L. A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15-- 37, 1984. 23  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 to i o streams and other time measures. Conclusions are given in Section 9. 2. Levin Search Levin search is one of the few rather general speed up algorithms. Within a (typically large) factor, it is the fastest algorithm for inverting a function g : Y X , if g can be evaluated quickly [11, 12]. Given x, an inversion algorithm p tries to nd a y 2 Y , called g witness for x, with g(y) x. Levin search just runs and veri es the result of all algorithms p in parallel with relative computation time ; i.e. a time fraction 2 is devoted to execute p, where l(p) is the length of program ....

L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15-37, 1984.

....j t) H( I(t : This is, in essence, our entropy nondecrease formula since we will see that the term I(t : is generally very small. It can also be regarded as a special case of a more general randomnessconservation property formulated by L.A. Levin in several ways, see its latest form in [8]. Other forms can be found in [4] 5] Alas, it is also an entropy nonincrease formula. Indeed, the same inequality between H( and H(U ) can also be used between H(U and H( We get therefore I(t : H(U I(t : U ) 6.1) According to this, the only amount of ....

L. A. Levin, Randomness conservation inequalities: Information and independence in mathematical theories, Information and Control 61 (1984), no. 1, 15-37.

....and non uniform reductions. These sets are provably not complete under the usual many one reductions. Let RK , R Kt , RKS , RKT be the sets of strings x having complexity at least x 2, according to the usual Kolmogorov complexity measure K, Levin s time bounded Kolmogorov complexity Kt [26], a space bounded Kolmogorov measure KS, and the time bounded Kolmogorov complexity measure KT that was introduced in [3] respectively. Our main results are: 1. RKS and R Kt are complete for PSPACE and EXP, respectively, under P poly truth table reductions. 2. EXP = NP . 3. PSPACE = ....

....used beyond PSPACE. Using very different techniques, we provide much stronger results in the same direction: the set of random strings can be exploited by efficient reductions. For instance, we show that the set R Kt of strings with high complexity using Levin s time bounded Kolmogorov notion Kt [26] is complete for EXP under truth table reductions computable by polynomial size circuits. Thus we obtain natural examples that witness the difference in power of various reducibilities. In some instances, we are also able to provide completeness results under uniform reductions. By making use of ....

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L. A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

....defect that this measure is not computable. This has motivated several di#erent approaches to the task of defining resource bounded versions of Kolmogorov complexity. Again, a good survey of this material can be found in [LV97] The approach that we will follow is based on a definition of Levin [Lev84] as extended and adapted to other complexity measures in [All01,ABK 02,AKRR03] First, we present (an equivalent restatement of) Levin s Kt measure, along with the deterministic time and space bounded Kolmogorov measures KT and KS of [All01,ABK 02] as reformulated in [AKRR03] ....

L. A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

....turn our attention to that task in Section 2, after first laying some groundwork by presenting our basic definitions. In the earlier papers [2, 3] we focused on notions of resource bounded Kolmogorov complexity, including Kt and KT. The first of these was originally defined and studied by Levin [19]. In this paper we will be introducing several more notions of resource bounded Kolmogorov complexity. In order to have a uniform framework for these new definitions, we need to modify the definitions of Kt and KT in minor ways that affect none of the theorems proved in the earlier work. ....

L. A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

....branch. The leftmost infinite branch can be constructed recursively in the set of admissible words, which is # 1 ; hence this branch must itself be # 2 . 3.2.2.3 Lemma Let be a non atomic computable measure. Then R( contains # 2 , but no # 1 , definable sequences. Proof (See also Schnorr [88, 56]. By 3.2.2.1, R( is a # 2 set of measure 1. Pick a # 1 set A # R( such A 0 and apply the Basis Theorem. If x is recursive and computable and 68 non atomic, then # n [x(n) is a total recursive sequential test with respect to any non atomic computable ; cf. remark 3.2.3.11. Although # 2 ....

L.A. Levin, Randomness conservation inequalities: information and independence in mathematical theories, Inf. and Control 61 (1984), 15-37.

....the information content is maximized (and there is no redundancy at all) Thus, perfect randomness is associated with a unique distribution the uniform one. In particular, by definition, one cannot generate such perfect random strings from shorter random strings. The second theory (cf. [17, 18]) due to Solomonov [23] Kolmogorov [16] and Chaitin [4] is rooted in computability theory and specifically in the notion of a universal language (equiv. universal machine or computing device) It measures the complexity of objects in terms of the shortest program (for a fixed universal ....

L.A. Levin. Randomness Conservation Inequalities: Information and Independence in Mathematical Theories. Inform. and Control, Vol. 61, pages 15--37, 1984.

....for various complexity classes. In particular, in addition to the usual Kolmogorov complexity measure K, we consider the time bounded Kolmogorov complexity measure KT that was introduced in [All01] as well as a space bounded measure KS, and Levin s time bounded Kolmogorov complexity Kt [Lev84] Let RK,R KT,R KS,R Kt be the sets of strings x having complexity at least x 2, according to each of these measures. Our main results are: RKS and R Kt are complete for PSPACE and EXP, respectively, under P poly truthtable reductions. EXP = NP RKt . PSPACE = ZPP . The ....

