K.-I Ko, On the notion of infinite pseudorandom sequences, Theoretical Computer Science 48 (1986) 9--33.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that for infinitely many numbers m # m : f(A m # ) A(m # ) # m : f(A m # ) A(m # ) 1 #. 16) A set A is p predictable if there is a p prediction function that predicts A, and a class is p predictable if there is a p prediction function predicting every set in C. KO [10] proposed p unpredictability as a formalization of polynomial time randomness. In [2] where p unpredictable sets are called Ko p stochastic, it was shown that these sets are just the sets on which strict and simple p martingales cannot succeed. The following theorem which in part parallels the ....

K.-I. Ko. On the notion of infinite pseudorandom sequences. Theor. Comput. Sci., 48(1):9--33, 1986.

....The material in this section is taken from Terwijn [53] For notational convenience, in this section we denote the initial segment of length n of an infinite sequence x by x n . We abbreviate the phrases infinitely often and almost everywhere by i.o. and a.e. respectively. Ko [21] investigated the relations between polynomial time and space bounded versions of Martin Lof randomness. His notion of pspace randomness is obtained by defining a sequence to be non pspace random if it is covered by a pspace computable Martin Lof test su#ciently fast. The extra condition on the ....

....by a pspace computable Martin Lof test su#ciently fast. The extra condition on the speed with which the set is covered is necessary, since otherwise the defined notion equals that of Martin Lof. The following su#cient condition for pspace randomness was proved by Ko. Theorem 3.6.1 (K. I. Ko [21]) Let p be a polynomial. Let C denote the s space bounded generating complexity. If for all polynomials q it holds that p(log n) a.e. then x is pspace random. 23 We now turn our attention to computable randomness. The next lemma is analogous to [30, Claim 2.2, p122] with the time bounded ....

K.-I Ko, On the notion of infinite pseudorandom sequences, Theoretical Computer Science 48 (1986) 9--33.

....This does not, however, diminish the interest of the above problems. There appear to be two ways to reconcile the theoretical definition of randomness with the practicing mathematician s interest in this concept: either employing (variants of) generalized Kolmogorov complexity as in Ko [11], or else questioning the use of recursion theory in defining randomness altogether, as in [18] ....

KEN-I KO, On the notion of infinite pseudorandom sequences, Theoretical Computer Science, vol. 48 (1986), pp. 9-33.

....decision on the majority of algorithms. This democratic vote idea has been used in [LW94, Vov92] for sequence prediction, and is referred to as weighted majority there. 6. 1 Time Limited Probability Distributions In the literature one can find time limited versions of Kolmogorov complexity [Dal73, Dal77, Ko86] and the time limited universal semimeasure [LV91, LV97, Sch02b] In the following, we utilize and adapt the latter and see how far we get. One way to define a timelimited universal chronological semimeasure is as a sum over all enumerable chronological semimeasures with computation time at ....

.... VW98] Related topics are the Weighted Majority Algorithm invented by Littlestone and Warmuth [LW94] universal forecasting by Vovk [Vov92] Levin search [Lev73] pac learning introduced by Valiant [Val84] and Minimum Description Length [LV92a, Ris89] Resource bounded complexity is discussed in [Dal73, Dal77, FMG92, Ko86, PF97], resource bounded universal probability in [LV91, LV97, Sch02b] Implementations are rare and mainly due to Schmidhuber [Con97, Sch97, SZW97, Sch02a] Excellent reviews with a philosophical touch are [LV92b, Sol97] For an older, but general review of inductive inference see Angluin [AS83] ....

K.-I. Ko. On the notion of infinite pseudorandom sequences. Theoretical Computer Science, 48(1):9--33, 1986.

....for infinitely many numbers m m # m : f(A m # ) A(m # ) m # m : f(A m # ) #= A(m # ) 1 #. 16) A set A is p predictable if there is a p prediction function that predicts A, and a class C is p predictable if there is a p prediction function predicting every set in C. KO [10] proposed p unpredictability as a formalization of polynomial time randomness. In [2] where p unpredictable sets are called Ko p stochastic, it was shown that these sets are just the sets on which strict and simple p martingales cannot succeed. The following theorem which in part parallels the ....

K.-I. Ko. On the notion of infinite pseudorandom sequences. Theor. Comput. Sci., 48(1):9--33, 1986.

....in these terms. Using the martingale concept, Schnorr [11] introduced resource bounded randomness concepts, and later Lutz [7] introduced a kind of resource bounded measure theory. Resource bounded versions of stochasticity concepts were also introduced by several authors, e.g. Wilber [18] Ko [6], and Ambos Spies et al. 3] The notion of unsafe approximations was introduced by Yesha in [19] an unsafe approximation algorithm for a set A is just a standard polynomial time bounded deterministic Turing machine M with outputs 1 and 0. Duris and Rolim [5] further investigated unsafe ....

