Quantitative Relativizations of Complexity Classes

Book, Ronald V.; Long, Timothy J.; Selman, Alan L.

doi:10.1137/0213030

Cited by 124 publications

(53 citation statements)

References 22 publications

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“…Finally let us mention some relationships between two classes investigated in [2] and the classes ^£/(NP) and ^Ü(NP).…”

Section: 1: (F) Sav^ismentioning

confidence: 99%

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The difference and truth-table hierarchies for NP

Köbler¹,

Schöning²,

Wagner³

1987

RAIRO-Theor. Inf. Appl.

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“…Finally let us mention some relationships between two classes investigated in [2] and the classes ^£/(NP) and ^Ü(NP).…”

Section: 1: (F) Sav^ismentioning

confidence: 99%

“…From several equivalent définitions for the polynomial-time truth-table reducibility given in [8] and [2] we choose the following. (ii) A set A is polynomial-time fc-bounded truth- In the preceding définition of g£, ^£_"(fc^ 1) and ^* tt Boolean functions are represented by Boolean circuits.…”

Section: The Truth-table Hierarchymentioning

confidence: 99%

The difference and truth-table hierarchies for NP

Köbler¹,

Schöning²,

Wagner³

1987

RAIRO-Theor. Inf. Appl.

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“…We adapt the treatment of [28] to the sequent format of one-step rules used here. Recall that a nondeterministic polynomial time multivalued (NPMV) function [9] is a function f : Σ * → PΔ * , where Σ and Δ are alphabets, such that there exists a polynomial p such that |y| ≤ p(|x|) for all y ∈ f (x), where | · | denotes size, and the graph {(x, y) | y ∈ f (x)} of f is in NP . With a view to implementing proof search on an alternating turing machine [13], this leads to the following definition.…”

Section: Theorem 19 Let Gr γ Then γ Has a Proof Where At Most |N(γmentioning

confidence: 99%

Coalgebraic Hybrid Logic

Myers

Pattinson

Schröder

2009

Foundations of Software Science and Computational Structures

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Abstract. We introduce a generic framework for hybrid logics, i.e. modal logics additionally featuring nominals and satisfaction operators, thus providing the necessary facilities for reasoning about individual states in a model. This framework, coalgebraic hybrid logic, works at the same level of generality as coalgebraic modal logic, and in particular subsumes, besides normal hybrid logics such as hybrid K, a wide variety of logics with non-normal modal operators such as probabilistic, graded, or coalitional modalities and non-monotonic conditionals. We prove a generic finite model property and an ensuing weak completeness result, and we give a semantic criterion for decidability in PSPACE. Moreover, we present a fully internalised PSPACE tableau calculus. These generic results are easily instantiated to particular hybrid logics and thus yield a wide range of new results, including e.g. decidability in PSPACE of probabilistic and graded hybrid logics.

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“…The crucial requirement for this is that Steps 3 and 4 can be performed in polynomial time, i.e. by suitable nondeterministic polynomialtime multivalued functions (NPMV) [Book et al 1984]. We recall that a function f : Σ * → P(∆ * ), where Σ and ∆ are alphabets, is NPMV iff (NPMV1) there exists a polynomial p such that |y| ≤ p(|x|) for all y ∈ f (x), where | · | denotes size, and (NPMV2) the graph {(x, y) | y ∈ f (x)} of f is in NP.…”

Section: · 23mentioning

confidence: 99%

PSPACE bounds for rank-1 modal logics

Schröder

Pattinson

2009

ACM Trans. Comput. Logic

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For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE . This leads to a unified derivation of tight PSPACE -bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

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Quantitative Relativizations of Complexity Classes

Cited by 124 publications

References 22 publications

The difference and truth-table hierarchies for NP

The difference and truth-table hierarchies for NP

Coalgebraic Hybrid Logic

PSPACE bounds for rank-1 modal logics

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