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Playing with Boolean blocks, part I: Post’s lattice with applications to complexity theory
 SIGACT News
"... Let us imagine children playing with a box containing a large number of building blocks such as LEGO TM, fischertechnik ® , Polydron, or something similar. Each block belongs to a certain class (given by, e. g., color, shape, or size) and usually the number of different such classes is relatively sm ..."
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Cited by 44 (14 self)
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Let us imagine children playing with a box containing a large number of building blocks such as LEGO TM, fischertechnik ® , Polydron, or something similar. Each block belongs to a certain class (given by, e. g., color, shape, or size) and usually the number of different such classes is relatively small. It is amazing to see how involved the constructions are that can be built by the kids. From
The Complexity of Satisfiability Problems: Refining Schaefer’s Theorem
 J. COMPUT. SYS. SCI
"... ... problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomialtime isomorphism (and these isomorphism types are distinct if and ..."
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Cited by 17 (7 self)
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... problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomialtime isomorphism (and these isomorphism types are distinct if and only if P ̸ = NP). We show that if one considers AC 0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.
THE COMPLEXITY OF GENERALIZED SATISFIABILITY FOR LINEAR TEMPORAL LOGIC
 VOL. 5 (1:1) 2009, PP. 1–1–21
, 2009
"... ..."
On Generalized Constraints and Certificates
 DISCRETE MATHEMATICS
"... Ekin, Foldes, Hammer, and Hellerstein showed that a set of Boolean functions is characterizable by a (possibly infinite) set of Boolean equations iff it is closed under permutation of variables, addition of dummy variables, and identification of variables. Subsequently, Pippenger introduced cons ..."
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Cited by 9 (0 self)
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Ekin, Foldes, Hammer, and Hellerstein showed that a set of Boolean functions is characterizable by a (possibly infinite) set of Boolean equations iff it is closed under permutation of variables, addition of dummy variables, and identification of variables. Subsequently, Pippenger introduced constraint characterizations, which generalize previous characterizations of clones. He showed that a set of finite functions can be characterized by a set of constraints iff it is characterizable by Boolean equations. He also developed a Galois theory for sets of finite functions characterizable by constraints. In this paper we generalize Pippenger's results by considering sets of finite functions, such as the unate functions, that are closed under permutation of variables and addition of dummy variables but not necessarily under identification of variables. We introduce the notion of a generalized constraint, and consider the question of which sets of functions can be characterized by...
HColoring Dichotomy Revisited
"... The HColoring problem can be expressed as a particular case of the Constraint Satisfaction Problem (CSP) whose computational complexity has been intensively studied under various approaches inthe last several years. We show that the dichotomy theorem proved by P.Hell and J.Nesetril [12] for the com ..."
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Cited by 8 (1 self)
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The HColoring problem can be expressed as a particular case of the Constraint Satisfaction Problem (CSP) whose computational complexity has been intensively studied under various approaches inthe last several years. We show that the dichotomy theorem proved by P.Hell and J.Nesetril [12] for the complexity of the HColoring problem for undirected graphs can be obtained using general methods for studying CSP, and that the criterion distinguishing the tractablecases of the HColoring problem agrees with that conjectured in [5]for the complexity of the general CSP.
The tractability of modelchecking for LTL: The good, the bad, and the ugly fragments
, 2007
"... In a seminal paper from 1985, Sistla and Clarke showed that the modelchecking problem for Linear Temporal Logic (LTL) is either NPcomplete or PSPACEcomplete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decr ..."
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Cited by 7 (4 self)
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In a seminal paper from 1985, Sistla and Clarke showed that the modelchecking problem for Linear Temporal Logic (LTL) is either NPcomplete or PSPACEcomplete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper systematically studies the modelchecking problem for LTL formulae over restricted sets of propositional and temporal operators. For almost all combinations of temporal and propositional operators, we determine whether the modelchecking problem is tractable (in P) or intractable (NPhard). We then focus on the tractable cases, showing that they all are NLcomplete or even logspace solvable. This leads to a surprising gap in complexity between tractable and intractable cases. It is worth noting that our analysis covers an infinite set of problems, since there are infinitely many sets of propositional operators.
Quantified Constraints: The Complexity of Decision and Counting for Bounded Alternation
"... We consider constraint satisfaction problems parameterized by the set of allowed constraint predicates. We examine the complexity of quantified constraint satisfaction problems with a bounded number of quantifier alternations and the complexity of the associated counting problems. We obtain classifi ..."
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Cited by 5 (3 self)
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We consider constraint satisfaction problems parameterized by the set of allowed constraint predicates. We examine the complexity of quantified constraint satisfaction problems with a bounded number of quantifier alternations and the complexity of the associated counting problems. We obtain classification results that completely solve the Boolean case, and we show that hardness results carry over to the case of arbitrary finite domains.
Complexity of some problems concerning varieties and quasivarieties of algebras
 SIAM J. Comput
"... Abstract. In this paper we consider the complexity of several problems involving finite algebraic structures. Given finite algebras A and B, these problems ask the following. (1) Do A and B satisfy precisely the same identities? (2) Do they satisfy the same quasiidentities? (3) Do A and B have the ..."
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Cited by 5 (2 self)
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Abstract. In this paper we consider the complexity of several problems involving finite algebraic structures. Given finite algebras A and B, these problems ask the following. (1) Do A and B satisfy precisely the same identities? (2) Do they satisfy the same quasiidentities? (3) Do A and B have the same set of term operations? In addition to the general case in which we allow arbitrary (finite) algebras, we consider each of these problems under the restrictions that all operations are unary and that A and B have cardinality two. We briefly discuss the relationship of these problems to algebraic specification theory.
Preferred representations of Boolean relations
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 119 (2005)
, 2005
"... We introduce the notion of a plain basis for a coclone in Post’s lattice. Such a basis is a set of relations B such that every constraint C over a relation in the coclone is logically equivalent to a conjunction of equalities and constraints over B and the same variables as C; this differs from th ..."
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Cited by 5 (2 self)
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We introduce the notion of a plain basis for a coclone in Post’s lattice. Such a basis is a set of relations B such that every constraint C over a relation in the coclone is logically equivalent to a conjunction of equalities and constraints over B and the same variables as C; this differs from the usual notion of a basis in that existential quantification of auxiliary variables is not allowed. We give such a basis for every coclone and in particular for those in the infinite part of the lattice; it turns out that most of these bases correspond to sets of propositional clauses, thus providing a strong link between classes of formulas defined for CSP and CNF representations. We then show that a socalled preferred representation of a relation over one of its bases can be computed efficiently, as well as the minimal coclone including a given relation, which solves some open structure identification problem as well as the open expressibility problem from database theory.