T. Gundermann, N. A. Nasser, and G. Wechsung. A survey on counting classes. In Proceedings 5th Structure in Complexity Theory Conference, pages 140-153. IEEE Computer Society Press, Los Alamitos, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....but equivalent to, the usual definition; see Section 2) Gill noted that PP is closed under complementation, but stated that it was not known if PP is closed under intersection and union. Since Gill s paper, PP and related counting classes have been studied extensively by numerous researchers [2, 8, 16, 19, 25, 28, 29, 30, 31], though few closure properties have been shown for the class. In 1985 Russo [25] showed that the symmetric difference of two sets in PP is also in PP, and in 1991 Beigel, Hemachandra, The authors may be reached by writing to Department of Computer Science, P.O. Box 2158, New Haven, CT ....

....We will use jxj to mean the absolute value of the real number x. To avoid confusion, we will never use jXj to denote the length of the input X. All logarithms are base two logarithms. 2. Building Turing machines from rational functions Beigel and Gill [7] and Gundermann, Nasser, and Wechsung [16] have 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, ....

[Article contains additional citation context not shown here]

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proceedings of the 5th Annual Conference on Structure in Complexity Theory, pages 140--153. IEEE Computer Society Press, July 1990.

....of the difference hierarchy over PP has not been shown to translate upward. However, the class C= P, which is closely related to PP, is closed under disj m reductions, as we show in Theorem 15 (similar closure properties were obtained independently by Gundermann, Nasser, and Wechsung [15]) Applying our main result and a theorem of Toran, we find that the difference hierarchy over C=P does not collapse unless the polynomial hierarchy relative to PP collapses. Green [14] independently proved a similar theorem. However, our techniques give a stronger collapse of the polynomial ....

....collapse unless the polynomial hierarchy relative to PP collapses. Green [14] independently proved a similar theorem. However, our techniques give a stronger collapse of the polynomial hierarchy relative to PP. This structural consequence complements a result of Gundermann, Nasser, and Wechsung [15], who constructed oracles that make the difference hierarchy over C=P proper. 2. Preliminaries We assume that the reader is familiar with oracle Turing machines. PH denotes C [ NP [ NP [ Delta Delta Delta. We define the difference hierarchy over a class C. Definition 1. ffl DIFF 1 ....

[Article contains additional citation context not shown here]

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In pages 140--153. IEEE Computer Society Press, July 1990.

.... PhiP and BPP are not contained in P C=P (see section 5) A direct consequence of the separation of BPP from P C=P are separations of both PP and Sigma p 2 Pi p 2 from P C=P . The constructions are based on circuit lower bounds, building on a result of Gundermann, Nasser and Wechsung [12], as well as a new characterization of P C=P (easily proved using the analogies between C 6= P and NP) The circuit lower bounds are for depth 2 circuits consisting of a single equals gate over AND gates (called EQ circuits ) and are actually lower bounds on the fanin of the AND gates rather ....

....depth 2 circuits consisting of a single equals gate over AND gates (called EQ circuits ) and are actually lower bounds on the fanin of the AND gates rather than lower bounds on the size of the circuits. We finally turn to oracle results relating to question 2. Gundermann, Nasser and Wechsung [12] obtained an oracle separation of the Boolean hierarchy over C= P. Since the Boolean hierarchy is intertwined with the query hierarchy, in some relativized world, k 1 queries to C=P are more powerful than k. Here we consider the question of whether k 1 questions to NP cannot be answered by k ....

[Article contains additional citation context not shown here]

T. Gundermann, N. A. Nasser and G. Wechsung, "A Survey on Counting Classes", 5th Annual Conference on Structure in Complexity Theory (1990) 140-153.

....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, SAT rp m USAT with ....

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proceedings of the 5th Structure in Complexity Theory Conference, pages 140--153, July 1990.

....this natural identi cation. A counting problem in a general sense we de ne in the following way. Let a sequence (A n ) be given for which A n is a subset of f0; 2 n g. The counting problem for (A n ) is the set of all circuits c(x 1 ; x n ) such that # 1 (c) 2 A n , see [15] for an analogous de nition of (general) counting classes. In this way, absolute, gap and relative counting problems are counting problems. It is easy to give an example of a (general) counting problem which is nontrivial but in P, for example the set of all circuits with an odd arity (it is the ....

