M. Ogiwara and A. Lozano, Sparse hard sets for counting classes. Theoretical Computer Science 112, 255--276 (1993).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....about strings y # x. It is straightforward to check that the polynomially related self reducible sets introduced by Ko [23] as well as the length decreasing and word decreasing self reducible sets of Balcazar [6] are self reducible in our sense. Furthermore, it is well known (see, for example, [9, 6, 29]) that complexity classes like NP, # P k , # P k , PP, C=P, ModmP, PSPACE, and EXP have many one complete self reducible sets. Karp and Lipton [22] introduced the notion of advice functions in order to characterize nonuniform complexity classes. A function h : N # # # is called a ....

....that the left set of A (and since A is polynomial time many one reducible to its left set, also A) is low for ZPP(NP) Corollary 4.6. For every k # 1, if C=P # (# P k # # P k ) poly then CH = ZPP(# P k ) Proof. First, since C=P has complete word decreasing self reducible languages [29], C=P # (# P k # # P k ) poly implies C=P # ZPP(# P k ) # PH. Second, since PH # BPP(C=P) 37, 35] C=P # (# P k # # P k ) poly implies PH # (# P k # # P k ) poly and therefore PH collapses to ZPP(# P k ) by Corollary 4.3. Finally, since C=P(PH) # BPP(C=P) 37] ....

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M. Ogiwara and A. Lozano, On sparse hard sets for counting classes, Theoret. Comput. Sci., 112 (1993), pp. 255--275.

.... P mc(c log n) for some c 1, then C = P. Proof Note that NP is a subset of either C or co C. Since C and co C are both contained in P mc(c log n) by Theorem 5.1, P = NP. Hence P = Sigma p 2 . Since each of C= P, PP, and PSPACE has one word decreasing self reducible p m complete sets [OL93] and if a worddecreasing self reducible set A is in P=poly, then Sigma p 2 (A) Sigma p 2 [KL80] we have C Sigma p 2 . This establishes that C = P. 2 Let k 2. A set A is in MOD k P [CH90] if there is some machine M such that for every x, x 2 A if and only if #accM (x) 6j 0 ....

M. Ogiwara and A. Lozano. Sparse hard sets for counting classes. Theoret. Comput. Sci., 112:255--276, 1993.

....OE(x 1 ; x n 1 ; y 1 ; y n 2 ) returns jf b 2 f0; 1g n 2 : OE(x 1 ; x n 1 ; b) 2 USATgj: Recall that USAT is the set of formulas that have exactly one satisfying assignment. v. This is a generalization of the previous example. It is due to Lozano and Ogiwara [58]. If N is a nondeterministic polynomial time machine and x 2 Sigma then let #accN (x) be the number of accepting paths of N on x, and #rejN (x) be the number of rejecting paths of N . Let Q be a polynomial time decidable predicate on N Theta N. A set A is in the class C = QP if there exists a ....

M. Ogiwara and A. Lozano. On sparse hard sets for counting classes. Theoretical Comput. Sci., 112:255--275, 1993.

.... the concept of left sets and proved that sparse p btt hard sets for NP would collapse NP to P [OW91] Furthermore, Ogihara and Lozano generalized the notion and proved that the collapse C = P follows from the existence of sparse p btt hard sets for C for many complexity classes above P [OL93]. These two breakthrough results started a gold rush in the study of sparse hard sets for classes above P and many interesting theorems were proven (for a survey, see [HOW92, You92, CO96] 1.3 The Sparseness Conjecture for P and NL The question whether there are sparse hard sets applies to ....

M. Ogiwara and A. Lozano. Sparse hard sets for counting classes. Theoretical Computer Science, 112(2):255--276, 1993.

....for each fixed k truth table on any given level of the assignment tree. The theorem of Ogihara and Watanabe is proved. 4. 1 Extensions The breakthrough by Ogihara and Watanabe has provoked a flurry of results about sparse hard complete set problems [HL94, AHH 93, AKM92b, AKM92a, AA95, RR92, OL93] see [HOW92] for a survey) Below we state the best known results on polynomial time sparse hard sets for NP. Theorem 4.2 1. AKM92b] NP is included in the P btt reducibility closure of the languages that are P ctt reducible to sparse sets if and only if P = NP. 2. AKM92a] Sparse P ....

