S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148,1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....we present the following alternative ways that SBP can be looked at. When one abstains from the probability gap in the definition of BPP this yields the class PP (probabilistic polynomial time) Since PP can be defined via GapP functions and since these functions have different characterizations [FFK94] the following statements are equivalent to saying that L 2 PP. 1. There is a nondeterministic polynomial time machine M with x 2 L ( acc M (x) rej M (x) 2. There exist f 2 #P and g 2 FP such that x 2 L ( f(x) g(x) 3. There exist f; g 2 #P such that x 2 L ( f(x) g(x) Interestingly, ....

.... some f 2 #P such that for all x 2 If one weakens this definition and asks for some f 2 GapP one meets the GapP counterpart of UP, the class SPP (stoic PP because the machine doesn t change its behavior much between accept and reject) It was introduced in 1991 independently by Fenner et al. [FFK94], Gupta [Gup91] under the name ZUP) and Ogiwara and Hemachandra [OH93] under the name XP) Definition 4.1 ( FFK94, Gup91, OH93] The class SPP consists of all languages L there exists an f 2 GapP such that for all x 2 Theorem 4.2 ( FFK94] Few SPP In [FFK94] it is shown that SPP ....

[Article contains additional citation context not shown here]

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48:116--148, 1994.

....O(1) hardness. That is, they prove that every nontrivial counting property of circuits is UP O(1) hard. In the same vein, they ask if it is possible to improve the lower bound beyond UP O(1) and they show that relativizable techniques cannot raise the UP O(1) hardness lower bound to SPP [OH93,FFK94] the gap analog of UP. In particular, they note that if every nontrivial counting property of circuits is SPP hard then SPP . We prove that every nontrivial counting property of circuits is Few hard (equivalently, FewP hard) We thus raise Hemaspaandra and Rothe s constant ambiguity ....

....(# polynomial q) g # q C] 5. Val76] UP = # 1 g(x) 0] 6. AR88] FewP = g(x) 0] 7. CH90] Few = P P[1] i.e. the class of languages accepted by P machines that on each input are allowed at most one query to a function from # few P. 8. OH93,FFK94] SPP is the class of all languages such that there exist a function f #P and a polynomial computable function g : # # N such that, for all x, the following hold: a) x L =# f(x) g(x) and 3 (b) x L =# f(x) g(x) 1. We now define the standard reductions used in the paper. ....

[Article contains additional citation context not shown here]

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148, 1994.

....polynomial time many one reductions. Thus in particular there are globally unique logic games that are as hard to solve as factoring, which is actually equivalent to some decision problem in UP#co UP. It is open to determine which classes above UP and co UP, such as FewP, C=P#C= P, or SPP (see [FFK94] for the latter) may also reduce to GUQBF. 16 ....

S. Fenner, L. Fortnow, and S. Kurtz, Gap-definable counting classes, J. Comp. Sys. Sci., 48, 116-184, (1994).

....x 2 A ( M B accepts x. ffl Let C be a class of languages and f(n) be a function from N to N . P is the class of languages A such that for some B in C, A T B via M and M makes no more than f(n) queries on any input of length n. The GAP function was introduced by Fenner, Fortnow and Kurtz [8]. Definition 9. ffl Let M be a NTM. acc M (x) is the number of accepting paths of M on input x. rej M (x) is the number of rejecting paths of M on input x. GAP(M;x) acc M (x) Gamma rec M (x) ffl A language L is in PP (resp. PL) if there exists a polynomial time (resp. logspace) NTM M ....

....That is, we show how to construct a low degree rational approximation to any function in NC 1 or NC 1 . We assume all polynomials in this section are integer coefficients polynomials. 3.1. Rational Functions The following two lemmas can be verified by straightforward simulating the method in [8]. Lemma 17. Let f 1 be (s 1 ; t 1 ) generatable, f 2 be (s 2 ; t 2 ) generatable, and c 1 be a constant. There exists a constant c 0 such that ffl f 1 f 2 and f 1 Gamma f 2 are (max(s 1 ; s 2 ) c; max(t 1 ; t 2 ) c) generatable; and ffl f 1 Delta f 2 is (max(s 1 ; s 2 ) c; t 1 t 2 ....

