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24
Efficient Interpolant Generation in Satisfiability Modulo Theories ⋆
, 2007
"... Abstract. The problem of computing Craig Interpolants for propositional (SAT) formulas has recently received a lot of interest, mainly for its applications in formal verification. However, propositional logic is often not expressive enough for representing many interesting verification problems, whi ..."
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Cited by 26 (5 self)
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Abstract. The problem of computing Craig Interpolants for propositional (SAT) formulas has recently received a lot of interest, mainly for its applications in formal verification. However, propositional logic is often not expressive enough for representing many interesting verification problems, which can be more naturally addressed in the framework of Satisfiability Modulo Theories, SMT. Although some works have addressed the topic of generating interpolants in SMT, the techniques and tools that are currently available have some limitations, and their performace still does not exploit the full power of current stateoftheart SMT solvers. In this paper we try to close this gap. We present several techniques for interpolant generation in SMT which overcome the limitations of the current generators mentioned above, and which take full advantage of stateoftheart SMT technology. These novel techniques can lead to substantial performance improvements wrt. the currently available tools. We support our claims with an extensive experimental evaluation of our implementation of the proposed techniques in the MathSAT SMT solver. 1
Encodings of the SEQUENCE constraint
 In: CP. Volume 4741 of LNCS
, 2007
"... The SEQUENCE constraint is useful in modelling car sequencing, rostering, scheduling and related problems. We introduce half a dozen new encodings of the SEQUENCE constraint, some of which do not hinder propagation. We prove that down the whole branch of a search tree domain consistency can be enf ..."
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Cited by 13 (7 self)
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The SEQUENCE constraint is useful in modelling car sequencing, rostering, scheduling and related problems. We introduce half a dozen new encodings of the SEQUENCE constraint, some of which do not hinder propagation. We prove that down the whole branch of a search tree domain consistency can be enforced on the SEQUENCE constraint in just O(n 2 log n) time. This improves upon the previous bound of O(n 3) for each call down the tree. We also consider some generalizations including multiple SEQUENCE constraints, and cyclic SEQUENCE constraints. Our experiments suggest that, on very large and tight problems, domain consistency algorithms are best. However, on smaller or looser problems, much simpler encodings are better, even though these encodings hinder propagation. When there are multiple SEQUENCE constraints, a more expensive propagator shows promise.
FlowBased Propagators for the SEQUENCE and Related Global Constraints
, 2008
"... We propose new filtering algorithms for the SEQUENCE constraint and some extensions of the SEQUENCE constraint based on network flows. We enforce domain consistency on the SEQUENCE constraint in O(n 2) time down a branch of the search tree. This improves upon the best existing domain consistency al ..."
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Cited by 11 (3 self)
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We propose new filtering algorithms for the SEQUENCE constraint and some extensions of the SEQUENCE constraint based on network flows. We enforce domain consistency on the SEQUENCE constraint in O(n 2) time down a branch of the search tree. This improves upon the best existing domain consistency algorithm by a factor of O(log n). The flows used in these algorithms are derived from a linear program. Some of them differ from the flows used to propagate global constraints like GCC since the domains of the variables are encoded as costs on the edges rather than capacities. Such flows are efficient for maintaining bounds consistency over large domains and may be useful for other global constraints.
Stable models and difference logic
 Ann. Math. Artif. Intell
, 2008
"... on his 65th birthday The paper studies the relationship between logic programs with the stable model semantics and difference logic recently considered in the Satisfiability Modulo Theories framework. Characterizations of stable models in terms of level rankings are developed building on simple line ..."
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Cited by 6 (1 self)
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on his 65th birthday The paper studies the relationship between logic programs with the stable model semantics and difference logic recently considered in the Satisfiability Modulo Theories framework. Characterizations of stable models in terms of level rankings are developed building on simple linear integer constraints allowed in difference logic. Based on a characterization with level rankings a translation is devised which maps a normal program to a difference logic formula capturing stable models of the program as satisfying valuations of the resulting formula. The translation makes it possible to use a solver for difference logic to compute stable models of logic programs. 1
SDSAT: Tight Integration of Small Domain Encoding and Lazy Approaches in a Separation Logic Solver
 In Proc. TACAS’06, volume 3920 of LNCS
, 2006
"... Existing difference logic (DL) solvers can be broadly classified as eager or lazy, each with its own merits and demerits. We propose a novel difference logic solver SDSAT that combines the strengths of both these approaches and provides a robust performance over a wide set of benchmarks. The solver ..."
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Cited by 6 (2 self)
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Existing difference logic (DL) solvers can be broadly classified as eager or lazy, each with its own merits and demerits. We propose a novel difference logic solver SDSAT that combines the strengths of both these approaches and provides a robust performance over a wide set of benchmarks. The solver SDSAT works in two phases: allocation and solve. In the allocation phase, it allocates nonuniform adequate ranges for variables appearing in difference predicates. This phase is similar to previous small domain encoding approaches, but uses a novel algorithm NuSMOD with 12 orders of magnitude improvement in performance and smaller ranges for variables. Furthermore, the difference logic formula is not transformed into an equisatisfiable Boolean formula in a single step, but rather done lazily in the following phase. In the solve phase, SDSAT uses a lazy refinement approach to search for a satisfying model within the allocated ranges. Thus, any partially DLtheory consistent model can be discarded if it cannot be satisfied within the allocated ranges. Note the crucial difference: in eager approaches, such a partially consistent model is not allowed in the first place, while in lazy approaches such a model is never discarded. Moreover, we dynamically refine the allocated ranges and search for a feasible solution within the updated ranges. This combined approach benefits from both the smaller search space (as in eager approaches) and also from the theoryspecific graphbased algorithms (characteristic of lazy approaches). Experimental results show that our method is robust and always better than or comparable to stateofthe art solvers using similar eager or lazy techniques.
