J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal of Computing, 17(6):1263--1282, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of equality. Those intermediate results that are of interest in their own right are proven in Sections 3 and 4. We conclude this section with some comments and literature pointers. As to techniques, to study upward translations of equality resulting from collapses of the boolean hierarchy, Kadin [15] introduced what is known as the easy hard technique, and that technique was employed and strengthened in a long series of papers by many authors (see the survey [12] In particular, Hemaspaandra, Hemaspaandra, and Hempel [13] achieved Theorem 1.1 by introducing what can be called the no search ....

J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263-1282, 1988. Erratum appears in the same journal, 20(2):404.

....one minimal model. Note that this problem is similar to the well studied USAT problem, which asks if a Boolean expression E has a unique satisfying assignment [4] Another similar problem is UOASAT, which asks if the truth assignment that satis es the maximum number of a set of clauses is unique [16]. As for USAT and UOASAT, UMINSAT is easily proved co NP hard, but it is not clear how to reduce SAT to it. USAT is complete for the class D [24] under the randomized reduction vv m of Valiant Vazirani [31] and, as recently proved, USAT is not in co D , which contains NP[co NP, unless ....

....possible with O(log m) 1 = O(log m) oracle calls; from this the claim follows immediately. This result suggests that the deduction problem under GCWA and under CCWA is not m complete for 3 , since it seems rather unlikely that a problem in P is m complete for 3 , cf. [16, 34]. 4 Conclusion Our main results are summarized in Table 1. Answering the question in [5] we have shown that the deduction problem with ECWA and with circumscription is 2 complete for propositional theories, even for theories in 3XCNF and a single literal. Moreover, we proved the same ....

J. Kadin. The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses. SIAM J. Comp., 17(6):1262-1283, 1988.

....samik cse.buffalo.edu Krentel [Kre88] showed that if any function that can be computed by two queries to SAT can be computed by one query, then P = NP, i.e, if PF = PF P = NP. It is natural to ask whether we can obtain such collapse if we focus on languages instead of functions. Kadin [Kad88] showed that if P the polynomial time hierarchy collapses to 3 . Wagner [Wag87, Wag89] showed that the collapse can be improved to 3 = P . Beigel, Chang, and Ogihara [BCO93] building on the work of Wagner [Wag87, Wag89] and Chang and Kadin [CK96] obtained a stronger ....

J. Kadin. The polynomial-time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing, 17(1988), pp. 1263-- 1282.

....to compute one value of this function. With bounded query complexity we look at the set of functions that can be calculated if we put an upper bound on the number of queries that we allow the computer to ask the oracle. This notion has been extensively studied both in the resource bounded setting [2, 4, 5, 13, 12, 11, 17, 60, 75, 104] and in the recursive setting[15, 16] This notion and its variants has lead to a series of techniques and tools that are used throughout complexity theory. In this chapter we combine some of the bounded query notions with quantum computation. The main goal is to further as was done by Fortnow ....

....or non adaptive queries) and vice versa [13] In other words: Moreover, there is strong evidence that this trade off is optimal in the sense that every non adaptive class P is different for different values of k. For example if P , then the polynomial hierarchy collapses [60] (see also [27, 52] We will see that if we allow the query machine to make use of quantum mechanical effects such as superposition and interference the situation changes. In the nonadaptive case we will show that 2k classical queries can be simulated with only k non adaptive ones on a quantum ....

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Jim Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, 1988.

....tt and that the proofs of most of the upward collapse results involving P tt actually start with the argument that P tt implies that D is closed under complementation. Previous results have shown that under the weaker assumption that the Boolean Hierarchy collapses to D coNP NP=poly [Kad88]. In contrast, the results in this paper make the stronger assumption that PH collapses to D . The stronger assumption allows us to reduce the advice to just one bit and also allows us to prove the converse. Our results also generalize to the k bit case. We are able to show that: k ( coNP ....

