M. Blum and R. Impagliazzo. Generic oracles and oracle classes (extended abstract). In Proceedings of 28th FOCS, pages 118--126, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and familiar notions of decision trees from computational complexity theory. While the results in this paper pertain to models for computation in higher types, the techniques used are closely related to work in Boolean decision tree complexity. Namely, a technique known as Blum s trick ([2], 6] 21] which is used to show that Boolean functions with small nondeterministic and co nondeterministic complexity have smalldepth decision trees is generalized to show that in certain cases, sequential functionals can efficiently simulate continuous functionals. A lower bound on Boolean ....

....is itself C sequential, a construction which is efficient, relative to B and C, is possible. This implies that, for a certain natural class of moduli, continuity and sequentiality do coincide. 4. An efficient simulation The result presented in this section is a generalization of Blum s trick , [2], 6] 21] which relates certificate size and decision tree complexity for Boolean functions, to the case of the type two functionals considered here. A similar proof is given in [8] but for a special case which is described in detail below. We begin with the following simple fact about ....

M. Blum and R. Imagliazzo. Generic oracles and oracle classes. In Proc. 28th Ann. IEEE Symposium on Foundations of Computer Science, pages 118--126, 1987.

....Embedding Conjecture literally to posets. Proposition 28. There exists 3 posets (G; f) and (G but relative to some oracle, the polynomial hierarchy is in nite and NP(G; f) NP(G ) Proof. Let (G; f) be the left and (G ) be the right 3 poset in Figure 5. Blum and Impagliazzo [4] constructed an oracle with NP coNP = P and the polynomial hierarchy strict. In fact, their proof also shows that there is an oracle D such that the polynomial hierarchy is strict and for all disjoint sets A; B 2 NP there is a set C 2 P with A C B. Hence, given an NP homomorphism S ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings 28th Symposium on Foundations of Computer Science, pages 118-126. IEEE Computer Society Press, Washington, D.C., 1987.

.... Note that by our main results, if the negation of a statement L 2 DT IME(f) is not provable in PA 1 , then the statement itself is true (semantically) in N (up to adding the inverse of any Wainer function to f ) Let us also remark that, as any generic oracle (in the sense of Blum Impagliazzo [BI87]) is necessarily a meager language, the last lemma offers an easy proof to to their Theorem 1:5. 5.1 A concrete example of non provability Let us demonstrate the emergence of non provability of a lower bound, by translating the Paris Harrington version of Ramsey Theorem into a decision problem ....

Blum M., and Impagliazzo R., "Generic Oracles and Oracle Classes", 28`th Symposium on Foundations of Computer Science, (1987),118-126.

....NP, and that these generics exist in exponential time. Thus the typical exponential time set (in the sense of polynomial time bounded category) separates P and NP. This is not true for Lutz s original definition of resource bounded category based on strings. We also prove a negative result. In [BI87], Blum and Impagliazzo show that if P = NP (unrelativized) then with respect to a (full) generic set, P = NP co NP and P = UP. We show that the degree of genericity needed for these results is essentially tight; no reasonable notion of resourcebounded genericity is strong enough to guarantee ....

....21 and theorem 17 together show that the notions of Delta St;St meagerness and Delta Lp;Lp meagerness are incomparable. This together with proposition 5 implies that the notion of Delta St;Lp meagerness is strictly more inclusive than both. 13 5 P versus UP and NP co NP In [BI87], Blum and Impagliazzo proved a rather counterintuitive result about generic sets as oracles. They showed, given the assumption P = NP (unrelativized) that P G = UP G = NP co NP) G for any generic G. 2 This result runs against the common wisdom that a generic oracle separates ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science, pages 118--126, 1987.

....meets S if there is a oe OE A with oe 2 S. A strongly avoids S if there is a oe OE A such that for all oe, 62 S. A set G is 1 generic if G either meets or strongly avoids every r.e. set of strings. For more on 1 generic and generic sets and their uses in recursion and complexity theory, see [Joc80, Soa87, BI87, FFKL93]. If M is a time bounded accepting rejecting oracle Turing machine (OTM) and f a partial characteristic function, then M f (x) is defined just in case all of M s oracle queries on input x are in domain(f) M f then denotes the partial characteristic function computed by M with oracle f . The ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, pages 118--126, New York, 1987. IEEE.

....are too weak to resolve these questions. In particular, it is shown that for any set X there is an aw2 generic set G such that NP G co NP G 6 P G PhiX . On the other hand, if G is 1 generic, then NP G co NP G P G PhiSAT , where SAT is the NP complete Satisfiability problem [6]. This result runs counter to the fact that most finite extension constructions in complexity theory can be made effective. These results imply that any finite extension construction that ensures any of the Friedberg analogs must be noneffective, even relative to an arbitrary incomplete r.e. ....

