On truth-table reducibility to SAT

Buss, Samuel R.; Hay, Louise

doi:10.1016/0890-5401(91)90075-d

Cited by 92 publications

(56 citation statements)

References 10 publications

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“…The latter class was first studied by Papadimitriou & Zachos (1983) and consists of the decision problems that can be solved in polynomial time by O(log n) queries to some NP language. It is known (Buss & Hay 1991;Hemachandra 1989) that equivalently, P NP [log] can also be characterized as the set of languages in P NP whose queries are nonadaptive. Several natural complete problems for P NP[log] are known; see for instance Krentel (1986) and Hemaspaandra et al (1997).…”

Section: Resultsmentioning

confidence: 99%

The complexity of semilinear problems in succinct representation

2006

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We prove completeness results for twenty-three problems in semilinear geometry. These results involve semilinear sets given by additive circuits as input data. If arbitrary real constants are allowed in the circuit, the completeness results are for the Blum-Shub-Smale additive model of computation. If, in contrast, the circuit is constant-free, then the completeness results are for the Turing model of computation. One such result, the P NP[log] -completeness of deciding Zariski irreducibility, exhibits for the first time a problem with a geometric nature complete in this class.

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Section: Resultsmentioning

confidence: 99%

The complexity of semilinear problems in succinct representation

2006

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“…This actually shows the stronger result that the first-order closure of monadic NP is contained in P NP & , defined as the class of properties that are recognized by some deterministic polynomial-time oracle machine with an NP oracle where all of the oracle queries must be made in parallel (unlike P NP where each query can depend on the answers to previous queries); for definitions and results concerning P NP & see, for example, [Wag90]. In fact, by combining a syntactic characterization of P NP given by Buss and Hay [BH91] with the fact that 7 Once again, as we now discuss, all of these inclusions are proper, even when we restrict our attention to undirected graph properties. The properness of the first inclusion follows from the fairly simple-to-show fact (Theorem 8.1) that the property``There are exactly two connected components'' belongs to the Boolean closure of monadic NP but does not belong to monadic NP nor to monadic co-NP.…”

Section: Beyond Monadic Npmentioning

confidence: 99%

The Closure of Monadic NP

Ajtai

Fagin

Stockmeyer

2000

Journal of Computer and System Sciences

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It is a well-known result of Fagin that the complexity class NP coincides with the class of problems expressible in existential second-order logic (7 1 1 ), which allows sentences consisting of a string of existential second-order quantifiers followed by a first-order formula. Monadic NP is the class of problems expressible in monadic 7 1 1 , i.e., 7 1 1 with the restriction that the second-order quantifiers are all unary and hence range only over sets (as opposed to ranging over, say, binary relations). For example, the property of a graph being 3-colorable belongs to monadic NP, because 3-colorability can be expressed by saying that there exists three sets of vertices such that each vertex is in exactly one of the sets and no two vertices in the same set are connected by an edge. Unfortunately, monadic NP is not a robust class, in that it is not closed under first-order quantification. We define closed monadic NP to be the closure of monadic NP under first-order quantification and existential unary second-order quantification. Thus, closed monadic NP differs from monadic NP in that we allow the possibility of arbitrary interleavings of firstorder quantifiers among the existential unary second-order quantifiers. We show that closed monadic NP is a natural, rich, and robust subclass of NP. As evidence for its richness, we show that not only is it a proper extension of monadic NP, but that it contains properties not in various other extensions of monadic NP. In particular, we show that closed monadic NP contains an undirected graph property not in the closure of monadic NP under first-order quantification and Boolean operations. Our lower-bound proofs require a number of new game-theoretic techniques. Academic Press

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“…The equivalence of P (NP) with each of these is proved by Buss and Hay in [7] (see also [14,31,32]), who also provided the following very useful characterization: A problem L lies in P (NP) if and only if for some fixed natural number n, a deterministic Turing machine can solve L by making n rounds of parallel queries to an oracle for some NP-complete problem. Here, the queries in the later rounds may depend on answers to queries in earlier rounds; i.e., intuitively P (NP) allows a finite, but fixed, amount of adaptivity.…”

Section: Iterating the Frattini Constructionmentioning

confidence: 99%

Computational Complexity of Generators and Nongenerators in Algebra

Bergman

Slutzki

2002

Int. J. Algebra Comput.

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Abstract. We discuss the computational complexity of several problems concerning subsets of an algebraic structure that generate the structure. We show that the problem of determining whether a given subset X generates an algebra A is P-complete, while determining the size of the smallest generating set is NP-complete. We also consider several questions related to the Frattini subuniverse, Φ(A), of an algebra A. We show that the membership problem for Φ(A) is co-NP-complete, while the membership problems for Φ(Φ(A)), Φ(Φ(Φ(A))),... all lie in the class P (NP).In the analysis of any algebraic structure, determining those subsets that generate the structure frequently plays a key role. This is evident in linear algebra for example, where the discussion of bases and spanning sets forms a central element of the subject. The same is true in other branches of algebra such as group and lattice theory.Knowledge of the generating subsets of an algebra gives us information on its subalgebras, homomorphic images, automorphism group etc. As computer algebra systems become commonplace in the toolboxes of scientists (see for example GAP [11], and the "algebra calculator" [21]), questions of efficiency in the determination of generating subsets arise. In this paper we address this fundamental issue by providing completeness results for several variants of the basic question: does the subset X generate the algebra A? In particular, we consider the question of minimal generating sets and the existence of a generating set of a given cardinality.In addition to questions such as these, it is sometimes of interest to ask about the role an individual element can play in the generation of an algebra. Schmid [28] suggests classifying elements as generators, nongenerators and irreducibles. To explain these, we need the notion of the Frattini subuniverse of an algebra.The Frattini subgroup has played an important role in group theory since it was first considered by Giovanni Frattini in 1885 [10]. It was observed quite some time ago that most of the basic properties of the Frattini subgroup hold more generally in any algebraic structure. Numerous papers have discussed Frattini sublattices. For a sample, see [1,2,9,27]. There are 2000 Mathematics Subject Classification. 68Q17, 08A30, 20D25.

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On truth-table reducibility to SAT

Cited by 92 publications

References 10 publications

The complexity of semilinear problems in succinct representation

The complexity of semilinear problems in succinct representation

The Closure of Monadic NP

Computational Complexity of Generators and Nongenerators in Algebra

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