....queries are made in parallel, we use the notation P tt . We will use P tt when the machines are allowed polynomial many queries. The connections between the Boolean Hierarchy and the bounded queries to SAT is rich and varied. We ask the reader to consult the literature for a full accounting [WW85, KSW87, Wag88, Kad88, CGH 89, ABG90, Wag90, Bei91, Cha92, HN93, BCO93, CGL97, CK96, HHH99, BF99, Cha01]. For this paper, we make use of the following facts about the Boolean Hierarchy and bounded queries to SAT. SAT[k 1] tt BH k coBH k BH k [ coBH k P tt [KSW87; Bei91] tt [Bei91] BH k = coBH k = BH = BH k [CGH 88; CGH 89] BH k = coBH k = SAT 2 NP=poly [Kad88] 3 Proof ....

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J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263-1282, December 1988.

....(2) and the negative result (3) do not match. It is an open problem to improve either of them. The simplest open question along these lines is 2 tt (4) In this paper we also give a partial answer to that question: 2 tt =poly: 5) The proof uses a delicate hard easy argument (cf. [2, 12]) to exploit very convenient representations of 2 ary Boolean functions. It appears very difficult to generalize even to 3 ary Boolean functions. Question (4) can be understood informally as follows: Consider an unknown assignment ff to a set of Boolean variables. Suppose that we are given a black ....

J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM J. Comput., 17(6):1263--1282, Dec. 1988.

....it is fairly easy to see that NPSV t , the class of all total NPSV functions, is contained in coNPMV. We also note that Theorem 4. 2 extends to higher levels of the difference hierarchies over NPMV and NP, that is, 12 NPMV(k) coNPMV(k) NPMV(k) c coNPMV(k) NP(k) coNP(k) By a result of Kadin [Kad88], a collapse of the Boolean hierarchy implies a collapse of the polynomial time hierarchy. Hence, there is likely to be a whole hierarchy between coNPMV and NPMV . 4.4 Theorem. For all k 2, we have Furthermore, all of the inclusions are strict unless the polynomial time hierarchy ....

....equivalent to FP by Corollary 5.3, which implies P S 2 . But then P = S = PH. For the other inclusions, suppose . Then FP [k 1] By a theorem of Fenner et al. FHOS97] this implies that FP 2 [k] 2 [k 1] which, by a relativization of Kadin s theorem [Kad88], implies that the polynomial hierarchy collapses. Thus we see, combining Theorem 4.4 and Corollary 5.5, that all classes of the difference hierarchy over NPMV are included in the query hierarchy over coNPMV, in fact already in its first level. There are (under reasonable assumptions) no ....

J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, December 1988.

....B;P) The next result states that if extra queries to an NP complete set do not allow us to solve extra decision problems in polynomial time, then all NP hard sets are Psuperterse, unless P = NP. We tend to disbelieve the hypothesis (because it implies that the polynomial time hierarchy collapses [Kad88]) and we tend to believe the conclusion (because it is true under almost all relativizations [Bei87b] However, neither belief has been proven true, and it is reassuring to know that at least one of them must be true, unless P = NP. Theorem 5.7.3 Assume that P 6= NP, and let B be a ....

....closely related to the Boolean Hierarchy. Cai and Hemachandra [CH86] have constructed oracles that make the Boolean hierarchy collapse at arbitrary levels. They have also constructed oracles that make the hierarchy proper; Cai [Cai86] has shown that almost oracles make the hierarchy proper. Kadin [Kad88] has shown that if the Boolean hierarchy collapses then the polynomial time hierarchy collapses. Book and Ko [BK88] have constructed, for each k 0, a sparse set A such that for every sparse set B it is true that Q k (k; A; P) ae Q k (k 1; B;P) This result does not hold if we remove either ....

Jim Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SICOMP, 17(6):1263--1282, December 1988.

