J. Kobler, U. Schoning, S. Toda, and J. Toran. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272--286, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....polynomial ambiguity nondeterministic Turing machines operating under any polynomial time computable counting acceptance mechanism (see [CH90] for full details, or see Definition 2. 1 for a simple alternate definition characterization of the class) It is known that UP O(1) FewP Few SPP ( KSTT92,FFK94] In this section we prove that every nontrivial counting property of circuits is Few hard, thus raising the lower bound. We first prove that for any nontrivial property A, there exists a predicate Q such that at least one of Counting(A) and Counting(A) is m hard for USATQ , where, ....

J. Kobler, U. Schoning, S. Toda, and J. Toran. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272--286, 1992.

....[1] the class FewP was introduced (see also [4] the class was called FNP in [1] FewP is the class of all sets accepted by NP machines such that, if input x is accepted, the number of accepting paths is polynomial in the length of x. FewP has also been studied by a number of other authors (e.g. [7, 11, 21, 26]) One of the main results in [4] and [1] was Theorem 11 [4] Every sparse set in P is P printable i# there are no sparse sets in FewP P. Since the upward separation technique is useful in showing that questions about sparse sets in NP are equivalent to questions about NE, it was natural to ....

J. Kobler, U. Schoning, S. Toda, and J. Toran, Turing machines with few accepting computations and low sets for PP, Proc. 4th Structure in Complexity Theory Conference, 1989, pp. 208--215.

....Corollary 4 is due to Gundermann, Nasser, and Wechsung [6] Both of those corollaries also follow from Beigel, Reingold, and Spielman s (personal communication, 1990) very recent discovery that P PP[log ] PP. Corollary 5 is related to a result of Kobler, Schoning, Toda, and Tor an [9], who showed that the class Few [4] is contained in C= P. These two results have been unified by Aspnes, Beigel, Furst, and Rudich (personal communication, 1990) who show that FewEnum is contained in C= P. Finally, it should be noted that our result contrasts in an interesting way with Toda s ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. JCSS, 44(2):272--286, 1992.

....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 ....

Johannes Kobler, Uwe Schoning, Seinosuke Toda, and Jacobo Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44, 1992. To appear.

....2 BPP path . Corollary 3.3 BPP path is closed under complementation, intersection, and union. Since NP is contained in BPP path , it follows that the closure of NP under truth table reductions is contained in BPP path . Corollary 3.4 P NP[log] BPP path . It is known that BPP is low for PP [KSTT92] and for itself [Ko82,Zac82] i.e. PP BPP = PP and BPP BPP = BPP. We show in the next theorem that BPP is also low for BPP path . Observe that relative to Beigel s previously mentioned oracle making P NP not contained in PP, we must also have that NP, and hence BPP path , cannot be low for ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272--286, 1992.

.... PP L PP UP A = PP A . This contradicts the fact that Delta P;A 2 6 PP A . Note that PP UP A = PP A follows from the fact that UP A is low for PP A for all sets A. This is just a relativization of the recent result by Kobler, Schoning, Toda, and Tor an that UP is low for PP[KSTT92]. We extend this level 1 result by constructing an oracle set A relative to which no languages in UP are in any level of the high hierarchy. On the other hand, considering the fact that UP is low for PP, it is not entirely unreasonable to think that UP might belong to the low hierarchy for NP; ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2), 1992.

....Hence, if UP R p b (R p c (SPARSE) then Left (A) is in R p b (R p c (SPARSE) and by Theorem 3.4 it follows that A is in P. Theorem 3.6 If FewP is contained in R p b (R p c (SPARSE) then P = Few. Proof By a similar proof as above it can be inferred that P = FewP. Since Few P FewP [KSTT] it follows that P = Few. Theorem 3.7 If PP is contained in R p b (R p c (SPARSE) then P = PP. Proof Consider the PP complete set fhx; mi j there are at least m satisfying assignments for xg which has exactly the required properties of left sets. Under the assumption that this set is in R p ....

