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37
A Fast LinearArithmetic Solver for DPLL(T
, 2006
"... Abstract. We present a new Simplexbased linear arithmetic solver that can be integrated efficiently in the DPLL(T) framework. The new solver improves over existing approaches by enabling fast backtracking, supporting a priori simplification to reduce the problem size, and providing an efficient for ..."
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Cited by 183 (7 self)
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Abstract. We present a new Simplexbased linear arithmetic solver that can be integrated efficiently in the DPLL(T) framework. The new solver improves over existing approaches by enabling fast backtracking, supporting a priori simplification to reduce the problem size, and providing an efficient form of theory propagation. We also present a new and simple approach for solving strict inequalities. Experimental results show substantial performance improvements over existing tools that use other Simplexbased solvers in DPLL(T) decision procedures. The new solver is even competitive with stateoftheart tools specialized for the difference logic fragment. 1
Lazy Satisfiability Modulo Theories
 Journal on Satisfiability, Boolean Modeling and Computation
, 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 74 (32 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 of acquiring a comprehensive background knowledge in lazy SMT, is of simple solution. In this paper we present an extensive survey of SMT, with particular focus on the lazy approach. We survey, classify and analyze from a theoryindependent perspective the most effective techniques and optimizations which are of interest for lazy SMT and which have been proposed in various communities; we discuss their relative benefits and drawbacks; we provide some guidelines about their choice and usage; we also analyze the features for SAT solvers and Tsolvers which make them more suitable for an integration. The ultimate goals of this paper are to become a source of a common background knowledge and terminology for students and researchers in different areas, to provide a reference guide for developers of SMT tools, and to stimulate the crossfertilization of techniques and ideas among different communities.
Propositional Satisfiability and Constraint Programming: a Comparative Survey
 ACM Computing Surveys
, 2006
"... Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, crossfertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms ..."
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Cited by 32 (4 self)
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Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, crossfertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms that are most successful at solving both kinds of problems. They also exhibit differences in the way they are used to state and solve problems, since SAT’s approach is in general a blackbox approach, while CP aims at being tunable and programmable. This survey overviews the two areas in a comparative way, emphasising the similarities and differences between the two and the points where we feel that one technology can benefit from ideas or experience acquired
Fast and Flexible Difference Constraint Propagation for DPLL(T)
 IN PROC. SAT, VOLUME 4121 OF LNCS
, 2006
"... In the context of DPLL(T), theory propagation is the process of dynamically selecting consequences of a conjunction of constraints from a given set of candidate constraints. We present improvements to a fast theory propagation procedure for difference constraints of the form x − y ≤ c. These improve ..."
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Cited by 22 (1 self)
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In the context of DPLL(T), theory propagation is the process of dynamically selecting consequences of a conjunction of constraints from a given set of candidate constraints. We present improvements to a fast theory propagation procedure for difference constraints of the form x − y ≤ c. These improvements are demonstrated experimentally.
Delayed theory combination vs. NelsonOppen for satisfiability modulo theories: A comparative analysis
 IN PROC. LPAR’06, VOLUME 4246 OF LNAI
, 2006
"... Many approaches for Satisfiability Modulo Theory (SMT(T)) rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory T (Tsolver). When T is the combination T1 ∪ T2 of two simpler theories, the approach is typically handled by means of Nelson ..."
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Cited by 21 (7 self)
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Many approaches for Satisfiability Modulo Theory (SMT(T)) rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory T (Tsolver). When T is the combination T1 ∪ T2 of two simpler theories, the approach is typically handled by means of NelsonOppen’s (NO) theory combination schema in which two specific Tsolvers deduce and exchange (disjunctions of) interface equalities. In recent papers we have proposed a new approach to SMT(T1 ∪ T2), called Delayed Theory Combination (DTC). Here part or all the (possibly very expensive) task of deducing interface equalities is played by the SAT solver itself, at the potential cost of an enlargement of the boolean search space. In principle this enlargement could be up to exponential in the number of interface equalities generated. In this paper we show that this estimate was too pessimistic. We present a comparative analysis of DTC vs. NO for SMT(T1 ∪T2), which shows that, using stateoftheart SATsolving techniques, the amount of boolean branches performed by DTC can be upper bounded by the number of deductions and boolean branches performed by NO on the same problem. We prove the result for different deduction capabilities of the Tsolvers and for both convex and nonconvex theories.
veriT: an open, trustable and efficient SMTsolver
 Proc. Conference on Automated Deduction (CADE), volume 5663 of Lecture Notes in Computer Science
, 2009
"... Abstract. This article describes the first public version of the satisfiability modulo theory (SMT) solver veriT. It is opensource, proofproducing, and complete for quantifierfree formulas with uninterpreted functions and difference logic on real numbers and integers. 1 ..."
