H. Buhrman, L. Fortnow, and S. Laplante. Resource-bounded Kolmogorov complexity revisited. SIAM J. Comput., 31(3):887-- 905, 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of memory. Stated another way, we show that short descriptions are often su#cient for NL machines to reproduce large objects of interest. Although the technique is not really new it is nearly two decades old, and was used again recently to prove results about time bounded Kolmogorov complexity [BFL02] it seems that its usefulness in NL is not as widely known as it should be. Supported in part by NSF grant CCR 0104823. allender cs.rutgers.edu #2003 Published by Elsevier Science B. V. A more general goal of this paper is to examine di#erent notions of spacebounded Kolmogorov ....

....KNS U (x) min d s : As above, we define KNS as KNS U , such that for all U # ,wehaveKNS U (x) KNS U # (x) for some constant c. One of the first types of resource bounded Kolmogorov complexity to be studied was distinguishing complexity. For more on the history of this notion, see [BFL02] In [AKRR03] a version of distinguishing complexity was introduced that has the same flavor as Levin s Kt measure: Definition 3.3 Let U be a deterministic Turing machine. Define KDt U (x) to be min d logt : # x U(d, y) runs in time t and accepts i# x = y Again, we have to be careful ....

H. Buhrman, L. Fortnow, and S. Laplante. Resource-bounded Kolmogorov complexity revisited. SIAM J. Comput., 31(3):887--905, 2002.

.... time bounded Kolmogorov complexity measures (KNt and KNT) and examine the properties of these measures using constructions of hitting set generators for nondeterministic circuits [22, 26] We observe that KNt bears many similarities to the nondeterministic distinguishing complexity CND of [8]. This motivates the definition of a new notion of time bounded distinguishing complexity KDt, as an intermediate notion with connections to the class FewEXP.ThesetofKDt random strings is complete for EXP under P poly reductions. Most of the notions of resource bounded Kolmogorov complexity ....

....is a SNP computable hitting set generator for linear size strong nondeterministic circuits and threshold # . 3 Distinguishing Complexity One of the first types of resource bounded Kolmogorov complexity to be studied was distinguishing complexity. For more on the history of this notion, see [8], where the following notion of nondeterministic distinguishing complexity was introduced. Definition 11 Let p be a polynomial, and let U be a universal nondeterministic Turing machine. CND #x# is defined to be the minimum #d# such that U#d; y# accepts in time p##x## if and only if y # x. KNt ....

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H. Buhrman, L. Fortnow, and S. Laplante. Resourcebounded Kolmogorov complexity revisited. SIAM Journal on Computing, 31(3):887--905, 2002.

....class PP. A similar statement would be also useful in our context, since current upper bounds for formulas in 3 CNF with a unique satisfying assignment [PPSZ98] see section 3) are better than the bounds for arbitrary formulas [Sch99] see section 5) There are many di erent proofs of the lemma [BF97, Cha94, CRS93, MVV87]. Also, there are several proofs of a close result: the existence of a reduction to formulas with odd (or zero) number of satisfying assignments [Gup93, NRS95] In this section we give two new proofs. Both our proofs are based on the idea used in [BF97] In [BF97] this idea is combined with the ....

....proofs of the lemma [BF97, Cha94, CRS93, MVV87] Also, there are several proofs of a close result: the existence of a reduction to formulas with odd (or zero) number of satisfying assignments [Gup93, NRS95] In this section we give two new proofs. Both our proofs are based on the idea used in [BF97]. In [BF97] this idea is combined with the use of the Kolmogorov complexity. In our proofs, we get rid of the application of the Kolmogorov complexity. This simpli es the proof in [BF97] and displays its number theoretic essence. Remark. As we mentioned above, a statement similar to the ....

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H. Buhrman, L. Fortnow, Resource-bounded Kolmogorov complexity revisited, Proceedings of STACS'97, LNCS 1200, Springer-Verlag, pp. 105-116.