[Article contains additional citation context not shown here]

S. Fenner, L. Fortnow, and S. Kurtz. Gapdefinable counting classes. In Proceedings of the 6th Annual Conference on Structure in Complexity Theory, pages 30--42. IEEE Computer Society Press, 1991.

....used polynomials to prove closure properties of various counting classes. In this section we extend the techniques of [7] where they used a single polynomial, we use a sequence of rational functions. These new twists appear to be crucial to obtaining our closure properties for PP. Fenner, et al. [10] provide a convenient notation for studying counting classes like PP. Definition 1. 10] For a nondeterministic Turing machine N and input X, let Gap(N;X) denote the number of accepting paths of N on input X minus the number of rejecting paths of N on input X. Definition 2. A language L is in ....

....we extend the techniques of [7] where they used a single polynomial, we use a sequence of rational functions. These new twists appear to be crucial to obtaining our closure properties for PP. Fenner, et al. 10] provide a convenient notation for studying counting classes like PP. Definition 1. [10] For a nondeterministic Turing machine N and input X, let Gap(N;X) denote the number of accepting paths of N on input X minus the number of rejecting paths of N on input X. Definition 2. A language L is in PrTIME(t(n) if there exists a t(n) time bounded nondeterministic Turing machine N such ....

Steven Fenner, Lance Fortnow, and Stuart Kurtz. Gap-definable counting classes. In Proc. of the 6th Annual Conference on Structure in Complexity Theory, pages 30--42. IEEE Computer Society Press, July 1991.

....An operator H is a closure property of a class G of functions if, for all g in G, Hg belongs to G. An operator H is a relativizable closure property of a class G of functions if, for all oracles A, for all g in G , Hg belongs to G . Closure properties of #P and GapP, studied in [4, 5], yield important closure properties of various counting classes. Hertrampf, Vollmer, and Wagner [6] considered the special case where Hg = f ffi g for some function f of a single variable. It is known [4, 5] that #P and GapP are closed under addition and under f(n) k GapP is also closed ....

....A, for all g in G , Hg belongs to G . Closure properties of #P and GapP, studied in [4, 5] yield important closure properties of various counting classes. Hertrampf, Vollmer, and Wagner [6] considered the special case where Hg = f ffi g for some function f of a single variable. It is known [4, 5] that #P and GapP are closed under addition and under f(n) k GapP is also closed under subtraction. Therefore, if a univariate function f is a linear combination of binomial coefficients then GapP is closed under f ; if, in addition, f is a positive linear combination of binomial ....

[Article contains additional citation context not shown here]

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. In Proceedings of the 6th Annual Conference on Structure in Complexity Theory, pages 30--42. IEEE Computer Society Press, 1991.

....on complexity classes of sets (num Delta in Toda s notion [Tod90b] as a generalization of Valiant s class of counting functions #P [Val79a, Val79b] such that # Delta P = #P. Subsequently other function classes such as OptP [Kre88, Kre92, GKR95] SpanP [Kob89, KST89] MidP [Tod90a] or GapP [FFK94] were generalized by operators in several manners (see e.g. VW93, Vol94, HV95, VW95, HW97] We will focus our attention to the operators # Delta; max Delta and min Delta [Tod90b, HW97] considered under restriction of cluster sets. Such operators, called cluster operators, will be denoted by ....

....= minfyjjyj = p(jxj) hx; yi 2 Bg. Analogously, the converse can be proven. 2) is a simple consequence of the first proof. q Remark 4.6. In Theorem 4. 5 the class P can be replaced by every complexity class closed under complementation, intersection and P m reductions such as by PP, XP [OH93, FFK94] or PhiP. Using Theorem 3.4 one can conclude a simple statement about the position c# Delta P with respect to cmax Delta NP and cmin Delta NP. The inclusion c# Delta P c# Delta NP is a direct consequence of the monotonicity of c# Delta. Proposition 4.7. c# Delta P (c# Delta NP cmax ....

