Schmidhuber, J. (2000). Algorithmic theories of everything (Technical Report IDSIA-20-00 (Version 2.0)). Istituto Dalle Molle di Studi sull'Intelligenza Artificiale, Manno-Lugano, Switzerland.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....sense that it is uniquely de ned up to an additive constant. K(x) can be approximated from above (is coenumerable) but not nitely computable. See [13] for an excellent introduction to Kolmogorov Complexity and [22] for a review of Kolmogorov inspired prediction schemes. Recently, Schmidhuber [15] has generalized Kolmogorov complexity in various ways to the limits of computability and beyond. In the following, we also need a generalization, but of a di erent kind. We need a short description of a function, rather than a string. The following de nition of the complexity of a function f ....

J. Schmidhuber. Algorithmic theories of everything. Report IDSIA-20-00, quantph /0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

....what constraints are already known to be due to the Anthropic Principle. The Multiverse is the result of unfolding one particular mathematical theory namely that of Hilbert spaces. The Tegmark ensemble is already bigger than the Multiverse. An alternative ensemble was proposed by Schmidhuber[11, 12], namely the ensemble of all programs, as interpreted by a particular universal Turing machine (UTM) a sort of model abstract computer) If we interpret the Tegmark ensemble as the ensemble of all consistent formal axiomatic systems, then a formal system can be represented by a program that that ....

Jurgen Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA-20-00, IDSIA, Galleria 2, 6928 Manno (Lugano), Switzerland, 2000. arXiv:quant-ph/0011122.

.... by a Turing machine which enumerates all enumerable semimeasures [Sol64, Sol78, LV97] In this case, sometimes called the number of wisdom) has interesting properties in itself [Cal98, Cha75, Cha91] Recently, M has been further enlarged to include all cumulatively enumerable semi measures [Sch00, Sch02]. In the enumerable and cumulatively enumerable cases, is not nitely computable, but can still be approximated to arbitrary but not pre speci able precision. If we consider all approximable (i.e. asymptotically computable) distributions, then the universal distribution , although still well ....

....enumerable cases, is not nitely computable, but can still be approximated to arbitrary but not pre speci able precision. If we consider all approximable (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 ....

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J. Schmidhuber. Algorithmic theories of everything. Report IDSIA-20-00, quantph /0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

.... Cha91] If we enlarge M to include all enumerable semimeasures, we attain Solomonoff s universal probability, apart from normalization, which has to be treated differently in this case [Sol64, Sol78] Recently, M has been further enlarged to include all cumulatively enumerable semi measures [Sch00]. In all cases, is not finitely computable, but can still be approximated to arbitrary but not pre specifiable precision. If we consider all approximable (i.e. asymptotically computable) distributions, then the universal distribution , although still well defined, is not even approximable ....

....[Sch00] In all cases, is not finitely computable, but can still be approximated to arbitrary but not pre specifiable precision. If we consider all approximable (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 ....

[Article contains additional citation context not shown here]

J. Schmidhuber. Algorithmic theories of everything. Report IDSIA-20-00, quantph /0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

....that it is uniquely de ned up to an additive constant. K(x) can be approximated from above (is co enumerable) but not nitely computable. See [LV97] for an excellent introduction to Kolmogorov Complexity and [VL00] for a review of Kolmogorov inspired prediction schemes. Recently, Schmidhuber [Sch00] has generalized Kolmogorov complexity in various ways to the limits of computability and beyond. In the following, we also need a generalization, but of a di erent kind. We need a short description of a function, rather than a string. The following de nition of the complexity of a function f K ....

J. Schmidhuber. Algorithmic theories of everything. Report IDSIA-20-00, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

....the number of wisdom) has interesting properties in itself (Calude et al. 1998; Chaitin, 1991) If we enlarge M to include all enumerable semi measures, we attain Solomono s (1964; 1978) universal probability, apart from normalization, which has to be treated di erently in this case. Recently, Schmidhuber (2000) has further enlarged M to include all cumulatively enumerable semi measures. In all cases, is not nitely computable, but can still be approximated to arbitrary but not pre speci able precision. If we consider all approximable (i.e. asymptotically computable) distributions, then the universal ....

....In all cases, is not nitely computable, but can still be approximated to arbitrary but not pre speci able precision. If we consider all approximable (i.e. asymptotically computable) 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 ....

[Article contains additional citation context not shown here]

Schmidhuber, J. (2000). Algorithmic theories of everything (Report IDSIA-20-00). IDSIA, Manno (Lugano), Switzerland.

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J. Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA-20-00, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

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J. Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA-20-00, quantph /0011122, IDSIA, Manno (Lugano), Switzerland, 2000. Sections 1-5: see [37]; Section 6: see

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J. Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA-2000, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

....algorithm computing PRG(n) for all n, otherwise the halting problem would be solvable, which it is not [43] Hence in general there is no computer that outputs x and only x without ever editing some previously computed history. That is, if we want to study the set of all programmable universes [36] then we should not limit ourselves to MTMs but consider GTMs as well. Note, however, that the output of a GTM might not stabilize in the sense that certain output bits might ip from 0 to 1 and back forever. Enumerable Output Machines (EOMs) EOMs embody the important concept of computable ....

....to a history without any short description (given the appropriate TM type) is necessarily unlikely. To a certain extent, this justi es Occam s razor (e.g. 4] which expresses the ancient preference of simple solutions over complex ones. A more detailed analysis can be found elsewhere [36]. Acknowledgments Thanks to Marcus Hutter and Sepp Hochreiter for independently checking all theorems, to Ray Solomono , Christof Schmidhuber, Leonid Levin and Peter G acs, for useful comments, and to Marcus Hutter for the unsolicited proof of Theorem 4.2. ....

J. Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA-20-00, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

.... general loss bounds and showed that the expected loss of the universal scheme does not exceed by much the loss of the optimal scheme [9] Recent research also generalized Solomono s approach to the case of much less restrictive universal nonenumerable priors that are computable in the limit [22, 23]. One might say that Solomono s restriction of recursiveness leads to a slightly more computable approach than the more general case. However, while M is enumerable, it is not recursive, and thus practically infeasible. This drawback inspired less general yet practically more feasible ....

....Section 3 will derive a near optimal computable strategy for making predictions, given past observations. 2 Speed Prior S Let us assume that the observed data sequence is generated by a computational process, and that any possible sequence of observations is therefore computable in the limit [22]. This assumption is stronger and more radical than the traditional one: Solomono just insists that the probability of any sequence pre x is recursively computable, but the (in nite) sequence itself may still be generated probabilistically. Under our starting assumption that data are ....

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J. Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA-2000, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

....that the observations are statistically independent. 4 Super Omegas and Generalizations of Kolmogorov Complexity Algorithmic Probability Our recent research generalized Solomono s approach to the case of less restrictive nonenumerable universal priors that are still computable in the limit [50, 53]. An object X is formally describable if a nite amount of information completely describes X and only X . More to the point, X should be representable by a possibly in nite bitstring x such that there is a nite, possibly never halting program p that computes x and nothing but x in a way that ....

....Turing machines with one way write only output tape. This leads to the universal enumerable Solomono Levin (semi) measure. We introduced more general, nonenumerable, but still limit computable measures and a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity [50, 53], suggesting that the true information content of some (possibly in nite) bitstring x actually is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. In fact, this true content is often smaller than the ....

[Article contains additional citation context not shown here]

J. Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA-2000, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

.... general loss bounds and showed that the expected loss of the universal scheme does not exceed by much the loss of the optimal scheme [9] Recent research also generalized Solomonoff s approach to the case of much less restrictive univer sal nonenumerable priors that are computable in the limit [22, 23]. One might say that Solomonoff s restriction of recursiveness leads to a slightly more computable approach than the more general case. However, while M is enumerable, it is not recursive, and thus practically infeasible. This drawback inspired less general yet practically more feasible prin ....

....Section 3 will derive a near optimal cornputable strategy for making predictions, given past observations. 2 Speed Prior Let us assume that the observed data sequence is generated by a computational process, and that any possible sequence of observations is therefore computable in the limit [22]. This assumption is stronger and more radical than the traditional one: Solomonoff just insists that the probability of any sequence prefix is recursively computable, but the (infinite) sequence itself may still be generated probabilis tically. Under our starting assumption that data are ....

[Article contains additional citation context not shown here]

J. Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA-2000, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

.... speed ups due to halting programs if there are any) Nonbinary, nonuniversal variants of Osearch were used to solve machine learning toy problems unsolvable by traditional methods [58, 47] Probabilistic alternatives based on probabilistically chosen maximal program runtimes in Speed Prior style [41, 45] also outperformed traditional methods on certain toy problems [39, 40] 2.4 Incremental Search Since Newell Simon s early attempts at building a General Problem Solver [32, 35] much work has been done to develop mostly heuristic machine learning algorithms that solve new problems based on ....

....most recent code and prolongations thereof. Yet other oops variants will also assign fractions of the total time to the second most recent program and its prolongations, the third most recent program and its prolongations, etc. We may also consider probabilistic oops variants in Speed Prior style [41, 45]. One not necessarily useful idea: Suppose the number of tasks to be solved by a single program is known in advance. Now we might think of an OOPS variant that works on all tasks in parallel, again spending half the search time on programs starting at a last , half on programs starting at a ....

J. Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA-20-00, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

....Report IDSIA 20 00, Version 2.0; 20 Dec 2000 Minor revision of Version 1. 0 [75], quant ph 0011122 ALGORITHMIC THEORIES OF EVERYTHING Jurgen Schmidhuber IDSIA, Galleria 2, 6928 Manno (Lugano) Switzerland juergen idsia.ch http: www.idsia.ch juergen Abstract The probability distribution P from which the history of our universe is sampled represents a theory of ....

....universal priors, speed prior, universal search, inductive inference, Occam s razor, computable universes, theory of everything, collapse of the wave function, many worlds interpretation of quantum mechanics, countable vs uncountable. Note: This is a slightly revised version of a recent preprint [75]. The essential results should be of interest from a purely theoretical point of view independent of the motivation through formally describable universes. To get to the meat of the paper, skip the introduction and go immediately to Subsection 1.1 which provides a condensed outline of the main ....

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J. Schmidhuber. Algorithmic theories of everything. Technical Report IDSIA20 -00, Version 1.0, IDSIA, Manno (Lugano), Switzerland, November 2000. http://arXiv.org/abs/quant-ph/0011122.

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Schmidhuber, J. (2000). Algorithmic theories of everything (Technical Report IDSIA-20-00 (Version 2.0)). Istituto Dalle Molle di Studi sull'Intelligenza Artificiale, Manno-Lugano, Switzerland.

No context found.

J. Schmidhuber. Algorithmic theories of everything. Report IDSIA-20-00, quantph /0011122, IDSIA, Manno (Lugano), Switzerland, 2000.

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