C. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and de nitions. It also contains some basic upward scaling results that are proven using a padding Note that this bound on the rate of growth is satis ed by the function log n. This is similar to the optimal algorithm for NP, that uses the self reducibility of NP complete languages [Lev73, Sch76]. argument. Section 3 contains the observation that the existence of a BPP complete language implies a hierarchy theorem for BPtime, along with some corollaries. Section 4 contains our main result, which is a hierarchy theorem for slightly non uniform probabilistic Turing machines. 2 ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In S. Michaelson and R. Milner, editors, Third International Colloquium on Automata, Languages and Programming, pages 322-337, University of Edinburgh, 20-23 July 1976. Edinburgh University Press.

....The special case k = 1 of the preceding corollary was obtained independently by Amir and Gasarch [2] The result above follows for all sets that are NP hard under 1 tt reductions, by Observation 29(ii) Below, we see that the same method of proof applies to all ttself reducible sets. In [26], Schnorr defined self reducibility. Definition 37 A set A is self reducible if there is a polynomial time bounded oracle Turing machine M such that ffl All strings queried by M are strictly shorter than the input string. ffl The language accepted by M (machine M computing with oracle A) is ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

....B;P) In particular FQ(2 ; B;P) FQ(k; B;P) so F k 2 FQ(k; B;P) ii. Using the same proof as of Observation 2.2. 15(ii) we see that (8n k) FQ k (n; B;P) FQ k (k; B;P) B;P) FQ k (k; B;P) so k 2 FQ k (k; B;P) FQ(k; B;P) Self reducible sets were defined by Schnorr in [Sch76]: Definition 5.4.4 A set B is self reducible if there exists a polynomial time bounded oracle machine M such that for every string x the machine M determines whether x is in B by querying only strings that are shorter than x. We say that a set B is self tt reducible if we can determine in ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

....algorithms. Nor does our technique apply to more general self reductions, such as those used for QBF. 5.1. Implementing Recursion with Bounded Nondeterminism Recursive algorithms for NP problems usually take the form of d self reductions ( d for disjunctive) Self reductions were defined in [27] and d self reductions were defined in [28] Definition 3. Let jyj denote the size of the problem instance y. A partial order OE is polynomial well founded if there exists a polynomial bounded function p such that 5 ffl ym OE Delta Delta Delta OE y 1 ) m p(jy 1 j) ffl ym OE Delta Delta ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

....if we can compute the size of subtrees, then we can balance nondeterministic search trees, which reduces the amount of nondeterminism needed. Meyer and Paterson [21] formalized the notion of self reductions. Recursive algorithms for NP problems often take the form of disjunctive self reductions [21, 34] (henceforth, simply dsr ) We will show how to convert T (n) time recursive algorithms of this type to T (n) volume, polynomial time molecular algorithms. In an earlier version of this paper [6] we favored tight exposition over tight time bounds. In order to firm up the time bounds in this ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

....to Proposition 2.7. 6 Corollary 2.10 1. IP[ PhiP] MIP[ PhiP] BPP PhiP . 2. IP[PP] MIP[PP] BPP PP . As shown in [LT92] the graph isomorphism problem GI is low for MA if it is contained in P=poly. This follows from the result of Lozano and Tor an mentioned above and the fact [GMW85, Schn76] that GI and its complement are both contained in IP[GI] Using Theorem 2.8 we get the following improvement. Corollary 2.11 If GI is contained in P=poly then GI is low for MIP[P=poly] Finally, we observe that MIP[GI] IP[GI] In general, if both L and L have an IP[L] protocol, then every ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In 3rd Int. Colloq. Automata, Languages and Programming, Edinburgh, University Press, 322-337.

....time is only polynomial in j( G)j. 8 5.1 Almost uniform sampling and approximate counting Jerrum, Valiant and Vazirani [31] have shown that there is a close connection between almost uniform sampling and approximate counting. In particular, for self reducible combinatorial structures [46], a fully polynomial almost uniform sampler exists if and only if an FPRAS exists. We will not give a formal de nition of self reducible but intuitively it means that outputs corresponding to a given input can be expressed in terms of outputs corresponding to smaller inputs. That is, the ....

