csl02 - Limit-Computable Mathematics and its Applications...

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Limit-Computable Mathematics and its Applications Sep, 22, 2002 CSL’02, Edinburgh, Scotland, UK
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LCM : Limit-Computable Mathematics Constructive mathematics is a mathematics based on 0 1 -functions , i.e. recursive functions. In the same sense, LCM is a mathematics based on 0 2 - functions .
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The aim of the talk The talk aims to present basic theoretical ideas of LCM and a little bit of the intended application as the motivation. Thus, in this talk THEORY APPLICATION (Proof Animation) although the original project was application oriented and still the motto is kept.
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Why 0 2 -functions? (1) 0 2 -functions are used as models of learning processes, and, in a sense, semi- computable. The original and ultimate goal of LCM project is materialization of Proof Animation Proof Animation is debugging of proofs. See http://www.shayashi.jp/PALCM/ for details of Proof Animation.
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Why 0 2 -functions? (2) The 0 2 -functions are expected to be useful for Proof Animation as learning theoretic algorithms were useful in E. Shapiro’s Algorithmic Debugging of Prolog programs Shapiro’s debugger debugged Prolog programs, i.e. axiom systems in Horn logic. In a similar vein, an LCM proof animator is expected to debug axiom systems and proofs of LCM logic, which is at least a super set of predicate constructive logic.
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An example of semi-computable learning process (1) MNP (Minimal Number Principle): Let f be a function form Na t to Nat . Then, there is n : Nat such that f(n) is the smallest value among f(0), f(1), f(2),… Nat : the set of natural numbers
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An example of semi-computable learning process (2) Such an n is not Turing-computable from f . However, the number n is obtained in finite time from f by a mechanical “computation”.
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A limit-computation of n (1) Regard the function f as a stream f(0), f(1), f(2), … Have a box of a natural number. We denote the content of the box by x .
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A limit-computation of n (2) Initialize the box by setting x =0 . Compare f( x ) with the next element of the stream, say f( n ). If the new one is smaller than f( x ) , then put n in the box. Otherwise, keep the old value in the box. Repeat the last step forever.
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A limit-computation of n (3) The process does not stop. But your box will eventually contain the correct answer and after then the content x will never change. In this sense, the non-terminating process
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csl02 - Limit-Computable Mathematics and its Applications...

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