J.-y. Cai and L. Hemachandra, The Boolean hierarchy: Hardware over NP, Tech. Rep. #TR85-724, Cornell University, Computer Science Department, December 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... a study of intermediate partitions into a nite number k of parts (such partitions are called k partitions in the forthcoming) the boolean hierarchy BH k (NP) of k partitions over NP for k 3 was introduced in [32] as a generalization of the well studied boolean hierarchy of NP sets (cf. e.g. [8 11, 28, 43]) Whereas the latter hierarchy has a very clear structure with no more than two incomparable classes, for k 3 the situation turned out be much more opaque: Already the hierarchy BH 3 (NP) does not have bounded width with respect to set inclusion unless the polynomial hierarchy collapses [32] ....

....A, kAk denotes its cardinality. A set A is k elementary if and only if kAk = k. IN = f0; 1; 2; g and IN = f1; 2; g. The classes K(i) and coK(i) de ned by K(1) def K and K(i 1) def K(i) K for i 1 build the boolean hierarchy over K that has many equivalent de nitions (see [9, 11, 28, 43] or the case k = 2 in De nition 1) The class BC(K) is the boolean closure of K, that is the smallest class which contains K and which is closed under intersection, union, and complementation. Let us make some notational conventions about partitions. Partitions will be considered with respect to ....

J.-Y. Cai and L. Hemachandra. The Boolean hierarchy: Hardware over NP. In Proceedings 1st Structure in Complexity Theory Conference, volume 223 of Lecture Notes in Computer Science, pages 105-124. Springer-Verlag, Berlin, 1986.

....In a similar way, the two bounded query hierarchies of sets provide natural measures of the complexity of NP hard decision problems. Those two hierarchies are closely related to the Boolean hierarchy, which has been studied by many people including Cai, Hemachandra, Kobler, Schoning, and Wagner [14, 17]. The second level of the Boolean hierarchy was studied by Papadimitriou and Yannakakis [22] In Section 4. we determine how the bounded NP query hierarchies of sets interleave with the Boolean hierarchy, and then we show that either all three hierarchies of sets collapse at some level, or else ....

....function f and a polynomial p such x 2 A ( 9y:jyj p(jxj) f(x; y) 2 B] 3. The Boolean Hierarchy Several papers [23, 25, 29] have discussed the Boolean Hierarchy. Cai and Hemachandra studied it extensively, and they proved that several definitions of the Boolean Hierarchy are equivalent [14]. We prefer the following: Definition 9 The ith level of the Boolean Hierarchy is NP(i) where NP(0) P; NP(i 1) fL 1 Gamma L 2 : L 1 2 NP;L 2 2 NP(i) g The function # k defined below determines how many of k strings are elements of A. Definition 10 We define an integer valued function ....

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J. Cai and L. A. Hemachandra. The Boolean hierarchy: Hardware over NP. In A. L. Selman, editor, Structure in Complexity Theory, pages 105--124. SpringerVerlag, June 1986. Lecture Notes in Computer Science 223. 27

....A, 1 query reducibility to an NP complete set is identical with Turing reducibility to an NP complete set. Thus, there is a relativized world in which P 6= NP, but extra queries to an NP complete set do not allow us to solve extra decision problems. If B is NP complete, Cai and Hemachandra [CH86] have constructed relativizations for each value of k that make Q k (k; B;P) ae Q k (k 1; B;P) Q k ( 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 ....

....all NP hard sets are superterse. Thus we have found a relativization that makes all NP hard sets P superterse. In fact all NP complete sets are P self encoding if and only if the Boolean Hierarchy of Wagner and Wechsung [WW85] collapses. Thus any oracle that collapses the Boolean Hierarchy [CH86] makes all NP complete sets P superterse. In [Bei87b] we have shown that all NP hard sets are P superterse under almost all relativizations. Open Question 5.7.6 ffl If P 6= NP are all NP hard problems P superterse ffl Does there exist an oracle A such that P and some NP hard problem ....

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Jin-yi Cai and Lane A. Hemachandra. The Boolean hierarchy: Hardware over NP. In Alan L. Selman, editor, Structure in Complexity Theory, pages 105--124. Springer-Verlag, June 1986. Lecture Notes in Computer Science 223.

....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 ....

J. Cai and L. A. Hemachandra. The Boolean hierarchy: Hardware over NP. In Structure in Complexity Theory, pages 105--124. Springer-Verlag, June 1986. Volume 223 of Lecture Notes in Computer Science.