.... unfortunately false, as can be shown by a straightforward forcing argument (see [28] for example) There are, however, a number of results in complexity theory which approximate equation (3) and whose proofs are similar to that of equation (1) The best known of these is due to Blum Impagliazzo [6], which in essence states that for every 1 generic set G, NP G co NP G P G PhiSAT : 4) See Appendix A for a list of the known results of this type. Since equation (4) relates to polynomial time bounded computations, it is natural to ask if it holds for generic sets in some weaker, ....

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M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science, pages 118--126, 1987.

....randomized complexities of f are also called Monte Carlo complexity, and the zero error randomized complexity is called Las Vegas complexity. For every p; we have the following obvious relationships between these complexity measures: R p 2 (f) R p 1 (f) R(f) D(f) Blum and Impagliazzo [1], Hartmanis and Hemachandra [3] and Tardos [11] observed independently that there is at most a quadratic gap between the deterministic complexity and the Las Vegas complexity. More precisely, for every f we have D(f) R(f) 2 : Nisan [7] showed that the deterministic complexity can not ....

M. Blum and R. Impagliazzo (1987), Generic oracles and oracle classes, Proceedings of 18th IEEE FOCS, 118-126.

....there is an oracle set A relative to which P = NP, suggesting that diagonalization reduction cannot be used to separate these two classes. There are nonrelativizing results in complexity theory, as will be mentioned below. It is interesting to note that relative to ageSeL oracle, P #= NP [BI87, SCY97]. A Boole an circuit is a finite acyclic graph in which each non input node, or gate , is labelled with a Boolean connective; typically from AND, OR, NOT . The input nodes are labeled with variables x 1, x n , and for each assignment of 0 or 1 to each variable the circuit computes ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Ashok K. Chandra, editor, Proce0 ings of

....We also identify some of the main remaining open questions. The complexity measures discussed here also have interesting relations with circuit complexity [47,4,7] parallel computing [10,41,31,47] communication complexity [33,9] and the construction of oracles in computational complexity theory [6,43,15,16], which we will not discuss here. The paper is organized as follows. In Section 2 we introduce some notation concerning Boolean functions and multivariate polynomials. In Section 3 we define the three main variants of decision trees that we discuss: deterministic decision trees, randomized ....

....bs(f) 2 . This would be optimal, since the function f of Example 2 has bs(f) p n and D(f) n. Open problem 2 Is D(f) 2 O(bs(f) 2 ) Of course, Theorem 11 also holds with C (0) instead of C (1) Since bs(f) maxfC (0) f) C (1) f)g, we also obtain the following result, due to [6,21,43]. Corollary 2 D(f) C (0) f)C (1) f) Now we will show that D(f) is upper bounded by deg(f) 4 and g deg(f) 6 . The first result is due to Nisan and Smolensky, below we give their (previously unpublished) proof. It improves the earlier result D(f) 2 O(deg(f) 8 ) of Nisan and Szegedy ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes (extended abstract). In Proceedings of 28th FOCS, pages 118--126, 1987.

....complexity or the nondeterministic complexity of f in the black box model. i.e. the minimum number of bits of the input that must be revealed (by someone who knows all the input bits) to convince a deterministic algorithm about the value of f(x) A key result, that was first discovered by Blum [BI], shows that in the black box model the deterministic and nondeterministic (certificate) complexity of a function are polynomially related. Recall that in the black box model we only count the number of queries made by the algorithm, not the number of steps of computation performed by the ....

Blum, M. and Impagliazzo, R., "Generic oracles and oracle classes", 28th Annual Symposium on Foundations of Computer Science, IEEE computer society press, 1987.

....At exponentially far apart lengths, these have exactly one string and are empty elsewhere. Note that theorem 1.3 still works when G is UP coUP generic. Generic oracles turn out to be enormously useful in a variety of situations. For more about them, see the papers by Blum and Impagliazzo [BI87]; Fenner, Fortnow, and Kurtz [FFK92] and Fenner, Fortnow, Kurtz, and Li [FFKL93] For more about UP generic and UP coUP generic oracles, see the paper by Fortnow and Rogers [FR94] 2 Results Theorem 2.1 There is an oracle C such that: 1. NP C = EXP C , so the complete m degree of NP C ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, pages 118--126. IEEE, New York, 1987.

....on all coordinates not in S and such that f(x) 6= f(y) Then, by the minimality of S, for each i 2 S, fig is a set of coordinates to which f is sensitive of input y. Clearly ffigji 2 Sg is a disjoint collection of sets. Since the block sensitivity of f is k, it follows that jSj k. THEOREM 21.28 [BI87, HH86, Tar88] For any function f : D 1 Theta Delta Delta Delta Theta Dn R, decision tree complexity(f) certificate complexity(f) 2 . PROOF First note that if f(x) 6= f(y) then every certificate C for f on input x intersects every certificate C 0 for f on input y. Otherwise, it ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In FOCS, pages 118--126, 1987.

....and familiar notions of decision trees from computational complexity theory. While the results in this paper pertain to models for computation in higher types, the techniques used are closely related to work in Boolean decision tree complexity. Namely, a technique known as Blum s trick ([2], 6] 21] which is used to show that Boolean functions with small nondeterministic and co nondeterministic complexity have smalldepth decision trees is generalized to show that in certain cases, sequential functionals can efficiently simulate continuous functionals. A lower bound on Boolean ....

....is itself C sequential, a construction which is efficient, relative to B and C, is possible. This implies that, for a certain natural class of moduli, continuity and sequentiality do coincide. 4. An efficient simulation The result presented in this section is a generalization of Blum s trick , [2], 6] 21] which relates certificate size and decision tree complexity for Boolean functions, to the case of the type two functionals considered here. A similar proof is given in [8] but for a special case which is described in detail below. We begin with the following simple fact about ....

M. Blum and R. Imagliazzo. Generic oracles and oracle classes. In Proc. 28th Ann. IEEE Symposium on Foundations of Computer Science, pages 118--126, 1987.

....levels in PH are distinct. It follows that all levels in PH are absolutely distinct. Generic sets were introduced by Cohen [4] as a tool for proving independence results in set theory. A general treatment of complexity theory relative to a generic oracle was developed by Blum and Impagliazzo [3], and Fenner et al. [8] contains a recent survey of the subject. 4 In general if two complexity classes coincide relative to a generic oracle, then they coincide absolutely. This follows from a result in [3] reproduced as Theorem 2.2 below. The converse does not always hold, since for example IP ....

....of complexity theory relative to a generic oracle was developed by Blum and Impagliazzo [3] and Fenner et al. [8] contains a recent survey of the subject. 4 In general if two complexity classes coincide relative to a generic oracle, then they coincide absolutely. This follows from a result in [3], reproduced as Theorem 2.2 below. The converse does not always hold, since for example IP = PSPACE [19] 24] but the classes are distinct relative to a generic oracle [9] However, the question of whether a generic oracle separates two classes is natural and robust . In general, one generic ....

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Blum, M., and Impagliazzo, R. (1987), Generic oracles and oracle classes, in "Proceedings, 28th IEEE Symposium on Foundations of Computer Science", pp. 118--126.

....to the functions F , G, and H used in the definition of many one reducibility in the previous subsection. Notice that (CQ) CQ when A 2 P. The following theorem shows that the problem of separating relativized NP search classes is equivalent to separating them relative to any generic oracle [BI87] and also equivalent to showing that there is no reduction between the corresponding type 2 problems. Theorem 1: Let Q 1 ; Q 2 2 TFNP . The following are equivalent: i) Q 1 is many one reducible to Q 2 ; ii) For all oracles A, CQ 1 ) iii) There exists a generic oracle G such that ....

....to an argument due to Riis [Rii93] which is itself similar to the proof that if a Boolean function and its negation both can be written in disjunctive normal form with terms of size d, then the function has a Boolean decision tree of height d . This last result was implicit in [HH87, HH91] BI87] Tar89] and appears explicitly in [IN88] Fix j N 0 1 and let P j be the set of all paths in pigeon trees with leaf label j. Since Case II does not hold, the paths in P j are mutually inconsistent. We describe H j implicitly as a strategy for querying the purported matching ff. The strategy ....

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Manuel Blum and Russell Impagliazzo. Generic oracles and oracle classes. In 28th Annual Symposium on Foundations of Computer Science, pages 118--126, Los Angeles, CA, October 1987. IEEE.

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M. Blum and R. Impagliazzo. Generic oracles and oracle classes (extended abstract). In Proceedings of 28th FOCS, pages 118--126, 1987.

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M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proc. of 28th IEEE FOCS, 1987, 118--126.

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M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the Twenty-Eighth Annual IEEE Symposium on Foundations of Computer Science, pages 118-126, 1987.

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M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the 28th IEEE Symposium on Foundations of Computer science, pages 118-126, New York, 1987. IEEE Computer Societey Press.

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M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the Twenty-Eighth Annual IEEE Symposium on Foundations of Computer Science, pages 118--126, 1987.

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M. Blum and R. Imagliazzo. Generic oracles and oracle classes. In Proc. 28th Ann. IEEE Symposium on Foundations of Computer Science, pages 118--126, 1987.

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M. Blum and R. Impagliazzo (1987). Generic oracles and oracle classes. In: Proc. of 28th IEEE FOCS, 118--126.

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M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, pages 118-126. IEEE, New York, 1987.

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M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In 28th Annual IEEE Symp. Foundations of Computer Science, October 1987.

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M. Blum and R. Impagliazzo, Generic Oracles and Oracle Classes, in Proc. 28 th IEEE Symposium on Foundations of Computer Science, pages 118-126. IEEE, 1987.

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