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Decompositions of All Different, Global Cardinality and Related Constraints
"... We show that some common and important global constraints like ALLDIFFERENT and GCC can be decomposed into simple arithmetic constraints on which we achieve bound or range consistency, and in some cases even greater pruning. These decompositions can be easily added to new solvers. They also provide ..."
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We show that some common and important global constraints like ALLDIFFERENT and GCC can be decomposed into simple arithmetic constraints on which we achieve bound or range consistency, and in some cases even greater pruning. These decompositions can be easily added to new solvers. They also provide other constraints with access to the state of the propagator by sharing of variables. Such sharing can be used to improve propagation between constraints. We report experiments with our decomposition in a pseudoBoolean solver. 1
Boolean Equipropagation for Concise and Efficient SAT Encodings of Combinatorial Problems
"... We present an approach to propagationbased SAT encoding of combinatorial problems, Boolean equipropagation, where constraints are modeled as Boolean functions which propagate information about equalities between Boolean literals. This information is then applied to simplify the CNF encoding of the ..."
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We present an approach to propagationbased SAT encoding of combinatorial problems, Boolean equipropagation, where constraints are modeled as Boolean functions which propagate information about equalities between Boolean literals. This information is then applied to simplify the CNF encoding of the constraints. A key factor is that considering only a small fragment of a constraint model at one time enables us to apply stronger, and even complete, reasoning to detect equivalent literals in that fragment. Once detected, equivalences apply to simplify the entire constraint model and facilitate further reasoning on other fragments. Equipropagation in combination with partial evaluation and constraint simplification provide the foundation for a powerful approach to SATbased finite domain constraint solving. We introduce a tool called BEE (BenGurion Equipropagation Encoder) based on these ideas and demonstrate for a variety of benchmarks that our approach leads to a considerable reduction in the size of CNF encodings and subsequent speedups in SAT solving times. 1.
A New Look at BDDs for PseudoBoolean Constraints
"... PseudoBoolean constraints are omnipresent in practical applications, and thus a significant effort has been devoted to the development of good SAT encoding techniques for them. Some of these encodings first construct a Binary Decision Diagram (BDD) for the constraint, and then encode the BDD into a ..."
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Cited by 6 (0 self)
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PseudoBoolean constraints are omnipresent in practical applications, and thus a significant effort has been devoted to the development of good SAT encoding techniques for them. Some of these encodings first construct a Binary Decision Diagram (BDD) for the constraint, and then encode the BDD into a propositional formula. These BDDbased approaches have some important advantages, such as not being dependent on the size of the coefficients, or being able to share the same BDD for representing many constraints. We first focus on the size of the resulting BDDs, which was considered to be an open problem in our research community. We report on previous work where it was proved that there are PseudoBoolean constraints for which no polynomial BDD exists. We also give an alternative and simpler proof assuming that NP is different from CoNP. More interestingly, here we also show how to overcome the possible exponential blowup of BDDs by coefficient decomposition. This allows us to give the first polynomial generalized arcconsistent ROBDDbased encoding for PseudoBoolean constraints. Finally, we focus on practical issues: we show how to efficiently construct such ROBDDs, how to encode them into SAT with only 2 clauses per node, and present experimental results that confirm that our approach is competitive with other encodings and stateoftheart PseudoBoolean solvers. 1.
Answer Set Solving with Lazy Nogood Generation
 UNDER CONSIDERATION FOR PUBLICATION IN THEORY AND PRACTICE OF LOGIC PROGRAMMING
, 2012
"... Although Answer Set Programming (ASP) systems are highly optimised, their performance is sensitive to the size of problem encodings. We address this deficiency by introducing a new extension to ASP solving. The idea is to integrate external propagators to represent parts of the encoding implicitly, ..."
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Cited by 5 (0 self)
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Although Answer Set Programming (ASP) systems are highly optimised, their performance is sensitive to the size of problem encodings. We address this deficiency by introducing a new extension to ASP solving. The idea is to integrate external propagators to represent parts of the encoding implicitly, rather than generating it apriori. To match the stateoftheart in conflictdriven solving, however, external propagators can generate an encoding of their inference on demand. We demonstrate the applicability of our approach in a novel Constraint Answer Set Programming system that can seamlessly integrate Constraint Programming techniques without sacrificing the advantages of conflictdriven techniques. Experiments provide evidence for computational impact.
Generalising and unifying SLUR and unitrefutation completeness
 SOFSEM 2013: Theory and Practice of Computer Science, volume 7741 of Lecture Notes in Computer Science (LNCS
, 2013
"... Abstract. The class SLUR (Single Lookahead Unit Resolution) was introduced in [22] as an umbrella class for efficient SAT solving. [7,2] extended this class in various ways to hierarchies covering all of CNF (all clausesets). We introduce a hierarchy SLURk which we argue is the natural “limit ” of ..."
