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124
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 194 (8 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
A decision procedure for bitvectors and arrays
 In Computer Aided Verification, number 4590 in LNCS
, 2007
"... Abstract. STP is a decision procedure for the satisfiability of quantifierfree formulas in the theory of bitvectors and arrays that has been optimized for large problems encountered in software analysis applications. The basic architecture of the procedure consists of wordlevel preprocessing alg ..."
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Cited by 133 (7 self)
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Abstract. STP is a decision procedure for the satisfiability of quantifierfree formulas in the theory of bitvectors and arrays that has been optimized for large problems encountered in software analysis applications. The basic architecture of the procedure consists of wordlevel preprocessing algorithms followed by translation to SAT. The primary bottlenecks in software verification and bug finding applications are large arrays and linear bitvector arithmetic. New algorithms based on the abstractionrefinement paradigm are presented for reasoning about large arrays. A solver for bitvector linear arithmetic is presented that eliminates variables and parts of variables to enable other transformations, and reduce the size of the problem that is eventually received by the SAT solver. These and other algorithms have been implemented in STP, which has been heavily tested over thousands of examples obtained from several realworld applications. Experimental results indicate that the above mix of algorithms along with the overall architecture is far more effective, for a variety of applications, than a direct translation of the original formula to SAT or other comparable decision procedures. 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 85 (34 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.
SCIP: solving constraint integer programs
, 2009
"... Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), wh ..."
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Cited by 57 (0 self)
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Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and noncommercial use and can be downloaded in source code. This paper gives an overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs. To illustrate the performance and flexibility of SCIP, we apply it to two different problem classes. First, we consider mixed integer programming and show by computational experiments that SCIP is almost competitive to specialized commercial MIP solvers, even though SCIP supports the more general constraint integer programming paradigm. We develop new ingredients that improve current MIP solving technology. As a second application, we employ SCIP to solve chip design verification problems as they arise in the logic design of integrated circuits. This application goes far beyond traditional MIP solving, as it includes several highly nonlinear constraints, which can be handled nicely within the constraint integer programming framework. We show anecdotally how the different solving techniques from MIP, CP, and SAT work together inside SCIP to deal with such constraint classes. Finally, experimental results show that our approach outperforms current stateoftheart techniques for proving the validity of properties on circuits containing arithmetic.
The yices smt solver
, 2006
"... Abstract. SMT stands for Satisfiability Modulo Theories. An SMT solver decides the satisfiability of propositionally complex formulas in theories such as arithmetic and uninterpreted functions with equality. SMT solving has numerous applications in automated theorem proving, in hardware and software ..."
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Cited by 49 (1 self)
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Abstract. SMT stands for Satisfiability Modulo Theories. An SMT solver decides the satisfiability of propositionally complex formulas in theories such as arithmetic and uninterpreted functions with equality. SMT solving has numerous applications in automated theorem proving, in hardware and software verification, and in scheduling and planning problems. This paper describes Yices, an efficient SMT solver developed at SRI International. Yices supports a rich combination of firstorder theories that occur frequently in software and hardware modeling: arithmetic, uninterpreted functions, bit vectors, arrays, recursive datatypes, and more. Beyond pure SMT solving, Yices can solve weighted MAXSMT problems, compute unsatisfiable cores, and construct models. Yices is the main decision procedure used by the SAL model checking environment, and it is being integrated to the PVS theorem prover. As a MAXSMT solver, Yices is the main component of the probabilistic consistency engine used in SRI’s CALO system. 1
DPLL(T) with Exhaustive Theory Propagation and its Application to Difference Logic
 In Proc. CAV’05, volume 3576 of LNCS
, 2005
"... Abstract. At CAV’04 we presented the DPLL(T) approach for satisfiability modulo theories T. It is based on a general DPLL(X) engine whose X can be instantiated with different theory solvers SolverT for conjunctions of literals. Here we go one important step further: we require SolverT to be able to ..."
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Cited by 43 (6 self)
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Abstract. At CAV’04 we presented the DPLL(T) approach for satisfiability modulo theories T. It is based on a general DPLL(X) engine whose X can be instantiated with different theory solvers SolverT for conjunctions of literals. Here we go one important step further: we require SolverT to be able to detect all input literals that are Tconsequences of the partial model that is being explored by DPLL(X). Although at first sight this may seem too expensive, we show that for difference logic the benefits compensate by far the costs. Here we describe and discuss this new version of DPLL(T), the DPLL(X) engine, and our SolverT for difference logic. The resulting very simple DPLL(T) system importantly outperforms the existing techniques for this logic. Moreover, it has very good scaling properties: especially on the larger problems it gives improvements of orders of magnitude w.r.t. the existing stateoftheart tools. 1
Efficient Ematching for SMT solvers
, 2007
"... Satisfiability Modulo Theories (SMT) solvers have proven highly scalable, efficient and suitable for integrating theory reasoning. However, for numerous applications from program analysis and verification, the ground fragment is insufficient, as proof obligations often include quantifiers. A well ..."