K. Ko. On the notion of infinite pseudorandom sequences. Theoret. Comput. Sci., 48:9--33, 1986.

.... [6] The extraordinary power and scope of this notion have recently been surveyed by Kolmogorov and Uspenskii [19] and Li and Vitanyi [21] In this paper we are primarily concerned with resource bounded Kolmogorov complexities, which have been investigated by Hartmanis [10] Sipser [39] Ko [17], Longpr e [22] Balc azar and Book [3] Huynh [13] Lutz [24] Allender and Watanabe [2] and many others. Martin Lof [29] showed that K(xjn) the conditional Kolmogorov complexity of infinite binary sequences x, exhibits a strong Shannon effect. Specifically, Martin Lof proved that if the ....

....sequences. Following work by Yao [45] Blum and Micali [5] Goldreich, Goldwasser, and Micali [8] Levin [20] Allender [1] and others on the generation of finite pseudorandom sequences from shorter random sequences, and following work by Schnorr [34,36] Wilber [43] Huynh [12,13] Ko [17], and others on pseudorandom properties of infinite sequences, Lutz [25,27] gave a measure theoretic definition of infinite pseudorandom sequences. This definition of pseudorandomness is analogous to the MartinL of [28] definition of randomness, but is based on resource bounded measure theory and ....

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K. Ko, On the notion of infinite pseudorandom sequences, Theoretical Computer Science 48 (1986), pp. 9--33.

....(resp. pair hprogram; yi) which, when given as input to U , will lead U to write down w as output. Hartmanis [14] and Sipser [25] modified the original idea of Kolmogorov complexity to include the running time or space used by the Universal Turing machine, in order to produce an output. Ko, in [18] 1 , followed the same approach but applying it to the notion of infinite sequences with respect to polynomial time and space complexity. The sets of bounded Kolmogorov complexity strings K[f(n) g(n) is defined as follows: Definition 7 K[f(n) g(n) fx j 9y; jyj f(jxj) U(y) x in at ....

....T (P=log) P=poly, since tally sets are in P log (see [16] for more details) Since P=log 6= P=poly (see for instance [11] we have: Theorem 9 P T (P=log) is not included in P=log. Moreover, one can see that even Pm(P=log) is not included in P=log. see again [16] 1 Preliminary versions of [18] circulated simultaneously to [14] and [25] 3 The classes Full P=log and Pref P=log Since the logarithmic analog to P poly is not closed under most usual reducibilities, an alternative approach was introduced by Ko [19] Ko s class, although with a different name, is introduced in the ....

K. Ko. On the Notion of Infinite Pseudorandom Sequences. Theoretical Computer Science, 48:9--13, 1986.

....in complexity theory. 1.1 Complexity of Strings Before going any further, it is necessary to define the sort of time bounded Kolmogorov complexity that we will be considering. Many alternate approaches exist for adding a time complexity component to Kolmogorov complexity. Sipser [Sip83] and Ko [Ko86] proposed essentially identical definitions, allowing one to define, for each function f , a f(n) time bounded Kolmogorov complexity measure K f , where K f (x) is the length of the shortest description of x from which x can be produced in f(jxj) steps. A related (and much more influential) ....

....complexity to measure the complexity of a language L is to consider the characteristic sequence of L: the sequence a 1 ; a 2 ; where a i is zero or one, according to whether or not x i 2 L, where x 1 ; x 2 ; is an enumeration of 6 3 . Investigations of this sort may be found in [Ko86, Huy85, Huy86, BDG87, MS90, Lut91]. For example, in [BDG87] it was shown that PSPACE poly is the class of all languages L such that each finite prefix of the characteristic sequence of L has small space bounded Kolmogorov complexity. It is often useful, however, to consider the complexity of the individual strings in a language ....

K. Ko. On the notion of infinite pseudorandom sequences. Theoret. Comput. Sci. 48:9--33, 1986.

....it admits random strings which do not satisfy the law of iterated logarithm. A stronger, succesful definition of randomness was introduced by Martin Lof [Martin Lof 1966] This and closely related work on Kolmogorov complexity [Kolmogorov 1965, Solomonoff 1964, Chaitin 1966] have been extended by Ko [Ko 1986], Huynh [Huynh 1986a, Huynh 1986b] and Lutz [Lutz 1988] to define randomness relative to complexity classes, but these definitions seem to be too strong to admit a hierarchy theorem as sharp as the one reported here. 1.1 Preliminaries First we give some notations and the necessary definitions. ....