....the nic measure. Intuitively, we consider x to be a hard instance if there is no easier way for an A consistent nondeterministic program to decide x than to explicitly encode x into the program. To de ne this formally, we consider a nondeterministic version of Kolmogorov complexity rst de ned in [3]. More precisely, given a time bound t and a string x 2 , CND t (x) minf jM j j M is a t time bounded nondeterministic Turing machine with L(M) fxg g is the nondeterministic t time bounded decision Kolmogorov complexity of x w.r.t. M . In the standard way (see [9] we can consider ....

....we can consider the CND measure to be de ned w.r.t. a xed universal machine. Notice that the CND measure is a nondeterministic generalization of Sipser s CD measure [14] We note in passing that there is no di erence between the nondeterministic Kolmogorov complexity of checking and generating [3] (unlike in the deterministic case where the C and CD time bounded measures appear to be di erent [14] The CND measure gives an immediate upper bound to nondeterministic instance complexity. The next proposition will provide an upper bound on the size of a sound proof system that for a given ....

H. Buhrman and L. Fortnow, Resource-bounded Kolmogorov complexity revisited, In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, LNCS, 1200: 105-116, Springer, 1997.

....nic measure. Intuitively, we consider x to be a hard instance if there is no easier way for an A consistent nondeterministic program to decide x than to explicitly encode x into the program. To define this formally, we consider a nondeterministic version of Kolmogorov complexity first defined in [3]. More precisely, given a time bound t and a string x 2 Sigma , CND t (x) minf jM j j M is a nondeterministic Turing machine such that L(M) fxg, and for all y, timeM (y) t(jyj)g is the nondeterministic t time bounded decision Kolmogorov complexity of x w.r.t. M . In the standard way ....

....we can consider the CND measure to be defined w.r.t. a fixed universal machine. Notice that the CND measure is a nondeterministic generalization of Sipser s CD measure [14] We note in passing that there is no difference between the nondeterministic Kolmogorov complexity of checking and generating [3] (unlike in the deterministic case where 3 the C and CD time bounded measures appear to be different [14] The CND measure gives an immediate upper bound to nondeterministic instance complexity. The next proposition will provide an upper bound on the size of a sound proof system that for a ....

H. Buhrman and L. Fortnow, Resource-bounded kolmogorov complexity revisited, In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, LNCS, 1200: 105--116, Springer, 1997.

.... Gamma c for a constant c 1, then the size is at least (roughly) 2 n=4 . If programs have length bounded by a constant c, then the size goes all the way up to 2 n q(n) where q is a polynomial. We can get a similar result for time bounded computation using a result of Buhrman and Fortnow [BF97]. Their theorem allows us to bound the CD complexity of instances in sets of bounded size. Theorem 3.3 (Buhrman Fortnow) For any set A (accessible as an oracle) there is a polynomial p and a constant c such that for all strings x 2 A Sigma n , CD p;A (x) 2 log(jA Sigma n j) c log ....

....such as the Valiant Vazirani randomized reduction from boolean formulas to formulas which have at most one satisfying assignment, enumerating satisfying assignments, and showing that BPP is in the polynomial hierarchy. These applications and more can be found in a paper of Buhrman and Fortnow [BF97]. Clearly, the best upper bound one can hope for is to have programs of length log d, since d programs are required for sets of size d. In fact, log d log n length is required to account for the string length n. Theorem 3.3 gives a construction for programs of length 2 log d O(log n) and ....

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H. Buhrman and L. Fortnow. "Resource-bounded Kolmogorov complexity revisited," Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, LNCS, Springer, 1997.

....for each nondeterministic linear exponential time (2 o(n) machine accepting an input, it is possible to compute, deterministically, in linear exponential time, one accepting computation 1 . Note that trivially Q implies that E = NE; the converse is known to fail under relativization [17] see [11] for a proof of this based on Kolmogorov complexity) A similar statement at the NP level is well known to characterize P = NP, via the computation of witnesses provided by the self reducibility of SAT. It is well known that there exist Universal Turing machines able to simulate any other Turing ....

