Title: The Polynomial Hierarchy collapses. PDF Classical Simulation of Commuting Quantum Computations ... the polynomial time hierarchy | at least with currenttechniques. CiteSeerX — Does the Polynomial Hierarchy Collapse if Onto ... 3. During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. Let L2 j. Graph Nonisomorphism Has Subexponential Size Proofs Unless The Polynomial-Time Hierarchy Collapses. PDF The Polynomial Hierarchy and Alternations the area is whether or not this hierarchy collapses. 1.1 Alternating Turing Machines al. The polynomial hierarchy for some structures over the ... This result is re-proved here by a different technique. References 1 ANGLUIN, D. On counting problems and the polynomial-time hierarchy. If PH=PSPACE, then PH collapses, because then TQBF sits at some finite level of the hierarchy and is PH-complete. Polynomial hierarchy - Wikipedia If the polynomial hierarchy does collapse this means that there is some isuch that Σp i = ∪ jΣ p j = PH. Derandomzing Arthur-Merlin Games. PDF The polynomial and linear time hierarchies in V0 SIAM Journal on Computing, 28(2):383-393, 1999. What was the first OS with the type-ahead capability from a dumb terminal? L2 i+1P if and only if there exists a language R2 iP and polynomial p: N ! In the same paper, they showed a stronger hypothesis P 2P 2. The result even holds when the first player is conondeter-ministic, and is tight as there exists a trivial protocol for ǫ = 0. By computing a multivalued function in deterministic polynomial-time we mean . [1] Jim Kadin, The polynomial time hierarchy collapses if the Boolean hierarchy collapses, SIAM Journal on Computing 17 (1988), no. De nition 1.3. 340-354, doi . Given n bit string, w, the circuit C w is uniformly generated (in poly n time . In particular, if the theory is PV, then the collapse is to the Boolean hierarchy (i.e., the bounded query hierarchy). Proof We show that for all j>i, we have j= j 1. IQP is easy theorem : If the output of uniform (poly-time/size) IQP circuits is restricted to O(log n) may be sampled (without approximation) by a classical randomized process that runs in time O(poly n). 1263-1282, doi: 10.1137/0217080. the polynomial-time hierarchy collapses to its third level. The result follows from a lower bound for Any such instance of $|V|$ variables and $|C|$ clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with Equality, of size at most $2/3|V|$ variables and at most $|C|$ clauses. Kadin's result was improved by Wagner [25], who showed that if [2] Richard Chang and Jim Kadin, The Boolean hierarchy and the polynomial hierarchy: a closer connection , SIAM Journal on Computing 25 (1996), no. Fourth, a major challenge for quantum computing research is to get better evidence that quantum computers cannot solve NP-complete problems in polynomial time. is bounded by a polynomial in n, unless NP coNP=poly, and the Polynomial-Time Hierarchy collapses. Keywords E.g. 2 and that PH collapses to the P 2 \ P 2 level, finishing the proof. It is shown that if the Boolean hierarchy (BH) collapses, then there exists a sparse set S such that $ {\text {co-NP}} \subseteq {\text { NP}}^S $, and therefore the polynomial time hierarchy (PH) collapses to $ {\text {P}}^ { {\text {NP}}^ { {\text {NP}}} [O (\log n)]} $, a subclass of $\Delta _3^ {\text {P}} $. ReviewofDefinitions Definitionof NP AlanguageL isinNP ifforanyx 2f0;1g,9TM M andp() x 2L ()9u 2f0;1gp(jxj) s:t: M(x;u) = 1 Definitionof coNP AlanguageL isincoNP ifforanyx 2f0;1g,9TM M andp() To support this belief, consider the three conditions: (fi)The polynomial hierarchy collapses(fl)S2 proves that the polynomial hierarchy collapses(°)S2 is flnitely axiomatizedOur results show that (fl) and (°) are equivalent; however, we do not expectto show that (fi) is equivalent to (°) using current techniques. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hierarchy . This then, to the best of our knowledge, is the only natural computational problem where determining the existence of an object (the annihilating polynomial in our case) can be done efficiently but the actual . Conjecture 2 Polynomial hierarchy does not collapse. Evidence for $\mathsf{P} \neq \mathsf{PP}$ if the polynomial hierarchy collapses? 10 E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. Title: The Polynomial Hierarchy collapses. If the theory is S\, then the collapse is to PNP. 6, pp. Theorem 5.6 p i = Π p i then PH = Σ p NP-hard unless the polynomial hierarchy collapses. • Corollary: If PH does not collapse, then PH⊂PSPACE. We mention some further results on the classical simulation properties of IQP circuit families, in particular showing that if the output distribution results from measurements on only 0(log n) lines then it may, in fact, be classically efficiently sampled. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to . P (complexity) 100% (1/1) P polynomial time PTIME. Over five hundred languages have been confirmed ΣP 2-decidable, and rescue work- ers fear the count may reach as high as ℵ0. Infinite Hierarchy Conjecture. the structure formerly known as the Polynomial Hierarchy collapses to the level above $\text{P}=\text{NP}$. The polynomial time hierarchy collapses if the boolean hierarchy collapses. Ask Question Asked 6 years, 10 months ago. If for almost every language A, the polynomial-time hierarchy relative to A collapses, then the (unrelativized) polynomial-time hierarchy collapses. Modified 6 years, 10 months ago. Hot Network Questions Least squares fit versus averaging How can my pirate haven remain hidden and safe while running illegal activities? For 1-in-3 SAT instances with non-negated literals we obtain considerably smaller equivalent instances of 0 . (3) If Z can prove the polynomial hierarchy is inflnite then for all i, Si 2 ' § p i 6= ƒ p i. We give a relativized negative answer to this question by exhibiting an oracle relative to which Q holds and the polynomial-time hierarchy is in nite. We show that there is no reduction from OR-SAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the Polynomial-Time Hierarchy collapses. As a consequence, sparse sets are not strong nondeterministic polynomial time Turing complete in NP unless the polynomial time hierarchy collapses to DELTA //2**p. View full-text. Polynomial Hierarchy does not collapse. [LFKN92] have shown that #SAT, the #P-complete function that outputs the number of satisfying assignments of a Boolean for-mula, can be computed by a linear-round interactive protocol. He showed that if the difference hierarchy over NP collapses, then the polynomial hierarchy is equal to ∆p 3. We show that if there is an NP function that, when given a satisfiable formula as input, outputs one satisfying assignment uniquely, then the polynomial hierarchy collapses to its second level. Questions about the polynomial-time hierarchy are studied. In [1] it is proved that PB = NPB for some oracle B, hence the relativized Polynomial Hierarchy collapses to PB. Published online: 13 July 2006. Define PH k 0 ∑P k It is straightforward to prove that PH PSPACE, but it is not known whether the inclusion is strict. The polynomial time hierarchy of Meyer and Stockmeyer has several equivalent characterizations | in particular it can be deflned either in terms of polynomial time oracle Turing machines [Sto76], or in terms of polyno- . Indeed the annihilating polynomial A does not even admit a small circuit representation unless the polynomial hierarchy collapses. Proof. This result settles an open problem proposed by Bodlaender et. The smaller iis, the weaker, and hence more plausible, is the conjecture that PH does not collapse to the ith level. Whether or not the problem is NP-hard (or 2-hard) for smaller approximation factors, or even for the exact version, remains an open (It collapses to P i . tial hierarchy is an exponential-time analogue of the polynomial-time hierarchy. an NP-complete problem (unless the polynomial hierarchy collapses). This collapse has since been improved ([7, 2, 3] Kobler & Watanabe improved it to ZPPNP, Cai, with Sengupta, improved it to SP 2, and Chakaravarthy & Roy im-proved it further to OP 2). This extended technique allows each perfect matching in a bipartite graph on 2n nodes to be expressed as a . We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level k, then PH collapses to i P NP (k\Gamma1)-tt j NP , the class of sets recognized in polynomial time with k \Gamma 1 nonadaptive queries to a set in NP NP and an unlimited number of queries to a set in NP. Computational Complexity, by Fu Yuxi Polynomial Hierarchy 35 / 45 We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to ZPP NP (and thus, in particular, to NP NP ). What is especially interesting is that if P=NP, then we get "the great collapse", where the entire polynomial hierarchy $$\Sigma_k^P$$ (which includes problems with more and more nested quantifiers to infinity) all collapses to P. This is one of the reasons why people think it's so unlikely, because there are really really deep problems in .
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