....1 get a low index. For x 2 f0; 1g n , let r(x) #fk j k n; bit k (x) 1g be the number of bits equal to 1 in a string x. Then we define the order OE on Sigma n Theta Sigma n as follows: s OE t if (r(x) r(y) or (r(x) r(y) and x precedes y in the lexicographical order) Lemma 2. 5 [Ko 1986] The function dec, defined by dec(n; m) the m th string in f0; 1g n in the order OE, is computable in time O(n 2 ) and in space O(n) Lemma 2.6 [Chaitin 1966] Let ffl be a real number between 0 and 1 2 . Then, for any n 1, log n b( 1 2 Gamma ffl)nc n Delta H(ffl) c for ....

K.I. Ko. On the notion of infinite pseudorandom sequences. Theoretical Computer Sciences, 48(1986):9--13.

....in complexity theory. 1.1 Complexity of Strings Before going any further, it is necessary to define the sort of time bounded Kolmogorov complexity that we will be considering. Many alternate approaches exist for adding a time complexity component to Kolmogorov complexity. Sipser [Sip83] and Ko [Ko86] proposed essentially identical definitions, allowing one to define, for each function f , a f(n) time bounded Kolmogorov complexity measure K f , where K f (x) is the length of the shortest description of x from which x can be produced in f(jxj) steps. A related (and much more influential) ....

....complexity to measure the complexity of a language L is to consider the characteristic sequence of L: the sequence a 1 ; a 2 ; where a i is zero or one, according to whether or not x i 2 L, where x 1 ; x 2 ; is an enumeration of Sigma . Investigations of this sort may be found in [Ko86, Huy85, Huy86, BDG87, MS90, Lut91]. For example, in [BDG87] it was shown that PSPACE poly is the class of all languages L such that each finite prefix of the characteristic sequence of L has small space bounded Kolmogorov complexity. It is often useful, however, to consider the complexity of the individual strings in a language ....

K. Ko. On the notion of infinite pseudorandom sequences. Theoret. Comput. Sci. 48:9--33, 1986.

.... we denote the output of M on input (p 1 ; p 2 ; x) by M(p 1 ; p 2 ; x) and the number of steps in the computation by timeM (p 1 ; p 2 ; x) We assume that t(n) n for all time bounds t = t(n) The following notion of time bounded Kolmogorov complexity was introduced by Hartmanis [6] Ko [9], and Sipser [21] Intuitively, the t bounded Kolmogorov complexity of x is the length of the shortest program which computes x in t(jxj) steps from the empty input. Definition 1 For any time bound t and x; y 2 Sigma the t bounded Kolmogorov complexity of x conditional to y using M is defined ....

K. Ko. On the notion of infinite pseudorandom sequences. Theoretical Computer Science, 48:9--33, 1986.

....we study applications of randomness concepts in complexity theory. For computational complexity classes, several definitions of pseudorandom sequences have been proposed. Blum and Micali [5] and Yao [28] gave a relatively weak definition of resource bounded random sequences. Schnorr [20] and Ko [10] introduced resource bounded versions of the notions of MartinL of and Kolmogorov randomness. More recently, Lutz [12,13] further pursued these ideas and systematically developed a resource bounded measure theory. Especially, he introduced a feasible measure concept, of which he and others have ....

....name of person. 3.4 Resource bounded Ko randomness In the previous sections, we have studied the resource bounded randomness concepts based on martingales. In this section, we discuss resource bounded Ko randomness concept which is based on the constructive null covers. Definition 18 (see Ko [10]) A Ko (C 1 ; C 2 ) test is a pair (U; g) where U 2 C 1 is a subset of Sigma (notice that we identify a set with its characteristic function) and g 2 C 2 is an unbounded, nondecreasing function from N to N such that the following conditions hold. 1) U [0] Sigma . 2) U [k 1] U ....

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K. Ko, On the notion of infinite pseudorandom sequences, Theoret. Comput. Sci. 48 (1986) 9--33.

....by Levin [Lev74] and G acs [G ac74] see also [LV88] Using resource bounded Kolmogorov complexity, we address the issue of the validity of symmetry of information in a resource bounded environment. Resource bounded versions of Kolmogorov complexity have recently been studied (see for example [Har83, Sip83, Ko86, Lon86]) For a specific (universal) Turing machine M , time bound T (n) and integer m, define KT (x; T (n) minfl j l = jyj and M(y) x; using at most T (jxj) timeg KT (x j m; T (n) minfl j l = jyj and M(hy; mi) x; using at most T (jxj) timeg S(n) space bounded Kolmogorov complexity KS(x; ....

