95 research outputs found
Compilation for QCSP
We propose in this article a framework for compilation of quantified
constraint satisfaction problems (QCSP). We establish the semantics of this
formalism by an interpretation to a QCSP. We specify an algorithm to compile a
QCSP embedded into a search algorithm and based on the inductive semantics of
QCSP. We introduce an optimality property and demonstrate the optimality of the
interpretation of the compiled QCSP.Comment: Proceedings of the 13th International Colloquium on Implementation of
Constraint LOgic Programming Systems (CICLOPS 2013), Istanbul, Turkey, August
25, 201
Generalizing Consistency and other Constraint Properties to Quantified Constraints
Quantified constraints and Quantified Boolean Formulae are typically much
more difficult to reason with than classical constraints, because quantifier
alternation makes the usual notion of solution inappropriate. As a consequence,
basic properties of Constraint Satisfaction Problems (CSP), such as consistency
or substitutability, are not completely understood in the quantified case.
These properties are important because they are the basis of most of the
reasoning methods used to solve classical (existentially quantified)
constraints, and one would like to benefit from similar reasoning methods in
the resolution of quantified constraints. In this paper, we show that most of
the properties that are used by solvers for CSP can be generalized to
quantified CSP. This requires a re-thinking of a number of basic concepts; in
particular, we propose a notion of outcome that generalizes the classical
notion of solution and on which all definitions are based. We propose a
systematic study of the relations which hold between these properties, as well
as complexity results regarding the decision of these properties. Finally, and
since these problems are typically intractable, we generalize the approach used
in CSP and propose weaker, easier to check notions based on locality, which
allow to detect these properties incompletely but in polynomial time
Positional Games and QBF: The Corrective Encoding
Positional games are a mathematical class of two-player games comprising
Tic-tac-toe and its generalizations. We propose a novel encoding of these games
into Quantified Boolean Formulas (QBF) such that a game instance admits a
winning strategy for first player if and only if the corresponding formula is
true. Our approach improves over previous QBF encodings of games in multiple
ways. First, it is generic and lets us encode other positional games, such as
Hex. Second, structural properties of positional games together with a careful
treatment of illegal moves let us generate more compact instances that can be
solved faster by state-of-the-art QBF solvers. We establish the latter fact
through extensive experiments. Finally, the compactness of our new encoding
makes it feasible to translate realistic game problems. We identify a few such
problems of historical significance and put them forward to the QBF community
as milestones of increasing difficulty.Comment: Accepted for publication in the 23rd International Conference on
Theory and Applications of Satisfiability Testing (SAT2020
QBF with Soft Variables
QBF formulae are usually considered in prenex form, i.e. the quantifierblock is completely separated from the propositional part of the QBF.Among others, the semantics of the QBF is defined by the sequence ofthe variables within the prefix, where existentially quantifiedvariables depend on all universally quantified variables stated to theleft.In this paper we extend that classical definition and consider a newquantification type which we call soft variable. The idea is toallow a flexible position and quantifier type for these variables.Hence the type of quantifier of the soft variable can also bealtered. Based on this concept, we present an optimization problemseeking an optimal prefix as defined by user-given preferences. We statean algorithm based on MaxQBF, and present several applications – mainlyfrom verification area – which can be naturally translated into theoptimization problem for QBF with soft variables. We further implementeda prototype solver for this formalism, and compare our approach toprevious work, that differently from ours does not guarantee optimalityand completeness
The number of clones determined by disjunctions of unary relations
We consider finitary relations (also known as crosses) that are definable via
finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite
parameter set . We prove that whenever contains at least one
non-empty relation distinct from the full carrier set, there is a countably
infinite number of polymorphism clones determined by relations that are
disjunctively definable from . Finally, we extend our result to
finitely related polymorphism clones and countably infinite sets .Comment: manuscript to be published in Theory of Computing System
Set Constraint Model and Automated Encoding into SAT: Application to the Social Golfer Problem
On the one hand, Constraint Satisfaction Problems allow one to declaratively
model problems. On the other hand, propositional satisfiability problem (SAT)
solvers can handle huge SAT instances. We thus present a technique to
declaratively model set constraint problems and to encode them automatically
into SAT instances. We apply our technique to the Social Golfer Problem and we
also use it to break symmetries of the problem. Our technique is simpler, more
declarative, and less error-prone than direct and improved hand modeling. The
SAT instances that we automatically generate contain less clauses than improved
hand-written instances such as in [20], and with unit propagation they also
contain less variables. Moreover, they are well-suited for SAT solvers and they
are solved faster as shown when solving difficult instances of the Social
Golfer Problem.Comment: Submitted to Annals of Operations researc
Generalized Craig Interpolation for Stochastic Boolean Satisfiability Problems with Applications to Probabilistic State Reachability and Region Stability
The stochastic Boolean satisfiability (SSAT) problem has been introduced by
Papadimitriou in 1985 when adding a probabilistic model of uncertainty to
propositional satisfiability through randomized quantification. SSAT has many
applications, among them probabilistic bounded model checking (PBMC) of
symbolically represented Markov decision processes. This article identifies a
notion of Craig interpolant for the SSAT framework and develops an algorithm
for computing such interpolants based on a resolution calculus for SSAT. As a
potential application area of this novel concept of Craig interpolation, we
address the symbolic analysis of probabilistic systems. We first investigate
the use of interpolation in probabilistic state reachability analysis, turning
the falsification procedure employing PBMC into a verification technique for
probabilistic safety properties. We furthermore propose an interpolation-based
approach to probabilistic region stability, being able to verify that the
probability of stabilizing within some region is sufficiently large
Counting of Teams in First-Order Team Logics
We study descriptive complexity of counting complexity classes in the range from P to NP. A corollary of Fagin's characterization of NP by existential second-order logic is that P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. In this paper we extend this study to classes beyond P and extensions of first-order logic with team semantics. These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of FO in Tarski's semantics. Our results show that the class NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of NP and P , respectively. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean -formulae is NP-complete as well as complete for the function class generated by dependence logic.Peer reviewe
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