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Cited by 16 (5 self)
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Abstract. This article describes the first public version of the satisfiability modulo theory (SMT) solver veriT. It is opensource, proofproducing, and complete for quantifierfree formulas with uninterpreted functions and difference logic on real numbers and integers. 1
SMT techniques for fast predicate abstraction
 In Computer Aided Verification (CAV
, 2006
"... Abstract. Predicate abstraction is a technique for automatically extracting finitestate abstractions for systems with potentially infinite state space. The fundamental operation in predicate abstraction is to compute the best approximation of a Boolean formula ϕ over a set of predicates P. In this ..."
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Cited by 15 (1 self)
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Abstract. Predicate abstraction is a technique for automatically extracting finitestate abstractions for systems with potentially infinite state space. The fundamental operation in predicate abstraction is to compute the best approximation of a Boolean formula ϕ over a set of predicates P. In this work, we demonstrate the use for this operation of a decision procedure based on the DPLL(T) framework for SAT Modulo Theories (SMT). The new algorithm is based on a careful generation of the set of all satisfying assignments over a set of predicates. It consistently outperforms previous methods by a factor of at least 20, on a diverse set of hardware and software verification benchmarks. We report detailed analysis of the results and the impact of a number of variations of the techniques. We also propose and evaluate a scheme for incremental refinement of approximations for predicate abstraction in the above framework. 1
On SAT Modulo Theories and Optimization Problems
 In Theory and Applications of Satisfiability Testing (SAT), LNCS 4121
, 2006
"... Abstract. Solvers for SAT Modulo Theories (SMT) can nowadays handle large industrial (e.g., formal hardware and software verification) problems over theories such as the integers, arrays, or equality. Here we show that SMT approaches can also efficiently solve problems that, at first sight, do not h ..."
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Cited by 13 (4 self)
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Abstract. Solvers for SAT Modulo Theories (SMT) can nowadays handle large industrial (e.g., formal hardware and software verification) problems over theories such as the integers, arrays, or equality. Here we show that SMT approaches can also efficiently solve problems that, at first sight, do not have a typical SMT flavor. In particular, here we deal with SAT and SMT problems where models M are sought such that a given cost function f(M) is minimized. For this purpose, we introduce a variant of SMT where the theory T DPLL Modulo Theories framework. We discuss two different examples of applications of this SMT variant: weighted MaxSAT and weighted MaxSMT. We show how, with relatively little effort, one can obtain a competitive system that, in the case of weighted MaxSMT in the theory of Difference Logic, can even handle wellknown hard radio frequency assignment problems without any tailored heuristics. These results seem to indicate that MaxSAT/SMT techniques can already be used for realistic applications. 1
An efficient nelsonoppen decision procedure for difference constraints over rationals
, 2005
"... Abstract. Nelson and Oppen provided a methodology for modularly combining decision procedures for individual theories to construct a decision procedure for a combination of theories. In addition to providing a check for satisfiability, the individual decision procedures need to provide additional fu ..."
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Cited by 12 (3 self)
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Abstract. Nelson and Oppen provided a methodology for modularly combining decision procedures for individual theories to construct a decision procedure for a combination of theories. In addition to providing a check for satisfiability, the individual decision procedures need to provide additional functionalities, including equality generation. In this paper, we propose a decision procedure for a conjunction of difference constraints over rationals (where the atomic formulas are of the form x ≤ y + c or x < y + c). The procedure extends any negative cycle detection algorithm (like the BellmanFord algorithm) to generate (1) equalities between all pair of variables, (2) produce proofs and (3) generates models that can be extended by other theories in a NelsonOppen framework. All the operations mentioned above can be performed with only a linear overhead to the cycle detection algorithm. 1
Design and results of the 1st satisfiability modulo theories competition (SMTCOMP
 Journal of Automated Reasoning
, 2005
"... Abstract. The Satisfiability Modulo Theories Competition (SMTCOMP) is intended to spark further advances in the decision procedures field, especially for applications in hardware and software verification. Public competitions are a wellknown means of stimulating advancement in automated reasoning. ..."
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Cited by 12 (8 self)
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Abstract. The Satisfiability Modulo Theories Competition (SMTCOMP) is intended to spark further advances in the decision procedures field, especially for applications in hardware and software verification. Public competitions are a wellknown means of stimulating advancement in automated reasoning. Evaluation of SMT solvers entered in SMTCOMP took place while CAV 2005 was meeting. Twelve solvers were entered, 1352 benchmarks were collected in seven different divisions.