## On Kolmogorov machines and related issues (1988)

Venue: | Bull. of Euro. Assoc. for Theor. Computer Science |

Citations: | 6 - 2 self |

### BibTeX

@ARTICLE{Gurevich88onkolmogorov,

author = {Yuri Gurevich},

title = {On Kolmogorov machines and related issues},

journal = {Bull. of Euro. Assoc. for Theor. Computer Science},

year = {1988},

pages = {71--82}

}

### OpenURL

### Abstract

I felt honored and uncertain when Grzegorsz Rozenberg, the president of EATCS, proposed that I write a continuing column on logic in computer science in this Bulletin. Writing essays wasn’t my favorite subject in high school. After some hesitation, I decided to give it a try. I’ll need all the help I can get from you: criticism, comments, queries, suggestions, etc. Andrei Nikolayevich Kolmogorov died a few months ago. In recent years he chaired the Department of Mathematical Logic at the Moscow State University. In a later article or articles, I hope to discuss Kolmogorov’s ideas on randomness and information complexity; here let me take up the issue of Kolmogorov machines and their close relatives, Schönhage machines. I believe, we are a bit too faithful to the Turing model. It is often easier to explain oneself in a dialog. To this end, allow me to introduce my imaginary student Quizani. • Quizani: I think you should introduce yourself too. Don’t assume everyone knows you. • Author: All right. I grew up in the Soviet Union and started my career in the Ural University as an algebraist and self-taught logician. In 1973, I emigrated to Israel where I did logic and taught at Ben-Gurion

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Citation Context ...king, the Schönhage model provides a good measure of time complexity at the current state of art (though I would prefer something along the lines of the random access computers of Angluin and Valiant =-=[AV]-=-). On the theoretical side, all good things said above about the KU model apply also to the Schönhage model. Still, I don’t think that Schönhage’s suggestion has sufficient theoretical justifications.... |

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Citation Context ...for simulating p on input x. By analogy with what is known as Kolmogorov or information complexity (introduced explicitly or implicitly by Kolmogorov [Ko2], Solomonoff [So] and somewhat later Chaitin =-=[Ch]-=-), define the Levin complexity (this is my term) of a string w relative to x (and F ): L(w/x) = min{|p|+log(T (p, x)) : Given x, p computes w and then runs F on w and finds out whether F (w) = x} Here... |

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Citation Context ...ept that the tape can change its topology. I will use an unorthodox presentation of KU machines that seems more convenient [L2]; it is somewhat influenced by Schönhage’s presentations of his machines =-=[Sh]-=-. The tape is a finite connected graph with a distinguished (active) node. The graph is directed but symmetric: If there is an edge from u to v then there is an edge from v to u. The edges are colored... |

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Citation Context ...t only being simulated by, but actually being) the computation of an appropriate KU machine (in the more general form). In a sense, this is stronger than Turing’s thesis. The only 3theorem proved in =-=[KU]-=- is that every partial recursive function is KU computable. • Q: Is there any evidence that the KU model is indeed more powerful than the Turing model? • A: Grigoriev [Gr] exhibited a function real-ti... |

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Citation Context ...t the universal machine U needs for simulating p on input x. By analogy with what is known as Kolmogorov or information complexity (introduced explicitly or implicitly by Kolmogorov [Ko2], Solomonoff =-=[So]-=- and somewhat later Chaitin [Ch]), define the Levin complexity (this is my term) of a string w relative to x (and F ): L(w/x) = min{|p|+log(T (p, x)) : Given x, p computes w and then runs F on w and f... |

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Citation Context ...eneral, the notion of linear-time computability is very dependent on the computational model. A closely related class NL of functions KU computable in nearly-linear time n · log n O(1) is very robust =-=[GS]-=-. Instead of the KU model, one can use the Schönhage model, numerous RAM models, and so on and so forth. It 4is conjectured in [GS] that not all NL functions are Turing computable in nearly linear ti... |

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Citation Context ...x) be the time that the universal machine U needs for simulating p on input x. By analogy with what is known as Kolmogorov or information complexity (introduced explicitly or implicitly by Kolmogorov =-=[Ko2]-=-, Solomonoff [So] and somewhat later Chaitin [Ch]), define the Levin complexity (this is my term) of a string w relative to x (and F ): L(w/x) = min{|p|+log(T (p, x)) : Given x, p computes w and then ... |

1 |
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Citation Context ...olmogorov-Uspensky, machines [Ko1, KU, US] are similar to Turing machines except that the tape can change its topology. I will use an unorthodox presentation of KU machines that seems more convenient =-=[L2]-=-; it is somewhat influenced by Schönhage’s presentations of his machines [Sh]. The tape is a finite connected graph with a distinguished (active) node. The graph is directed but symmetric: If there is... |

1 |
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Citation Context ...mbol, then outputs a symbol within c steps, and so on. In the case of more powerful machines, one may want to speak about more complicated data items (than just symbols) as one-step inputs or outputs =-=[PS]-=-. In applications, the situation is much more difficult. There is no consensus among experts on real-time systems what real-time computability is. • Q: How does Grigoriev exploit changing topology? • ... |

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Citation Context ...t this or another formalization of Schönhage’s thesis. • Q: Are Schönhage machines more powerful than KU machines with respect to real time? • A: That is a hard question; some progress is reported in =-=[St]-=-. Schönhage machines seem more powerful with respect to real time and linear time. They can multiply integers in linear time [Sh]; KU machines probably cannot do that. It is known that Turing machines... |