C.P. Schnorr, Optimal algorithms for self-reducible problems, in Proceedings of the 3rd International Colloquium on Automata Theory, Languages and Programming (1976), 322-337.

....(x)g is infinite. We also say that P holds infinitely often in S. Time Constructible As in [6] we say that a computable function f : N N is time constructible if and only if there is a program p which computes f(n) using some reasonable representation of numbers) in time O(f(n) Schnorr, in [10], terms such functions linearly honest . Constant Slowdown Timed Interpreter An interpreter is a program int such that given a program p and a d 2 D we have [ int] p:d) p] d) A timed interpreter is a program tint takes as input a program p, a value d 2 D and a time bound t 2 N (in some ....

....satisfiable the extend A with the assignment x : true, set E : E true and go to step 3. Else extend A with the assignment x : false, set E : E false and go to step 3. It is not hard to see that this algorithm solves the search problem for SAT in time O(jEj Theta time pSAT (E) Schnorr, in [10], formalizes the above idea and applies it to other problems. In the rest of this section we shall review some of his results. Schnorr s notation is very different from ours, so to avoid complicated technicalities we will restrict our definitions and theorems to important special cases of his ....

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C. P. Schnorr. Optimal Algorithms for Self-Reducible Problems. In S. Michaelson and R. Milner, editors, Third International Colloquium on Automata, Languages and Programming, pages 322--337, University of Edinburgh, 20--23 July 1976. Edinburgh University Press.

....a RAM in polynomial time p there is a RAM M inverting h such that for every RAM M 0 inverting h there is a constant c 0 with time M (y) c Delta(time M 0 (y) p(jM 0 (y)j) for every y in the range of h. Note that the brief note [9] contains no proof. Proofs of Theorem 1 can be found in [14, 6] (where the latter article uses the Kolmogorov Uspensky machine model also used in [9] It is noted in [14] that one can transfer the result to the Turing machine model if one replaces the term c Delta ( in the above theorem by c 0 Delta ( Delta log( To study the ....

....constant c 0 with time M (y) c Delta(time M 0 (y) p(jM 0 (y)j) for every y in the range of h. Note that the brief note [9] contains no proof. Proofs of Theorem 1 can be found in [14, 6] where the latter article uses the Kolmogorov Uspensky machine model also used in [9] It is noted in [14] that one can transfer the result to the Turing machine model if one replaces the term c Delta ( in the above theorem by c 0 Delta ( Delta log( To study the existence of optimal algorithms in a more machine independent fashion it is suitable to use the following ....

[Article contains additional citation context not shown here]

Claus-Peter Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the third International Colloquium on Automata, Languages, and Programming (ICALP'76), pages 322--337. Edinburgh University Press, 1976.

....from G(r; n) then whp GENERATE is a polynomial time generator for HAM(G) Given a polynomial time generator for a set X one can usually estimate its size. This notion is made precise in Jerrum, Valiant and Vazirani [10] The results there are based on the notion of self reducibility (Schnorr [14]) which we do not have here. On the other hand, our method of proof does lead to an FPRAS (Fully Polynomial Randomised Approximation Scheme) for almost every G 2 G(r; n) An FPRAS for HAM(G) is a randomised algorithm which on input ; 0 produces an estimate Z such that Pr Z ....

C.P.Schnorr, Optimal algorithms for self-reducible problems, Proceedings of the Third International Colloquium on Automata, Languages and Programming (1976) 322-337.

....from G(r; n) then whp GENERATE is a polynomial time generator for HAM(G) Given a polynomial time generator for a set X one can usually estimate its size. This notion is made precise in Jerrum, Valiant and Vazirani [10] The results there are based on the notion of self reducibility (Schnorr [14]) which we do not have here. On the other hand, our method of proof does lead to an FPRAS (Fully Polynomial Randomised Approximation Scheme) for almost every G 2 G(r; n) An FPRAS for HAM(G) is a randomised algorithm which on input ffl; ffi 0 produces an estimate Z such that Pr fi fi fi fi ....

C.P.Schnorr, Optimal algorithms for self-reducible problems, Proceedings of the Third International Colloquium on Automata, Languages and Programming (1976) 322-337.

