S. Toda, "On Polynomial-Time Truth-Table Reducibility to C=P Sets", Colloquium, Department of Computer Science, University of Chicago, October 26, 1990. 18  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the role of NP. For the first result, recall that truth table reductions to NP are exactly as powerful as Turing reductions with logarithmically many queries, i.e. P NP O(log(n) GammaT = P NP tt ( 7] 10] Here we find the same is true of C= P. This result has also been found by Toda [19]. The result and the proof reported here were obtained independently 1 . The proof we give is typical of those that use the closure properties shared by NP and C 6= P. Theorem 5 P C=P O(log(n) GammaT = P C=P tt . Proof: The inclusion from left to right is straightforward: the query tree ....

S. Toda, "On Polynomial-Time Truth-Table Reducibility to C=P Sets", Colloquium, Department of Computer Science, University of Chicago, October 26, 1990. 18

.... 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. ....

S. Toda. On polynomial-time truth-table reducibility to C=P sets. Seminar, University of Chicago, October 26, 1990.

.... time nondeterministic Turing machines (for 2 the precise de nition of C= P, see Section 4) It is well known that C=P is closed under coNP m reductions and p dtt reductions, so it is of type (II) We obtain the equivalence results concerning C= P, which was rst proved by Toda[26], and recently observed by Green[10] Furthermore, we prove that C=P and coC=P are closed under polynomial time positive Turing reductions. This paper is organized as follows: In section 2, we will investigate reducibility notions equivalent to polynomial time constant round truth table ....

....p logT (C= P) L tt[O(1) C= P) L logT (C= P) L T (C= P) Corollary 4.7 C=P is closed under p pos reductions. Corollary 4.8 coC=P is closed under p pos reductions. The equivalence p tt[O(1) C= P) p logT (C= P) was independently proven by Green[10] and by Toda[26]. The last two corollaries are extensions of Proposition 4.5 (2) 5 Conclusion In this paper, we gave sucient conditions for closure under positive Turing reductions and for equivalence among p tt[O(1) reducibility, p logT reducibility, L tt[O(1) reducibility, L logT ....

S. Toda, On polynomial-time truth-table reducibility to C=P sets, colloquium, at Dept. of Computer Science, Univ. of Chicago, October, 1990.

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