....Supported in part by NSF Research Grant CCR 88 23053. Dept. of Computer Science, University of Electro Communications, Chofu si, Tokyo 182, Japan. This work was done while the author was at Dept. of Information Science, Tokyo Institute of Technology. 1. Introduction Numerous researchers [3, 5, 8, 9, 10, 11, 16, 17, 24, 25, 26, 27] have studied the Boolean hierarchy over NP. This hierarchy intertwines the query hierarchies over NP, and is identical to the Haussdorf and the difference hierarchies over NP. Similar relations hold among hierarchies over many classes other than NP [7] A central question is whether these ....

....NP. Similar relations hold among hierarchies over many classes other than NP [7] A central question is whether these hierarchies collapse. Because they stand or fall together, it is sufficient to study a single one. We find that the difference hierarchy is the most amenable to analysis. Kadin [16] was the first to discover non trivial structural consequences of the collapse of the difference hierarchy over NP. He showed that if the difference hierarchy over NP collapses, then the polynomial hierarchy is equal to Delta 3 . Kadin s result can be understood as translating a collapse of ....

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J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM J. Comput., 17(6):1263--1282, Dec. 1988.

....N with h 2 NPMV as the oracle. Then we can easily construct a machine that computes f by making k queries to h and m queries to g. Therefore, f 2 PF NPMV[k m] tt . This proves the lemma. The Boolean hierarchy over NP is defined by Wagner and Wechsung [19] and has been studied extensively [6, 7, 8, 12]. We denote the k th level of the Boolean hierarchy as NP(k) By definition, 1. NP(1) NP, and 2. for every k 2, NP(k) NP Gamma NP(k Gamma 1) The Boolean hierarchy over NP, denoted by BH is the union of all NP(k) k 1. Kadin [12] proved that the Boolean hierarchy collapses only if the ....

....Wechsung [19] and has been studied extensively [6, 7, 8, 12] We denote the k th level of the Boolean hierarchy as NP(k) By definition, 1. NP(1) NP, and 2. for every k 2, NP(k) NP Gamma NP(k Gamma 1) The Boolean hierarchy over NP, denoted by BH is the union of all NP(k) k 1. Kadin [12] proved that the Boolean hierarchy collapses only if the polynomial time hierarchy collapses. Theorem 3.11. Let k 1. If PF NPMV[k 1] PF NPMV[k] then BH collapses to its 2 k th level. Proof. Suppose that PF NPMV[k 1] PF NPMV[k] By Lemma 3.9 and Theorem 3.5, for every m k, ....

J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, December 1988. ORACLES THAT COMPUTE VALUES 23

....di#erent; intuition suggests that you can gain more information from asking k 1 questions than from asking only k questions. This has been shown to be the case when considering a random oracle set [TB91] but here we are considering reductions to any set in NP (or to any NP complete set) Kadin [Ka88] provided evidence that this intuition is correct by establishing the following result. The reader will find more recent result in [CK90] Theorem 2.2. If the polynomial time hierarchy extends to infinitely many levels, then for every k 0, P NP k tt # #= P NP (k 1) tt , that is, there ....

J. Kadin, The polynomial time hierarchy collapses if the Boolean hierarchy collapses, in "Proc. 3rd Structure in Complexity Theory Conference", IEEE (1988), 278-292.

....it is fairly easy to see that NPSV t , the class of all total NPSV functions, is contained in coNPMV. We also note that Theorem 4. 2 extends to higher levels of the difference hierarchies over NPMV and NP, that is NPMV(k) coNPMV(k) NPMV(k) c coNPMV(k) NP(k) coNP(k) By a result of Kadin [Kad88], a collapse of the Boolean hierarchy implies a collaps of the polynomial time hierarchy. Hence, there is a whole hierarchy between coNPMV and NPMV NP . 4.4 Theorem. For all k 2, we have coNPMV NPMV(2) NPMV(k) NPMV(k 1) NPMV(n O(1) NPMV NP : Furthermore, all of the ....

