L. Adleman and K. Manders. Reducibility, randomness, and intractibility. In Proceedings of the 9th ACM Symposium on the Theory of Computing, pages 151-163. ACM, New York, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the eciently computable problems with the class of problems solve in probabilistic polynomial time. A whole new area of complexity theory was developed to help understand the power of probabilistic computation. Gill [Gil77] de ned the class BPP to capture this new notion. Adleman and Manders [AM77] de ned the class R that represented the set of problems with one sided randomness the machine only accepts if the instance is guaranteed to be in the language. The Solovay Strassen algorithm puts compositeness in R. Babai introduced the concept of a Las Vegas probabilistic algorithm that ....

L. Adleman and K. Manders. Reducibility, randomness, and intractibility. In Proceedings of the 9th ACM Symposium on the Theory of Computing, pages 151-163. ACM, New York, 1977.

....functions q. ffl A is q p superterse if PF (q(n) Gamma1) tt for all X. ffl A is p superterse if A is q p superterse for all constant functions q. Obviously, if A is p superterse then ffl A is p terse. ffl If A 1 tt B then B is p superterse. The classes UP and R were defined in [55] and [2, 48] respectively. Definition 3. ffl UP is the class of languages accepted by a polynomial time bounded nondeterministic Turing machine such that at most one path accepts each input. ffl R is the class of languages accepted by a polynomial time bounded nondeterministic Turing machine such that ....

L. Adleman and K. Manders. Reducibility, randomness, and intractibility. In Proceedings of the 9th Annual ACM Symposium on Theory of Computing, pages 151--153, 1977.

....reductions. However, the definition of the randomized reduction was never quite satisfying because the probability of a correct reduction can approach zero as the length of the formula increases. The discrepancy between the Valiant Vazirani definition and the earlier Adleman Manders [AM77] definition has been noted previously [Joh85] This column reflects on recent results about the complexity of USAT and of D P which shed a new light on the meaning of completeness under randomized reductions. For example, it is pointed out that, under randomized reductions, USAT is complete ....

....under randomized reductions 1 . However, this variety of randomized reduction is not quite satisfying, because the probability of the reduction being correct can approach zero as the length of x increases. One would have expected a probability bound of 1=2 (in keeping with the AdlemanManders [AM77] definition) The justification for the Valiant Vazirani definition is that in many situations the probability bound can be amplified, in which case, the definitions would be equivalent. Before we continue, we need to introduce some notation and terminology to facilitate our discussion of ....

L. M. Adleman and K. Manders. Reducibility, randomness, and intractibility

....is stated in terms of a parameter c 2 N. While it seems inappropriate to spend a great deal of time giving rigorous definitions of the complexity theoretic notions used in this paper, it seems worthwhile to provide some guidance. More information on these notions may be found in [Gil77] AHU74] [AM77], and [GJ79] We assume the concept of a polynomial time computable function is understood. A computational problem C is thought of as a set of pairs hx; S x i, where x is an input for which an output is desired and S x is the set of possible correct outputs on input x. For example C1 = fhn; ....

....gcd(a; n) 1. Hence if C1 is in P, then one can solve C4 in deterministic polynomial time for almost all inputs. See also Rem20 86 . Rem4 94 Since C1 is now known to be in R (see Rem1 94 ) it follows that C4 is also in R. C4 also has cryptographic applications [Sch91] BM92] oC91] Ref4 [AM77] 5 Integer factoring C5 Input n 2 N. Output p 1 ; p 2 ; p k 2 Primes and e 1 ; e 2 ; e k 2 N such that n = k Y i=1 p e i i if n 1 : O5a Is C5 in P O5b Is C5 in R Rem5 86 Another classical problem, mentioned by Gauss in his Disquisitiones Arithmeticae (see Rem1 86 ) ....

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Leonard M. Adleman and K. Manders. Reducibility, randomness, and intractibility. In Proc. 9th Annual ACM Symposium On Theory Of Computing, pages 151--163, New York, 1977. Association for Computing Machinery.

....ffl A is q p superterse if PF A q(n) T 6 PF X (q(n) Gamma1) tt for all X. ffl A is p superterse if A is q p superterse for all constant functions q. Obviously, if A is p superterse then ffl A is p terse. ffl If A 1 tt B then B is p superterse. The classes UP and R were defined in [55] and [2, 48] respectively. Definition 3. ffl UP is the class of languages accepted by a polynomial time bounded nondeterministic Turing machine such that at most one path accepts each input. ffl R is the class of languages accepted by a polynomial time bounded nondeterministic Turing machine such that ....

L. Adleman and K. Manders. Reducibility, randomness, and intractibility. In Proceedings of the 9th Annual ACM Symposium on Theory of Computing, pages 151--153, 1977.

....coNP(B) holds. This is a generalization of so called fl reducibility, which can be understood as a strong nondeterministic many one reducibility. Several (number theoretic) problems have been shown to be fl complete (hence SNP T complete) and are not known to be NP complete in the usual sense [2, 3, 32, 33] (see also [14] Long [30] has systematically studied these strong nondeterministic reducibilities. It is easily seen that A SNP T B if and only if NP(A) NP(B) see [39, 30] Therefore, our first comparison with highness immediately follows: SNP T hardness is exactly the same as ....

....that SAT RP m UniqueSAT (UniqueSAT contains all Boolean formulas that have a unique satisfying assignment; in fact, SAT reduces to every solution A of the promise problem (1SAT; SAT) see [28] for a more extensive discussion of this topic) Therefore, UniqueSAT is RP m hard for NP . In [2, 3, 42] some problems are shown to be coRP T complete (in fact, coRP m complete) for NP . These problems are not known to be NP complete in the usual sense. Furthermore, in [38] a tally language is shown to be coRP T complete (in fact, coRP m complete) for all sparse sets in NP . The ....

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L. Adleman and K. Manders. Reducibility, randomness, and intractibility. In Proceedings of the 9th ACM Symposium on Theory of Computing, 151--163. ACM Press, 1977.

....3.7 1. NPSV t = FP NP guardedly strong . 2. NP = NP NPMV guardedly total, guardedly single valued . 3. NPSV t = NPSV t ) NPMV guardedly total, guardedly single valued . We will use the fl reductions of Adleman and Manders, which are the same as manyone strong nondeterministic reductions [AM77,Lon82] We say that A fl B if there is a nondeterministic Turing machine N (which on each accepting path p outputs some value, call it output(x; p) such that (i) for each x, N(x) has at least one accepting path, and (ii) for each x, and each accepting path p of N(x) it holds that x 2 A ( ....

L. Adleman and K. Manders. Reducibility, randomness, and intractibility. In Proceedings of the 9th ACM Symposium on Theory of Computing, pages 151-- 153, 1977.

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