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Symbolic Decision Procedures for QBF
 Proceedings of 10th Int. Conf. on Principles and Practice of Constraint Programming (CP 2004
, 2004
"... Much recent work has gone into adapting techniques that were originally developed for SAT solving to QBF solving. In particular, QBF solvers are often based on SAT solvers. Most competitive QBF solvers are searchbased. In this work we explore an alternative approach to QBF solving, based on symb ..."
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Much recent work has gone into adapting techniques that were originally developed for SAT solving to QBF solving. In particular, QBF solvers are often based on SAT solvers. Most competitive QBF solvers are searchbased. In this work we explore an alternative approach to QBF solving, based on symbolic quantifier elimination. We extend some recent symbolic approaches for SAT solving to symbolic QBF solving, using various decisiondiagram formalisms such as OBDDs and ZDDs. In both approaches, QBF formulas are solved by eliminating all their quantifiers. Our first solver, QMRES, maintains a set of clauses represented by a ZDD and eliminates quantifiers via multiresolution. Our second solver, QBDD, maintains a set of OBDDs, and eliminate quantifier by applying them to the underlying OBDDs. We compare our symbolic solvers to several competitive searchbased solvers. We show that QBDD is not competitive, but QMRES compares favorably with searchbased solvers on various benchmarks consisting of nonrandom formulas.
Constraint Propagation as a Proof System
 10th Int.Conf. on Principles and Practice of Constraint Programing, LN in Computer Science vol.3258
, 2004
"... Refutation proofs can be viewed as a special case of constraint propagation, which is a fundamental technique in solving constraintsatisfaction problems. ..."
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Cited by 27 (1 self)
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Refutation proofs can be viewed as a special case of constraint propagation, which is a fundamental technique in solving constraintsatisfaction problems.
On Preservation under Homomorphisms and Unions of Conjunctive Queries
, 2006
"... Unions of conjunctive queries, also known as selectprojectjoinunion queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive firstorder formulas and are preserved under homomorphisms. A classical result of mathematical log ..."
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Cited by 25 (4 self)
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Unions of conjunctive queries, also known as selectprojectjoinunion queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive firstorder formulas and are preserved under homomorphisms. A classical result of mathematical logic asserts that the existential positive formulas are the only firstorder formulas (up to logical equivalence) that are preserved under homomorphisms on all structures, finite and infinite. After resisting resolution for a long time, it was eventually shown that, unlike other classical preservation theorems, the homomorphismpreservation theorem holds for the class of all finite structures. In this paper, we show that the homomorphismpreservation theorem holds also for several restricted classes of finite structures of interest in graph theory and database theory. Specifically, we show that this result holds for all classes of finite structures of bounded degree, all classes of finite structures of bounded treewidth, and, more generally, all classes of finite structures whose cores exclude at least one minor.
Search vs. symbolic techniques in satisfiability solving
 in Proceedings 7th International Conference on Theory and Applications of Satisfiability Testing
, 2004
"... Abstract. Recent work has shown how to use OBDDs for satisfiability solving. The idea of this approach, which we call symbolic quantifier elimination, is to view an instance of propositional satisfiability as an existentially quantified propositional formula. Satisfiability solving then amounts to q ..."
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Abstract. Recent work has shown how to use OBDDs for satisfiability solving. The idea of this approach, which we call symbolic quantifier elimination, is to view an instance of propositional satisfiability as an existentially quantified propositional formula. Satisfiability solving then amounts to quantifier elimination; once all quantifiers have been eliminated we are left with either 1 or 0. Our goal in this work is to study the effectiveness of symbolic quantifier elimination as an approach to satisfiability solving. To that end, we conduct a direct comparison with the DPLLbased ZChaff, as well as evaluate a variety of optimization techniques for the symbolic approach. In comparing the symbolic approach to ZChaff, we evaluate scalability across a variety of classes of formulas. We find that no approach dominates across all classes. While ZChaff dominates for many classes of formulas, the symbolic approach is superior for other classes of formulas. Once we have demonstrated the viability of the symbolic approach, we focus on optimization techniques for this approach. We study techniques from constraint satisfaction for finding a good plan for performing the symbolic operations of conjunction and of existential quantification. We also study various variableordering heuristics, finding that while no heuristic seems to dominate across all classes of formulas, the maximumcardinality search heuristic seems to offer the best overall performance. 1
On the power of kconsistency
 In Proceedings of ICALP2007
, 2007
"... Abstract. The kconsistency algorithm for constraintsatisfaction problems proceeds, roughly, by finding all partial solutions on at most k variables and iteratively deleting those that cannot be extended to a partial solution by one more variable. It is known that if the core of the structure encod ..."