...., and PH = Sigma p 2 . For the other inclusions, suppose FP coNPMV[k] FP coNPMV[k 1] Then FP NPMV NP [k] FP NPMV NP [k 1] By a theorem of Fenner et al. FHOS93] this implies that FP Sigma p 2 [k] FP Sigma p 2 [k 1] which, by a relativization of Kadin s theorem [Kad88], implies that the polynomial hierarchy collapses. 2 Thus we see, combining Theorems 4.4 and 5.5, that all classes of the difference hierarchy over NPMV are included in the query hierarchy over coNPMV, in fact already in its first level. There are (under reasonable assumptions) no inclusions in ....

J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM J. on Comput., 17(6):1263--1282, December 1988.

....the P m complete languages Recent results about the complexity of USAT and of D P shed a new light on these questions. Under the assumption that the Polynomial Hierarchy has infinitely many levels, much has been discovered about SATSAT, the D P P m complete set. For example, Kadin [Kad88, CK90a] showed that D P = co D P = PH collapses. So, SATSAT 62 co D P unless PH collapses. Similarly, one can show that SATSAT cannot have OR unless PH collapses [CK90b] These results show that SATSAT does not have the same robust properties of SAT. So, how does USAT, the vv m ....

J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, December 1988.

....bounded truth table hierarchy at level j k implies a collapse of the boolean hierarchy at level j k 1 which in turn implies a collapse of the polynomial hierarchy. This follows from the intertwined structure of the bounded truth table hierarchy and the boolean hierarchy and a result by Kadin [Kad88] In order to specify the depth of the collapse of the polynomial hierarchy, we mention that one can conclude a collapse of the polynomial hierarchy to BH j k 1 DeltaDIFF j k ( Sigma p 2 ) fL 1 DeltaL 2 j L 1 2 BH j k 1 L 2 2 DIFF j k ( Sigma p 2 )g, where Delta denotes the symmetric ....

J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, 1988. Erratum appears in the same journal, 20(2):404.

.... follows that P SAT[log n] P MaxClique[1] We will show that P 2MC[1] P SAT[log log n 1] Thus, we have P SAT[log n] P MaxClique[1] P 2MC[1] P SAT[log log n 1] which collapses the (extended) Boolean Hierarchy which in turn collapses the Polynomial Hierarchy [Wag88, Kad88] Now, it is not obvious that P 2MC[1] P SAT[log log n 1] because the computation of the P 2MC[1] machine depends on the value of 2MC(G) for some query string G and there is no obvious way for a P SAT[log log n 1] machine to compute 2MC. However, we can find a number z that ....

J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, December 1988.

.... NPMV P NP [log] NPMV NP . This establishes the inclusions. Suppose coNPMV = NPMV(2) Take a set A 2 NP(2) Interpret A as a graph of a function f 2 NPMV(2) By assumption, f 2 coNPMV, thus A 2 coNP. Thus, we have NP(2) coNP, which implies the collapse of the polynomial time hierarchy [Kad88]. The proofs for the other inclusions are completely analogous. 2 8 Theorem 4.5 is striking. Even though coNPMV is very close to NPMV NP , there is an infinite hierarchy of complexity classes between them. 4.6 Theorem. NPMV NP FP coNPMV[1] Delta Delta Delta FP coNPMV[k] ....

....and PH = Sigma p 2 . For the other inclusions, suppose FP coNPMV[k] FP coNPMV[k 1] Then FP NPMV NP [k] FP NPMV NP [k 1] By a theorem of Fenner et al. FHOS93] this implies that FP Sigma p 2 [k] FP Sigma p 2 [k 1] which, by a relativization of Kadin s theorem [Kad88], implies that the polynomial hierarchy collapses. 2 Thus we see, combining Theorems 4.5 and 4.6, that all classes of the difference hierarchy over NPMV are included in the query hierarchy over coNPMV, in fact already in its first level. There are (under reasonable assumptions) no inclusions in ....

J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM J. on Comput., 17(6):1263--1282, December 1988.

