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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.
Mathematics based on Incremental Learning AbstractExcluded middle and Inductive inference
"... Learning theoretic aspects of mathematics and logic have been studied by many authors. They study how mathematical and logical objects are algorithmically \learned " (inferred) from nite data. Although they study mathematical objects, the objective of the studies is learning. In this paper, ..."
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Learning theoretic aspects of mathematics and logic have been studied by many authors. They study how mathematical and logical objects are algorithmically \learned &quot; (inferred) from nite data. Although they study mathematical objects, the objective of the studies is learning. In this paper, a mathematics of which foundation itself is learning theoretic will be introduced. It is called LimitComputable Mathematics. Itwas originally introduced as a means for \Proof Animation, &quot; which is expected to make interactive formal proof development easier. Although the original objective was not learning theoretic at all, learning theory is indispensable for our research. It suggests that logic and learning theory are related in a still unknown but deep new way. 1 Mathematics based on Learning? Mathematical or logical concepts seem to be one of the main research targets of learning theory and its applications. Shapiro [25] investigated how axioms systems are inductively inferred by ideas of learning theory. Wemaysaythat Shapiro studied how logical systems (axiom systems) are learned. Stephan and Ventsov [27] investigated how algebraic structures are learned and have given some interesting learning theoretic characterizations of fundamental algebraic notions. We maysay that they investigated learnability of the mathematical concepts. Contrary to them, we arenowdeveloping a mathematics whose semantics and reasoning system are in uenced by ideas from computational learning theory. Let us compare these two lines of research.