Efficient Generation of Craig Interpolants in Satisfiability Modulo Theories
"... The problem of computing Craig Interpolants has recently received a lot of interest. In this paper, we address the problem of efficient generation of interpolants for some important fragments of firstorder logic, which are amenable for effective decision procedures, called Satisfiability Modulo The ..."
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Cited by 5 (1 self)
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The problem of computing Craig Interpolants has recently received a lot of interest. In this paper, we address the problem of efficient generation of interpolants for some important fragments of firstorder logic, which are amenable for effective decision procedures, called Satisfiability Modulo Theory solvers. We make the following contributions. First, we provide interpolation procedures for several basic theories of interest: the theories of linear arithmetic over the rationals, difference logic over rationals and integers, and UTVPI over rationals and integers. Second, we define a novel approach to interpolate combinations of theories, that applies to the Delayed Theory Combination approach. Efficiency is ensured by the fact that the proposed interpolation algorithms extend stateoftheart algorithms for Satisfiability Modulo Theories. Our experimental evaluation shows that the MathSAT SMT solver can produce interpolants with minor overhead in search, and much more efficiently than other competitor solvers.
Global Difference Constraint Propagation for Finite Domain Solvers
 In PPDP ’08
, 2008
"... Difference constraints of the form x − y ≤ d are well studied, with efficient algorithms for satisfaction and implication, because of their connection to shortest paths. Finite domain propagation algorithms however do not make use of these algorithms, and typically treat each difference constraint a ..."
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Cited by 3 (0 self)
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Difference constraints of the form x − y ≤ d are well studied, with efficient algorithms for satisfaction and implication, because of their connection to shortest paths. Finite domain propagation algorithms however do not make use of these algorithms, and typically treat each difference constraint as a separate propagator. Propagation does guarantee completeness of solving but can be needlessly slow. In this paper we describe how to build a (bounds consistent) global propagator for difference constraints that treats them all simultaneously. SAT modulo theory solvers have included theory solvers for difference constraints for some time. While a theory solver for difference constraints gives the basis of a global difference constraint propagator, we show how the requirements on the propagator are quite different. We give experiments showing that treating difference constraints globally can substantially improve on the standard propagation approach.
Incremental Satisfiability and Implication for UTVPI Constraints
"... Unit twovariableperinequality (UTVPI) constraints form one of the largest class of integer constraints which are polynomial time solvable (unless P=NP). There is considerable interest in their use for constraint solving, abstract interpretation, spatial databases, and theorem proving. In this pap ..."
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Cited by 3 (0 self)
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Unit twovariableperinequality (UTVPI) constraints form one of the largest class of integer constraints which are polynomial time solvable (unless P=NP). There is considerable interest in their use for constraint solving, abstract interpretation, spatial databases, and theorem proving. In this paper we develop new incremental algorithms for UTVPI constraint satisfaction and implication checking that require O(m + n log n + p) time and O(n + m + p) space to incrementally check satisfiability of m UTVPI constraints on n variables and check implication of p UTVPI constraints. The algorithms can be straightforwardly extended to create nonincremental implication checking and generation of all (nonredundant) implied constraints, as well as generate minimal unsatisfiable subsets and minimal implicants. Key words: unit two variable per inequality constraints; satisfaction; implication 1.
Encodings of the sequence constraint
 In: CP. Volume 4741 of LNCS
, 2007
"... Abstract. The SEQUENCE constraint is useful in modelling car sequencing, rostering, scheduling and related problems. We introduce half a dozen new encodings of the SEQUENCE constraint, some of which do not hinder propagation. We prove that, down a branch of a search tree, domain consistency can be e ..."
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Cited by 2 (0 self)
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Abstract. The SEQUENCE constraint is useful in modelling car sequencing, rostering, scheduling and related problems. We introduce half a dozen new encodings of the SEQUENCE constraint, some of which do not hinder propagation. We prove that, down a branch of a search tree, domain consistency can be enforced on the SEQUENCE constraint in just O(n 2 log n) time. This improves upon the previous bound of O(n 3) for each call down the tree. We also consider a generalization of the SEQUENCE constraint – the Multiple SEQUENCE constraint. Our experiments suggest that, on very large and tight problems, domain consistency algorithms are best. However, on smaller or looser problems, much simpler encodings are better, even though these encodings hinder propagation. When there are multiple SEQUENCE constraints, a more expensive propagator shows promise. 1
Open Contractible Global Constraints
"... Open forms of global constraints allow the addition of new variables to an argument during the execution of a constraint program. Such forms are needed for difficult constraint programming problems where problem construction and problem solving are interleaved. However, in general, filtering that is ..."
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Cited by 1 (0 self)
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Open forms of global constraints allow the addition of new variables to an argument during the execution of a constraint program. Such forms are needed for difficult constraint programming problems where problem construction and problem solving are interleaved. However, in general, filtering that is sound for a global constraint can be unsound when the constraint is open. This paper provides a simple characterization, called contractibility, of the constraints where filtering remains sound when the constraint is open. With this characterization we can easily determine whether a constraint is contractible or not. In the latter case, we can use it to derive the strongest contractible approximation to the constraint. We demonstrate how specific algorithms for some closed contractible constraints are easily adapted to open constraints. 1