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Abstract. The class SLUR (Single Lookahead Unit Resolution) was introduced in [22] as an umbrella class for efficient SAT solving. [7,2] extended this class in various ways to hierarchies covering all of CNF (all clausesets). We introduce a hierarchy SLURk which we argue is the natural “limit ” of such approaches. The second source for our investigations is the class UC of unitrefutation complete clausesets introduced in [10]. Via the theory of (treeresolution based) “hardness ” of clausesets as developed in [19,20,1] we obtain a natural generalisation UCk, containing those clausesets which are “unitrefutation complete of level k”, which is the same as having hardness at most k. Utilising the strong connections to (tree)resolution complexity and (nested) input resolution, we develop fundamental methods for the determination of hardness (the level k in UCk). A fundamental insight now is that SLURk = UCk holds for all k. We can thus exploit both streams of intuitions and methods for the investigations of these hierarchies. As an application we can easily show that the hierarchies from [7,2] are strongly subsumed by SLURk. We conclude with a discussion of open problems and future directions. 1
Decomposition of the NVALUE constraint
"... Abstract. We study decompositions of the global NVALUE constraint. Our main contribution is theoretical: we show that there are propagators for global constraints like NVALUE which decomposition can simulate with the same time complexity but with a much greater space complexity. This suggests that t ..."
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Abstract. We study decompositions of the global NVALUE constraint. Our main contribution is theoretical: we show that there are propagators for global constraints like NVALUE which decomposition can simulate with the same time complexity but with a much greater space complexity. This suggests that the benefit of a global propagator may often not be in saving time but in saving space. Our other theoretical contribution is to show for the first time that range consistency can be enforced on NVALUE with the same worstcase time complexity as bound consistency. Finally, the decompositions we study are readily encoded as linear inequalities. We are therefore able to use them in integer linear programs. 1
On unitrefutation complete formulae with existentially quantified variables
 In Proc. of KR’12
, 2012
"... We analyze, along the lines of the knowledge compilation map, both the tractability and the succinctness of the propositional language URCC of unitrefutation complete propositional formulae, as well as its disjunctive closure URCC[∨, ∃], and a superset of URCC where variables can be existentia ..."
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We analyze, along the lines of the knowledge compilation map, both the tractability and the succinctness of the propositional language URCC of unitrefutation complete propositional formulae, as well as its disjunctive closure URCC[∨, ∃], and a superset of URCC where variables can be existentially quantified and unitrefutation completeness concerns only consequences built up from free variables.
Perfect Hashing and CNF Encodings of Cardinality Constraints
 In Cimatti and Sebastiani [55
"... Abstract. We study the problem of encoding cardinality constraints (threshold functions) on Boolean variables into CNF. Specifically, we propose new encodings based on (perfect) hashing that are efficient in terms of the number of clauses, auxiliary variables, and propagation strength. We compare t ..."
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Abstract. We study the problem of encoding cardinality constraints (threshold functions) on Boolean variables into CNF. Specifically, we propose new encodings based on (perfect) hashing that are efficient in terms of the number of clauses, auxiliary variables, and propagation strength. We compare the properties of our encodings to known ones, and provide experimental results evaluating their practical effectiveness.
Exploiting Constraints
"... Abstract. Constraints can be exploited in paradigms outside of constraint programming. In particular, powerful global constraints can often be decomposed into small primitives and these decompositions can simulate complex propagation algorithms that perform sophisticated inference about a problem. W ..."
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Abstract. Constraints can be exploited in paradigms outside of constraint programming. In particular, powerful global constraints can often be decomposed into small primitives and these decompositions can simulate complex propagation algorithms that perform sophisticated inference about a problem. We illustrate this approach with examples of exploiting constraints in propositional satisfiability (SAT), pseudoBoolean (PB) solving, integer linear programming (ILP) and answer set programming (ASP). 1
Connections and Integration with SAT Solvers: A Survey and a Case Study in Computational Biology
"... Boolean constraints play a fundamental rôle in optimization and constraint satisfaction. The resolution of these constraints has been the subject of intense and successful work during the past decade, and SAT solvers have reached a spectacular maturity. This chapter gives a brief overview of the rel ..."
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Boolean constraints play a fundamental rôle in optimization and constraint satisfaction. The resolution of these constraints has been the subject of intense and successful work during the past decade, and SAT solvers have reached a spectacular maturity. This chapter gives a brief overview of the relevant literature on modern SAT solvers and on the recent efforts to better integrate Boolean reasoning with other constraint satisfaction techniques. As a case study that illustrates the use of SAT and CP we consider an application in computational biology: the task to build gene regulatory networks (GRNs). We report on experiments made on this problem with a combined SAT/CP approach.