Delineating classes of computational complexity via second order theories with weak set existence principles. I

1995 ◽  
Vol 60 (1) ◽  
pp. 103-121 ◽  
Author(s):  
Aleksandar Ignjatović
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Second Order ◽  
Complexity Classes ◽  
Bounded Arithmetic ◽  
Recursive Functions ◽  
Speed Up ◽  
Bounded Complexity ◽  
Polynomial Time Hierarchy ◽  
Comprehension Axiom

AbstractIn this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

ON HIGHER ARTHUR-MERLIN CLASSES

2004 ◽  
Vol 15 (01) ◽  
pp. 3-19
Author(s):  
JIN-YI CAI ◽  
DENIS CHARLES ◽  
A. PAVAN ◽  
SAMIK SENGUPTA
Keyword(s):  
Polynomial Time ◽  
Upper Bounds ◽  
Complexity Classes ◽  
Polynomial Time Hierarchy

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.

scholarly journals Approximate counting by hashing in bounded arithmetic

2009 ◽  
Vol 74 (3) ◽  
pp. 829-860 ◽  
Author(s):  
Emil Jeřábek
Keyword(s):  
Polynomial Time ◽  
Hash Functions ◽  
Bounded Arithmetic ◽  
Pigeonhole Principle ◽  
Approximate Counting ◽  
Arithmetic Hierarchy ◽  
Relationship Of ◽  
Polynomial Time Hierarchy

AbstractWe show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.

scholarly journals Notes on polynomially bounded arithmetic

1996 ◽  
Vol 61 (3) ◽  
pp. 942-966 ◽  
Author(s):  
Domenico Zambella
Keyword(s):  
Polynomial Time ◽  
General Model ◽  
Bounded Arithmetic ◽  
Theoretical Investigations ◽  
Polynomial Time Hierarchy

AbstractWe characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic.

scholarly journals Polynomial-Time Random Oracles and Separating Complexity Classes

2021 ◽  
Vol 13 (1) ◽  
pp. 11-16
Author(s):  
John M. Hitchcock ◽  
Adewale Sekoni ◽  
Hadi Shafei
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Random Oracle ◽  
Complexity Class ◽  
Complexity Classes ◽  
Random Oracles ◽  
Class Separation ◽  
Polynomial Time Hierarchy

Bennett and Gill [1981] showed that P A ≠ NP A ≠ coNP A for a random oracle A , with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We consider two notions of random oracles: p-random oracles in the sense of martingales and resource-bounded measure [Lutz 1992; Ambos-Spies et al. 1997], and p-betting-game random oracles using the betting games generalization of resource-bounded measure [Buhrman et al. 2000]. Every p-betting-game random oracle is also p-random; whether the two notions are equivalent is an open problem. (1) We first show that P A ≠ NP A for every oracle A that is p-betting-game random. Ideally, we would extend (1) to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation: (2) If P A ≠ NP A relative to every p-random oracle A , then BPP ≠ EXP. (3) If P A ≠ NP A relative to some p-random oracle A , then P ≠ PSPACE. Rossman, Servedio, and Tan [2015] showed that the polynomial-time hierarchy is infinite relative to a random oracle, solving a longstanding open problem. We consider whether we can extend (1) to show that PH A is infinite relative to oracles A that are p-betting-game random. Showing that PH A separates at even its first level would also imply an unrelativized complexity class separation: (4) If NP A ≠ coNP A for a p-betting-game measure 1 class of oracles A , then NP ≠ EXP. (5) If PH A is infinite relative to every p-random oracle A , then PH ≠ EXP. We also consider random oracles for time versus space, for example: (6) L A ≠ P A relative to every oracle A that is p-betting-game random.