[Article contains additional citation context not shown here]

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48:116--148, 1994.

....Note on SpanP Functions Meena Mahajan Thomas Thierauf y Vinodchandran N.V. z Keywords. Computational Complexity. 1 Introduction Valiant [10] introduced the class #P to count the number of solutions of NP sets. Recently, Fenner, Fortnow, and Kurtz [3] considered the class GapP, the closure of #P under subtraction, and showed many interesting properties of this class. Kobler, Schoning, and Tor an introduced the class SpanP that counts the number of distinct outputs produced by a nondeterministic Turing machine. With this concept, they could ....

....and NSF grant CCR 8957604. z The Institute of Mathematical Sciences, Madras 600 113, India. Part of the work done when the author was at the Department of Computer Science and Engineering, Indian Institute of Technology, Madras 600 036, India. 1 number of rejecting paths. #P [10] and GapP [3] are the classes of functions f for which there exists a polynomial time bounded counting machine M such that f = acc M and f = gap M , respectively. Another extension of #P is as follows. For a class C of sets, # Delta C is the class of functions f for which there exist a set A 2 C and a ....

[Article contains additional citation context not shown here]

S. A. Fenner, L. J. Fortnow, and S. A. Kurtz. Gap-definable counting classes. In Proceedings of the Sixth Annual Conference on Structure in Complexity Theory, pages 30--42, 1991. 5

....for AC 0 coincide if and only if the polynomial time hierarchy equals PSPACE . 1 Introduction Counting classes have been studied extensively since the introduction of #P by Valiant [22] A number of such classes were first defined in the context of polynomial time computation [16, 10]. Then, the logspace Work supported by NSERC of Canada and by FCAR du Qu ebec. y Work supported by the Alexander von Humboldt Foundation under a Feodor Lynen scholarship. counting class #L was investigated [3] together with many logspace variants adapted from the polynomial time case. ....

....4.10 GapNC 1 is closed under subtraction, weak sum, weak product, and binomial coefficients. Proof sketch. The only nontrivial point is closure under binomial coefficients, but this follows from the other closure properties by expressing binomial coefficients involving negative numbers as in [10]. Corollary 4.11 PNC 1 is closed under union and intersection. Proof. Follows from the established closure properties by reproducing the corresponding proof for PP from [6] or for PL from [1] essentially word for word. But we also have closure properties which #P and #L probably don t share: ....

S. Fenner, L. Fortnow, and S. Kurtz. Gapdefinable counting classes. Journal of Computer and System Sciences, 48:116--148, 1994.

....As a consequence, none of these classes can be contained in the polynomial hierarchy unless the polynomial hierarchy collapses. Thus, none of these classes can be contained in BPP path unless the polynomial hierarchy collapses. Ogiwara and Hemachandra [OH93] and Fenner, Fortnow, and Kurtz [FFK94] independently defined the counting class SPP as follows. Definition 3.14 [OH93,FFK94] SPP is the class of all sets L such that there exist a nondeterministic polynomial time Turing machine M and an FP function f such that for all x 2 Sigma it holds that x 2 L = acc M (x) f(x) 1; and x ....

....unless the polynomial hierarchy collapses. Thus, none of these classes can be contained in BPP path unless the polynomial hierarchy collapses. Ogiwara and Hemachandra [OH93] and Fenner, Fortnow, and Kurtz [FFK94] independently defined the counting class SPP as follows. Definition 3. 14 [OH93,FFK94] SPP is the class of all sets L such that there exist a nondeterministic polynomial time Turing machine M and an FP function f such that for all x 2 Sigma it holds that x 2 L = acc M (x) f(x) 1; and x 62 L = acc M (x) f(x) Fenner, Fortnow, and Kurtz [FFK94] argue that SPP is, in ....

