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.

....both can be computed by Sigma p 3 circuits whereas the minimal 1 b.p. computing f has quasipolynomial size (Theorem 3.4) 1 This result is not stated explicitly in [6] so we describe it more precisely in Theorem 2.1 below. For the analogous result about the depth of decision trees see [4, 9, 17]. 2 2 Decision Trees In this section we establish the announced bounds for decision trees. Recall that a (deterministic) decision tree is a binary tree whose internal nodes have labels from f1; ng and whose leaves have labels from f0; 1g. If a node has label i then one of the outgoing ....

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

....time Turing machines with access to an oracle (see, e.g. 16, page 294] and [11, Section 5. 3] for formal treatment) Bounds on the boolean decision trees complexity are useful tools in constructing oracles with desired relations between Turing complexity classes and in proving conditional results [2, 7, 8]. Conversely, all the facts proven for the corresponding Turing complexity classes that hold true under any oracle can be directly carried over decision trees. We mention three examples. 1. Arthur Merlin games are as powerful as a general interactive proof system [6] 2. The error in an ....

....1 , the maximum number of zeroes of f that differ from some one of f in disjoint blocks of variables. This is a simple extension of the bound r(f) Omega Gamma35 (f) from [12] Note that bound (4) together with relations nd(f) bs(f) bs( f) 5) and d(f) nd(f) nd( f) 6) proven in [12] and [2, 7, 14], respectively, implies the relation d(f) O(ip(f) 2 ip( f) 2 ) which is a qualitative generalization of (1) and (6) We suggest also a bound that is in a sense tighter. Namely, ip(f) sep(f) 2; 7) where sep(f) is a combinatorial characteristic of a boolean function that we call ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proc. of the 28th IEEE Ann. Symp. on Foundations of Computer Science (FOCS), pages 118--126, 1987.

....this conjecture for posets. Proposition 8. Let (G; f) and (G 0 ; f 0 ) be the 3 posets in Fig. 2. Then (G; f) 6 (G 0 ; f 0 ) but there exists an oracle where the polynomial time hierarchy does not collapse to any level and NP(G; f) NP(G 0 ; f 0 ) Proof. Blum and Impagliazzo [BI87] constructed an oracle with NP coNP = P and the polynomial time hierarchy is strict. In fact, their proof also shows that there is an oracle D such that the polynomial time hierarchy is strict and for all disjoint sets A; B 2 NP D there is a set C 2 P D with A C B. Hence, given an NP D ....

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

....function identically 1. Similarly, a maxterm is a set of variables such that a partial assignment to the variables in the set makes the function identically 0, but no partial assignment to a subset of the set makes the function identically 0. A useful fact, which was independently discovered in [BI,HH, Ta], and explicitly stated in [LMN] states that if all the minterms and maxterms of a boolean function f have size at most s and t respectively, then f can be evaluated by a decision tree of depth at most st. Since each branch of the decision tree corresponds to a monomial over the reals, we see ....

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

....make membership queries to an arbitrary oracle set A. However in [BGS75] it was shown that there is an oracle set A relative to which P = NP, suggesting that diagonalization reduction cannot be used to separate these two classes. It is interesting to note that relative to a generic oracle, P 6= NP [BI87, CIY97]. A Boolean circuit is a finite acyclic graph in which each non input node, or gate, is labelled with a Boolean connective; typically from fAND;OR;NOTg. The input nodes are labeled with variables x 1 ; x n , and for each assignment of 0 or 1 to each variable the circuit com7 putes a bit ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Ashok K. Chandra, editor, Proceedings of the 28th Annual Symposium on Foundations of Computer Science, pages 118--126, Los Angeles, CA, October 1987. IEEE Computer Society Press.

....of a boolean function f have size at most t, then for any subset S with jSj t 2 , the Fourier coefficient of f on S, f(S) is equal to 0. 14 Therefore, f = X Sae[n] jSjt 2 f (S) S : The above lemma is proved by using decision trees. The following fact was independently discovered in [BI,HH, Ta], and explicitly stated in [LMN] A relevant fact was observed in [Ha] and our proof is a simple adaptation of the proof there. Lemma 8 If all the minterms and maxterms of a boolean function f have size at most s and t respectively, then f can be evaluated by a decision tree of depth at most ....

M. Blum and R. Impagliazzo. Generic Oracles and Oracle Classes. Proceedings of the 28th Annual Symposium on Foundations of Computer Science, pages 118-126, October 1987

....type: if DNFs F 0 ; F 1 are small and the formula F 0 F 1 is not satisfiable, then given an assignment, we can either certify that F 0 is false on that assignment or certify that F 1 is false on that assignment by probing only a small number of variables. In the known result of this kind [1, 2, 3] the number of probed variables is at most mn, where m;n are maximum fanins of ANDs in F 0 ; F 1 , respectively. Theorem 1 yields a sometimes better bound of 2m ln N 2n ln M , where M;N are the numbers of ANDs in F 0 ; F 1 respectively. Theorem 2. Assume that F 0 is a DNF that is an OR of M ....

M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proc. of the 28th IEEE Ann. Symp. on Foundations of Computer Science (FOCS), pages 118--126, 1987.

.... is the complexity of the optimal tree and is denoted by N T (g) We define similarly, the complexity for ; queries and denote it by N; g) In the classical decision tree model it is folklore that the deterministic complexity is polynomially related to the non deterministic complexity (see also [8, 21, 18]) Here, again we have a complete separation as will follow from the next claim. Claim 5.3 For every Boolean function g : f0; 1g n f0; 1g, N; g) 2. Proof: The proof follows directly form the fact that for 1 inputs of g it is enough to examine a satisfied term in its DNF representation. ....

M. Blum and R. Impagliazzo. Generic oracles and oracles classes. In 28th Annual IEEE Symp. on Foundations Of Computer Science, pages 118--126, 1987.

....one leaf in a tree we can then throw away that leaf and start using the next one. We arrange the tree so that we have more leaves than injuries. Also, this now recursive method can be combined with techniques developed by Racko [Rac82] Hartmanis and Hemachandra [HH91] and Blum and Impagliazzo [BI87] to get P = UP = NP coNP, while still having P NP = NEXP relative to a recursive oracle. We eliminate machines that do not categorically accept on at most one path by forcing the acceptance on one of the leaves of our tree and then using a future leaf for encoding. The relativized world we ....

....super polynomial by virtue of the Cleanup phases. We can compress however by only retaining the single coding string and omitting the other strings added in the Cleanup phase. 16 standard technique developed and used by Racko [Rac82] Hartmanis and Hemachandra [HH91] and Blum and Impagliazzo [BI87] to compute N A i (x) ALGORITHM FOR N A i (x) Compute and w as above Repeatjxj i times If(9 ) w N i (x) 1]Then pick such an extension extend to include all strings in Q(N i (x) answered according to A Endif End Repeat Accept if N i (x) 1 END The algorithm ....

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.

....In this section we will show that relative to sp generics, acceptance of nondeterministic machines with a small number of accepting paths can be decided in polynomial time. Theorem 4.1 If A is an sp generic oracle then P A = FewP A . This proof will build on ideas from Blum and Impagliazzo [BI87], Hartmanis and Hemachandra [HH91] and Rackoff [Rac82] An immediate corollary is: Corollary 4.2 For any sp generic oracle A, P A = UP A . Let R i be the requirement: Either there is some input x such that M A i (x) has more than n i accepting paths, or L(M A i ) 2 P A . By our ....

....at most jxj i accepting paths. By Lemma 3.2 there is an sp condition oe extending such that oe forces SAT 2 P A . Suppose A extends oe. We will show that L(M A ) 2 P A . Consider the following algorithm for computing M A (x) using A as an oracle. The idea is the same as that used in [BI87]. We repeatedly look for some extension ff of the partial oracle (not necessarily compatible with A) which makes M have the maximum possible number of accepting paths. To ensure consistency with A, we then answer all queries in the domain of ff according to A. In the algorithm below, we maintain ....

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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.

....at what happens when we try to combine all of these constructions. Generic oracles as defined in recursion theory (see [15] do exactly this. Generics take all possible definable finite extensions. Thus, relative to any generic G, the polynomial time hierarchy is infinite. Blum and Impagliazzo [3] showed that generics can also collapse some classes. They showed how to use techniques from Rackoff [12] and Hartmanis and Hemachandra [8] to show that if P = NP in the unrelativized universe, then P = UP = NP coNP relative to generic oracles. Though the P = NP assumption seems unlikely, all ....

....class separations. Thus our result also shows that U Delta 0;p k Pi 0;p k 6= Delta 0;p k for all k 2. 2 Definitions In this paper, we will give the definitions necessary to describe our main result. For an overview of standard complexity classes, see [2] for generic oracles, see [3, 6], and for circuits, see [4] We need circuit complexity in order to show our result for k 2. 2.1 Complexity Classes For this paper, fix Sigma = f0; 1g and let Sigma be the set of all finite strings over Sigma. By Sigma n (resp. Sigma n ) we denote the set of strings of length n ....

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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.

....of the main remaining open questions. The complexity measures discussed here also have interesting relations with circuit complexity [Weg87, Bei93, Bop97] parallel computing [CDR86, Sim83, Nis91, Weg87] communication complexity [NW95, BW99] and the construction of oracles in complexity theory [BI87, Tar89, FFKL93, FR98] 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 decision trees, ....

....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 [BI87, HH87, Tar89] 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 ....

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

....input x 2 f0; 1g is either at most 1=4 or at least 3=4. Theorem 21. There is an oracle A such that BPP A = P A , but [1=4 CAPP] A is not solvable by any deterministic polynomial time oracle Turing machine D A . For our oracle construction, we use standard techniques, as found, e.g. in [BGS75, Rac82, Sip82, HH87, BI87, IN88, MV96]. A complete proof of Theorem 21 is given in Appendix B. 6 R.E. or Not R.E. In this section, we consider the problem of recursively enumerating classes AP and APP. We say that AP is recursively enumerable (r.e. if there is a deterministic Turing machine M(x; 1 k ; y) such that, for every ....

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.

....many n. Since G is in P G , we have that L G is not uniformly hard relative to G. Since Yao s proof [27] that the polynomial time hierarchy is infinite relative to some oracle is a finite extension argument, the polynomial time hierarchy is infinite relative to all generics (see also [9]) 2 Uniformly hardness also has an important role in the area of hardness versus randomness [22, 5, 16] One way to accurately state the recent result of Impagliazzo and Wigderson [16] is as follows. Theorem 2.3 (Impagliazzo Wigderson) If a language in E cannot be accepted by any 2 o(n) ....

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.

....of non deterministic query and communication complexities in the classical case, is the tight relation of these complexities with deterministic complexity. In the query complexity (decision tree) setting we have maxfN q (f) N q (f)g D q (f) N q (f)N q (f) This was independently shown in [BI87, HH87, Tar89] We conjecture that something similar holds in the quantum case: max ( ndeg(f) 2 ; ndeg(f ) 2 ) deg(f) 2 Q q (f) D q (f) O(NQ q (f)NQ q (f) O(ndeg(f)ndeg(f ) Here the part is open. This conjecture would imply D q (f) 2 O(Q 0 (f) 2 ) Q 0 (f) is zero error ....

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

....to the functions F , G, and H used in the definition of many one reducibility in the previous subsection. Notice that (CQ) A = 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 2 . The following are equivalent: i) Q 1 is many one reducible to Q 2 ; ii) For all oracles A, CQ 1 ) A (CQ 2 ) A ; iii) There exists a generic ....

....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 2 . This last result was implicit in [HH87, HH91] BI87] Tar89] and appears explicitly in [IN88] Fix j N Gamma 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 ....

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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.

....of size exp(O(log n log 2 N) where N is the total number of monomials in the minimal DNFs for f and :f . This result is not stated explicitly in Ehrenfeucht Haussler (1989) so we describe it more precisely in Theorem 2.1 below. For the analogous result about the depth of decision trees see Blum Impagliazzo (1987), Hartmanis Hemachandra (1987) Tardos (1989) Since nondeterministic decision trees are essentially equivalent to DNFs, this upper bound states that for decision trees we have NP co NP P where P stands for quasipolynomial time. Ehrenfeucht Haussler (1989) asked whether their bound can ....

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

....in order to find an appropriate certificate. This approach will not work for time bounded computability. All is not lost, however. A converse of this implication may be obtained by extending results techniques which deal with 0 1 valued oracles, developed independently by a number of researchers (Blum and Impagliazzo 1987, Hartmanis and Hemachandra 1990, Tardos 1989) to the case of N valued oracles. We will now consider this approach. 4 Decision tree algorithms Continuity is essentially a non deterministic notion. That is, the value of a functional on some input depends on the existence of a certificate. ....

M. Blum and R. Impagliazzo, Generic oracles and oracle classes, in Proc. 28th Ann. IEEE Symp. Found. Comput. Sci., 1987, 118--126.

....needs to query (in the worst case) in order to compute f , R 0 (f) the number of queries for a zero error classical algorithm, and R 2 (f) for bounded error. There is a monotone function g with R 0 (g) 2 O(D(g) 0:753: 40, 37] and it is known that R 0 (f) p D(f) for any function f [5, 18]. It is a longstanding open question whether R 0 (f) p D(f) is tight. We solve the analogous question for monotone functions for the quantum case. Let QE (f) Q 0 (f) Q 2 (f) respectively be the number of queries that an exact, zero error, or bounded error quantum algorithm must make to ....

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

....Separations Lance Fortnow Tomoyuki Yamakami y University of Chicago University of Toronto Department of Computer Science Department of Computer Science 1100 E. 58th St. Toronto, Ontario Chicago, IL 60637 Canada M5S 1A4 1 Introduction In 1987, Blum and Impagliazzo [BI87], using techniques of Hartmanis and Hemachandra [HH91] and Rackoff [Rac82] show that if P = NP then P(G) NP(G) coNP(G) UP(G) where G is a generic oracle. They leave open the question as to whether these collapses occur at higher levels of the polynomial time hierarchy. We give a surprising ....

....oracle A, SAT A is NP A complete. Email: fortnow cs.uchicago.edu. Partially supported by NSF grant CCR 92 53582. y Email: yamakami cs.toronto.edu. In this paper we will just give the definitions necessary to describe generic oracles and circuits. For an overview of generic oracles see [BI87, FFKL93], and for circuits see [BS90] We need circuit complexity in order to show our result for k 2. 2.1 Generic Oracles A condition oe is any function mapping Sigma to f0; 1g with finite domain. A condition extends a condition oe if the domain of oe is contained in the domain of and oe(x) ....

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.

.... extension functions and G is 1 generic, then P G 6= NP G 6= coNP G (because the standard diagonalization argument showing the existence of oracles satisfying this property [BGS75] can be modelled in this way) Different notions of genericity (for different classes 1) have been studied by [Maa82, AFH87, Dow82, Poi86, BI87, Fen91]. Observe that if 1 1 0 , then G 1 0 generic implies G 1 generic. In [BI87] Blum and Impagliazzo promoted the study of complexity classes relative to generic oracles specifically as an alternative to random oracles. They focused primarily on the notion of genericity that results when 1 is ....

....showing the existence of oracles satisfying this property [BGS75] can be modelled in this way) Different notions of genericity (for different classes 1) have been studied by [Maa82, AFH87, Dow82, Poi86, BI87, Fen91] Observe that if 1 1 0 , then G 1 0 generic implies G 1 generic. In [BI87] Blum and Impagliazzo promoted the study of complexity classes relative to generic oracles specifically as an alternative to random oracles. They focused primarily on the notion of genericity that results when 1 is the class of extension functions expressible in the first order theory of ....

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M. Blum and R. Impagliazzo. Generic oracles and oracle classes. In Proc. 28th IEEE Symp. Foundations of Computer Science, pages 118--126, 1987. Time-Bounded Kolmogorov Complexity in Complexity Theory 17

....needs to query (in the worst case) in order to compute f , R 0 (f) the number of queries for a zero error classical algorithm, and R 2 (f) for bounded error. There is a monotone function g with R 0 (g) 2 O(D(g) 0:753: Sni85, SW86] and it is known that R 0 (f) p D(f) for any function f [BI87, HH87] It is a longstanding open question whether R 0 (f) p D(f) is tight. We solve the analogous question for monotone functions for the quantum case. Let Q 0 (f) be the number of queries that a zero error quantum algorithm must make to compute f . For zero error quantum algorithms, there ....

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 FOCS, 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 ....

[Article contains additional citation context not shown here]

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.

....to the functions F , G, and H used in the definition of many one reducibility in the previous subsection. Notice that (CQ) A = 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 Q1 ; Q2 2 TFNP 2 . The following are equivalent: i) Q1 is many one reducible to Q2 ; ii) For all oracles A, CQ1 ) A (CQ2) A ; iii) There exists a generic oracle G ....

....to an argument due to Riis [ 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 2 . This last result was implicit in [HH87] HH90] [BI87], Tar] and appears explicitly in [IN88] We describe H j implicitly as a strategy for querying the purported matching ff. The strategy proceeds in at most k stages, and makes at most 2k queries in each stage. Let s represent the set of known edges of G at the beginning of stage s. Then in stage ....

[Article contains additional citation context not shown here]

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

.... give inclusions or alternative characterizations of the classes defined in [Pap94] All separations we exhibit hold even against Turing reductions so they show oracle separations between the Turing closures of the related type 1 search classes and these separations apply to all generic oracles ([BI87], CY95] 2.3 Some simple reductions It is easy to see that SOURCE:OR:SINK m LEAF , by ignoring the direction information on the input graph. Also it is immediate that SOURCE:OR:SINK m SINK . It is not hard to see that SINK m PIGEON : Let G be the input graph for SINK. The corresponding input ....

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

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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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