....can be similarly proved. Theorem 5.4 If there is a solution of (1SAT; SAT) in R p b (R p c (SPARSE) then there is a solution of (1SAT; SAT) in P. We need the following lemma for the next corollary. Lemma 5.5 Let L be a solution of (1SAT; SAT) then Few P L . Proof Since Few P FewP [KSTT] it suffices to show that FewP is contained in P L . Let A be a set in FewP via some nondeterministic machine M . Let p be the polynomial bounding the number of accepting paths of M . Consider the following NP set B. B = fhx; ii j M(x) has at least i accepting paths g Let acc M (x) denote the ....

J. Kobler, U. Schoning, S. Toda, J. Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences. To appear. Preliminary version appears as [KSTT89].

....values of i and j, h(x; i; j) is equal to 0. It now follows easily that any LogFew language is in L SPL . It was observed in [AR98] that L SPL is equal to SPL. It is perhaps worth noting that Theorem 5. 2 is in some sense the logspace analog of the inclusion Few SPP, which was proved in [KSTT92]. Their proof relies on the fact that, for any #P function f and any polynomial time function g that is bounded by a polynomial in n, the function i f(x) g(x) j is in #P. Note that, in contrast, this closure property is not known to hold for #L or GapL functions (although it is shown in ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences 44(2): 272--286, 1992.

....i; j) is equal to 1. For all other values of i and j, h(x; i; j) is equal to 0. It now follows easily that any LogFew language is in L SPL , which is equal to SPL. It is perhaps worth noting that Theorem 5. 2 is in some sense the logspaceanalog of the inclusion Few SPP, which was proved in [KSTT92]. Their proof relies on the fact that, for any #P function f and any polynomial time function g that is bounded by a polynomial in n, the function Gamma f(x) g(x) Delta is in #P. Note that, in contrast, this closure property is not known to hold for #L or GapL functions (but compare this ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for pp. Journal of Computer and System Sciences, 44:272--286, 1992.

....3.11 (implicit in [KST89] spanP 1 = #P 1 if and only if every tally NP set is in UP. Using Lemma 3.11, we show that spanP 1 and #P 1 are different classes unless NE = UE, or unless every sparse set in NP is low for SPP. A set S is said to be C low for some class C if C S = C (see, e.g. [Sch83,KS85,Sch87,KSTT92] for a number of important lowness results) In particular, it is known that every sparse NP set is low for P NP [KS85] and for PP [KSTT92] but it is not known whether all sparse NP sets are low for SPP. Tor an s result that in some relativized world there exists some sparse NP set that is not ....

....unless NE = UE, or unless every sparse set in NP is low for SPP. A set S is said to be C low for some class C if C S = C (see, e.g. Sch83,KS85,Sch87,KSTT92] for a number of important lowness results) In particular, it is known that every sparse NP set is low for P NP [KS85] and for PP [KSTT92], but it is not known whether all sparse NP sets are low for SPP. Tor an s result that in some relativized world there exists some sparse NP set that is not contained in PhiP [Tor88] and thus not in SPP, may be taken as some evidence that not all sparse NP sets are SPP low. Since Corollary 3.12 ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272--286, 1992.

....Corollary 4 is due to Gundermann, Nasser, and Wechsung [6] Both of those corollaries also follow from Beigel, Reingold, and Spielman s (personal communication, 1990) very recent discovery that P PP[log ] PP. Corollary 5 is related to a result of Kobler, Schoning, Toda, and Tor an [9], who showed that the class Few [4] is contained in C= P. These two results have been unified by Aspnes, Beigel, Furst, and Rudich (personal communication, 1990) who show that FewEnum is contained in C= P. Finally, it should be noted that our result contrasts in an interesting way with Toda s ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. JCSS, 44(2):272--286, 1992.

....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 ....

Johannes Kobler, Uwe Schoning, Seinosuke Toda, and Jacobo Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44, 1992. To appear.

....follows from the proof of [VV 86] showing that SAT is reducible via probabilistic reductions to the unique satisfiability problem . 8. Phi Delta Phi Delta C = Phi Delta C. 9. Phi DeltaP Phi DeltaP = Phi DeltaP. PZ 83] Thus Phi DeltaP is closed under p T . 10. PP BPP = PP. [KSTT 89] If the underlying class C is closed under positive reducibility (see [Sc 87] then the usual techniques of amplification can be used to exponentially reduce the error probability for sets in the class BP Delta C. Thus for such classes C, the following inclusions and equalities hold. It ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an, Turing machines with few accepting computations and low sets for PP, Proc. 4th IEEE Structure in Complexity Theory Conference, pp. 208--215.

