## Hierarchies Of Generalized Kolmogorov Complexities And Nonenumerable Universal Measures Computable In The Limit (2000)

Venue: | INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE |

Citations: | 39 - 20 self |

### BibTeX

@ARTICLE{Schmidhuber00hierarchiesof,

author = {Jürgen Schmidhuber},

title = {Hierarchies Of Generalized Kolmogorov Complexities And Nonenumerable Universal Measures Computable In The Limit},

journal = {INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE},

year = {2000},

volume = {13},

number = {4},

pages = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with one-way write-only output tape. This naturally leads to the universal enumerable Solomono-Levin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly in nite) bitstring x 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. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briey mention consequences for universes sampled from such priors.

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Citation Context |

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Citation Context ...er, and therefore is indeed EOM-computable. Obviously the following holds: CP T (x) = CP T (x) = C 0 (x) and P T (x) = X z2B ] P T (xz) = 0 (x): 2 Summary. The traditional universal c.e. measure [40,=-= 45, 29, 16, 17, 41, 14, 30]-=- derives from universal MTMs with random input. What is the nature of our novel generalization here? We simply replace the MTMs by EOMs. As shown above, this leads to universal cumulatively enumerable... |

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Citation Context ...he complexity of the c.e. number 0:s 2 I(x) with minimal K T (s). The facts P x P M (x) = 1, P x P E (x) = 1, P x P G (x)s1, as well as intuition and wishful thinking inspired by Shannon-Fano Theorem =-=[38]-=- and Coding Theorem 5.1 suggest there might indeed be tighter bounds: Conjecture 5.1 For x 2 B ] with P M (x) > 0: P M (x) = O(2 K M (x) ). Conjecture 5.2 For x 2 B ] with P E (x) > 0: P E (x) = O(2 K... |

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Citation Context ... 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 mightsip from 0 to 1 and back f... |

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Citation Context ...er, and therefore is indeed EOM-computable. Obviously the following holds: CP T (x) = CP T (x) = C 0 (x) and P T (x) = X z2B ] P T (xz) = 0 (x): 2 Summary. The traditional universal c.e. measure [40,=-= 45, 29, 16, 17, 41, 14, 30]-=- derives from universal MTMs with random input. What is the nature of our novel generalization here? We simply replace the MTMs by EOMs. As shown above, this leads to universal cumulatively enumerable... |

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Citation Context ... Given a GTM, it is not weakly decidable whether asnite bitstring is a converging program, or whether some of the output bits willsuctuate forever. Proof. (Based on the standard diagonalization trick =-=[11, 18, 43]-=-; compare a related result for analytic TMs [12, 21]). Let us write T (x) # if there is a z 2 B ] such that T (x) ; z. Let us write T (x) l if T 's outputsuctuates forever in response to x (e.g., bysi... |

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Citation Context ...niversal Turing Machines (TMs) more general than those considered in previous related work: unlike traditional TMs, General TMs or GTMs may edit their previous outputs (compare Burgin's inductive TMs =-=[6-=-]), and Enumerable Output Machines (EOMs) may do this provided the output does not decrease lexicographically. In the spirit of computability in the limit [19, 33, 15] we will dene: a formally describ... |

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Citation Context ...d of the output tape (if it is not the initial square or above the scanning head); (c) write 1 or 0 on square above output scanning head. Compare Burgin's inductive TMs and super-recursive algorithms =-=[6, 8]-=-. Example 2.1 (Pseudorandom universe based on halting problem) Why consider GTMs at all? Because in a certain sense they represent the non-plus-ultra of constructive computability. For instance, consi... |

3 |
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Citation Context ...are selected randomly. Since some of the possible input sequences cause nonhalting computations, the individual probabilities do not add up to 1. This and related issues inspired work on semimeasures =-=[40, 45, 41, 17, 3-=-0] as opposed to classical measures considered in traditional statistics. The next three denitions concerning semimeasures on B are almost but not quite identical to those of discrete semimeasures [3... |

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Citation Context ... GTMs may occasionally rewrite parts of their output, they are computationally more expressive than MTMs in the sense that they permit much more compact descriptions of certain objects | compare also =-=[7-=-]. For instance, K(x) K G (x) is unbounded, as will be shown next. This will later have consequences for predictions, given certain observations. Theorem 3.2 K(x) K G (x) is unbounded. Proof. Dene h 0... |

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1 |
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Citation Context ...eralized Kolmogorov Complexity for EOMs and GTMs Denition 3.2 (Generalized K T ) Given any TM T, dene K T (x) = min p fl(p) : T (p) ; xg Compare Schnorr's less general \process complexity" for MT=-=Ms [37, 44-=-]. Proposition 3.1 (K M ; K E ; K G based on Invariance Theorem) Consider Def. 2.4. Let C denote a set of TMs with universal TM U C (T 2 C). We drop the index T , writing K C (x) = K U C (x) K T (x) ... |