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Lazy Satisfiability Modulo Theories
 JOURNAL ON SATISFIABILITY, BOOLEAN MODELING AND COMPUTATION 3 (2007) 141Â224
, 2007
"... Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a firstorder formula with respect to some decidable firstorder theory T (SMT (T)). These problems are typically not handled adequately by standard automated theorem provers. SMT is being recognized as increasingl ..."
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Cited by 189 (50 self)
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Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a firstorder formula with respect to some decidable firstorder theory T (SMT (T)). These problems are typically not handled adequately by standard automated theorem provers. SMT is being recognized as increasingly important due to its applications in many domains in different communities, in particular in formal verification. An amount of papers with novel and very efficient techniques for SMT has been published in the last years, and some very efficient SMT tools are now available. Typical SMT (T) problems require testing the satisfiability of formulas which are Boolean combinations of atomic propositions and atomic expressions in T, so that heavy Boolean reasoning must be efficiently combined with expressive theoryspecific reasoning. The dominating approach to SMT (T), called lazy approach, is based on the integration of a SAT solver and of a decision procedure able to handle sets of atomic constraints in T (Tsolver), handling respectively the Boolean and the theoryspecific components of reasoning. Unfortunately, neither the problem of building an efficient SMT solver, nor even that
Strategies for Combining Decision Procedures
, 2003
"... Implementing efficient algorithms for combining decision procedures has been a challenge and their correctness precarious. In this paper we describe an inference system that has the classical NelsonOppen procedure at its core and includes several optimizations: variable abstraction with sharing, ca ..."
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Cited by 10 (2 self)
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Implementing efficient algorithms for combining decision procedures has been a challenge and their correctness precarious. In this paper we describe an inference system that has the classical NelsonOppen procedure at its core and includes several optimizations: variable abstraction with sharing, canonization of terms at the theory level, and Shostak's streamlined generation of new equalities for theories with solvers. The transitions of our system are finegrained enough to model most of the mechanisms currently used in designing combination procedures. In particular, with a simple language of regular expressions we are able to describe several combination algorithms as strategies for our inference system, from the basic NelsonOppen to the very highly optimized one recently given by Shankar and Rueß. Presenting the basic system at a high level of generality and nondeterminism allows transparent correctness proofs that can be extended in a modular fashion when new features are introduced in the system. Similarly, the correctness proof of any new strategy requires only minimal additional proof effort.
Superposition modulo a Shostak theory
 AUTOMATED DEDUCTION (CADE19), VOLUME 2741 OF LNAI
, 2003
"... We investigate superposition modulo a Shostak theory T in order to facilitate reasoning in the amalgamation of T and a free theory F. Free operators occur naturally for instance in program verication problems when abstracting over subroutines. If their behaviour in addition can be specied axiomatic ..."
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Cited by 2 (0 self)
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We investigate superposition modulo a Shostak theory T in order to facilitate reasoning in the amalgamation of T and a free theory F. Free operators occur naturally for instance in program verication problems when abstracting over subroutines. If their behaviour in addition can be specied axiomatically, much more of the program semantics can be captured. Combining the Shostakstyle components for deciding the clausal validity problem with the ordering and saturation techniques developed for equational reasoning, we derive a refutationally complete calculus on mixed ground clauses which result for example from CNF transforming arbitrary universally quantied formulae. The calculus works modulo a Shostak theory in the sense that it operates on canonizer normalforms. For the Shostak solvers that we study, coherence comes for free: no coherence pairs need to be considered.
Combination of Convex Theories: Modularity, Deduction Completeness, and Explanation
, 2008
"... ..."
Canonized Rewriting and Ground AC Completion Modulo Shostak Theories
, 2001
"... ACcompletion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground ACcompletion for deciding formulas in th ..."
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Cited by 1 (1 self)
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ACcompletion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground ACcompletion for deciding formulas in the combination of the theory of equality with userdefined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our algorithm, called AC(X), is obtained by augmenting in a modular way ground ACcompletion with the canonizer and solver present for the theory X. This integration rests on canonized rewriting, a new relation reminiscent to normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is proved sound, complete and terminating, and is implemented to extend the core of the AltErgo theorem prover.
Ground Associative and Commutative Completion Modulo Shostak Theories
"... ACcompletion efficiently handles equality modulo associative and commutative function symbols. In the ground case, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground ACcompletion for deciding formulas in the comb ..."
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ACcompletion efficiently handles equality modulo associative and commutative function symbols. In the ground case, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground ACcompletion for deciding formulas in the combination of the theory of equality with userdefined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. The main ideas of our algorithm are first to adapt the definition of rewriting in order to integrate the canonizer of X and second, to replace the equation orientation mechanism found in ground ACcompletion with the solver for X. 1
Author manuscript, published in "TACAS Tools and Algorithms for the Construction and Analysis of Systems (2011)" Canonized Rewriting and Ground AC Completion Modulo Shostak Theories ⋆
, 2013
"... Abstract. ACcompletion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground ACcompletion for deciding formu ..."
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Abstract. ACcompletion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground ACcompletion for deciding formulas in the combination of the theory of equality with userdefined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our algorithm, called AC(X), is obtained by augmenting in a modular way ground ACcompletion with the canonizer and solver present for the theory X. Thisintegration rests on canonized rewriting, a new relation reminiscent to normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is proved sound, complete and terminating, and is implemented to extend the core of the AltErgo theorem prover.
Strategies for combining decision procedures �
"... www.elsevier.com/locate/tcs Implementing efficient algorithms for combining decision procedures has been a challenge and their correctness precarious. In this paper we describe an inference system that has the classical Nelson–Oppen procedure at its core and includes several optimizations: variable ..."
Abstract
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www.elsevier.com/locate/tcs Implementing efficient algorithms for combining decision procedures has been a challenge and their correctness precarious. In this paper we describe an inference system that has the classical Nelson–Oppen procedure at its core and includes several optimizations: variable abstraction with sharing, canonization of terms at the theory level, and Shostak’s streamlined generation of new equalities for theories with solvers. The transitions of our system are finegrained enough to model most of the mechanisms currently used in designing combination procedures. In particular, with a simple language of regular expressions we are able to describe several combination algorithms as strategies for our inference system, from the basic Nelson–Oppen to the very highly optimized one recently given by Shankar and Rueß. Presenting the basic system at a high level of generality and nondeterminism allows transparent correctness proofs that can be extended in a modular fashion when new features are introduced in the system.