H. Buhrman and L. Fortnow. Resource bounded Kolmogorov complexity revisited. In Reischuk and Morvan, editors, 14th Annual Symposium on Theoretical Computer Science, volume 1200 of Lecture Notes in Computer Science, pages 105--116. Springer, 1997.

....with the amount of time spent in recovering the string. Sipser [Sip83] showed that the string can be recognized with high probability in polynomial time with log s O(log n) bits. Buhrman and Fortnow removed the use of randomized algorithms but their solution requires 2 log s O(log n) bits [BF97]. Using current knowledge about extractors, we bring this back down to log s log O(1) n= at the price of having the result hold only for almost every string in the set. Our results are subject to improvement if better extractor constructions are found (up to a theoretical limit of log s ....

....central a role it plays in computational complexity. Recall from Chapter 2 that the CD complexity of a string, given a time bound t, is the length of the smallest program which can recognize that string, and that string only, in time t. As a recent paper of Buhrman and Fortnow has established [BF97], techniques similar to those used by Sipser can be used to show many important results in complexity. Applications include questions about computing and isolating satisfying assignments. They also give illuminating new proofs of oracle results showing that there are relativized worlds where EXP = ....

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H. Buhrman and L. Fortnow. Resource-bounded Kolmogorov complexity revisited. In 14th Annual Symposium on Theoretical Aspects of Computer Science, volume 1200 of Lecture Notes in COmputer Science, L#beck, Germany, 1997. Springer.

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H. Buhrman, L. Fortnow, and S. Laplante. Resource-bounded Kolmogorov complexity revisited. SIAM Journal on Computing, 31(3):887--905, 2002.

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H. Buhrman and L. Fortnow. Resource bounded kolmogorov complexity revisited. In Reischuk and Morvan, editors, 14th Annual Symposium on Theoretical Computer Science, volume 1200 of Lecture Notes in Computer Science, pages 105-- 116. Springer, 1997.

....Theorem 1.1 follows directly from Lemma 3.1. Valiant 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 ....

H. Buhrman and L. Fortnow. Resource-bounded kolmogorov complexity revisited. In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, volume 1200 of Lecture Notes in Computer Science, pages 105--116. Springer, Berlin, 1997.

....( Sip83] For every set A 2 P; there is a polynomial p and a constant c such that for every n and for most r of length p(n) and for every x 2 A n ; CD p (xjr) log jA n j c log n: Buhrman and Fortnow showed how to eliminate r at the cost of doubling the complexity. Theorem 2. 3 ([BF97]) For every set A 2 P; there is a polynomial p and a constant c such that for all strings x 2 A n ; CD p (x) 2 log jA n j c log n: Recently Buhrman, Laplante and Miltersen [BLM00] have proved that the constant factor 2 in the last theorem is optimal in relativized worlds. ....

H. Buhrman and L. Fortnow. Resource-bounded kolmogorov complexity revisited. In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, pages 105-116, Berlin, 1997. Springer.

....that this is not as 6 straightforward as it might look. The randomness used by Arthur in interactive protocols is used for hiding and can in general not be substituted by computational randomness. The idea of using strings of high CD complexity and Zuckerman s sampler derandomization stems from [BF00] Section 8) which is the full version of [BF97] Though they do not explicitly de ne the set R CD t , they use the same approach to derandomize BPP computations there. The proof needs a string of high CD p respectively CND p complexity for p some polynomial. We rst show that we can ....

....On the other hand, we chose r such that: CND q (r) jrj = 1 ) 1 k)m 2 m 2 km 4m which gives a contradiction whenever q p. 2 The following corollary shows that a string of high enough CD poly complexity can be used to derandomize a BPP machine (See also Theorem 8. 2 in [BF00] Corollary 3.11 Let A be a set in BPP. For any 0 there exists a polynomial time Turing machine M a polynomial q such that if CD q (r) jrj with jrj = q(n) then for all x of length n it holds that x 2 A ( M(x; r) 1. Proof of Theorem 3.1. Let A be a language in MA. Let q, M , and q ....

