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# Topic: Polynomial hierarchy

###### In the News (Mon 21 Apr 14)

 PlanetMath: polynomial hierarchy is a hierarchy Unlike the arithmetical hierarchy, the polynomial hierarchy is not known to be proper. "polynomial hierarchy is a hierarchy" is owned by uzeromay. This is version 2 of polynomial hierarchy is a hierarchy, born on 2002-09-06, modified 2004-03-27. planetmath.org /encyclopedia/PolynomialHierarchyIsAHierarchy.html   (149 words)

 PlanetMath: polynomial hierarchy The polynomial hierarchy is a hierarchy of complexity classes generalizing the relationship between The polynomial hierarchy is closely related to the arithmetical hierarchy; indeed, an alternate definition is almost identical to the definition of the arithmetical hierarchy but stricter rules on what quantifiers can be used. This is version 3 of polynomial hierarchy, born on 2002-09-06, modified 2003-12-02. www.planetmath.org /encyclopedia/PolynomialHierarchy.html   (97 words)

 Arithmetical hierarchy - Wikipedia, the free encyclopedia In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene hierarchy classifies the set of arithmetic formulas (or arithmetic sets) according to their degree of solvability. Layers in the hierarchy are defined as those formulas which satisfy a proposition (description) of a certain complexity. The polynomial hierarchy is a "feasible resource-bounded" version of the arithmetical hierarchy, in which polynomial length bounds are placed on the strings involved, or equivalently, polynomial time bounds are placed on the Turing machines involved. en.wikipedia.org /wiki/Arithmetical_hierarchy   (333 words)

 Encyclopedia: Polynomial-hierarchy   (Site not responding. Last check: 2007-11-06) This definition reflects the close connection between the polynomial hierarchy and the arithmetical hierarchy, where DEC (decidable languages) and CE (computably enumerable languages) play roles analogous to P and NP, respectively. In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy: PH is contained in the complexity classes PPP (the class of problems that are decidable by a polynomial time Turing machine with an access to PP oracle) and PSPACE. In computational complexity theory, the exponential hierarchy is a hierarchy of complexity classes, starting with EXP: and continuing with and so on. www.nationmaster.com /encyclopedia/Polynomial_hierarchy   (1398 words)

 PhD Thesis of Harald Hempel   (Site not responding. Last check: 2007-11-06) It is known that a collapse of the boolean hierarchy implies a collapse of the polynomial hierarchy (Kadin 1988). In contrast to upward collapse results (most hierarchies such as the polynomial hierarchy and the boolean hierarchy display upward collapse) downward collapse results are rather rare. We show that (in almost all cases) the order of queries to levels of the boolean hierarchy is important for the computational power of the resulting class unless the polynomial hierarchy collapses. www.eccc.uni-trier.de /eccc-local/ECCC-Theses/hempel.html   (681 words)

 Complexity Classes PH: the union of classes known as the polynomial-time hierarchy BH: the union of classes known as the Boolean hierarchy QH: the union of classes known as the query hierarchy "polynomial time" means there is a fixed polynomial p(n) and for each string x, accepted by a machine, the number of steps the machine takes is bounded by p(n). P is a set of languages where the restriction is a deterministic Turing machine with moves bounded by a polynomial in the length of x, x www.cs.umbc.edu /~squire/reference/classes.shtml   (284 words)

 Read about Polynomial hierarchy at WorldVillage Encyclopedia. Research Polynomial hierarchy and learn about Polynomial ...   (Site not responding. Last check: 2007-11-06) In computational complexity theory, the polynomial hierarchy is a hierarchy of The union of all classes in the polynomial hierarchy is the complexity class The polynomial hierarchy is an analogue (at much lower complexity) of the encyclopedia.worldvillage.com /s/b/Polynomial_hierarchy   (591 words)

 Arthur-Merlin protocol - Wikipedia, the free encyclopedia The complexity class AM (or AM[1]) is the set of decision problems that can be decided in polynomial time by an Arthur-Merlin protocol where Arthur communicates first, Merlin replies and both of them can only send one message to the other party. Both MA and AM are contained in the polynomial hierarchy. MA is also contained in NP/poly, the class of decision problems computable with in non-deterministic polynomial time with a polynomial size advice. www.wikipedia.org /wiki/Arthur-Merlin_protocol   (321 words)