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Cited by 36 (7 self)
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Satisfiability Modulo Theories (SMT) solvers have proven highly scalable, efficient and suitable for integrating theory reasoning. However, for numerous applications from program analysis and verification, the ground fragment is insufficient, as proof obligations often include quantifiers. A well known approach for quantifier reasoning uses a matching algorithm that works against an Egraph to instantiate quantified variables. This paper introduces algorithms that identify matches on Egraphs incrementally and efficiently. In particular, we introduce an index that works on Egraphs, called Ematching code trees that combine features of substitution and code trees, used in saturation based theorem provers. Ematching code trees allow performing matching against several patterns simultaneously. The code trees are combined with an additional index, called the inverted path index, which filters Egraph terms that may potentially match patterns when the Egraph is updated. Experimental results show substantial performance improvements over existing stateoftheart SMT solvers.
Efficient satisfiability modulo theories via delayed theory combination
 In Proc. CAV 2005, volume 3576 of LNCS
, 2005
"... Abstract. The problem of deciding the satisfiability of a quantifierfree formula with respect to a background theory, also known as Satisfiability Modulo Theories (SMT), is gaining increasing relevance in verification: representation capabilities beyond propositional logic allow for a natural model ..."
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Cited by 33 (15 self)
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Abstract. The problem of deciding the satisfiability of a quantifierfree formula with respect to a background theory, also known as Satisfiability Modulo Theories (SMT), is gaining increasing relevance in verification: representation capabilities beyond propositional logic allow for a natural modeling of realworld problems (e.g., pipeline and RTL circuits verification, proof obligations in software systems). In this paper, we focus on the case where the background theory is the combination T1 £ T2 of two simpler theories. Many SMT procedures combine a boolean model enumeration with a decision procedure for T1 £ T2, where conjunctions of literals can be decided by an integration schema such as NelsonOppen, via a structured exchange of interface formulae (e.g., equalities in the case of convex theories, disjunctions of equalities otherwise). We propose a new approach for SMT¤T1 £ T2¥, called Delayed Theory Combination, which does not require a decision procedure for T1 £ T2, but only individual decision procedures for T1 and T2, which are directly integrated into the boolean model enumerator. This approach is much simpler and natural, allows each of the solvers to be implemented and optimized without taking into account the others, and it nicely encompasses the case of nonconvex theories. We show the effectiveness of the approach by a thorough experimental comparison. 1
Solving quantified verification conditions using satisfiability modulo theories
 In CADE
, 2007
"... Abstract. First order logic provides a convenient formalism for describing a wide variety of verification conditions. Two main approaches to checking such conditions are pure first order automated theorem proving (ATP) and automated theorem proving based on satisfiability modulo theories (SMT). Trad ..."
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Cited by 31 (1 self)
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Abstract. First order logic provides a convenient formalism for describing a wide variety of verification conditions. Two main approaches to checking such conditions are pure first order automated theorem proving (ATP) and automated theorem proving based on satisfiability modulo theories (SMT). Traditional ATP systems are designed to handle quantifiers easily, but often have difficulty reasoning with respect to theories. SMT systems, on the other hand, have builtin support for many useful theories, but have a much more difficult time with quantifiers. One clue on how to get the best of both worlds can be found in the legacy system Simplify which combines builtin theory reasoning with quantifier instantiation heuristics. Inspired by Simplify and motivated by a desire to provide a competitive alternative to ATP systems, this paper describes a methodology for reasoning about quantifiers in SMT systems. We present the methodology in the context of the Abstract DPLL Modulo Theories framework. Besides adapting many of Simplify’s techniques, we also introduce a number of new heuristics. Most important is the notion of instantiation level which provides an effective mechanism for prioritizing and managing the large search space inherent in quantifier instantiation techniques. These techniques have been implemented in the SMT system CVC3. Experimental results show that our methodology enables CVC3 to solve a significant number of benchmarks that were not solvable with any previous approach. 1
Proofproducing Congruence Closure
 16th International Conference on Rewriting Techniques and Applications
, 2005
"... www.lsi.upc.es/{~roberto,~oliveras} Abstract. Many applications of congruence closure nowadays require the ability of recovering, among the thousands of input equations, the small subset that caused the equivalence of a given pair of terms. For this purpose, here we introduce an incremental congruen ..."
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Cited by 31 (2 self)
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www.lsi.upc.es/{~roberto,~oliveras} Abstract. Many applications of congruence closure nowadays require the ability of recovering, among the thousands of input equations, the small subset that caused the equivalence of a given pair of terms. For this purpose, here we introduce an incremental congruence closure algorithm that has an additional Explain operation. First, two variations of unionfind data structures with Explain are introduced. Then, these are applied inside a congruence closure algorithm with Explain, whereakstep proof can be recovered in almost optimal time (quasilinear in k), without increasing the overall O(n log n)runtime of the fastest known congruence closure algorithms. This nontrivial (ground) equational reasoning result has been quite intensively sought after (see, e.g., [SD99, dMRS04, KS04]), and moreover has important applications to verification.