H. Buhrman and L. Fortnow. Resource bounded kolmogorov complexity revisited. Electronic manuscript, obtainable from http://www.neci.nj.nec.com/homepages/fortnow/, 2000.

....a xed polynomial is hard for MA under nondeterministic reductions. MA is the class of MerlinArthur games introduced by Babai [Bab85] As an immediate consequence we obtain that BPP and NP BPP are in NP R CD t . Next we shift our attention to the nondeterministic distinguishing complexity [BF97] CND t (x) which is de ned as the size of the smallest nondeterministic algorithm that runs in time t(n) and accepts only x. We de ne R CND t = fx : CND t (x) jxjg, for t a xed polynomial. We show that AM NP R CND t where AM is the class of Arthur Merlin games [Bab85] It ....

....that prints x, given y and uses at most s(jxj jyj) tape cells [Har83] Likewise we de ne cR CS s = f x; y : CS s (xjy) jxjg for s(n) a polynomial. We show that PSPACE NP cR CS s . For the rst two results we use the oblivious sampler construction of Zuckerman [Zuc96] a lemma [BF97] that measures the size of sets in terms of CD complexity, and we prove a Lemma that shows that the rst bits of a random string are in a sense more random than the whole string. For the last result we make use of the interactive protocol [LFKN92, Sha92] for QBF. To show optimality of our ....

[Article contains additional citation context not shown here]

H. Buhrman and L. Fortnow. Resource bounded kolmogorov complexity revisited. In Reischuk and Morvan, editors, 14th Annual Symposium on Theoretical Computer Science, volume 1200 of Lecture Notes in Computer Science, pages 105-116. Springer, 1997. 25

....counting there has to be a string x 2 A n such that its time unbounded Kolmogorov complexity, C(x) log(jjA n jj) Theorem 1 has one drawback and that is the requirement of the polynomial size random advice string r. Is it possible to eliminate this advice string r Buhrman and Fortnow [BF97] prove that this is possible at the cost of a factor of 2: Theorem 2 [BF97] For any A n and for all x 2 A n : CD p;A n (x) 2 log(jjA n jj) O(log(n) for some polynomial p. In many applications of resource bounded Kolmogorov complexity it is desirable to have Theorem 2 without ....

....Kolmogorov complexity, C(x) log(jjA n jj) Theorem 1 has one drawback and that is the requirement of the polynomial size random advice string r. Is it possible to eliminate this advice string r Buhrman and Fortnow [BF97] prove that this is possible at the cost of a factor of 2: Theorem 2 [BF97] For any A n and for all x 2 A n : CD p;A n (x) 2 log(jjA n jj) O(log(n) for some polynomial p. In many applications of resource bounded Kolmogorov complexity it is desirable to have Theorem 2 without the factor of 2. See for example [BF97] and [BT98] for applications of ....

[Article contains additional citation context not shown here]

H. Buhrman and L. Fortnow. Resource bounded kolmogorov complexity revisited. In Reischuk and Morvan, editors, 14th Annual Symposium on Theoretcal Computer Science, volume 1200 of Lecture Notes in Computer Scienc, pages 105-- 116. Springer, 1997.

....simple counting there has to be a string x 2 A n such that its time unbounded Kolmogorov complexity, C(x) log(jjA n jj) Theorem 1 has one drawback and that is the requirement of the polynomial size random advice string r. Is it possible to eliminate this advice string r Buhrman and Fortnow [BF97] prove that this is possible at the cost of a factor of 2: Theorem 2 [BF97] For any A n and for all x 2 A n : CD p;A n (x) 2 log(jjA n jj) O(log(n) for some polynomial p. In many applications of resource bounded Kolmogorov complexity it is desirable to have Theorem 2 without the ....

....Kolmogorov complexity, C(x) log(jjA n jj) Theorem 1 has one drawback and that is the requirement of the polynomial size random advice string r. Is it possible to eliminate this advice string r Buhrman and Fortnow [BF97] prove that this is possible at the cost of a factor of 2: Theorem 2 [BF97] For any A n and for all x 2 A n : CD p;A n (x) 2 log(jjA n jj) O(log(n) for some polynomial p. In many applications of resource bounded Kolmogorov complexity it is desirable to have Theorem 2 without the factor of 2. See for example [BF97] and [BT98] for applications of Theorem 2. ....