....configuration is encoded by the string of all 1 s. By using this type of encoding it is not hard to see that E 2 A PS Delta T (F ) The converse is easier and it is left to the reader. b) From Theorem 3. 4(b) and the existence of relativized worlds in which the following classes BH [6], NP co NP [23] RP [23] and BPP [8] lack complete languages, it follows from the representability of T that reducibilities btt , T , m , and m cannot be represented by complementary dot operators. c) For all the reducibilities of (c) but rpos it is easy to see that P = NP ....

J. Cai, L. A. Hemachandra. The Boolean hierarchy: hardware over NP, Proc. 1st Structure in Complexity Theory Conference, LNCS 223, 1994, pp. 105--124.

.... A B A 2 K; B 2 K 0 ; K K 0 = def A [ B A 2 K; B 2 K 0 ; K K 0 = def A4B A 2 K; B 2 K 0 : The classes K(i) and coK(i) de ned by K(0) def f;g and K(i 1) K(i) K build the boolean hierarchy over K that has many equivalent de nitions (see [WW85,CH86,KSW87,CGH 88] 1 Some of them can be found in the following theorem. 1 Usually for K = NP, a level 0 is not considered in the way we do. The zero level there is P. However for our purposes it is more helpful to regard P not as an element of the boolean hierarchy (unless P = NP) 3 ....

J.-Y. Cai and L. Hemachandra. The Boolean hierarchy: Hardware over NP. In Proceedings 1st Structure in Complexity Theory Conference, volume 223 of Lecture Notes in Computer Science, pages 105-124. Springer-Verlag, Berlin, 1986.

....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 ....

J. Cai and L. Hemachandra. The Boolean hierarchy: Hardware over NP. In Structure in Complexity Theory, Lecture Notes in Computer Science 223, pages 105--124, Berlin, 1986. Springer-Verlag.

....configuration is encoded by the string of all 1 s. By using this type of encoding it is not hard to see that E 2 A PS Delta PS T (F ) The converse is easier and it is left to the reader. b) From Theorem 3. 4(b) and the existence of relativized worlds in which the following classes BH [6], NP co NP [23] RP [23] and BPP [8] lack complete languages, it follows from the representability of p T that reducibilities p btt , SN T , BPP m , and RP m cannot be represented by complementary dot operators. c) For all the reducibilities of (c) but p lpos and p rpos it ....

J. Cai, L. A. Hemachandra. The Boolean hierarchy: hardware over NP, Proc. 1st Structure in Complexity Theory Conference, LNCS 223, 1994, pp. 105--124.

.... K m 2 Q jj (2; K) Gamma Q(1; K) From [24] If PH does not collapse then (8m 2) V SAT m 2 P jjSAT [2] Gamma P SAT [1] The second result can be stated more precisly: V SAT 2 2 P SAT [1] iff BHNP [2] P SAT [1] 24] BHNP [2] is the second level of the boolean hierarchy, [17]) This consequence of V SAT m 2 P SAT [1] implies that BHNP [2] co BHNP [2] which by [11] implies that P NP NP [1] PhiNP = PH. 12 The proofs of both upper bounds are similar and easy. The lower bounds were easier to obtain in recursion theory than in complexity theory. 6) Behavior ....

J. Cai and L. A. Hemachandra. The Boolean hierarchy: Hardware over NP. In Structure in Complexity Theory, volume 223 of Lecture Notes in Computer Science, pages 105--124, Berlin, June 1986. Springer-Verlag.

....this question, we return to USAT. Suppose someone were to construct a randomized reduction from SAT to USAT with probability 1=2 1=poly. Then, USAT would be complete for D P in a much stronger sense. In fact, such a theorem would answer the frequently posed question of whether USAT has OR 2 [CH86, GW86, CGH 89, GNW90] It is known that SATSAT does not have OR 2 unless PH collapses [CK90b] Corollary 4. If SAT rp m USAT with probability 1=2 1=p(n) for some polynomial bound p, then USAT does not have OR 2 unless PH collapses. Proof: We know that SAT P m USAT. By assumption, ....

.... they called d BP Delta in which the error probability is required to be very small: A 2 d BP DeltaC = 8p; 9B 2 C; Prob z [ x 2 A ( x; z) 2 B ] 1 Gamma 2 Gammap(n) We know that if the class C is closed under majority reductions, then all of the reductions above are equivalent [Sch86] However, for classes not known to be closed under majority reductions, it makes sense to consider d BP Delta operations separately. In their paper, Toda and Ogiwara showed that PH d BP DeltaK where K is any of the counting classes: PP, Phi P, C =P, or MOD g P. Moreover, they use a ....

