Eric Allender. Applications of time-bounded kolmogorov complexity in complexity theory. In Osamu Watanabe (Ed.), Kolmogorov Complexity and Computational Complexity, pages 4--22. Springer-Verlag, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... and G acs independent papers on prefix complexity and m [35, 27] Solomonoff s work on inductive inference helped to inspire less general yet practically more feasible principles of minimum description length [95, 66, 44] as well as time bounded restrictions of Kolmogorov complexity, e.g. [42, 2, 96, 56], as well as the concept of logical depth of x, the runtime of the shortest program of x [8] Equation (15) makes predictions of the entire future, given the past. This seems to be the most general approach. Solomonoff [83] focuses just on the next bit in a sequence. Although this provokes ....

A. Allender. Application of time-bounded Kolmogorov complexity in complexity theory. In O. Watanabe, editor, Kolmogorov complexity and computational complexity, pages 6--22. EATCS Monographs on Theoretical Computer Science, Springer, 1992.

.... was partly inspired by presentations found in (Chaitin, 1987) Li and Vit anyi, 1993) and (Solomonoff, 1986) 3 PROBABILISTIC SEARCH FOR USEFUL SELF SIZING PROGRAMS WITH LOW LEVIN COMPLEXITY Levin s universal search algorithm was considered of interest for theoretical purposes (see, e.g. (Allender, 1992) and (Li and Vit anyi, 1993) However, it seems that nobody implemented it for experimental applications, perhaps in fear of the ominous constant factor which may be large. To my knowledge, general universal search was implemented for the first time during the project that led to this paper. ....

Allender, A. (1992). Application of time-bounded Kolmogorov complexity in complexity theory. In Watanabe, O., editor, Kolmogorov complexity and computational complexity, pages 6--22. EATCS Monographs on Theoretical Computer Science, Springer.

....see that using f we can generate an infinite subset of L that is P printable. The following paragraphs introduce the notion of time bounded Kolmogorov complexity that we use. The definitions below were introduced in [Lev84, All89c] more formal definitions and background may be found there and in [All92] Definition 2.20 Kt(x) minfjyj log t : M u (y) x in at most t stepsg where M u is a universal Turing machine. Let L f0; 1g 3 . Then KL (n) minfKt(x) x 2 L =n g. If L =n = OE then KL (n) is undefined. It turns out that questions about the difficulty of NE predicates can ....

....the KL complexity of sets L in P. This is made precise in the following proposition. Proposition 2.3 ffl Every NE predicate is E solvable iff for every set L in P, KL (n) O(log n) ffl No NE predicate is E immune iff for every set L in P, KL (n) 6= log n) Proof. Follows from Theorem 6 in [All92] see also [AW88, Theorem 4] The above proposition holds relative to an arbitrary oracle, and this fact is used in the proof of Proposition 6.3. 2.5 Rudimentary Reductions In the following paragraphs, we present the definitions of Jones [Jon75] to define the class of space bounded rudimentary ....

E. Allender. Applications of time-bounded Kolmogorov complexity in complexity theory. In O. Watanabe, editor, Kolmogorov Complexity: Theory and Relations to Computational Complexity. Springer-Verlag, 1992. To appear.

.... above is partly inspired by presentations found in (Chaitin, 1987) Li and Vit anyi, 1993) and (Solomonoff, 1986) 3 PROBABILISTIC SEARCH FOR USEFUL SELF SIZING PROGRAMS WITH LOW LEVIN COMPLEXITY Levin s universal search algorithm was considered of interest for theoretical purposes (see e.g. (Allender, 1992) and (Li and Vit anyi, 1993) However, it seems that nobody implemented it for experimental applications, perhaps in fear of the ominous constant factor which may be large. To my knowledge, general universal search was implemented for the first time during the project that led to this paper ....

Allender, A. (1992). Application of time-bounded Kolmogorov complexity in complexity theory. In Watanabe, O., editor, Kolmogorov complexity and computational complexity, pages 6--22. EATCS Monographs on Theoretical Computer Science, Springer.

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Eric Allender. Applications of time-bounded kolmogorov complexity in complexity theory. In Osamu Watanabe (Ed.), Kolmogorov Complexity and Computational Complexity, pages 4--22. Springer-Verlag, 1992.

.... logt : U(d) x in at most t steps . Note that log x #Kt(x) # x O(log x ) The elements of # # can be enumerated in order of increasing Kt(x) and Levin observed that this ordering yields essentially the fastest way to search for accepting computations. The following definitions (from [All89,All92]) will be useful in stating certain correspondences. Definition 2. Let L # # # . Define KtL (n) to be min Kt(x) x # L =n . In [All89,All92] the function KtL was called KL . Here and elsewhere, L =n is the set of all strings in L of length n. If there is no string in L of length ....

....Kt(x) and Levin observed that this ordering yields essentially the fastest way to search for accepting computations. The following definitions (from [All89,All92] will be useful in stating certain correspondences. Definition 2. Let L # # # . Define KtL (n) to be min Kt(x) x # L =n . In [All89,All92] the function KtL was called KL . Here and elsewhere, L =n is the set of all strings in L of length n. If there is no string in L of length n,then KtL (n) is undefined. When considering the rate of growth of a function KtL (n) the undefined values are not taken into consideration. Observe ....

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Eric Allender. Applications of time-bounded kolmogorov complexity in complexity theory. In Osamu Watanabe (Ed.), Kolmogorov Complexity and Computational Complexity, pages 4--22. Springer-Verlag, 1992.

....complexity of NE predicates and a version of time bounded Kolmogorov complexity. The following paragraphs introduce the notion of time bounded Kolmogorov complexity that we use. The definitions below were introduced in [Lev84, Al89] more formal definitions and background may be found there and in [Al92]. Since it will sometimes be necessary to speak about Kolmogorov complexity relative to an oracle, we present the definitions relative to an arbitrary oracle A. Definition: Kt A (x) minfjyj log t : M A u (y) x in at most t stepsg where M u is a universal Turing machine. Let L f0; ....

....A i on input xg. Proposition 7 For any oracle A, the following are equivalent: a) Every NE A predicate is E A solvable. b) For every set L in P A , K A L (n) O(log n) c) K A S(A) n) O(log n) Proof: The implication (a) b) is proved (in the unrelativized case) as Theorem 6 in [Al92] (see also [AW90, Theorem 4] Since S(A) is clearly in P A , it follows that (b) implies (c) For the remaining implication, suppose K A S(A) n) c log n for some constant c. To solve the NE A predicate defined by the NE A machine M A i , we can use the following algorithm: On input x, ....

E. Allender. Applications of time-bounded Kolmogorov complexity in complexity theory. In Kolmogorov Complexity and Computational Complexity, Osamu Watanabe, ed., EATCS Monograph Series, Springer-Verlag, 1992.

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A. Allender. Application of time-bounded Kolmogorov complexity in complexity theory. In O. Watanabe, editor, Kolmogorov complexity and computational complexity, pages 6-22. EATCS Monographs on Theoretical Computer Science, Springer, 1992.

No context found.

A. Allender. Application of time-bounded Kolmogorov complexity in complexity theory. In O. Watanabe, editor, Kolmogorov complexity and computational complexity, pages 6--22. EATCS Monographs on Theoretical Computer Science, Springer, 1992.

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