 Complexity classes P and NP - Wikipedia, the free encyclopedia In fact, by the time hierarchy theorem, they cannot be solved in significantly less than exponential time. The languages in the polynomial hierarchy, PH, correspond to all of second order logic. The proof y can be verified in polynomial time by a competent computer scientist, the existence of which we assert. en.wikipedia.org /wiki/Complexity_classes_P_and_NP   (2614 words)

 Chris Pollett > Publications, Reviews, and Talks In Pollett~\cite{cpollett00} a bounded arithmetic theory \$Z\$ was shown not to be able to prove the collapse of the polynomial hierarchy. Namely, it can "reason" about all the functions in the log-time hierarchy, it can prove the log-time hierarchy differs from NP, and, in fact, we give some evidence it might be able to show the log-time hierarchy is infinite. We then use these results together with recent results about bounded query classes to derive tighter collapses of the polynomial hierarchy under the assumption that various bounded arithmetic theories are equal. www.cs.sjsu.edu /faculty/pollett/papers   (2908 words)

 [No title]   (Site not responding. Last check: 2007-11-06) A functional of type two is a polynomial-time computable if it is computed by a deterministic function-oracle Turing machien whose runtime is bounded by a polynomial that does not depend on the choice of oracle functions. Townsend also introduced a boldface polynomial hierarchy of type two by a relativization method, and gave a ``topological'' characterization of the first level of this hierarchy. We reformulate Townsend's topological notions associated with time bounded computations of function-oracle Turing machines, and further extend his ``topological'' characterization to all levels of the boldface polynomial hierarchy of type two. theory.lcs.mit.edu /~iandc/Abstracts/yamakami95.ltx   (142 words)

 Complexity Zoo References   (Site not responding. Last check: 2007-11-06) The collapsing hierarchies, Bulletin of the EATCS 33, October 1987. Hierarchies of memory-limited computations, Proceedings of the 6th Annual IEEE Symposium on Switching Circuit Theory and Logic Design, pp. Hilbert's Nullstellensatz is in the polynomial hierarchy, Journal of Complexity 12(4):273-286, 1996, DIMACS TR 96-27. www.complexityzoo.com /zooref.html   (5760 words)

 Computational Complexity: Favorite Theorems: The Polynomial-Time Hierarchy Alternation characterizations of the hierarchy using quantifiers and second-order logic. If this happens for some k we say the polynomial-time hierarchy collapses, otherwise the we say the hierarchy is infinite. I seem to recall that the SIGACT News complexity theory column was going to run a column on problems complete for "high" levels of the polynomial hierarchy. weblog.fortnow.com /2005/06/favorite-theorems-polynomial-time.html   (454 words)

 What's Up with Downward Collapse: Using the Easy-Hard Technique to Link Boolean and Polynomial Hierarchy Collapses - ...   (Site not responding. Last check: 2007-11-06) The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what beforehand were thought to be) lower complexity levels. The Boolean Hierarchy of NP-Partitions - Kosub, Wagner (2000) 37 The boolean hierarchy and the polynomial hierarchy: A closer.. citeseer.ist.psu.edu /hemaspaandra98whats.html   (702 words)

 Bounded queries to SAT and the Boolean hierarchy   (Site not responding. Last check: 2007-11-06) We also consider the similarly-defined hierarchies of functions that can be computed by a polynomial-time Turing machine that makes a bounded number of queries to an NP oracle. In addition we show that the Boolean hierarchy and the bounded query hierarchies (of languages) either stand or collapse together. Finally we show that if the Boolean hierarchy collapses to any level but the zeroth (deterministic polynomial time), then for all k there are functions computable in polynomial time with k parallel queries to an NP set that are not computable in polynomial time with www.cis.temple.edu /~beigel/papers/queries-SAT-tcs.html   (347 words)

 Comparing the Space Efficiency of PKR Formalisms The higher a formalism is in the model hierarchy, the more its efficiency in representing models is -- and analogously for theorems. In Theorems 5 and 7 we assume that the polynomial hierarchy does not collapse. For example, as a consequence of Theorems 3 and 7 there is no poly-size reduction from PL to the syntactic restriction of PL allowing only Horn clauses that preserves the theorems, unless the polynomial hierarchy collapses. www.cs.cmu.edu /afs/cs.cmu.edu/project/jair/pub/volume13/cadoli00a-html/node9.html   (774 words)