[Article contains additional citation context not shown here]

H. Buhrman and L. Fortnow. Resource bounded kolmogorov complexity revisited. In Reischuk and Morvan, editors, 14th Annual Symposium on Theoretcal Computer Science, volume 1200 of Lecture Notes in Computer Scienc, pages 105-- 116. Springer, 1997.

....complexity, Kolmogorov complexity. Abstract. We show two sets of results applying the theory of extractors to resource bounded Kolmogorov complexity: Most strings in easy sets have nearly optimal polynomial time CD complexity. This extends work of Sipser [Sip83] and Buhrman and Fortnow [BF97]. We use extractors to extract the randomness of strings. In particular we show how to get from an arbitrary string, an incompressible string which encodes almost as much polynomial time CND complexity as the original string. 1 Introduction The Kolmogorov complexity of a string x, denoted ....

.... jyj jzj jwj) When y is the empty string, we write CND t (x) For more information on Kolmogorov complexity we recommend the comprehensive book by Li and Vit#nyi [LV97] 3 Complexity Bounds on Easy Sets Our theorems improve upon the results of Sipser [Sip83] and Buhrman and Fortnow [BF97] in the sense that our bounds are stronger and the strings we obtain have high mutual information. The price we pay for these improvements is that our bounds apply only to imostj strings, not to all strings as in the work of Sipser, and Buhrman and Fortnow [Sip83,BF97] Theorem 4 (Sipser) For ....

[Article contains additional citation context not shown here]

H. Buhrman and L. Fortnow. Resource-bounded kolmogorov complexity revisited. In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, volume 1200 of Lecture Notes in Computer Science, pages 105116. Springer, Berlin, 1997.

....complexity, Kolmogorov complexity. Abstract. We show two sets of results applying the theory of extractors to resource bounded Kolmogorov complexity: Most strings in easy sets have nearly optimal polynomial time CD complexity. This extends work of Sipser [Sip83] and Buhrman and Fortnow [BF97]. We use extractors to extract the randomness of strings. In particular we show how to get from an arbitrary string, an incompressible string which encodes almost as much polynomial time CND complexity as the original string. 1 Introduction The Kolmogorov complexity of a string x, denoted ....

.... jyj jzj jwj) When y is the empty string, we write CND t (x) For more information on Kolmogorov complexity we recommend the comprehensive book by Li and Vit anyi [LV97] 3 Complexity Bounds on Easy Sets Our theorems improve upon the results of Sipser [Sip83] and Buhrman and Fortnow [BF97] in the sense that our bounds are stronger and the strings we obtain have high mutual information. The price we pay for these improvements is that our bounds apply only to most strings, not to all strings as in [Sip83,BF97] Theorem 4 (Sipser) For every set A 2 P, there is a polynomial p and a ....

[Article contains additional citation context not shown here]

H. Buhrman and L. Fortnow. Resource-bounded kolmogorov complexity revisited. In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, volume 1200 of Lecture Notes in Computer Science, pages 105--116. Springer, Berlin, 1997.

....moves are in P. Theorem 5.2 (Jenner Toran) If FP NP jj = FP NP[log] then 1. NP DTIME(2 n O(1= log log(n) 2. NP(log k (n) P. Buhrman and Fortnow showed that the FP NP jj = FP NP[log] question can be phrased as a question on resource bounded Kolmogorov complexity [BF97] Theorem 5.3 (Buhrman Fortnow) The following are equivalent: 1. CND poly (x j y) C poly (x j y) O(log(jxj) 2. CND poly (x j y) CD poly (x j y) O(log(jxj) 3. FP NP jj = FP NP[log] The connection with Kolmogorov complexity enables one to use Theorem 5.2 to prove: ....

