Limit-Computable Mathematics

and Its Applications

Susumu Hayashi1 and Yohji Akama2

1 Kobe UniversityRokko-dai Nada Kobe 657-8501, Japan

shayashi@kobe-u.ac.jp

http://www.shayashi.jp2 Tohoku University

Sendai Miyagi 980-8578, Japanakama@math.tohoku.ac.jp

http://www.math.tohoku.ac.jp/~akama

Abstract. Limit-Computable Mathematics (LCM) is a fragment ofclassical mathematics in which classical principles are restricted so thatthe existence theorems are realized by limiting recursive functions. LCMis expected to be a right means for “Proof Animation,” which was intro-duced by the first author. In the lecture, some mathematical foundationsof LCM will be given together with its relationships to various areas.

LCM is constructive mathematics augmented with some classical principles “ex-ecutable” by the limiting recursive functions of the computational learning the-ories. It may be said based on the notion of learning in the same sense thatconstructive mathematics is based on the notion of computation.

It was introduced to materialize the idea of Proof Animation by the firstauthor, which is a technique to animate formal proofs for validation in the samesense as formal specifications are animated for validation. Proof animation re-sembles Shapiro’s algorithmic debugging of logic programs, which is also basedon learning theory.

LCM was conceived through a fact that David Hilbert’s original proof ofhis famous finite basis theorem in 1888 is realized by Gold’s idea of learning.Hilbert’s proof is known to be a “first” non-computational proof of the area.

However, Hilbert’s proof gives a limiting recursive process by which the so-lutions are learned (computable in the limit). This is because he used only thelaws of excluded middle limited to Σ0

1-formulas. LCM is a mathematics whoseproofs are restricted to this kind of proofs. A remarkable thing is that a wideclass of classical proofs of concrete mathematics falls in the scope of LCM.

Some different approaches of mathematical foundations of LCM have beengiven by the authors and Berardi. Hayashi, Kohlenbach et al. have shown thatthere is a hierarchy of the laws of excluded middle and their equivalent theoremsin mathematics, resembling the hierarchy of reverse mathematics. Relationshipsof LCM to learning theory, computability theory over real numbers and oth-ers have been known. Information including manuscripts on LCM and ProofAnimation are available at http://www.shayashi.jp/PALCM/.

J. Bradfield (Ed.): CSL 2002, LNCS 2471, p. 1, 2002.c© Springer-Verlag Berlin Heidelberg 2002