Thomas Gundermann, Nasser A. Nasser, Gerd Wechsung, A survey on counting classes, Procceedings 5th Annual Conference on Structure in Complexity Theory, 1990, 140-153. 15

....hierarchy over PP has not been shown to translate upward. However, the class C= P, 2 which is closely related to PP, is closed under p disj and co NP m reductions, as we show in Theorem 15 (similar closure properties were obtained independently by Gundermann, Nasser, and Wechsung [15]) Applying our main result and a theorem of Toran, we nd that the di erence hierarchy over C=P does not collapse unless the polynomial hierarchy relative to PP collapses. Green [14] independently proved a similar theorem. However, our techniques give a stronger collapse of the polynomial ....

....collapse unless the polynomial hierarchy relative to PP collapses. Green [14] independently proved a similar theorem. However, our techniques give a stronger collapse of the polynomial hierarchy relative to PP. This structural consequence complements a result of Gundermann, Nasser, and Wechsung [15], who constructed oracles that make the di erence hierarchy over C=P proper. 2. Preliminaries We assume that the reader is familiar with oracle Turing machines. PH C denotes C [ NP C [ NP NP C [ We de ne the di erence hierarchy over a class C. De nition 1. DIFF 1 (C) C, ....

[Article contains additional citation context not shown here]

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proceedings of the 5th Annual Conference on Structure in Complexity Theory, pages 140-153. IEEE Computer Society Press, July 1990.

.... sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NP complete sets and the PSPACE complete sets) For the study of the complexity of closure properties, a recent urry of results [21,33,49,6,7, 16] has established an analog of (1) these papers explicitly demonstrate many closure properties possessed by PP and C=P (and the proofs implicitly give closure properties of the function class #P) The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an ....

....: g i 2 DF = h 2 DF ; where h(x) f(g 1 (x) g i (x) 4. Let DF and CF be classes of functions. We say that DF is CF closed if DF has every CF closure property. 1 For example, in [38,9,5] note also the striking recent closure results for the language classes PP [7,6,16] and C=P [21,33,49]. 1 5. Let DF and CF be classes of functions. Let f be a CF closure property. We say that f is hard for the CF closure properties of DF (for short, a DF hard CF closure property, or, in the case that CF = PF, simply DF hard ) if it holds that: DF has closure property f ( DF is CF closed. ....

[Article contains additional citation context not shown here]

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proceedings of the 5th Structure in Complexity Theory Conference, pages 140-153. IEEE Computer Society Press, July 1990.

....observation is immediate from the de nition. Observation 2.6 For every two sets A and B, 1. if A is P 0 tt reducible to B, then A is in P and 2. if A is SN 0 tt reducible to B, then A is in NP co NP. Now we de ne the counting classes in a general setting, using the notation in [12], and the main complexity classes that we will use, in terms of the functions #accN ( and #rej N ( De nition 2.7 For a polynomial time decidable binary predicate 1 Q on N N, a set L is in fQgP if there exists a polynomial time nondeterministic Turing machine N such that for every x 2 ....

T. Gundermann, N. A. Nasser, and G. Wechsung, A survey on counting classes, Proceedings of the 5th Annual Conference on Structure in Complexity Theory (IEEE, 1990) 140-153.

....respects this natural identification. A counting problem in a general sense we define the following way. Let a sequence (A n ) be given for which A n is a subset of f0; 2 n g. The counting problem for (A n ) is the set of all circuits c(x 1 ; x n ) such that # 1 (c) 2 A n , see [12] for an analogous definition of (general) counting classes. In this way, absolute, gap and relative counting problems are counting problems. It is easy to give an example of a (general) counting problem which is nontrivial but in P, for example the set of all circuits with an odd arity (it is the ....

T. Gundermann, N. A. Nasser, G. Wechsung. A survey on counting classes, Proc. 5th Annual IEEE Conference on Computational Complexity, 1990, pp. 140--153.

....we just add up the number of accepting paths, but mapping all numbers greater than k to k. Filtering out those cases where this number is in subset A of f1; kg results in counting classes of finite acceptance type. These classes were introduced by Gundermann, and Wechsung [GW87] see also [GNW90]. In [GW87, GNW90] the inclusionship structure of such classes was investigated for cases where the used sets A consisted of intervals. In the current paper we are interested in the general case, where A may be any finite set of natural numbers. We give a translation of the question, whether a ....

....up the number of accepting paths, but mapping all numbers greater than k to k. Filtering out those cases where this number is in subset A of f1; kg results in counting classes of finite acceptance type. These classes were introduced by Gundermann, and Wechsung [GW87] see also [GNW90] In [GW87, GNW90] the inclusionship structure of such classes was investigated for cases where the used sets A consisted of intervals. In the current paper we are interested in the general case, where A may be any finite set of natural numbers. We give a translation of the question, whether a given class is ....