LINDSTRÖM QUANTIFIERS AND LEAF LANGUAGE DEFINABILITY

1998 ◽  
Vol 09 (03) ◽  
pp. 277-294 ◽  
Author(s):  
HANS-JÖRG BURTSCHICK ◽  
HERIBERT VOLLMER
Keyword(s):  
Polynomial Time ◽  
Second Order ◽  
New Model ◽  
First Order ◽  
Theoretic Characterization ◽  
Polynomial Time Hierarchy ◽  
The Way

We introduce second-order Lindström quantifiers and examine analogies to the concept of leaf language definability. The quantifier structure in a second-order sentence defining a language and the quantifier structure in a first-order sentence characterizing the appropriate leaf language correspond to one another. Under some assumptions, leaf language definability and definability with second-order Lindström quantifiers may be seen as equivalent. Along the way we tighten the best up to now known leaf language characterization of the classes of the polynomial time hierarchy and give a new model-theoretic characterization of PSPACE.

On the Decidability and Complexity of Autoepistemic Reasoning1

1992 ◽  
Vol 17 (1-2) ◽  
pp. 117-155
Author(s):  
Ilkka N.F. Niemelä
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Classical Logic ◽  
General Setting ◽  
Decision Problems ◽  
Consequence Relation ◽  
Autoepistemic Logic ◽  
Complete Problems ◽  
Decidability And Complexity ◽  
Polynomial Time Hierarchy

The decidability and computational complexity of autoepistemic reasoning is investigated in a general setting where the autoepistemic logic CLae built on top of a given classical logic CL is studied. Correct autoepistemic conclusions from a set of premises are defined in terms of expansions of the premises. Three classes of expansions are studied: Moore style stable expansions, enumeration based expansions, and L-hierarchic expansions. A simple finitary characterization for each type of expansions in CLae is developed. Using the characterizations conditions ensuring that a set of premises has at least one or exactly one stable expansion can be stated and an upper bound for the number of stable expansions of a set of premises can be given. With the aid of the finitary characterizations results on decidability and complexity of autoepistemic reasoning are obtained. E.g., it is shown that autoepistemic reasoning based on each of the three types of expansions is decidable if the monotonic consequence relation given by the underlying classical logic is decidable. In the propositional case decision problems related to the three classes of expansions are shown to be complete problems with respect to the second level of the polynomial time hierarchy. This implies that propositional autoepistemic reasoning is strictly harder than classical propositional reasoning unless the polynomial time hierarchy collapses.

scholarly journals A PSPACE Subclass of Dependency Quantified Boolean Formulas and Its Effective Solving

2019 ◽  
Vol 33 ◽  
pp. 1584-1591
Author(s):  
Christoph Scholl ◽  
Jie-Hong Roland Jiang ◽  
Ralf Wimmer ◽  
Aile Ge-Ernst
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
State Of The Art ◽  
Linear Increase ◽  
Experimental Results ◽  
Decision Problems ◽  
General Strategy ◽  
Quantified Boolean Formulas ◽  
Boolean Formulas ◽  
Polynomial Time Hierarchy

Dependency quantified Boolean formulas (DQBFs) are a powerful formalism, which subsumes quantified Boolean formulas (QBFs) and allows an explicit specification of dependencies of existential variables on universal variables. This enables a succinct encoding of decision problems in the NEXPTIME complexity class. As solving general DQBFs is NEXPTIME complete, in contrast to the PSPACE completeness of QBF solving, characterizing DQBF subclasses of lower computational complexity allows their effective solving and is of practical importance.Recently a DQBF proof calculus based on a notion of fork extension, in addition to resolution and universal reduction, was proposed by Rabe in 2017. We show that this calculus is in fact incomplete for general DQBFs, but complete for a subclass of DQBFs, where any two existential variables have either identical or disjoint dependency sets over the universal variables. We further characterize this DQBF subclass to be ΣP3 complete in the polynomial time hierarchy. Essentially using fork extension, a DQBF in this subclass can be converted to an equisatisfiable 3QBF with only a linear increase in formula size. We exploit this conversion for effective solving of this DQBF subclass and point out its potential as a general strategy for DQBF quantifier localization. Experimental results show that the method outperforms state-of-the-art DQBF solvers on a number of benchmarks, including the 2018 DQBF evaluation benchmarks.

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