## Toward Formalizing Non-Monotonic Reasoning in Physics: the Use of Kolmogorov Complexity to Formalize the Notions "Typically" and "Normally" (2004)

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Venue: | Proceedings of the Workshops on Intelligent Computing WIC’04 associated with the Mexican International Conference on Artificial Intelligence MICAI’04 |

Citations: | 10 - 8 self |

### BibTeX

@INPROCEEDINGS{Kreinovich04towardformalizing,

author = {Vladik Kreinovich},

title = {Toward Formalizing Non-Monotonic Reasoning in Physics: the Use of Kolmogorov Complexity to Formalize the Notions "Typically" and "Normally"},

booktitle = {Proceedings of the Workshops on Intelligent Computing WIC’04 associated with the Mexican International Conference on Artificial Intelligence MICAI’04},

year = {2004},

pages = {187--194}

}

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### Abstract

When a physicist writes down equations, or formulates a theory in any other terms, he usually means not only that these equations are true for the real world, but also that the model corresponding to the real world is "typical" among all the solutions of these equations. This type of argument is used when physicists conclude that some property is true by showing that it is true for "almost all" cases. There are formalisms that partially capture this type of reasoning, e.g., techniques based on the Kolmogorov-Martin-Lof definition of a random sequence.

### Citations

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(Show Context)
Citation Context ...it statement – that events with probability 0 cannot occur – is the basis of Kolmogorov-Martin-Löf formalization of the notion of a random sequence (and, more generally, a random object); see, e.g., =-=[13]-=-. The need for such a definition comes from the fact that in traditional statistics, there is no definition of a random sequence, while from the physics viewpoint, some sequences are random and some a... |

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Citation Context ...oduce the entire sequence. Thus, when K(x) is approximately equal to the length len(x) of a sequence, this sequence is random, otherwise it is not. The best source for Kolmogorov complexity is a book =-=[10]-=-. Physicists assume that initial conditions and values of parameters are not abnormal.sTo a mathematician, the main contents of a physical theory is the equations. The fact that the theory is formulat... |

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Citation Context .... However, anyone can toss a coin 100 times, and this prove that some sequences are physically possible. Historical comment. This problem was first noticed by Kyburg under the name of Lottery paradox =-=[9]-=-: in a big (e.g., state-wide) lottery, the probability of winning the Grand Prize is so small, then a reasonable person should not expect it. However, some people do win big prizes. How to formalize t... |

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Citation Context ...t of all heads. In plain words, if we have flipped a coin n times, and the results are n heads, then this coin is biased: it will always fall on heads. Let us describe this idea in mathematical terms =-=[4, 8]-=-. To make formal definitions, we must fix a formal theory: e.g., the set theory ZF (the definitions and results will not depend on what exactly theory we choose). A set S is called 5 definable if ther... |

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(Show Context)
Citation Context ...t of all heads. In plain words, if we have flipped a coin n times, and the results are n heads, then this coin is biased: it will always fall on heads. Let us describe this idea in mathematical terms =-=[4, 8]-=-. To make formal definitions, we must fix a formal theory: e.g., the set theory ZF (the definitions and results will not depend on what exactly theory we choose). A set S is called 5 definable if ther... |

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Citation Context ...nsions, even though no accurate estimates on the coefficients on these terms is known [6]. In particular, such methods are used in quantum field theory, where we add up several first Feynman diagrams =-=[4]-=-; in celestial mechanics [17], etc. Chaos naturally appears. Restriction to not abnormal also explains the origin of chaotic behavior of physical systems; see, e.g., [10]. Application: justification o... |

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Citation Context ...ate estimates on the coefficients on these terms is known [6]. In particular, such methods are used in quantum field theory, where we add up several first Feynman diagrams [4]; in celestial mechanics =-=[17]-=-, etc. Chaos naturally appears. Restriction to not abnormal also explains the origin of chaotic behavior of physical systems; see, e.g., [10]. Application: justification of physical induction. From th... |

8 |
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Citation Context ...physics, explicitly stated that such inverse processes "may be regarded as impossible, even though from the viewpoint of probability theory that outcome is only extremely improbable, not impossib=-=le." [1]-=-. Example 3. If we flip a fair coin 100 times in a row, and get heads all the time, then a person who is knowledgeable in probability would say that it is possible -- since the probability is still po... |

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4 |
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(Show Context)
Citation Context ...nduction). The general physical induction is difficult to justify, to the extent that a prominent philosopher C. D. Broad has called the unsolved problems concerning induction a scandal of philosophy =-=[2]. We can s-=-ay that the notion of "not abnormal" justifies physical induction by making it a provable theorem (and thus resolves the corresponding scandal). Acknowledgments. This work was supported in p... |

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4 |
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(Show Context)
Citation Context ...To describe this result – originally proven by V. Lifschitz [14] – in precise terms, let us recall the definitions of computable numbers, computable functions, and computable compact sets; see, e.g., =-=[16, 18]-=- (see also [1–6, 11, 12]). Denition 5. A real number x is called computable if there exists an algorithm (program) that transforms an arbitrary natural number k into a rational number rk which is 2 −... |

2 |
Ethics and th history of philosophy, Routledge and Kegan
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(Show Context)
Citation Context ...g ideas, the general physical induction is difficult to formalize, to the extent that a prominent philosopher C. D. Broad has called the unsolved problems concerning induction a scandal of philosophy =-=[2]-=-. We can say that the notion of “not abnormal” justifies physical induction (and thus resolves the corresponding scandal). Acknowledgments. This work was supported in part by the NASA grant NCC5-209, ... |

2 |
A.: Investigation of constructive functions by the method of
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(Show Context)
Citation Context ..., . . . , fm(s)) = (r1, . . . , rm) = r, then as the desired function F (x) we can take the sum of the squares F (x) = m∑ i=1 (fi(x)− ri)2. To describe this result – originally proven by V. Lifschitz =-=[14]-=- – in precise terms, let us recall the definitions of computable numbers, computable functions, and computable compact sets; see, e.g., [16, 18] (see also [1–6, 11, 12]). Denition 5. A real number x ... |