.... Gamma1 ) produces a random feasible solution Z for x such that the distribution of Z is within variation distance ffl from the uniform distribution on the feasible solutions for x. 9 Jerrum, Valiant and Vazirani proved [11] the following result: Theorem 4 If relation R is self reducible[12] then the following two conditions are equivalent: ffl There exist a f.p. almost uniform generator for R. ffl There exist a f.p. randomized approximate counter for the counting version of R. 6 Notes A reference for the material in section 1 is [13] Section 3.2 is from [5] To bound the ....

C.P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the Third International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

....; B;P) FQ(k; B;P) so F B 2 k 2 FQ(k; B;P) ii. Using the same proof as of Observation 2.2. 15(ii) we see that (8n k) FQ k (n; B;P) FQ k (k; B;P) In particular FQ k (2 k ; B;P) FQ k (k; B;P) so F B 2 k 2 FQ k (k; B;P) FQ(k; B;P) Self reducible sets were defined by Schnorr in [Sch76]: Definition 5.4.4 A set B is self reducible if there exists a polynomial time bounded oracle machine M such that for every string x the machine M B determines whether x is in B by querying only strings that are shorter than x. We say that a set B is self tt reducible if we can determine in ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

....Paths. A non simple path in a graph is a path which may visit any vertex or any edge more than once. In this section an RNC algorithm for uniformly generating polynomial length non simple paths based on a counter is given. Crucially the problem is shown to be self reducible (in the sense of [21]) Let A be the adjacency matrix of G. It is well known (see [10] for example) that the (i; j) th entry of A k , the k th power of A, is the number of non simple paths from vertex i to vertex j of length k (A k may be found in time S(n) lg k using n 3 = lg n processors where S(n) is the ....

C.P. Schnorr. Optimal Algorithms for Self-Reducible Problems. 3rd International Colloquium on Automata, Languages and Programming, pages 322--337, 1976.

....The special case k = 1 of the preceding corollary was obtained independently by Amir and Gasarch [2] The result above follows for all sets that are NP hard under 1 tt reductions, by Observation 29(ii) Below, we see that the same method of proof applies to all ttself reducible sets. In [26], Schnorr defined self reducibility. Definition 37 A set A is self reducible if there is a polynomial time bounded oracle Turing machine M such that ffl All strings queried by M are strictly shorter than the input string. ffl The language accepted by M A (machine M computing with oracle A) is ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

....to any set B such that A p T B. 2 This result was first stated (without proof) in [10] crediting the current authors. We subsequently learned that [36] obtained the result independently. Many natural sets, e.g. all NP complete sets in [31] are self reducible, as defined by Schnorr in [62]. We use an alternative definition which is more general and is implicit in the literature. It was first introduced by [56] and a variation of it is in [48] Definition 3.14. Let p be a function and OE be an ordering on Sigma . The ordering OE has p bounded chains if whenever xm OE xm Gamma1 ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proc. of the 3th ICALP, pages 322--337, 1976.

....inefficient. However, if we can compute the size of subtrees, then we can balance nondeterministic search trees, which reduces the amount of nondeterminism needed. Recursive algorithms for NP problems usually take the form of d selfreductions ( d for disjunctive) Self reductions were defined in [28] and d selfreductions were defined in [29] Definition3. Let jyj denote the size of the problem instance y. A partial order OE is polynomial well founded if there exists a polynomial bounded function p such that ym OE Delta Delta Delta OE y 1 ) m p(jy 1 j) y m OE Delta Delta ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proc. 3rd ICALP, pp. 322--337, 1976.

....if we can compute the size of subtrees, then we can balance nondeterministic search trees, which reduces the amount of nondeterminism needed. Meyer and Paterson [21] formalized the notion of self reductions. Recursive algorithms for NP problems often take the form of disjunctive self reductions [21, 34] (henceforth, simply dsr ) We will show how to convert T (n) time recursive algorithms of this type to T (n) volume, polynomial time molecular algorithms. In an earlier version of this paper [6] we favored tight exposition over tight time bounds. In order to firm up the time bounds in this ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

....be recognized in polynomial time. We begin with a review of two constructions from the literature. The first one shows that the Graph Isomorphism problem is self computable, in the sense that given GI as an oracle, we can construct an isomorphism between two isomorphic graphs in polynomial time [18, 25]. We reproduce the proof of this well known theorem because we need to make references to the construction in the proof and because we need to estimate the sizes of the graphs queried. Lemma 4.1 There exists a polynomial time Turing machine using GI as an oracle which finds an isomorphism between ....