 The Boolean Hierarchy and the Polynomial Hierarchy: a Closer Connection - Chang, Kadin (ResearchIndex)   (Site not responding. Last check: 2007-11-06) Abstract: We show that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH 3 (k), where BH 3 (k) is the k th level of the Boolean hierarchy over \Sigma P 2. This is an improvement over the known results [3], which show that the polynomial hierarchy would collapse to P NP NP [O(log n)]. The Boolean hierarchy and the polynomial hierarchy: A closer connection. citeseer.ist.psu.edu /chang93boolean.html   (466 words)

 Citations: The polynomial-time hierarchy - STOCKMEYER (ResearchIndex)   (Site not responding. Last check: 2007-11-06) This hierarchy is de ned by iterating the notion of polynomial jump, in analogy with the Turing jump operator. 4.2 The Polynomial Hierarchy While numerous hard decision problems have been proved NP complete, a small number are outside NP and have escaped this classi cation. The existence of a polynomial time algorithm is surprising, given that mere decidability was once in question. citeseer.ist.psu.edu /context/160800/0   (2817 words)

 Abstracts for Publications of Prof. J. Maurice Rojas Polynomials with real coefficients are also considered, and bounds for the expected number of real roots and for the condition number are given. Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms appearing in at least one g_i is exactly m. As a consequence, we obtain new complexity bounds for polynomial system solving which are (a) expressible in terms of Newton polytope volumes, and (b) completely general, free of non-degeneracy assumptions. www.math.tamu.edu /~rojas/abstracts.html   (5211 words)

 Nat' Academies Press, Probability and Algorithms (1992) The scheme for the multivariate polynomial is an example of a multi-oracle instance-hiding scheme. Interestingly, this low-degree polynomial trick, which was devised in order to construct instance-hiding schemes, became a crucial ingredient in the characterization of the set-recognition power of interactive proof systems, both one-prover and multiprover. The zero'th level of the hierarchy is polynomial time, and the first is nondeterministic polynomial time. www.nap.edu /books/0309047765/html/60.html   (589 words)

 (Hemaspaandra E., Hemaspaandra L.A., Hempel H.) Query Order and the Polynomial Hierarchy   (Site not responding. Last check: 2007-11-06) The present paper studies, for the first time, query order as it applies to the levels of the polynomial hierarchy. We prove that the levels of the polynomial hierarchy are order-oblivious: Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. www.jucs.org /jucs_4_6/query_order_and_the   (177 words)

 NP-easy   (Site not responding. Last check: 2007-11-06) In complexity theory the complexity class NP-easy is the set of function problems are solvable in polynomial time by a deterministic Turing machine with oracle for some decision problem in NP. In other words a problem X is if and only if there exists some problem Y in such that X is polynomial-time Turing reducible to Y. This means that given oracle for Y there exists an algorithm solves X in polynomial time (possibly by using that oracle). There are algorithms as Quicksort that can sort the list using a polynomial number of calls to the routine plus a polynomial amount of additional Therefore sorting is NP-easy. www.freeglossary.com /NP-easy   (213 words)

 ICALP'97: Accepted Papers We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The testers in the latter have the goal of determining whether a program computes a polynomial of given degree, whereas we are interested in checking the properties of a given polynomial. All current polynomial time methods (to our knowledge) are unlikely to recover the topology of the true tree from sequences of realistic lengths (bounded, perhaps, by 10,000 nucleotides) for large sets of widely divergent sequences, even if such methods are known to be able to reconstruct the correct topology given long enough sequences. www.cs.unibo.it /icalp/AccptAbstr.html   (9507 words)

 Polynomial hierarchy   (Site not responding. Last check: 2007-11-06) In computational complexitytheory, the polynomial hierarchy is a hierarchy of complexity classes that generalize the classes P, NP and co-NP to oracle machines. is the class of problems solvable in polynomial time withan oracle for some problem in NP. P) as the set of decision problems solvable in polynomial time on an alternating Turingmachine with k alternations starting in an existential (respectively, universal) state. www.therfcc.org /polynomial-hierarchy-221923.html   (370 words)

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