....2 6= Sigma p 2 and thus all of the six hypotheses are false. The six hypotheses also fail relative to generic and random oracles. Creating relativized worlds where some of the six hypotheses are true while others fail appears considerably more difficult. Recently Beigel, Buhrman, and Fortnow [BBF97] have made some progress in this direction. Theorem 7.1 (Beigel Buhrman Fortnow) There exists an oracle A such that P A = PhiP A 6= NP A = EXP A One can use PhiP to solve Unique SAT questions. Toda [Tod91b] uses this fact in his celebrated proof that PH P #P . Combined with ....

H. Buhrman and L. Fortnow. Resource bounded kolmogorov complexity revisited. In Reischuk and Morvan, editors, 14th Annual Symposium on Theoretcal Computer Science, volume 1200 of Lecture Notes in Computer Scienc, pages 105--116. Springer, 1997.

....t a fixed polynomial is hard for MA under nondeterministic reductions. MA is the class of MerlinArthur games introduced by Babai [Bab85] As an immediate consequence we obtain that BPP and NP BPP are in NP R CD t . Next we shift our attention to the nondeterministic distinguishing complexity [BF97], CND t (x) which is defined as the size of the smallest nondeterministic algorithm that runs in time t(n) and accepts only x. We define R CND t = fx : CND t (x) jxjg, for t a fixed polynomial. We show that AM NP R CND t where AM is the class of Arthur Merlin games [Bab85] It ....

....program that prints x, given y and uses at most s(jxj jyj) tape cells [Har83] Likewise we define R CS s = f x; y : CS s (xjy) jxjg for s(n) a polynomial. We show that PSPACE NP R CS s . For the first two results we use the oblivious sampler construction of Zuckerman [Zuc96] a Lemma [BF97] that measures the size of sets in terms of CD complexity, and we prove a Lemma that shows that the first bits of a random string are in a sense more random than the whole string. For the last result we make use of the interactive protocol [LFKN90, Sha92] for QBF . Last we construct an oracle ....

[Article contains additional citation context not shown here]

H. Buhrman and L. Fortnow. Resource bounded kolmogorov complexity revisited. In Reischuk and Morvan, editors, 14th Annual Symposium on Theoretical Computer Science, volume 1200 of Lecture Notes in Computer Science, pages 105-- 116. Springer, 1997.

....Theorem 1.1 follows directly from Lemma 3.1. Valiant and Vazirani s construction creates random subspaces of the assignments. Mulmuley, Vazirani and Vazirani [MVV87] give an alternate proof looking at the 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 ....

H. Buhrman and L. Fortnow. Resource-bounded kolmogorov complexity revisited. In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, volume 1200 of Lecture Notes in Computer Science, pages 105--116. Springer, Berlin, 1997.

....y University of Chicago December 4, 1996 Abstract We show two sets of results applying the theory of extractors to resource bounded Kolmogorov complexity: ffl Most strings in easy sets have nearly optimal polynomial time CD complexity. This extends work of Sipser [Sip83] and Buhrman and Fortnow [BF97]. ffl We use extractors to extract the randomness of strings. In particular we show how to get a random string of high polynomial time C complexity from a potentially nonrandom string of high polynomial time CND complexity. 1 Introduction Extractors denote a set of bipartite graphs designed to ....

....represents the length of the smallest program that nondeterministically distinguishes x in polynomial time. Sipser [Sip83] shows that for every set A in P, for all strings x of length n in A, the CD p complexity of x given a random string r is bounded by log jAj O(log n) Buhrman and Fortnow [BF97] remove the dependency on the random string but at a cost of only bounding the CD p complexity of x by 2 log jAj O(log n) We nearly achieve the optimal bound of Sipser without the random string by bounding the CD p complexity of most strings x in A by log jAj log O(1) n. Following ....

[Article contains additional citation context not shown here]

H. Buhrman and L. Fortnow. Resource-bounded kolmogorov complexity revisited. In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science. Springer, Berlin, 1997. To appear.

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H. Buhrman, L. Fortnow, and S. Laplante. Resource-bounded Kolmogorov complexity revisited. SIAM J. Comput., 31(3):887-- 905, 2002.

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