....1. Sparse Sets versus Complexity Classes 16 such a complexity class C includes either NP or coNP, then sparse P btt hard sets for C collapses C to P. Generally speaking, for every counting class in the sense of [GNW90] sparse P btt hard sets for the class collapse it within NP T coNP [OL93] The story for modulo based counting complexity classes such as PhiP is slightly different, for it is not known whether the class contain NP or coNP. Ogihara and Lozano [OL93] extend the notion of left sets and show that sparse P btt hard sets for a modulo based counting class collapse it to ....

[Article contains additional citation context not shown here]

M. Ogiwara and A. Lozano. Sparse hard sets for counting classes. Theoret. Comput. Sci., 112(2):255--276, 1993.

....for each fixed k truth table on any given level of the assignment tree. The theorem of Ogihara and Watanabe is proved. 4. 1 Extensions The breakthrough by Ogihara and Watanabe has provoked a flurry of results about sparse hard complete set problems [HL94, AHH 93, AKM92b, AKM92a, AA95, RR92, OL93] see [HOW92] for a survey) Below we state the best known results on polynomial time sparse hard sets for NP. Theorem 4.2 1. AKM92b] NP is included in the P btt reducibility closure of the languages that are P ctt reducible to sparse sets if and only if P = NP. 2. AKM92a] Sparse P ....

....self reducible structure, and thus, if such a complexity class C includes either NP or coNP, then sparse P btt hard sets for C collapses C to P. Generally speaking, for every counting class in the sense of [GNW90] sparse P btt hard sets for the class collapse it within NP T coNP [OL93] The story for modulo based counting complexity classes such as PhiP is slightly different, for it is not known whether the class contain NP or coNP. Ogihara and Lozano [OL93] extend the notion of left sets and show that sparse P btt hard sets for a modulo based counting class collapse it to ....

[Article contains additional citation context not shown here]

M. Ogiwara and A. Lozano. Sparse hard sets for counting classes. Theoret. Comput. Sci., 112(2):255--276, 1993.

....about strings y OE x. It is straightforward to check that the polynomially related self reducible sets introduced by Ko [23] as well as the length decreasing and word decreasing self reducible sets of Balc azar [6] are self reducible in our sense. Furthermore, it is well known (see, for example, [9, 6, 29]) that complexity classes like NP, Sigma P k , Pi P k , PP, C= P, Modm P, PSPACE, and EXP have many one complete self reducible sets. Karp and Lipton [22] introduced the notion of advice functions in order to characterize non uniform complexity classes. A function h : N Sigma is called ....

....the left set of A (and since A is polynomial time many one reducible to its left set, also A) is low for ZPP(NP) Corollary 4.6. For every k 1, if C=P ( Sigma P k Pi P k ) poly then CH = ZPP( Sigma P k ) Proof. First, since C=P has complete word decreasing self reducible languages [29], C=P ( Sigma P k Pi P k ) poly implies C=P ZPP( Sigma P k ) PH. Second, since PH 12 J. K OBLER AND O. WATANABE BPP(C= P) 37, 35] C=P ( Sigma P k Pi P k ) poly implies PH ( Sigma P k Pi P k ) poly and therefore PH collapses to ZPP( Sigma P k ) by Corollary ....

[Article contains additional citation context not shown here]

M. Ogiwara and A. Lozano, On sparse hard sets for counting classes, Theoretical Computer Science, 112 (1993), pp. 255--275.

....was introduced by J. Simon [Sim75, p. 94] who proved that exact counting can be simulated by general probabilistic computation (in current terminology, C =P PP) The exact counting model and its power, in Simon s setting, have been studied in many papers (e.g. Wag86,Tor91,TO92,KSTT92,OL93] Theorem 4.1 Universally serializable probabilistic polynomial time tasks passing only one bit between tasks and using the exact counting acceptance mechanism accept exactly those languages in NP PP (that is, ProbabilisticSSF 2 = NP PP ) The idea behind the ProbabilisticSSF 2 NP PP ....

M. Ogiwara and A. Lozano. On sparse hard sets for counting classes. Theoretical Computer Science, 112(2):255--275, 1993.

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M. Ogiwara and A. Lozano, Sparse hard sets for counting classes. Theoretical Computer Science 112, 255--276 (1993).

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