T. Gundermann, N. A. Nasser, G. Wechsung, A survey on counting classes, Proc. 5th Structure in Complexity Theory Conference (1990), pp. 140--153.

....machine which, on input x, has exactly f(x) accepting computation paths. Many complexity classes in the area between P and PSPACE can be defined by a condition on f(x) for example, NP is the class of sets A such that there is a function f in #P with x 2 A iff f(x) 0. For a good survey see [11] or [19] In analogy to this Alvarez and Jenner [2] introduced #L as the class of functions f , such that there is a nondeterministic logspace Turing machine which, on input x, has exactly f(x) accepting computation paths. Now the above characterization of NP becomes a characterization of NL, ....

Thomas Gundermann, Nasser Ali Nasser, and Gerd Wechsung. A survey on counting classes. In Proc. of 5th Conference on Structure in Complexity Theory, pages 140-- 153, 1990.

.... reductions, which provides in addition an alternative proof of Fenner, Fortnow, and Li s result that these classes are gap definable [FFL93] Furthermore, an elegant proof of Gundermann, Nasser, and Wechsung s insight that C=P is closed under polynomial time positive truth table reductions [GNW90] is given, via different techniques than theirs. 2 Preliminaries and Definitions In general, the standard notations of Hopcroft and Ullman [HU79] are adopted. We consider sets of strings over the alphabet Sigma : f0; 1g. For each string u 2 Sigma , juj denotes the length of u, i.e. u 2 ....

....A p c B : A p tt B via f such that (8x) x 2 A ( Ass(f(x) B] 4. A p ptt B : A p tt B via k tt condition f(x) hff; y 1 ; y k i which satisfies that for each s; t 2 Sigma k with s t, if ff(s) 1 then ff(t) 1. 9 For the following result, the reader is referred to [GNW90]. Even though neither a proof nor the result itself is explicitly given therein, Gundermann, Nasser, and Wechsung have noticed that their method also works to prove that C=P is closed under the p ptt reduction. Via the GAP operator, their insight is slightly more generally stated and proven in ....

T. Gundermann, N. A. Nasser, and G. Wechsung. A survey on counting classes. In Proceedings of the 5th IEEE Conference on Structure in Complexity Theory , 1990, 140--153.

....but equivalent to, the usual definition; see Section 2) Gill noted that PP is closed under complementation, but stated that it was not known if PP is closed under intersection and union. Since Gill s paper, PP and related counting classes have been studied extensively by numerous researchers [2, 8, 16, 19, 25, 28, 29, 30, 31], though few closure properties have been shown for the class. In 1985 Russo [25] showed that the symmetric difference of two sets in PP is also in PP, and in 1991 Beigel, Hemachandra, The authors may be reached by writing to Department of Computer Science, P.O. Box 2158, New Haven, CT ....

....We will use jxj to mean the absolute value of the real number x. To avoid confusion, we will never use jXj to denote the length of the input X. All logarithms are base two logarithms. 2. Building Turing machines from rational functions Beigel and Gill [7] and Gundermann, Nasser, and Wechsung [16] have 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, ....

[Article contains additional citation context not shown here]

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proceedings of the 5th Annual Conference on Structure in Complexity Theory, pages 140--153. IEEE Computer Society Press, July 1990.

....result has been extended by Toda Ogiwara (1992) and independently by Tarui (1993) Tarui shows that the polynomial hierarchy is probabilistically m reducible to PP with zero sided error. Beigel et al. 1991a) have shown that P NP[log] is contained in PP. Their result has been improved by Gundermann et al. 1990) who showed that P C= P[log ] PP, and by Beigel et al. 1994) who showed that P PP[log ] PP. People have asked whether some of those techniques can be extended to show that more of the polynomial hierarchy is contained in PP. Our lower bound for ODD MAX BIT yields an oracle relative to ....

....contained in PP iff f(n) O(log n) Independently, Fu 1992 has observed that Minsky and Papert s one in a box theorem yields an oracle relative to which a weaker separation holds: NP NP 6 PP. Since the techniques of Toda (1991) Beigel et al. 1991a) Toda Ogiwara (1992) Tarui (1993) Gundermann et al. 1990) , and Beigel et al. 1994) relativize, this means that other techniques will be needed in order to determine how much of the polynomial hierarchy is contained in PP. Perceptrons, PP, and PH 3 2. Threshold Circuits Throughout this paper we assume that the weights on the inputs to a threshold ....

