1,721,219 research outputs found
The polynomial-time hierarchy
No full textAbstractThe polynomial-time hierarchy is that subrecursive analog of the Kleene arithmetical hierarchy in which deterministic (nondeterministic) polynomial time plays the role of recursive (recursively enumerable) time. Known properties of the polynomial-time hierarchy are summarized. A word problem which is complete in the second stage of the hierarchy is exhibited. In the analogy between the polynomial-time hierarchy and the arithmetical hierarchy, the first order theory of equality plays the role of elementary arithmetic (as the ω-jump of the hierarchy). The problem of deciding validity in the theory of equality is shown to be complete in polynomial-space, and close upper and lower bounds on the space complexity of this problem are established
Complete sets and the polynomial-time hierarchy
No full textAbstractNew proofs of two properties of the polynomial-time hierarchy are given. The classes in the hierarchy are characterized using polynomially bounded quantifiers. Using this result, a sequence of complete sets for the hierarchy is exhibited
Bounding queries in the analytic polynomial-time hierarchy
No full textAbstractIn a previous paper the present authors (Baier and Wagner, 1996) investigated an ∃-∀-hierarchy over P using word quantifiers as well as two types of set quantifiers, the so-called analytic polynomial-time hierarchy. The fact that some constructions there result in a bounded number of oracle queries and the recent PCP results which can be expressed by set quantifiers with a bounded number of queries motivated us to examine a hierarchy which extends the analytic polynomial-time hierarchy by considering restrictions on the number of oracle queries. This hierarchy is called bounded analytic polynomial-time hierarchy. We show that every class from this hierarchy having a certain normal form coincides with one of the classes NP, coNP, PSPACE, Σkexp or Πkexp (k ⩾ 1). All these characterizations remain valid if the queries are asked in a nonadaptive form, i.e. in “parallel”
Control problems and the polynomial time hierarchy
Get PDFClassifies control problems by exhibiting their alternating quantifier structure. This classification allows the authors to relate these control problems to the computational complexity classes of the polynomial time hierarchy. A specific synthesis problem for uncertain systems is shown to be hard in the class II^p_2
The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses
No full textThe structure of the Boolean hierarchy (BH) is related to the polynomial time hierarchy (PH) by showing that if the BH collapses, then
Intuitionistic Deductive Databases And The Polynomial Time Hierarchy
No full textthis paper, we establish more comprehensive results by exploring the interaction of negation-as-failure with a natural syntactic restriction called linearity. The main result is a tight connection between intuitionistic logic, database queries, and the polynomial time hierarchy. A tight connection with second-order logic follows as a corollary. First, we show that rulebases in our language fit neatly into a well-established logical framework---intuitionistic logic. Second, we show that linearity reduces their data complexity from PSPACE to NP. Third, we show that negation-as-failure increases their complexity from NP to some level in the polynomial time hierarchy (PHIER). Specifically, linear rulebases with k strata are data complete for \Sigm
Optimization Problems in the Polynomial-Time Hierarchy
No full textThis talk surveys work on classifying the complexity and approximability of problems residing in the Polynomial-Time Hierarchy, above the first level. Along the way, we highlight some prominent natural problems that are believed – but not yet known – to be Σ^p₂-complete. We describe how strong inapproximability results for certain Σ^p₂ optimization problems can be obtained using dispersers to build error-correcting codes. Finally we adapt a learning algorithm to produce approximation algorithms for these problems
General Terms Theory
No full textThe complexity of algorithms tax even the resources of sixty billion gigabits--or of a universe full of bits; Meyer and Stockmeyer had proved, long ago, that, regardless of computer power, problems existed which could not be solved in the life of the universe
On the Hardness of Satisfiability with Bounded Occurrences in the Polynomial-Time Hierarchy
No full textIn 1991, Papadimitriou and Yannakakis gave a reduction implying the NP-hardness of approximating the problem 3-SAT with bounded occurrences [10]. Their reduction is based on expander graphs. We present an analogue of this result for the second level of the polynomial-time hierarchy, thereby resolving an open question of Ko and Lin [7]. More precisely, we show that given an instance of 89-3-SAT in which every variable occurs at most B times (for some absolute constant B), it is \Pi 2-hard to distinguish between the following two cases: YES instances, in which for any assignment to the universal variables there exists an assignment to the existential variables that satisfies all the clauses, and NO instances in which there exists an assignment to the universal variables such that any assignment to the existential variables satisfies at most 1- " fraction of the clauses.Our reduction is based on superconcentrator graphs, and should be useful in deriving inapproximability results in the polynomial-time hierarchy. We also generalize this result to any level of the polynomial-time hierarchy
On counting problems and the polynomial-time hierarchy
No full textAbstractWe consider the relation between the relativized polynomial time hierarchy and relativizations of Gill's class PP of sets recognizable in polynomial time by probabilistic Turing machines and of Valiant's class D≠P of sets polynomial time Turing reducible to functions that give the number of accepting computations of nondeterministic polynomial-time bounded Turing machines. The main result is that there exists an oracle set A such that PPA −(Π2P,A ∪ σ2P,A) ≠ ∅, with the corollary that also D ≠PA − (Π2P,A ∪ σ2P,A ≠ ∅. The proof is an application of Baker and Selman's technique for showing that σ2P,A ⊆ σ3P,A for some oracle set A
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