J. Cai and L. A. Hemachandra. The Boolean hierarchy: Hardware over NP. In Structure in Complexity Theory, Springer-Verlag Lecture Notes in Computer Science #223, pages 105--124, 1986.

.... polynomial time 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 ....

....knows that the easy NP algorithm recognizes all of SAT =m . In either case, the NP NP machine can use an NP algorithm for SAT =m to remove one level of oracle querying from a Sigma P 3 machine, and therefore recognize any Sigma P 3 language. Now, we show that Sigma P 3 P NP NP [2] . Since an NP NP machine can guess and verify hard formulas, T 2 NP NP . This implies that a P NP NP machine can tell with one query if there are any hard formulas of a given length. Since this is exactly what an NP NP machine needs to recognize a Sigma P 3 language, the P NP NP ....

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J. Cai and L. A. Hemachandra. The Boolean Hierarchy: Hardware over NP. In Structure in Complexity Theory, pages 105--124, 1986. Springer-Verlag Lecture Notes in Computer Science # 223.

....fi fi A 2 K; B 2 K 0 ; A B = Psi , and K Phi K 0 = def Phi A4B fi fi A 2 K; B 2 K 0 Psi . The classes K(i) and coK(i) defined by K(1) def K and K(i 1) def K(i) Phi K for i 1 build the boolean hierarchy over K. There are many equivalent definitions (see [WW85, Kob85, CH86, KSW87, CGH 88] Some of them can be found in the following theorem. Theorem 2.1. Let ; M 2 K, let K be closed under union and intersection, and let i 1. 1. K(2i Gamma 1) Phi A 1 [ S i Gamma1 j=1 (A 2j 1 n A 2j ) fi fi A 1 ; A 2i Gamma1 2 K and A 1 Delta Delta ....

J.-Y. Cai and L. A. Hemachandra. The Boolean hierarchy: Hardware over NP. In Proceedings 1st Structure in Complexity Theory Conference, volume 223 of Lecture Notes in Computer Science, pages 105--124, 1986.

....Supported in part by NSF Research Grant CCR 88 23053. z 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 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 di erence hierarchies over NP. Similar relations hold among hierarchies over many classes other than NP [7] A central question is whether these ....

J. Cai and L. A. Hemachandra. The Boolean hierarchy: Hardware over NP. In Structure in Complexity Theory, pages 105-124. Springer-Verlag, June 1986. Volume 223 of Lecture Notes in Computer Science.

....= def fA4B j A 2 K; B 2 K 0 g where A4B denotes the symmetric di erence of A and B. For a set A, kAk denotes its cardinality. The classes K(i) and coK(i) de ned by K(1) def K and K(i 1) def K(i) K for i 1 build the boolean hierarchy over K that has many equivalent de nitions (see [WW85,CH86,KSW87,CGH 88] or the case k = 2 in De nition 1) The class BC(K) is the boolean closure of K, that is the smallest class which contains K and which is closed under intersection, union, and complementation. Let us make some notational conventions about partitions. For any set M , a k tuple ....

J.-Y. Cai and L. Hemachandra. The Boolean hierarchy: hardware over NP. In Proceedings 1st Structure in Complexity Theory Conference, volume 223 of Lecture Notes in Computer Science, pages 105-124. Springer-Verlag, 1986.

....In a similar way, the two bounded query hierarchies of sets provide natural measures of the complexity of NP hard decision problems. Those two hierarchies are closely related to the Boolean hierarchy, which has been studied by many people including Cai, Hemachandra, Kobler, Schoning, and Wagner [14, 17]. The second level of the Boolean hierarchy was studied by Papadimitriou and Yannakakis [22] In Section 4. we determine how the bounded NP query hierarchies of sets interleave with the Boolean hierarchy, and then we show that either all three hierarchies of sets collapse at some level, or else ....

....function f and a polynomial p such that x 2 A ( 9y:jyj p(jxj) f(x; y) 2 B] 3. The Boolean Hierarchy Several papers [23, 25, 29] have discussed the Boolean Hierarchy. Cai and Hemachandra studied it extensively, and they proved that several definitions of the Boolean Hierarchy are equivalent [14]. We prefer the following: Definition 9 The ith level of the Boolean Hierarchy is NP(i) where NP(0) P; NP(i 1) fL 1 Gamma L 2 : L 1 2 NP;L 2 2 NP(i) g The function # A k defined below determines how many of k strings are elements of A. Definition 10 We define an integer valued function ....