T. Gundermann, N. Nasser, and G. Wechsung, A survey of counting classes. In Proceedings of the 5th Ann. Conf. Structure in Complexity Theory. IEEE Computer Society Press, 1990, 140--153.

....which gives analogous lowness results for the classes Mod k P, k 3, we observe one more consequence of the results in this section. Proposition 4.9 If C=P AmpMP, then CH = MP. Proof. Assume that C=P AmpMP. Since the class C=P is closed under disjunctive and conjunctive reductions ( Tor 88, GuNaWe 90, Gr 93, BeChOg 93] it follows from Theorem 4.6 that C=P would be low for MP. However, from the result of [Tor 88] that PP PP = PP C=P , this would give PP PP MP C=P = MP, implying that the entire counting hierarchy collapses to MP. 2 5 Lowness of Mod Classes for the Class MP In this ....

T. Gundermann, N. Nasser, and G. Wechsung, A survey on counting classes. In Proceedings of the 5th Annual Conference on Structure in Complexity Theory, (1990), 140-153.

....hierarchy over PP has not been shown to translate upward. However, the class C= P, 2 which is closely related to PP, is closed under p disj and co NP m reductions, as we show in Theorem 15 (similar closure properties were obtained independently by Gundermann, Nasser, and Wechsung [15]) Applying our main result and a theorem of Toran, we find that the difference hierarchy over C=P does not collapse unless the polynomial hierarchy relative to PP collapses. Green [14] independently proved a similar theorem. However, our techniques give a stronger collapse of the polynomial ....

....collapse unless the polynomial hierarchy relative to PP collapses. Green [14] independently proved a similar theorem. However, our techniques give a stronger collapse of the polynomial hierarchy relative to PP. This structural consequence complements a result of Gundermann, Nasser, and Wechsung [15], who constructed oracles that make the difference hierarchy over C=P proper. 2. Preliminaries We assume that the reader is familiar with oracle Turing machines. PH C denotes C [ NP C [ NP NP C [ Delta Delta Delta. We define the difference hierarchy over a class C. Definition 1. ffl ....

[Article contains additional citation context not shown here]

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proceedings of the 5th Annual Conference on Structure in Complexity Theory, pages 140--153. IEEE Computer Society Press, July 1990.

....that C =P is known to possess. Theorem 4.1 NP C =P ( P NP jj C = P. Proof: Assuming NP C =P and recalling coNP C = P, we have that DP (see Section 3) is contained in C = P, since each DP set is the intersection of an NP set and a coNP set, and Gundermann, Nasser, and Wechsung [GNW90] have shown that C =P is closed under intersection. Since C =P is known ( GNW90] see the discussion in [Rot93] and [BCO93] to also be closed under disjunctive truth table reductions [LLS75] and since the disjunctive truth table closure of DP is equal to P NP jj it follows that P NP jj C = ....

....= P. Proof: Assuming NP C =P and recalling coNP C = P, we have that DP (see Section 3) is contained in C = P, since each DP set is the intersection of an NP set and a coNP set, and Gundermann, Nasser, and Wechsung [GNW90] have shown that C =P is closed under intersection. Since C =P is known ([GNW90], see the discussion in [Rot93] and [BCO93] to also be closed under disjunctive truth table reductions [LLS75] and since the disjunctive truth table closure of DP is equal to P NP jj it follows that P NP jj C = P. Regarding the claim we just made that fL j (9A 2 DP) L p dtt A]g = P NP ....

T. Gundermann, N. Nasser, and G. Wechsung. A survey on counting classes. In Proceedings of the 5th Structure in Complexity Theory Conference, pages 140-- 153. IEEE Computer Society Press, July 1990.

....having complete sets with some reasonable self reducible structure, and thus, if 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 ....

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proc. 5th Conf. on Structure in Complex. Theory, pages 140--153. IEEE, July 1990.

....Theorem 2.2 holds for every complexity class having complete sets with some reasonable 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 ....

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proc. 5th Conf. on Structure in Complex. Theory, pages 140--153. IEEE, July 1990.

....k (K) 2. The nested difference hierarchy over K: D 1 (K) K; D k (K) K Gamma D k Gamma1 (K) k 2; DH(K) k1 D k (K) 3. The Hausdorff ( union of differences ) hierarchy over K: 2 E 1 (K) K; E 2 (K) K Gamma K; 2 Hausdorff hierarchies ( Hau14] see [CGH 88, BBJ 89, GNW90] respectively, for applications to NP, E k (K) E k Gamma1 (K) K if k odd E k Gamma2 (K) E 2 (K) if k even ; k 2; EH(K) k1 E k (K) 4. The symmetric difference hierarchy over K: SD 1 (K) K; SD k (K) SD k Gamma1 (K) DeltaK; k 2; SDH(K) k1 SD k (K) It is easily ....