K. Ko. On the notion of infinite pseudorandom sequences. Theoretical Computer Science, 48:9--13, 1986.

....in these terms. He criticized Martin Lof s concept as being too strong and proposed a less restrictive concept as an adequate formalization of a random sequence instead. For computational complexity classes, several definitions of pseudorandom sequences have been proposed. Schnorr [20] and Ko [10] introduced resource bounded versions of the notions of Martin Lof and Kolmogorov randomness, respectively. More recently, Lutz [12, 13] further pursued these ideas and systematically developed a resource bounded measure theory. In particular, he introduced a feasible measure concept, of which he ....

....such that F 0 (x) F (x) for all x 2 Sigma . Proof. See Ambos Spies et al. 3] Juedes and Lutz [8] or Mayordomo [15] The above p randomness concept is defined in terms of typicalness. A p randomness concept in terms of stochasticity was introduced by Wilber [27] and was strengthened by Ko [10] as follows. Definition 3.6 (Ko [10] A sequence 2 Sigma 1 is Ko p stochastic if, for every polynomial time computable total function f : Sigma Sigma, lim n 1 kfk n : f( 0: k Gamma 1] k]gk n = 1 2 : In other words, a sequence is Ko p stochastic if and only if, for each ....

[Article contains additional citation context not shown here]

K. Ko. On the notion of infinite pseudorandom sequences. Theoret. Comput. Sci., 48:9--33, 1986.

....in these terms. Using martingales concepts, Schnorr [11] introduced resource bounded randomness concepts, and later Lutz [7] introduced a kind of resource bounded measure theory. Resource bounded version of stochasticity concepts were also introduced by several authors, see, e.g. Wilber [18] Ko [6] and Ambos Spies et al. 2] The notion of unsafe approximations was introduced by Yesha in [19] An unsafe approximation algorithm for a set A is just a standard polynomial time bounded deterministic Turing machine M with outputs 1 and 0. Duris and Rolim [5] further investigated unsafe ....

K. Ko. On the notion of infinite pseudorandom sequences. Theoret. Comput. Sci., 48:9--33, 1986.

....and Huwig [2] and which in addition controls the frequency with which a condition is met. We show that this concept coincides with the resource bounded version of Church s stochasticity [6] By uniformly describing these concepts and weaker notions of stochasticity introduced by Wilber [19] and Ko [11] in terms of prediction functions, we clarify the relations among these resource bounded stochasticity concepts. Moreover, we give descriptions of these concepts in the framework of Lutz s resource bounded measure theory [13] based on martingales: We show that t(n) stochasticity coincides with a ....

....grant CHRX CT93 0415 (COLORET) Supported in part by the EC through the Esprit Bra project 7141 (ALCOM II) and by the Spanish government through project DGICYT PB94 0564 and Accion Integrada HA 119. Resource bounded randomness concepts can be found e.g. in Schnorr [15] Wilber [19] Ko [11] and Lutz [13] While an attempt to clarify the relations among the various genericity notions has been recently made by Ambos Spies in [1] it seems that the relations among the different resource bounded randomness notions have not yet been explored systematically, though some isolated results ....

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K. Ko. On the notion of infinite pseudorandom sequences. Theoret. Comput. Sci., 48:9--33, 1986.

.... can be viewed also as a special case of instance complexity, because KD t (x) ic t (x : fxg) An early variant of Kolmogorov complexity that is somehow close in spirit to instance complexity is Loveland s uniform complexity K(A; x) 20] A time bounded version of this is discussed in [15]. In our notation, Loveland s definition can be formulated as: K(A; x) minfjM j : for all y x; M(y) 6= and M(y) 1 iff y 2 Ag: In the following, we first in Section 2 formulate the proper definitions of instance complexity and related notions. Then, in Section 3, we map out some ....

Ko, K.-I. On the notion of infinite pseudorandom sequences. Theoret. Comput. Sci. 48 (1986), 9--33.

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K.-I Ko, On the notion of infinite pseudorandom sequences, Theoretical Computer Science 48 (1986) 9--33.

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K. Ko, On the notion of infinite pseudorandom sequences, Theoretical Computer Science 48 (1986), pp. 9--33.

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K. Ko, On the notion of infinite pseudorandom sequences, Theoretical Computer Science 48 (1986), pp. 9--33.

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K. I. Ko. On the notion of infinite pseudorandom sequences. Theoretical Computer Science 48:9--33, 1986.

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