....be used beyond PSPACE. Using very di#erent techniques, we provide much stronger results in the same direction: the set of random strings can be exploited by e#cient reductions. For instance, we show that the set R Kt of strings with high complexity using Levin s time bounded Kolmogorov notion Kt [Lev84] is complete for EXP under truth table reductions computable by polynomial size circuits. Thus we obtain natural examples that witness the di#erence in power of various reducibilities. In some instances, we are also able to provide completeness results under uniform reductions. By making use of ....

[Article contains additional citation context not shown here]

L. A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

.... (i.e. asymptotically computable) distributions, then the universal distribution , although still well de ned, is not even approximable [Sch00] An interesting and quickly approximable distribution is the Speed prior S de ned in [Sch00] It is related to Levin complexity and Levin search [Lev73, Lev84], but it is unclear for now, which distributions are dominated by S. If one considers only nite state automata instead of general Turing machines, is related to the quickly computable, universal nite state prediction scheme of Feder et al. FMG92] which itself is related to the famous ....

L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15-37, 1984.

.... (i.e. asymptotically computable) distributions, then the universal distribution , although still well defined, is not even approximable [Sch00] An interesting and quickly approximable distribution is the Speed prior S defined in [Sch00] It is related to Levin complexity and Levin search [Lev73, Lev84], but it is unclear for now which distributions are dominated by S. If one considers only finitestate automata instead of general Turing machines, one can attain a quickly computable, universal finite state prediction scheme related to that of Feder et al. FMG92] which itself is related to the ....

L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

.... distributions, then the universal distribution , although still well de ned, is not even approximable (Schmidhuber, 2000) An interesting and quickly approximable distribution is the Speed prior S de ned in (Schmidhuber, 2000) It is related to Levin complexity and Levin search (Levin, 1973; Levin, 1984), but it is unclear for now, which distributions are dominated by S. If one considers only nitestate automata instead of general Turing machines, one can attain a quickly computable, universal nitestate prediction scheme similar to that of Feder et al. 1992) which itself is related to the ....

Levin, L. A. (1984). Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61, 15-37.

....and probability are expressed in form of conjectures. Section 6 addresses issues of temporal complexity ignored in the previous sections on describable universe histories (whose computation may require excessive time without a traditionally computable bound) Levin s universal search algorithm [36, 38], which takes into account program runtime in an optimal fashion, is modified to obtain the fastest way of computing all universes computable within countable time; uncountably many other universes are ignored because they do not even exist from a constructive point of view. A natural ....

....shift its writing and scanning heads across large sections of its internal tapes. This may consume more time than necessary. To overcome potential slow downs of this kind, and to optimize the TM specific constant factor, we will slightly modify an optimal search algorithm called Levin search [36, 38, 1, 39] (see [52, 70, 54] for the first practical applications we are aware of) Essentially, we will strip Levin search of its search aspects and apply it to possibly infinite objects. This leads to the most efficient (up to a constant factor depending on the TM) algorithm for computing all computable ....

L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

....are expressed in form of Conjectures 5.1 5.3. Section 6 addresses issues of temporal complexity ignored in the previous sections on describable universe histories (whose computation may require excessive time without recursive bounds) In Subsection 6. 2, Levin s universal search algorithm [53, 55] (which takes into account program runtime in an optimal fashion) is modified to obtain the fastest way of computing all S describable universes computable within countable time (Def. 6.1, Section 6.3) uncountably many other universes are ignored because they do not even exist from a ....

....shift its writing and scanning heads across large sections of its internal tapes. This may consume more time than necessary. To overcome potential slow downs of this kind, and to optimize the TM specific constant factor, we will slightly modify an optimal search algorithm called Levin search [53, 55, 1, 56] (see [73, 97, 76] for the first practical applications we are aware of) Essentially, we will strip Levin search of its search aspects and apply it to possibly infinite objects. This leads to the most efficient (up to a constant factor depending on the TM) algorithm for computing all computable ....

L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

.... by are considered. The main goal is to understand how the problems are solved by AI . For more details see [ Hutter, 2000b ] Implementation and approximation: The AI t l model su ers from the same large factor 2 l in computation time as Levin search for inversion problems [ Levin, 1973; 1984 ] Nevertheless, Levin search has been implemented and successfully applied to a variety of problems [ Schmidhuber, 1997; Schmidhuber et al. 1997 ] Hence, a direct implementation of the AI t l model may also be successful, at least in toy environments, e.g. prisoner problems. The AI ....

L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15-37, 1984.

....Theorem 1 to i o streams and other time measures. Conclusions are given in Section 9. 2 Levin Search Levin search is one of the few rather general speed up algorithms. Within a (typically large) factor, it is the fastest algorithm for inverting a function g : Y X, if g can be evaluated quickly [Lev73, Lev84]. Given x, an inversion algorithm p tries to nd a y 2Y , called g witness for x, with g(y) x. Levin search just runs and veri es the result of all algorithms p in parallel with relative computation time 2 l(p) i.e. a time fraction 2 l(p) is devoted to execute p, where l(p) is the length of ....

L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15-37, 1984.

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L. A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15-- 37, 1984. 23

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L. A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15-37, 1984.

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L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

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Leonid A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

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Leonid A. Levin. Randomness conservation inequalities: information and independence in mathematical theories. Information and Control, 61:15-37, 1984.

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L. A. Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

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L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15-37, 1984.

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L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15--37, 1984.

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L. A. Levin. Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control, 61:15-37, 1984.

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Leonid A. Levin. Randomness conservation inequalities: information and independence in mathematical theories. Information and Control, 61:15-37, 1984.

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