T. Gundermann, N. Nasser, and G. Wechsung. A survey on counting classes. In Proceedings of the 5th Structure in Complexity Theory Conference, pages 140--153. IEEE Computer Society Press, July 1990.

....p(jxj) and x 2 L iff f(x) 2 p(jxj) Gamma1 . It has been observed that w.l.o.g. one can assume that for all x, f(x) is not larger than 2 p(jxj) Gamma1 . Hence, C=P is a subclass of PP, and as a further consequence, C=P is closed under (polynomially length bounded) universal quantification [GNW90, Tor91, BCO93, OH93, G93]. Ogiwara [O94] has shown that C=P is closed under positive Turing reductions. A standard many one complete set for C=P is the language ExactSAT = fhx; ii j #SAT(x) ig. Definition 2.3 A language L belongs to PhiP if there is a function f 2 #P such that for all x, x 2 L iff f(x) is odd. In ....

T. Gundermann, N. Nasser, and G. Wechsung, A survey on counting classes. In Proceedings of the 5th Annual Conference on Structure in Complexity Theory, (1990), 140-153.

....of the probabilities p i are greater than one half. In other words N of the strings x 1 ; x k are accepted by N . Thus N accepts the language A. We mention three results that follow from this. They are proved in the exposition that follows. Corollary 3. P PP. Corollary 4. [6] P C=P[log ] PP. Corollary 5. FewEnum PP. The first corollary refers to the class P , which sits near the bottom of the polynomial hierarchy. It is, by definition, the class of languages accepted by polynomial time Turing machines allowed O(log n) calls to an NP oracle, and was first ....

....the class of languages accepted by a nondeterministic, polynomial time bounded Turing machine that accepts when exactly half of its paths are accepting. Ogiwara [12] has shown that C=P is closed under co NP m reductions and under polynomial time disjunctive reductions (similar results are in [20, 19, 6]) If a class K has those two closure properties then K[log ] is equal to the closure of K under polynomialtime parity reductions [2] Since C=P is contained in PP [15, 20] Corollary 4 holds as in the preceding paragraph. Corollary 5 refers to the class FewEnum, which is a restriction of P #P ....

[Article contains additional citation context not shown here]

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proceedings of the 5th Annual Conference on Structure in Complexity Theory, pages 140--153. IEEE Computer Society Press, July 1990.

....In other words N 0 accepts x if and only if an odd number of the strings x 1 ; x k are accepted by N . Thus N 0 accepts the language A. We mention three results that follow from this. They are proved in the exposition that follows. Corollary 3. P NP[log] PP. Corollary 4. [6] P C=P[log ] PP. Corollary 5. FewEnum PP. The first corollary refers to the class P NP[log] which sits near the bottom of the polynomial hierarchy. It is, by definition, the class of languages accepted by polynomial time Turing machines allowed O(log n) calls to an NP oracle, and was ....

....the class of languages accepted by a nondeterministic, polynomial time bounded Turing machine that accepts when exactly half of its paths are accepting. Ogiwara [12] has shown that C=P is closed under co NP m reductions and under polynomial time disjunctive reductions (similar results are in [20, 19, 6]) If a class K has those two closure properties then P K[log ] is equal to the closure of K under polynomialtime parity reductions [2] Since C=P is contained in PP [15, 20] Corollary 4 holds as in the preceding paragraph. Corollary 5 refers to the class FewEnum, which is a restriction of P ....

[Article contains additional citation context not shown here]

T. Gundermann, N. Nasser, and G. Wechsung. A survey of counting classes. In Proceedings of the 5th Annual Conference on Structure in Complexity Theory, pages 140--153. IEEE Computer Society Press, July 1990.

No context found.

T. Gundermann, N. A. Nasser, and G. Wechsung. A survey on counting classes. In Proceedings 5th Structure in Complexity Theory Conference, pages 140-153. IEEE Computer Society Press, Los Alamitos, 1990.

No context found.

T. Gundermann, N. Nasser, and G. Wechsung, A survey on counting classes. In Proceedings of the 5th Annual Conference on Structure in Complexity Theory, (1990), 140-153.

First 50 documents

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