.... log log log a n log 1 # n 1 a log a log a log 1 # is complete for polynomial time functions which use only O(log n) queries, denoted PF NP[O(log n) Since Krentel s original work, many connections between bounded query classes and standard complexity classes have been discovered [1, 2, 6, 7, 13, 14, 20, 23, 32, 33]. In many circumstances, these results show that one cannot decrease the number of queries needed to solve a problem by even a single query unless P = NP or PH collapses. For example, Hoene and Nickelsen [21] showed that to determine how many of the formulas in F 1 , F r are satisfiable, ....

J. Kadin, The polynomial time hierarchy collapses if the Boolean hierarchy collapses, SIAM J. Comput., 17 (1988), pp. 1263--1282.

....Assume that the boolean hierarchy over K does not collapse. Let (G; f) and (G 0 ; f 0 ) be k lattices. Then K(G; f) K(G 0 ; f 0 ) if and only if (G; f) G 0 ; f 0 ) Since the collapse of the boolean hierarchy over NP implies the collapse of the polynomial time hierarchy (cf. Kad88] this conjecture implies the following one. Conjecture 5.3. Embedding Conjecture for NP. Assume that the polynomial time hierarchy does not collapse. Let (G; f) and (G 0 ; f 0 ) be k lattices. Then NP(G; f) NP(G 0 ; f 0 ) if and only if (G; f) G 0 ; f 0 ) In the next section ....

J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, 1988.

....is fairly easy to see that NPSV t , the class of all total NPSV functions, is contained in coNPMV. We also note that Theorem 3 extends to higher levels of the difference hierarchies over NPMV and NP, that is, NPMV(k) # coNPMV(k) ##NPMV(k)# c coNPMV(k) ##NP(k) coNP(k) By a result of Kadin [Kad88], a collapse of the Boolean hierarchy implies a collapse of the polynomial time hierarchy. Hence, there is likely to be a whole hierarchy between coNPMV and NPMV NP . Theorem 4 For all k # 2, we have coNPMV #NPMV(2) # NPMV(k) # NPMV(k 1) # NPMV(Poly) # NPMV NP . Furthermore, all ....

....But then # p 2 = # p 2 = PH. For the other inclusions, suppose FP coNPMV[k] FP coNPMV[k 1] Then FP NPMV NP [k] FP NPMV NP [k 1] By a theorem of Fenner et al. FHOS97] this implies that FP # p 2 [k] FP # p 2 [k 1] which, by a relativization of Kadin s theorem [Kad88], implies that the polynomial hierarchy collapses. Proof of Corollary 6 # Thus we see, combining Theorem 4 and Corollary 6, that all classes of the dif 5 11 ference hierarchy over NPMV are included in the query hierarchy over coNPMV, in fact, already in its first level. There are (under ....

J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, 1988.

....University of Maine Orono, ME 04469 July 7, 1993 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) This result is significant in two ways. First, the theorem says that a deeper collapse of the Boolean hierarchy implies a deeper collapse of the polynomial hierarchy. Also, this result points to some previously ....

.... Turing reductions, oracle access, nonuniform algorithms, sparse sets AMS (MOS) subject classifications: 68Q15, 03D15, 03D20 1 Introduction The Boolean hierarchy (BH) was defined as the closure of NP under Boolean operations and is identical to the difference hierarchy of NP sets [1, 2, 7] Kadin [3] showed that if the BH collapses at any level, the polynomial time hierarchy (PH) collapses to P NP NP [O(logn) the class of languages in P NP NP that are recognized by deterministic polynomial time machines that make O(log n) queries Supported in part by NSF research grants ....

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J. Kadin. The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses. SIAM Journal on Computing, 17(6):1263--1282, December 1988. 18

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J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal of Computing, 17(6):1263--1282, 1988.

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J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263-1282, December 1988.

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J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263-1282, 1988.

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J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, December 1988.

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J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263--1282, December 1988.

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