theories

Rouhling

Farooque

Graham-Lengrand

et al. 2015

Frontiers of Combining Systems

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“…We consider a (single) copy of the set L of literals, denoted L d , whose elements are called decision literals, which are just tagged clones of the literals in L. Decision literals are denoted 6 by l d . We use the possibly indexed symbol ∆ to denote a finite sequence of possibly tagged literals, with ∅ denoting the empty sequence.…”

Section: Definition 8 (Decision Literals and Sequences)mentioning

confidence: 99%

“…we still take formulae to be trees and inference rules to organise the root-first decomposition of their connectives, rather than using DPLL(T )'s more flexible structures), it would be interesting to capture some of the related systems that extend DPLL(T ) with e.g. full first-order logic and/or equality [4][5][6]. The full version of LK p (T ) is indeed designed for handling quantifiers and equalities, so we hope to relate it to other techniques such as unification, paramodulation, superposition, etc.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus

Farooque

Graham-Lengrand

Mahboubi

2013

Proceedings of the Eighth ACM SIGPLAN International Workshop on Logical Frameworks &Amp; Meta-Languages: Theory &Amp; Practice

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We describe how the Davis-Putnam-Logemann-Loveland procedure DPLL is bisimilar to the goal-directed proof-search 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 proof-trees.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 SMT-solver.

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“…Another line of research is concerned with adding black-box reasoners for specific background theories to first-order automated reasoning methods (resolution [5,17,1], sequent calculi [25], instantiation methods [14,8,9], etc). In both cases, a major unsolved research challenge is to provide reasoning support that is "reasonably complete" in practice, so that the systems can be used more reliably for both proving theorems and finding counterexamples.…”

Section: Introductionmentioning

confidence: 99%

“…The resolution calculus in [17] has built-in inference rules for linear (rational) arithmetic, but is complete only under restrictions that effectively prevent quantification over rationals. Earlier work on integrating theory reasoning into model evolution [8,9] lacks the treatment of background-sorted foreground function symbols. The same applies to the sequent calculus in [25], which treats linear arithmetic with built-in rules for quantifier elimination.…”

Section: Introductionmentioning

confidence: 99%

Hierarchic Superposition Revisited

Baumgartner

Waldmann

2019

Lecture Notes in Computer Science

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Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory. ⋆ The final authenticated version is available online at https://doi.org/TBD.

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Model Evolution with Equality Modulo Built-in Theories

Cited by 15 publications

References 13 publications

Axiomatic Constraint Systems for Proof Search Modulo Theories

Axiomatic Constraint Systems for Proof Search Modulo Theories

A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus

Hierarchic Superposition Revisited

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