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A Limiting First Order Realizability Interpretation
"... Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics ..."
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Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics in which any proof of an existence theorem is equipped with a function computing the solution of the theorem in the limit.
Towards Limit Computable Mathematics
"... The notion of LimitComputable Mathematics (LCM) will be introduced. LCM is a fragment of classical mathematics in which the law of excluded middle is restricted to 1 0 2 formulas. We can give an accountable computational interpretation to the proofs of LCM. The computational content of LCMp ..."
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The notion of LimitComputable Mathematics (LCM) will be introduced. LCM is a fragment of classical mathematics in which the law of excluded middle is restricted to 1 0 2 formulas. We can give an accountable computational interpretation to the proofs of LCM. The computational content of LCMproofs is given by Gold's limiting recursive functions, which is the fundamental notion of learning theory. LCM is expected to be a right means for "Proof Animation," which was introduced by the first author [10]. LCM is related not only to learning theory and recursion theory, but also to many areas in mathematics and computer science such as computational algebra, computability theories in analysis, reverse mathematics, and many others.
Limit Computable Mathematics and Interactive Computation
, 2001
"... We are now investigating an "executable" fragment of classical mathematics for testing formal proofs to make formal proof developments less laborious. Several theories of execution of full classical proofs are known. In these theories, some kind of abstract values such as continuations, ar ..."
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We are now investigating an "executable" fragment of classical mathematics for testing formal proofs to make formal proof developments less laborious. Several theories of execution of full classical proofs are known. In these theories, some kind of abstract values such as continuations, are necessary. It makes them illegible from computational point of view, although they are mathematically interesting. In contrast, we consider only a fragment of classical mathematics and give a simple and natural "computational" contents without such abstract values. The fragment appears to cover a rather large domain of practical mathematics. The point is that codes associated to proof by our method is not computable in Turing's sense, i.e., \Delta 0 1 , but "executable " in the sense of Gold's theory of machine learning, i.e., \Delta 0 2 . I will give a survey of this new executable mathematics LCM. I will also discuss a possible framework of "interactive computation" emerged from LCM research. ...
A Notion of a Computational Step for Partial Combinatory Algebras
"... Abstract. Working within the general formalism of a partial combinatory algebra (or PCA), we introduce and develop the notion of a step algebra, which enables us to work with individual computational steps, even in very general and abstract computational settings. We show that every partial applicat ..."
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Abstract. Working within the general formalism of a partial combinatory algebra (or PCA), we introduce and develop the notion of a step algebra, which enables us to work with individual computational steps, even in very general and abstract computational settings. We show that every partial applicative structure is the closure of a step algebra obtained by repeated application, and identify conditions under which this closure yields a PCA.