[Article contains additional citation context not shown here]

J. Cai and L. A. Hemachandra. The Boolean hierarchy: Hardware over NP. In A. L. Selman, editor, Structure in Complexity Theory, pages 105--124. SpringerVerlag, June 1986. Lecture Notes in Computer Science 223.

....Supported in part by NSF Research Grant CCR 88 23053. z 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 ....

J. Cai and L. A. Hemachandra. The Boolean hierarchy: Hardware over NP. In Structure in Complexity Theory, pages 105--124. Springer-Verlag, June 1986. Volume 223 of Lecture Notes in Computer Science.

....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. 2 The Boolean hierarchy over NP is defined by [WW85] and has been studied extensively in [CGH 88, CGH 89, CH86, Kad88] 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 [Kad88] proved that the Boolean hierarchy collapses only ....

J. Cai and L. Hemachandra. The Boolean hierarchy: Hardware over NP. In Structure in Complexity Theory, Lecture Notes in Computer Science 223, pages 105--124, Berlin, 1986. Springer-Verlag.

....and the third level of SDH(UP) are not subsumed by any level E k (UP) of the EH(UP) hierarchy. Consequently, neither the D nor the E normal forms of Definition 3.1 capture the Boolean closure of UP. Theorem 3.4 For every k 1, D k (UP) C k (UP) E k (UP) Proof. For the first inclusion, by [CH85, Proposition 2.1.2] each set L 2 D k (UP) can be represented as L = A 1 Gamma (A 2 Gamma ( Delta Delta Delta (A k Gamma1 Gamma A k ) Delta Delta Delta) where A i = T 1ji L j , 1 i k, and the L j s are the original UP sets representing L. Note that since the proof of [CH85, ....

....by [CH85, Proposition 2.1.2] each set L 2 D k (UP) can be represented as L = A 1 Gamma (A 2 Gamma ( Delta Delta Delta (A k Gamma1 Gamma A k ) Delta Delta Delta) where A i = T 1ji L j , 1 i k, and the L j s are the original UP sets representing L. Note that since the proof of [CH85, Proposition 2.1.2] only uses intersection, the sets A i are in UP. A special case of [CH85, Proposition 2.1.3] says that sets in D k (UP) via decreasing chains such as the A i are in C k (UP) and so L 2 C k (UP) The proof of the second inclusion is done by induction on the odd and even levels ....

[Article contains additional citation context not shown here]

J. Cai and L. Hemachandra. The boolean hierarchy: Hardware over NP. Technical Report 85-724, Cornell University, Department of Computer Science, Ithaca, NY, December 1985.

....(NP) otherwise. Our proof employs the mind change technique, which predates complexity theory. The mind change technique or equivalent manipulation was applied to complexity theory in each of the early papers on the boolean hierarchy, including the work of Cai et al. CGH 88] see also [Wec85,CH86] Kobler et al. KSW87] and Beigel [Bei91] These papers use mind changes for a number of purposes. Most crucially they use the maximum number of mind changes (what a mind change is will soon be made clear) of a class as an upper bound that can be used to prove that the class is contained in ....

J. Cai and L. Hemachandra. The boolean hierarchy: Hardware over NP. In Proceedings of the 1st Structure in Complexity Theory Conference, pages 105-- 124. Springer-Verlag Lecture Notes in Computer Science #223, June 1986.

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J.-y. Cai and L. Hemachandra, The Boolean hierarchy: Hardware over NP, Tech. Rep. #TR85-724, Cornell University, Computer Science Department, December 1985.

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J.-y. Cai and L. Hemachandra, The Boolean hierarchy: Hardware over NP, Tech. Rep. #TR85-724, Cornell University, Computer Science Department, December 1985. 24

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J.-Y. Cai and L. Hemachandra. The Boolean hierarchy: Hardware over NP. In Proceedings 1st Structure in Complexity Theory Conference, volume 223 of Lecture Notes in Computer Science, pages 105-124. Springer-Verlag, Berlin, 1986.

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J.-Y. Cai and L. Hemachandra. The Boolean hierarchy: Hardware over NP. In Proceedings 1st Structure in Complexity Theory Conference, volume 223 of Lecture Notes in Computer Science, pages 105-124. Springer-Verlag, Berlin, 1986.

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