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Abstract. The kconsistency algorithm for constraintsatisfaction problems proceeds, roughly, by finding all partial solutions on at most k variables and iteratively deleting those that cannot be extended to a partial solution by one more variable. It is known that if the core of the structure encoding the scopes of the constraints has treewidth at most k, then the kconsistency algorithm is always correct. We prove the exact converse to this: if the core of the structure encoding the scopes of the constraints does not have treewidth at most k, then the kconsistency algorithm is not always correct. This characterizes the exact power of the kconsistency algorithm in structural terms. 1
The structure of tractable constraint satisfaction problems
 In MFCS 2006
, 2006
"... Abstract We give a survey of recent results on the complexity of constraint satisfaction problems. Our main emphasis is on tractable structural restrictions. 1 ..."
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Abstract We give a survey of recent results on the complexity of constraint satisfaction problems. Our main emphasis is on tractable structural restrictions. 1
Affine systems of equations and counting infinitary logic
 In ICALP’07, volume 4596 of LNCS
, 2007
"... Abstract We consider the definability of constraint satisfaction problems (CSP) in various fixedpoint andinfinitary logics. We show that testing the solvability of systems of equations over a finite Abelian group, ..."
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Cited by 17 (7 self)
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Abstract We consider the definability of constraint satisfaction problems (CSP) in various fixedpoint andinfinitary logics. We show that testing the solvability of systems of equations over a finite Abelian group,
Constraint satisfaction with succinctly specified relations
, 2006
"... The general intractability of the constraint satisfaction problem (CSP) has motivated the study of the complexity of restricted cases of this problem. Thus far, the literature has primarily considered the formulation of the CSP where constraint relations are given explicitly. We initiate the system ..."
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Cited by 13 (6 self)
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The general intractability of the constraint satisfaction problem (CSP) has motivated the study of the complexity of restricted cases of this problem. Thus far, the literature has primarily considered the formulation of the CSP where constraint relations are given explicitly. We initiate the systematic study of CSP complexity with succinctly specified constraint relations.
The complexity of conservative valued CSPs
 in: Proceedings of the 23rd ACMSIAM Symposium on Discrete Algorithms (SODA'12), 2012
"... We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a constraint language, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimi ..."
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We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a constraint language, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimise the sum. Under the unique games conjecture, the approximability of finitevalued VCSPs is wellunderstood, see Raghavendra [FOCS’08]. However, there is no characterisation of finitevalued VCSPs, let alone generalvalued VCSPs, that can be solved exactly in polynomial time, thus giving insights from a combinatorial optimisation perspective. We consider the case of languages containing all possible unary cost functions. In the case of languages consisting of only f0;1gvalued cost functions (i.e. relations), such languages have been called conservative and studied by Bulatov [LICS’03] and
Collapsibility and Consistency in Quantified Constraint Satisfaction
 In Proceedings of AAAI04
, 2004
"... The concept of consistency has pervaded studies of the constraint satisfaction problem. We introduce two concepts, which are inspired by consistency, for the more general framework of the quantified constraint satisfaction problem (QCSP). We use these concepts to derive, in a uniform fashion, proo ..."
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Cited by 12 (3 self)
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The concept of consistency has pervaded studies of the constraint satisfaction problem. We introduce two concepts, which are inspired by consistency, for the more general framework of the quantified constraint satisfaction problem (QCSP). We use these concepts to derive, in a uniform fashion, proofs of polynomialtime tractability and corresponding algorithms for certain cases of the QCSP where the types of allowed relations are restricted. We not only unify existing tractability results and algorithms, but also identify new classes of tractable QCSPs.