[Article contains additional citation context not shown here]

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148, 1994.

....computation. Recently, various counting problems have received considerable attention (see, e.g. Sch90] and, in order to model them, there have been introduced and extensively studied complexity classes called counting classes , typified by function classes #P [Val79] spanP [KST89] and GapP [FFK94], and language classes PP [Gil77] PhiP [PZ83] C=P [Sim75, Wag86a] and the counting hierarchy CH [Tor91, Wag86b] Unfortunately, many of the questions regarding counting classes, even the ones about inclusion relation, are left open. Confronted with such difficulties in resolving problems ....

S. Fenner, L. Fortnow, and S. Kurtz, Gap-definable counting classes, J. Comput. System Sci. 48 (1994), 116--148.

....Consider a set L # PrTime(n) PrTime(n) that is accepted by a linear time probabilistic oracle machine M A , where A # PrTime(n) via a machine M 1 .Since PrTime(n) is closed under complement, there is a linear time probabilistic machine M 0 deciding A. By standard techniques (see [13]) we can assume that for every a # 0, 1 and for every string x, the probability that M a accepts on input x is not equal to 1 2 ; that is, the probability of accepting is always greater than one half, or less than one half. We show how to accept L on a probabilistic linear time machine N with ....

....C=Time(n) A : x # A iff # a prob. linear time machine M s.t. Pr[ x # L(M) 1 2 . It is straightforward to verify that language (q, a, w) M a accepts input q along computation path w and v : v w#M a accepts q along v =2 q 1 is in C=Time(n) Results of Fenner et al. [13] show that C=Time(n) is closed under conjunctive reductions, and thus C # C=Time(n) Next we will show how to reduce computation of the machine N C to a formula from MAJMAJSAT. We know that for any C # PrTime(n) there is a deterministic machine MC taking two inputs x and y such that x # C ....

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 30-- 42, 1991.

....case in theorem 4.1. 2 Verifying the Determinant 15 Theorem 5.3. a) v paths is complete for m V DET: b) v shortest paths is complete for m V DET: Proof. a) We will give an AC 0 many one reduction from v paths difference to v paths. The basic idea of our reduction is the proof in [12], which shows that GapP = #P Gamma FP: We just add some technical refinements in order to ensure that the reduction used in that proof can be performed in AC 0 : Let G 1 and G 2 be two directed acyclic graphs with vertices respectively labeled 1; 2; n 1 and 1; 2; n 2 . We will ....

....classes were defined as the analogs of the well known classes GapP and and C=P (Tor an[20] by replacing polynomial time with logarithmic space. Many properties of C=P remain true for C=L, including closure under AC 0 conjunctive, disjunctive and positive truth table reductions (Fenner et al.[12]) where positive refers to reductions via monotone circuit families. Since paths is complete in #L; the function paths difference is clearly complete in GapL: Therefore GapL coincides with the class m DET: Also, v paths difference is complete for C= L: This in turn implies that m V DET and C=L ....

S. Fenner, L. Fortnow and S. Kurtz. Gap-definable counting classes. In Proc. 6th IEEE Structure in Complexity Theory Conference, 1991, 30--42.

....complete in #P, whereas computing the number of paths in a directed graph between two specified vertices is complete in #L. The so called gap classes were defined subsequently to include functions taking also negative values into the above model. GapP was introduced by Fenner, Fortnow and Kurtz [FFK94] as the difference of two functions in #P. The analogous definition for GapL was made independently by Vinay [Vin91] Toda [Tod91] Damm [Dam91] and Valiant [Val92] This later class has received considerable attention, mostly because it characterizes the complexity of computing the determinant ....

S. A. Fenner, L. J. Fortnow, and S. A. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148, 1994.

....We show that these problems, over solvable blackbox groups, are in the counting class SPP. The proof of this result is built on a constructive version of the fundamental theorem of finite abelian groups. The class SPP is known to be low for the counting classes PP, C=P and Mod k P for k 2 [FFK94] Since it is unlikely that the class NP is contained in SPP, these upper bounds give evidence that these problems are unlikely to be hard for NP. In the second part of the thesis we study the problem of computing a generator set of an unknown group, given a membership testing oracle for the ....