.... DeltaP) O(1) P (# const DeltaP) 1] Thus, the definition of Const is more analogous to the definition of Few than one might realize at first glance. It is well known that UP = UP1 UP2 Delta Delta Delta UP O(1) FewP Few SPP (the final containment is due to Kobler et al. KSTT92] and clearly UP O(1) Const Few. SPP plays a central role in much of complexity theory (see [For97] and in particular is closely linked to the closure properties of #P [OH93] Regarding relationships with the polynomial hierarchy, P UP FewP NP, and Few P FewP (so Few P NP ) ....

....is UP O(1) hard (indeed, is even UP O(1) p 1 tt hard 4 ) Our proof applies a constant setting technique that Cai and Hemaspaandra (then Hemachandra) CH90] introduced to prove that FewP PhiP. Their approach has previously been used in a number of other papers about counting classes [KSTT92,BHR96] Proof of Theorem 2.4: Let A be a nonempty proper subset of N. The paper of Borchert and Stephan [BS96] see Theorem 1.4 above) and, using different nomenclature, earlier papers [GW87,CGH 89] have shown that (a) if A is finite and nonempty, then Counting(A) is p m hard for coNP, ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272--286, 1992.

....is a solution of (1SAT; SAT) in R p bT (R p c (SPARSE) then (1SAT; SAT) has a solution in P, implying that Few = P and USAT 2 co NP. Proof The first implication follows along the lines of the previous theorem. The consequence Few = P follows from [To91] using the containment Few P FewP [KSTT] and USAT 2 co NP follows from [JT93] ....

J. Kobler, U. Schoning, S. Toda, J. Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272-286, 1992.

....which is still large enough to contain Few. We also show that SPP consists of exactly those languages low for GapP, and thus SPP languages are low for any gap definable class. These results unify and improve earlier disparate results of Cai Hemachandra [7] and Kobler, Schoning, Toda, Tor an [15]. We show further that any countable collection of languages is contained in a unique minimum gap definable class, which implies that the gap definable classes form a lattice under inclusion. Subtraction seems necessary for this result, since nothing similar is known for the #P definable ....

....of an NP machine. In section 4 we will look at complexity classes defined in terms of the gap of an NP machine. Some classes such as PP , C=P , and PhiP have very simple characterizations in this manner. In particular, in section 5 we study a class SPP , alluded to but not specifically named in [15]. This class has also been studied independently by Ogiwara Hemachandra [19] under the name XP , and by Gupta [12] under the name ZUP . We show that SPP , the gap analog of UP , is the smallest of all reasonable gap definable classes. SPP languages are exactly the low sets for GapP (that is, L 2 ....

[Article contains additional citation context not shown here]

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272--286, 1992.

....This mechanism 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 ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272-- 286, 1992.

....closure under union. The remainder of the proof is the same as the proof of Theorem 30. 9. Fewness In [7] Cai and Hemachandra proved a surprising result: the class Few (defined below) is a subset of MOD k P for every k 2 (that result has been generalized by Kobler, Schoning, Toda, and Tor an [14]) In this section, we prove a similar result: FewP MODZ k P for every k 2. Though this result can also be obtained by a close inspection of Cai and Hemachandra s proof, our proof will be simpler. Finally we show that every language in the class Few is as easy as distinguishing categorical ....

J. Kobler, U. Schoning, S. Toda, and J. Tor'an. Turing machines with few accepting computations and low sets for PP. J. Comput. Syst. Sci., 44, 1992. To appear.

No context found.

J. Kobler, U. Schoning, S. Toda, and J. Toran. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272--286, 1992.

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J. Kobler, U. Schoning, S. Toda, J. Tor'an. Turing machines with few accepting computations and low sets for PP. In Proceedings of the 4th Structure in Complexity Theory Conference, pages 208--215. IEEE Computer Society Press, June/July 1989.

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J. Kobler, U. Schoning, J. Tor'an, and S. Toda. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences 44, 1992, 272--286.

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J. Kobler, U. Schoning, S. Toda, J. Toran, Turing machine with few accepting computations and low sets for PP, in Proc. 4th Ann. IEEE Structures in Complexity Theory Conf., 1989, 208-215.

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