C. P. Schnorr. Optimal Algorithms for Self-Reducible Problems. In Proceedings of the 3rd International Conference on Automata, Language, and Programming (ICALP), pp. 322-337, 1976.

....and consider the behaviour of the ic measure under polynomial time reductions. In Section 4 we study the very interesting class of sets IC[log, poly] fA : for some constant c and polynomial t, ic t (x : A) c log jxj c for all xg: We show that for any polynomially self reducible [23, 28] set A, and also for any set that is p btt hard for NP, A 2 IC[log, poly] if and only if A 2 P. We also relate the new class to the advice complexity classes P log and P poly defined by Karp and Lipton [14] by showing that P log ( IC[log, poly] P poly: Thus, our result about the instance ....

....instance complexity (w.r.t. polynomial time bounds) on the other hand, the instance complexity of any set can grow at most linearly. In this section, we study the class of sets with logarithmically bounded instance complexity. Our main result is that if a polynomially self reducible set [23, 28] has logarithmic instance complexity then it is in P. Consequently SAT, and by application of Propositions 3.5 and 3.6, any set that is p btt hard for the class NP, can have logarithmic instance complexity only if P = NP. We also show that our class of sets lies properly between the advice ....

[Article contains additional citation context not shown here]

Schnorr, C.P. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming (Edinburgh, July). Edinburgh Univ. Press, Edinburgh, 1976, pp. 322--337.

....in Structural Complexity is SAT. This set is defined as follows: SAT = fOE j OE is a satisfiable quantifier free boolean formulag When any variable x in a formula OE is substituted by i 2 f0; 1g, a new formula is obtained, denoted by OEj x: i . An important property of SAT is self reducibility [23, 25]. Informally, SAT is self reducible because the satisfiability problem of a boolean formula is reducible to the satisfiability problem of smaller formulas: if x occurs in OE, then OE is satisfiable if and only if at least one of OEj x: 0 and OEj x: 1 is satisfiable. Definition 2 A set A is ....

C. P. Schnorr. Optimal Algorithms for Self-reducible Problems. In Proc. 3rd Colloq. on Automata, Languages, and Programming, pages 322--337, Edinburgh, University Press, 1976.

....This approach to algorithm design is called self reducibility, and has been formulated in many ways in the literature. In its most limited form, an assortment of restrictions are placed on the decision algorithm, its input and the lexicographic position of the output produced (see, for example, [Sc]) In more general forms, input output limitations are eliminated and decision algorithms quite distant from the original problem are permitted (see, for example, FL2] Additional variations exist, some even incorporating randomness or parallelism (see, for example, FF, KUW] It is not ....

C. P. Schnorr, "Optimal Algorithms for Self-reducible Problems," Proceedings, 1976 International Conference on Automata, Programming and Languages (1976), 322-- 337.

....algorithms. Nor does our technique apply to more general self reductions, such as those used for QBF. 5.1. Implementing Recursion with Bounded Nondeterminism Recursive algorithms for NP problems usually take the form of d self reductions ( d for disjunctive) Self reductions were defined in [27] and d self reductions were defined in [28] Definition 3. Let jyj denote the size of the problem instance y. A partial order OE is polynomial well founded if there exists a polynomial bounded function p such that ffl ym OE Delta Delta Delta OE y 1 ) m p(jy 1 j) ffl ym OE Delta Delta ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

....algorithms. Nor does our technique apply to more general self reductions, such as those used for QBF. 3.3.1. Implementing Recursion with Bounded Nondeterminism Recursive algorithms for NP problems usually take the form of d self reductions ( d for disjunctive) Self reductions were defined in [52] and d self reductions were defined in [53] Definition 11. Let jyj denote the size of the problem instance y. A partial order OE is polynomial well founded if there exists a polynomial bounded function p such that ffl y m OE Delta Delta Delta OE y 1 ) m p(jy 1 j) ffl y m OE Delta Delta ....

C. P. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

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C. Schnorr. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming, pages 322--337, 1976.

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