....it is shown that P is different from UP if and only if one way functions exist. More recently, it is shown in [FK92] that the problem of deciding primality is in UP co UP. UP is also low for many counting classes like PP, C=P and Mod k P for k 2. The class GapP, studied by Fenner et al. in [FFK94] is an important function complexity class. The main motivation for defining this class is from the observation that the class #P cannot take negative values. This led to the introduction of the class GapP as the closure of #P under subtraction. The class GapP satisfies many algebraic closure ....

[Article contains additional citation context not shown here]

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48:116--148, 1994.

....on e er inis ic C co u ion 3 3 3 o o o . o o o o o , o o , o o o o o o o o . n ro uc ion Counting classes have been studied extensively since the introduction of #P by Valiant [35] A number of such classes were first defined in the context of polynomial time computation [24, 17]. Then, the logspace counting class #L was investigated [5] together with many logspace variants adapted from the polynomial time case. Recently, counting classes based on finite model theory and on one way logspace have been considered as well [31, 13] The starting point of the present paper ....

....making use of results from [26] A natural question now is of course about the relation between C NC 1 and C=NC 1 . We come back to this issue in the next section. Corollary 3.5 motivates defining PNC 1 as C NC 1 . Observe that PP is defined analogously in terms of GapP functions in [17]. oroll r . The following languages are C=NC 1 complete (resp. PNC 1 complete) under uniform pro ections (a) Gi en a se uence of constant dimension matrices o er the integers f01; 0; 1g, determine whether a speci c entry in the matrix product is ero (resp. nonnegati e) b) Gi en a se ....

[Article contains additional citation context not shown here]

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48:116--148, 1994.

....to those in the proof of Lemma 4.2. It is convenient to use the class GapP, which is the class of functions that are the difference of two #P functions. We make use of the fact that A 2 PP if and only if there exists a GapP function f such that for every x, x 2 A if and only if f(x) 0 (see [Fenner et al. 1994]) One can show that the function p from the proof of Lemma 4.2 is in GapP, because the respective matrix powering is in GapP (see the proofs of Lemmas 4.8 and 4.1) and GapP is closed under multiplication and summation. Finally, PP is closed under polynomial time disjunctive reducibility [Beigel ....

Fenner, S., Fortnow, L., and Kurtz, S. 1994. Gap-definable counting classes. Journal of Computer and System Sciences 48, 1, 116--148.

No context found.

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148,1994.

No context found.

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148, 1994. An earlier version appeared in Proceedings of the 6th Annual IEEE Structurein Complexity Theory Conference, 1991, pp. 30--42.

No context found.

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148, 1994.

....other techniques can be found in [12] We list most of the known results here; in the listing, G can be any 1 generic set, and SAT is the NP complete set of satisfiable Boolean formulae. Definitions of UP and FewP can be found, for example, in [15] and [1] respectively. The class SPP is defined in [11] and in [27] under the name XP. The class BPP is well studied; see [4] for example. ffl NP G co NP G P G PhiSAT [6] ffl UP G P G PhiSAT [6] ffl FewP P G PhiSAT [12] ffl Every pair of disjoint NP G sets is P G PhiSAT separable [6, 9] ffl BPP G P G PhiC , ....

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48:116--148, 1994.

No context found.

Stephen A. Fenner, Lance J. Fortnow, and Stuart A. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148, February 1994.

No context found.

Stephen A. Fenner, Lance J. Fortnow, and Stuart A. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148, 1994.

No context found.

S. A. Fenner, L. J. Fortnow, S. A. Kurtz, Gap-definable counting classes. Journal of Computer and System Science, 48(1):116--148, 1995.

No context found.

S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116--148, 1994.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC