Michael Sipser. A Complexity Theoretic Approach to Randomness. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pages 330--334, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the polynomial time hierarchy. We show that SBP is located exactly between Babai s [Bab85] Arthur Merlin classes MA and AM. In particular, it is contained in the class 2 of the polynomial time hierarchy. In the proof we use similar arguments on linear hash functions as in the proof for BPP PH [Lau83, Sip83]. Furthermore, we show that BPP SBP BPP path (cf. Figure 1) On the basis of collapse consequences for the polynomial time hierarchy and on the basis of oracle constructions we give evidence that SBP does not coincide with known complexity classes like BPP, BPP path , MA, AM and AWPP. A ....

....= h i (y i ) If Collision(X; H) then we say that X has a collision w.r.t. H . The set of all families H = fh 1 ; h l g of l linear hash functions from to is denoted by H(l; m; k) In 1983 Sipser proved the following useful theorems about linear hash functions. Theorem 3. 6 ([Sip83, Coding Lemma]) Let X be a set of cardinality at most 2 . If we choose a hash family H uniformly at random from H(k; m; k) then the probability that X has a collision w.r.t. H is at most 1=2. So if the set X is not to big then collision does not occur to often. An easy pigeon hole argument shows ....

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th Symposium on Theory of Computing, pages 330--335, 1983.

....to N 1 and right to N 2 . So, the promise R(N;x) states that on input x, N 1 and N 2 behave like an RP machine and that either N 1 or N 2 accepts but not both. Q(N;x) holds if N 1 has an accepting path on input x. Next we recall the definitions and basic properties of hashing that we need. Sipser [25] used universal hashing, originally invented by Carter and Wegman [11] to estimate (probabilistically) the size of a finite set X of strings. A linear hash function h from is given by a Boolean (k; m) matrix (a ij ) and maps any string x = x 1 : xm to a string y = y 1 : y k , ....

....in X can be tested in P . We denote the set of all families H = h 1 ; h k ) of k linear hash functions from by H(k; m) As observed by Sipser, the size of a set X can be estimated by checking for which values of k, X is hashable by some hash family H 2 H(k; m) Lemma 5.3. [25] No hash family H 2 H(k; m) can hash a set X of cardinality jX j k2 . Furthermore, if jX j 2 , then some hash family H 2 H(k; m) hashes X . The next two lemmas make use of Stockmeyer s refinement of the hashing technique [26] Their proofs are straightforward (see, e.g. ....

M. Sipser. A complexity theoretic approach to randomness. In Proc. 15th ACM Symposium on Theory of Computing, pp. 330--335. ACM Press, 1983.

.... on (x, r) If we can at least check membership in R , we can reduce the error probability to zero by picking a coin flip sequence r at random until we get one in R (of which there are many) and then running on (x, r) The first interesting application of this approach is due to Sipser [34]. His proof that BPP lies in the polynomial time hierarchy uses = KD poly polynomial time bounded distinguishing complexity. The corresponding set R lies in coNP. The hardness versus randomness tradeoffs by Babai et al. 5] and by Impagliazzo and Wigderson [17] can be cast as an application ....

M. Sipser. A complexity theoretic approach to randomness. In ACM Symposium on Theory of Computing (STOC), pages 330--335, 1983.

....Problems and solving it brings a 1,000,000 prize from the Clay Mathematics Institute [Cla00] Quite surprisingly, one of the earliest discussions of a particular NP complete problem and the implications of nding an ecient solution came from Kurt G odel. In a 1956 letter to von Neumann [Har86, Sip83] G odel asks von Neumann about the complexity of what is now known to be an NP complete problem concerning proofs in rst order logic and asks if the problem can be solved in linear or quadratic time. In fact, G odel seemed quite optimistic about nding an ecient solution. He fully realized that ....

....show that primality is in R and thus ZPP. Very recently, Agrawal, Kayal and Saxena [AKS02] gave a deterministic polynomial time algorithm for primality. If this result was known in the 70 s, perhaps the study of probabilistic algorithms would not have progressed as quickly. In 1983, Sipser [Sip83] showed that BPP is contained in the polynomial time hierarchy. G acs (see [Sip83] improves this result to show BPP is in the second level of the hierarchy and Lautemann [Lau83] gives a simple proof of this fact. One can also consider probabilistic space classes. Aleliunas, Karp, Lipton, ....

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 330-335. ACM, New York, 1983.

....computation takes # g( p ) steps . Natural choices for g would be: polynomials, functions of order flog 2 f, where f is a polynomial, or functions of order 2 cn etc. For information on the use of these complexity measures in computer science, the reader may consult the references [36] 59] and [90] 2a . 5.2 Kolmogorov s program In [50,34] Kolmogorov writes The idea that randomness consists in a lack of regularity is thoroughly traditional. But apparently only now has it become possible to found directly on this simple idea precise formulations of conditions for the applicability ....

* M. Sipser, A complexity theoretic approach to randomness, Proc. 15 ACM Symp. Th. Comp. (1983), 330-335.

....hierarchy theorems hold for fully uniform probabilistic time. We show (using the standard diagonalization techniques) that if BPP has a complete problem (under a speci c notion of completeness de ned below) then BPtime(n ) for every constant d. By applying the Sipser Lautenman Theorem [Sip83, Lau83], we obtain that if BPP NP then BPP does have a complete problem (under this notion) and so in this case BPtime(n ) As a corollary we obtain that BPtime(n) 6= NP. 1.2 Techniques Our main technique is the use of an instance checker [BK95] for an EXP complete language in order to ....

.... BPtime(f(t(n) for every time constructible super polynomial function f :N N (Because the language K we de ned will be in BPtime(f(t(n) but not in BPtime(t(n) In particular, under this assumption it holds that BPtime(n) Remark 3.8. The proof of the Sipser Lautemann Theorem [Sip83, Lau83], implies that the promise problem CAP is reducible to a language L in the polynomial hierarchy (actually in 2 ) This implies that if BPP NP then BPP has a complete language (by De nition 3.1) since BPP NP implies that BPP PH and so that L 2 BPP. Combining this with Theorem 3.6, we get ....

Michael Sipser. A complexity theoretic approach to randomness. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 330-335, Boston, Massachusetts, 25-27 April 1983.

....functions from A to B. H is called a universal family of hash functions if for every x 1 6= x 2 2 A and y 1 ; y 2 2 B we have that Apart from the value of such families of hash function for various hashing purposes as proposed in [9] many new applications have been found. For example, Sipser [20] used universal hashing to obtain simulation of BPP in the polynomial time hierarchy; Goldwasser and Sipser [13] used it for the simulation of Interactive Proof systems by Arthur Merlin games; Impagliazzo and Zuckerman [14] used universal hashing for the ampli cation of the probability of ....

M. Sipser. A complexity theoretic approach to randomness. In Proceedings Boston, Massachusetts, pages 330-335, May 1983.

....This class lies between Delta and naturally generates a symmetric hierarchy corresponding to the polynomial time hierarchy. We demonstrate, using the probabilistic method, new containment theorems for BPP. We show that MA (and hence BPP) lies within S 2 , improving the constructions of [10, 8] (which show that BPP ae Sigma 2 ) Symmetric alternation is shown to enjoy two strong structural properties which are used to prove the desired containment results. We offer some evidence that S 6= Sigma 2 by demonstrating an oracle so that S 2 6= Sigma 2 assuming that the ....

....E mail address: koods theory.lcs.mit.edu. Supported by grants NSF 92 12184, AFOSR F49620 92J 0125, and DARPA N00014 92 J 1799 less tractable: the only (non trivial) relationships depend on unproven complexity theoretic assumptions ( 7, 12] We continue the study initiated by Sipser [10] of the relationship between randomness and quantification, that is, the relationship between BPP and classes arising by appropriate quantification of polynomial time predicates (e.g. NP, coNP and other classes in the polynomial time hierarchy [11] BPP was first shown to lie in the ....

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 330--335, 1983.

....following theorem. Theorem 12: The following learning tasks can be accomplished with deterministic polynomialtime algorithms that have access to a Sigma 3 oracle: n) O(sn n log n) Our tools are: Lemma 1, from Section 3; Theorem 13, due to Sipser (who also credits P. G acs) [S83], stated below; and Theorem 14, due to Stockmeyer [S85] stated below. is an algorithm that, on input f and I, decides whether or not f 2 C I in time polynomial in jIj and s. Then there exists a polynomial p and a predicate P such that, on input I, a; b 2 f0; 1g , r 2 f0; 2 g, and ffi ....

Michael Sipser. A Complexity Theoretic Approach to Randomness. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pages 330--334, 1983.

....and Vazirani s construction creates random subspaces of the assignments. Mulmuley, Vazirani and Vazirani [MVV87] give an alternate proof looking at maximal weighted cliques after putting random weights on the edges. Buhrman and Fortnow [BF97] show how Lemma 3. 1 follows from earlier work by Sipser [Sip83] on Kolmogorov complexity. Gupta [Gup97] gives a construction for Lemma 3.1 that improves the probability to a constant if we only require f(OE) to have an odd number of assignments. Attempts at a relativized counterexample to Hypothesis 1.2 have a long history. Rackoff [Rac82] gives a ....

M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 330--335. ACM, New York, 1983.

....is a completely trivial derandomization of BPP. The best (to our taste) way to understand the origin of the iterated application of the function t G in the result above, is explained in the recent paper [3] which further simplifies the proof of [2] They remind the reader that Sipser s proof [8] putting BPP in Sigma Pi actually gives much more. In fact, viewed appropriately, it almost begs (with hindsight) the use of hitting sets The key is, that in both the 89 and 98 expressions for the BPP language, the witnesses for the existential quantifier are abundant. Put ....

M. Sipser. A complexity-theoretic approach to randomness. In 15th STOC, pages 330--335, 1983.

....s 2k Gamma1 j c: The following section deals with the aforementioned subprotocols which prove upper and lower bounds on the size of sets. 4. Upper and Lower Bound Protocols Let us first consider a subprotocol for proving lower bounds on the size of sets. It is based on a lemma of Sipser s [S] and uses universal hashing [CW] Suppose C Sigma where membership in C is testable in polynomial time. The protocol can easily be modified to work when membership in C is testable in nondeterministic polynomial time but this will not concern us here. Let H be a k Theta b Boolean matrix ....

....probability at least 1 Gamma (jCj Gamma 1) 2 since for a fixed c and w in Sigma , the probability that h(w) h(c) is 1=2 . The proof of (2) is as follows. Let S be the number of elements of C which map to z. Then by a simple argument P r[V accepts] Sigma j=1 P r[S = j] P r[S i] for all i. Again we will apply Chebychev s inequality. Note that (S) 1 2 b and oe . So, for i P r[S i] P r[ Gamma S Gamma i] Gamma i) For i = 3 we get P r[V accepts] 3 5)2 jCj Gamma 1 5. Proof of Main Theorem In this section we give the ....

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Sipser M., "A Complexity Theoretic Approach to Randomness," Proc. of 15th Symposium on Theory of Computing, pp 330--335, Boston, 1983. 18

.... r) If we can at least check membership in R , we can reduce the error probability to zero by picking a coin flip sequence r uniformly at random until we get one in R (of which there are many) and then running on (x, r) The first interesting application of this approach is due to Sipser [Sip83] His proof that BPP lies in the polynomial time hierarchy uses =KD poly , the polynomial time bounded distinguishing complexity. The corresponding set R lies in coNP. The hardness versus randomness tradeo#s by Babai et al. BFNW93] and by Impagliazzo and Wigderson [IW97] can be cast as an ....

M. Sipser. A complexity theoretic approach to randomness. In ACM Symposium on Theory of Computing (STOC), pages 330--335, 1983.

.... sufficiently long interval is fi, etc; there are, of course, more sophisticated tests like the chi square test see [Knu97] From the point of 1 There were some earlier instances of derandomization in a complexity theoretic setting: Sipser s simulation of BPP in the polynomial time hierarchy [Sip83] and Stockmeyer s approximate counting algorithm in the polynomialtime hierarchy [Sto83] both using pairwise independent hash functions. Another example of the use of pairwise independent sampling arises in the crypto complexity work of Chor and Goldreich [CG84] 2 view of algorithmic ....

M. Sipser. A complexity-theoretic approach to randomness. In Proc. 15th Annual ACM Symposium on the Theory of Computing, pages 330--335, 1983.

....hierarchy collapses. This follows from the result of Boppana, Hastad, and Zachos [BHZ87] that if coNP AM then the polynomial hierarchy collapses to its second level. Since coNP BPP path , we get the same consequence from the assumption that BPP path is contained in AM. Sipser and G acs ( Sip83] see also [Lau83] showed that BPP R NP . It is an open question whether the same inclusion holds for BPP path . However, we show that BPP path BPP NP . As a first step, we show that a BPP path set can be decided by a deterministic polynomial time Turing machine making logarithmically ....

....polynomial time Turing machine making logarithmically many queries to a Sigma p 2 oracle, and hence BPP path is in the polynomial hierarchy. A randomized version of this algorithm can decide a BPP path set with an NP oracle. The proof applies Sipser s Coding Lemma for universal hashing [Sip83] We mention that we could get a shorter proof by applying the results of Stockmeyer [Sto85] to approximate #P functions and of Jerrum, Valiant, and Vazirani [JVV86] who showed a probabilistic version of Stockmeyer s theorem. However, we prefer to give a self contained proof here, thereby ....

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 330--335, 1983.

....z from Z; and if x ## L then Z has irrefutable proof z which can withstand any challenge y from Y . The motivation by both Canetti [C96] and Russell and Sundaram [RS95] was to provide a refinement of the Sipser Lautemann Theorem (with contribution by Gacs) that BPP # # p 2 # # p 2 [Si83, L83]. Indeed, Canetti [C96] extended Lautemann s proof to show that BPP # S p 2 , whereas Russell and Sundaram [RS95] showed further that MA # S p 2 . Note that BPP # MA is direct from definition (the two sided error version) of MA, thus BPP # MA # S p 2 . Also it is known that P NP ....

....field GF[2 n ] Here is an outline of the proof of Lemma 1. First we will use hash functions and the SAT oracle to get an approximate count of the subset S =n . The estimation can be done in a number of ways; we give a self contained account using the notion of isolation of Sipser. See [Si83, St83, JVV86]. If this set is polynomially small, then we can handle it trivially. Suppose it is large. Then we will devise a simple sampling strategy based on an estimate of points with unique inverse images from S =n under a random hash function. The details follow. First we handle the trivial case where ....

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M. Sipser, A Complexity Theoretic Approach to Randomness. STOC 1983: 330-335.

....we make considerable new progress. Computing the number of distinct elements is a fundamental 2 problem in its own right, besides its role as a primitive in streaming algorithm design. Surprisingly, however, this problem has not been fully solved. Alon et al. AMS99] based on the hashing idea of [Sip83, Sto83, FM85], show how to produce an approximator f F 0 in the streaming model, so that F 0 =c f F 0 cF 0 , for any constant c 2. We rst show (Section 4) how to build on the algorithm of [AMS99] to obtain similar approximations for the more interesting case where c = 1 for any 0. Our ....

M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th Annual ACM Symposium on the Theory of Computing (STOC), pages 330-335, 1983.

....perfect and statistical knowledge complexity is close enough for our result to hold for statistical knowledge complexity as well. In our second result which relates the knowledge complexity and the error probability we also employ techniques for deterministic bounds on set sizes developed in [Si 83, St 83, 4 JVV 86, BP 92] 1.6 Organization In Section 2 we give the de nitions and notations we use in the paper. In Section 3 we present our AM protocol for proving the entropy of a samplable distribution. In Section 4 we provide an overview of the construction in [AH 91] and explain why it ....

....is simple, but we do not know how to approximate log(Prob M [c] Prob (PM ;V) c] in polynomial time. By Equation 8 in Section 6, this approximation comes down to approximating set sizes. Note that all these sets which need to be approximated are recognizable in polynomial time. It is shown in [Si 83, St 83, BP 92] how to approximate the cardinality of a set S, which is recognizable in polynomial time, using ecient probabilistic computation with access to an NP oracle. The approximation there fails with negligible probability to give an approximation with relative accuracy 1 1 poly . We ....

M. Sipser. A Complexity Theoretic Approach to Randomness. Proceedings of the 15th Annual ACM Symposium on the Theory of Computing, ACM (1983).

....important applications (see [LV97, Chapter 6] Early in the history of computational complexity theory, many people naturally looked at resource bounded versions of Kolmogorov complexity. This line of research was initially fruitful and led to some interesting results. In particular, Sipser [Sip83] invented a new variation of resourcebounded complexity, CD complexity, where one considers the size of the smallest program that accepts one specific string and no others. Sipser used CD complexity for the first proof that BPP is contained in the polynomial time hierarchy. Complexity theory has ....

....theory has matured a bit, we ought to look back at resource bounded Kolmogorov complexity and see what new results and applications we can draw from it. First, we use algebraic techniques to give a new upper bound lemma for CD complexity without the additional advice required of Sipser s lemma [Sip83]. With this lemma, we can approximately measure the size of a set using CD complexity. We obtain better bounds on CD complexity using extractor graphs. These graphs are usually used for derandomization. However these improved bounds only apply to most of the strings. We also give a new simpler ....

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 330--335, 1983.

....the circuit B 0 with the polynomially small error. Next, we describe the approximator construction of Buhrman and Fortnow [BF99] This construction is much simpler than the ones presented in [ACR98, ACRT99] Buhrman and Fortnow observed that Lautemann s proof [Lau83] for Sipser s famous result [Sip83] that BPP P 2 ( NP NP ) actually proves that BPP RP RP . 3 Therefore, a double de randomization of RP (one of the oracle and one of the oracle machine) using two hitting sets de randomizes BPP . Given a circuit B 0 on n inputs with a 1 4n error, Lautemann builds a new circuit B ....

M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15 th Annual ACM Symposium on Theory of Computing (STOC), pages 330-335, 1983.

....under the assumption NP P=poly. The proof is not difficult and just a combination of known techniques, but the result as such has not been observed before, and we think it has some significance. In both figures the relative position of the classes NP BPP and BPP NP is also outlined. By [La83, Si83] (used in a relativized version) BPP NP is included in the class ( Sigma P 2 Pi P 2 ) NP = Sigma P 3 Pi P 3 . By the fact that PH = Sigma P 2 = Pi P 2 holds under the assumption NP P=poly, the class BPP NP is a subset of Sigma P 2 = Pi P 2 in Figure 2. It is still open ....

M. Sipser. A complexity theoretic approach to randomness. Proc. 15th ACM Symp. Theory of Computer Science (1983) 330--335. 5

....predicates would be good deter 5 This explanation is a bit oversimplified: our idea works as described only if X is a samplable flat distribution. For non flat distributions, a more sophisticated reduction is needed, which involves the use of approximate counting algorithms with an NP oracle [Sip83, Sto85, JVV86] ministic extractors. This looks like the standard problem of worst case to average case reduction, as solved in [BFNW93, Imp95, IW97, STV99] and observed to extend to NP circuits in [KvM99] However, in all such results, one gets predicates that are hard to predict with an ....

....a multivariate polynomial code and a Hadamard code, and is analyzed by providing a listdecoding procedure for the polynomial code and using the Goldreich Levin [GL89] list decoding procedure for the Hadamard code. We show that the use of approximate counting (implementable with an NP oracle [Sip83, Sto85, JVV86] can greatly improve the efficiency of the list decoding algorithm for the polynomial code. But we do not know whether a similar improvement is possible for the Hadamard code. Instead, we show how to use approximate counting and uniform sampling (also using an NP oracle [JVV86, ....

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Michael Sipser. A complexity theoretic approach to randomness. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 330--335, Boston, Massachusetts, 25--27 April 1983.

....languages accepted by polynomial time probabilistic Turing machines that have zero error probability for inputs not in the language, and error probability bounded by some # 1 2 for inputs in the language. It follows immediately from the definitions that P#R#BPP#NP, and Lautemann [25] and Sipser [31] showed that BPP# # P 2 # # P 2 . 2 Unambiguous computation was introduced by Valiant as a moderate form of nondeterminism. Definition 2 ( 33] 1. A nondeterministic Turing machine is unambiguous if, for every input, the machine has at most one accepting computation (accepting path) 2. ....

M. Sipser. A complexity theoretic approach to randomness. In 15th ACM Symposium on Theory of Computing, pages 330--335, 1983.

....j Delta c: The following section deals with the aforementioned subprotocols which prove upper and lower bounds on the size of sets. 4. Upper and Lower Bound Protocols Let us first consider a subprotocol for proving lower bounds on the size of sets. It is based on a lemma of Sipser s [S] and uses universal hashing [CW] Suppose C Sigma k where membership in C is testable in polynomial time. The protocol can easily be modified to work when membership in C is testable in nondeterministic polynomial time but this will not concern us here. Let H be a k Theta b Boolean matrix ....

....at least 1 Gamma (jCj Gamma 1) 2 b since for a fixed c and w in Sigma k , the probability that h(w) h(c) is 1=2 b . The proof of (2) is as follows. Let S be the number of elements of C which map to z. Then by a simple argument P r[V accepts] Sigma jCj j=1 1 j P r[S = j] P r[S i] 1 i for all i. Again we will apply Chebychev s inequality. Note that (S) 1 jC j Gamma1 2 b and oe 2 . So, for i P r[S i] P r[ Gamma S Gamma i] Gamma i) 2 : For i = 2 3 p 5 we get P r[V accepts] 3 p 5)2 b jCj Gamma 1 : 5. Proof of Main Theorem ....

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Sipser M., "A Complexity Theoretic Approach to Randomness," Proc. of 15th Symposium on Theory of Computing, pp 330--335, Boston, 1983. 18

....is proved by a nonrelativizing technique, it is provable that MA exp # co MA exp (a subclass of ZPEXP(NP) contains non P poly sets [12, 36] Some explanation of how our work builds on prior techniques is in order. The proof of our lowness result heavily uses the universal hashing technique [13, 34] and builds on ideas from [2, 14, 24] For the design of a zero error probabilistic algorithm which, with the help of an NP oracle, simulates a given ZPP(NP(A) computation (where A is a self reducible set in P poly) we further make use of the newly defined concept of half collisions. More ....

....will heavily make use of the hashing technique which has been very fruitful in complexity theory. Here we review some notations and facts about hash families. We also extend the notion of collision by introducing the concept of a half collision which is central to our proof technique. Sipser [34] used universal hashing, originally invented by Carter and Wegman [13] to decide (probabilistically) whether a finite set X is large or small. A linear hash function h from # m to # k is given by a Boolean (k, m) matrix (a ij ) and maps 314 JOHANNES K OBLER AND OSAMU WATANABE any string x = ....

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M. Sipser, A complexity theoretic approach to randomness, in Proceedings of the 15th ACM Symposium on Theory of Computing, ACM Press, New York, 1983, pp. 330--335.

....that in fact many strings have large depth. Resource bounded Kolmogorov complexity gives us perhaps our most interesting notion of computational depth. Classical resource bounded Kolmogorov complexity considers the shortest program that produces a given string in a small amount of time. Sipser [Sip83] considers the shortest program that distinguishes a string from all other strings. We consider the di erence of these two measures which nicely captures the di erence of search versus decision. We show that honest easily computable injective functions preserve depth. We also show that if a large ....

....Complexity, we could try to nd the shortest program that accepts only x. In the unbounded world the two measures coincides, as, we could run through all possible strings until nd one accepted by the program, and print it out. This new measure of complexity was rst used in 1983 by Sipser [Sip83], who used it to show that BPP is in the polynomial hierarchy. De nition 2.5 ( Sip83] Let U be some xed universal Turing machine, for any pair x; y 2 f0; 1g , the t time bounded Kolmogorov distinguishing complexity of x relative to y is de ned as CD t (xjy) min p 8 : jpj : U(p; ....

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 330-335, 1983. 13

....parameter. Let h : f0; 1g n Theta f0; 1g n f0; 1g mn Gamma2e n be a universal hash function. Let X 2D f0; 1g n , Y 2U f0; 1g n , and Z 2U f0; 1g mn Gamma2e n . Then L 1 (hh Y (X) Y i; hZ; Y i) 2 Gamma(e n 1) This lemma is a generalization of a lemma that appears in [S83]. There, D is the uniform distribution on a set S f0; 1g n with ]S = 2 mn . The papers [McIn87] and [BBR88] also proved similar lemmas. For the special case of linear hash functions, this lemma can be derived from [GL89] by considering unlimited adversaries. A generalization to a broader ....

Sipser, M., A Complexity Theoretic Approach to Randomness, 15 th ACM Symp. on Th. of Comp., 1983, pp. 330--335. A pseudorandom generator from any one-way function 33

....the confidence of detecting disconnectivity. Actually, we use a slightly more efficient procedure for increasing the confidence in this test) 3 Approximating the Size of a Connected Component One method for approximating the size of a connected component is based on hashing, as used by Sipser [13] in a different context. Suppose we want to distinguish between the case that the component of edge e is small (has less then s edges) or large (more that 8s edges) 1. Select a random hash function h : E [2s] where E is the name space for edges, and [2s] denotes the integers from 1 to 2s) ....

....of outputting small is at least 1=2. If the component is large, we may use the result of [3] to show that with high probability, the random walk visits at least 8e edges (regardless of the size and shape of the connected component) If h is chosen from a universal family of hash functions [5, 13], then it is likely that the hashed value of some visited vertex agrees with r. A different method for approximating the size of a connected component uses the fact that the expected time for a random walk to return to an edge is 2s, where s is the number of edges of the connected component. ....

M. Sipser. "A Complexity Theoretic Approach to Randomness". In 15th STOC, 330-- 335, 1983.

....is a completely trivial derandomization of BPP . The best (to our taste) way to understand the origin of the iterated application of the function t G in the result above, is explained in the recent paper [3] which further simpli es the proof of [2] They remind the reader that Sipser s proof [8] putting BPP in 2 2 actually gives much more. In fact, viewed appropriately, it almost begs (with hindsight) the use of hitting sets The key is, that in both the 89 and 98 expressions for the BPP language, the witnesses for the existential quanti er are abundant. Put di erently, BPP ....

M. Sipser. A complexity-theoretic approach to randomness. In 15th STOC, pages 330-335, 1983.

....Let h : f0; 1g n Theta f0; 1g n f0; 1g mn Gamma2e n be a universal hash function. Let X 2D f0; 1g n , Y 2 U f0; 1g n , and Z 2 U f0; 1g mn Gamma2e n . Then L 1 (hh Y (X) Y i; hZ; Y i) 2 Gamma(e n 1) 21 This lemma is a generalization of a lemma that appears in [Sip : 83] There, D is the uniform distribution on a set S f0; 1g n with ]S = 2 mn . The papers [McIn : 87] and [BBR : 88] also proved similar lemmas. For the special case of linear hash functions, this lemma can be derived from [GL : 89] by considering unlimited adversaries. A generalization to a ....

Sipser, M., "A Complexity Theoretic Approach to Randomness", 15 th ACM Symposium on Theory of Computing, 1983, pp. 330--335.

....various complexity classes under nondeterministic polynomial time reductions. We consider three well studied measures of Kolmogorov complexity that lie in between C p (x) and C t (x) for p a polynomial and t(n) 2 n k . We consider the distinguishing complexity as introduced by Sipser [Sip83] The distinguishing complexity, CD t (x) is the size of the smallest program that runs in time t(n) and accepts x and nothing else. We show that the set of random strings R CD t = fx j CD t (x) jxjg, for t a xed polynomial is hard for MA under nondeterministic reductions. MA is the ....

....[LV97] by C(xjy) minfjpj : U(p; y) xg. We de ne unconditional Kolmogorov complexity by C(x) C(xj ) Hartmanis de ned a time bounded version of Kolmogorov complexity in [Har83] but resource bounded versions of Kolmogorov complexity date back as far as [Bar68] See also [LV97] Sipser [Sip83] de ned the distinguishing complexity CD t . We will need the following versions of resource bounded Kolmogorov complexity and distinguishing complexity. CS s (xjy) min 8 : jpj : 1) U(p; y) x (2) U(p; y) uses at most s(jxj jyj) tape cells 9 = See [Har83] CD t ....

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M. Sipser. A complexity theoretic approach to randomness. In Proc. 15th ACM Symposium on Theory of Computing, pages 330-335, 1983.

....generation procedure. The second sequence of works deals with the two related problems of approximating the size of sets and uniformly generating elements in them. These problems were related by Jerrum et al. JVV 86] Procedures for approximating the size of sets were invented by Sipser [Si 83] and Stockmeyer [St 83] and further improved in [GS 89, AH 87] all using the hashing paradigm . The same hashing technique, is the basis of the universal extrapolation procedures of [ILu 90, ILe 90] However, the output of these procedures deviates from the objective (i.e. uniform ....

....(Theorem 2) of the subsequent section, we get the Main Theorem. Theorem 1 PKC(O(log n) BPP NP 7 Our proof follows the procedure suggested in [BP 92] which in turn follows the approach of [F 89, BMO 90, Ost 91] while introducing a new uniform generation procedure which builds on ideas of [Si 83, St 83, GS 89, JVV 86] see introduction) Suppose that (P; V ) is an interactive proof of perfect knowledge complexity k( Delta) O(log n) for the languages L, and let M be the simulator guaranteed by the fraction formulation (i.e. Definition 2.2) We consider the conversations of the original ....

M. Sipser. A Complexity Theoretic Approach to Randomness. Proceedings of the 15th Annual ACM Symposium on the Theory of Computing, ACM (1983).

....is a completely trivial derandomization of BPP . The best (to our taste) way to understand the origin of the iterated application of the function t G in the result above, is explained in the recent paper [3] which further simplifies the proof of [2] They remind the reader that Sipser s proof [8] putting BPP in Sigma 2 Pi 2 actually gives much more. In fact, viewed appropriately, it almost begs (with hindsight) the use of hitting sets The key is, that in both the 89 and 98 expressions for the BPP language, the witnesses for the existential quantifier are abundant. Put ....

M. Sipser. A complexity-theoretic approach to randomness. In 15th STOC, pages 330--335, 1983.

.... the most important unsolved questions are the following: Does the polynomial time hierarchy have in nitely many levels , or Is any class de ned above included in the polynomial time hierarchy In an attempt to settle these questions, there have been found many relationships between them [3, 4, 9, 18, 29, 31, 32, 33, 34]. Nevertheless, until now, neither of the above questions is solved. On the other hand, it is widely known that we can classify sets into some categories by using di erent reducibilities to sets of small density [6, 15, 16] Especially, a set having a census function bounded above by some ....

....probability 1=3 [11] The following relationships between BP operator and the polynomial time hierarchy are widely known. Theorem 5.10 1. 27] For every k 1, BP P k P k 1 . 2. 31, 33] PH BP C=P, PH BP PP, and PH BP MOD k P, where k 2. 3. 27] BPP = BP P. 4. [18, 29] BPP P 2 P 2 . From Proposition 2.8, Theorem 5.1 and Theorem 5.10, we obtain the following corollaries. Corollary 5.11 Let K be any class chosen from fMOD 2 P, MOD 3 P, g. If K has sparse SN btt hard sets, then PH P 2 P 2 . Corollary 5.12 Let K be any class chosen ....

M. Sipser, A complexity theoretic approach to randomness, Proceedings of the 15th Annual Symposium on Theory of Computing (ACM, 1983) 330-335.

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Michael Sipser. A Complexity Theoretic Approach to Randomness. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pages 330--334, 1983.

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 330-335, 1983.

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M. Sipser. A Complexity Theoretic Approach to Randomness. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 330--335, Boston, Massachusetts, 25--27 Apr. 1983.

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M. Sipser. A complexity-theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 330--335, 1983.

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M. Sipser. A complexity-theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 330--335, 1983. 24

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M. Sipser, A complexity-theoretic approach to randomness, Proceedings of the 15th ACM Symposium on the Theory of Computing, 1983, pp. 330-335.

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M. Sipser. A Complexity Theoretic Approach to Randomness. In 15th ACM Symposium on the Theory of Computing, pages 330-335, 1983.

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Sipser, M., "A Complexity Theoretic Approach to Randomness", 15 pp. 330-335.

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M. Sipser. A Complexity Theoretic Approach to Randomness. Proceedings of the 15th Annual Symposium on the Theory of Computing, ACM, 1983.

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 330--335, 1983.

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M. Sipser. A complexity theoretic approach to randomness. In Proc. 15th ACM Symposium on Theory of Computing, pp. 330-335. ACM Press, 1983. 25

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M. Sipser,A complexity theoretic approach to randomness. In Proc. 15th ACM Symp. Theory of Computer Science 1983, 330-335. 19

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 330-335, 1983.

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M. Sipser, "A Complexity Theoretic Approach to Randomness", 15th STOC, 1983, pp. 330--335.

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M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on the Theory of Computing, pages 330--335. ACM, New York, 1983. 11

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M. Sipser. \A complexity theoretic approach to randomness", Proceedings of the 15th Annual ACM Symposium on the Theory of Computation, 1983, 330-335.

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