Limit Computable Mathematics and Interactive Computation

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., 0 1 , but \executable" in the sense of Gold's theory of machine learning, i.e., 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. It may serve a uni ed framework for practical scienti c computing such as numerical analysis and problem solving in computer science such as veri cation via re nements. Many materials of this paper is complied from the manuscript [6]. 1 Proof Animation: testing proofs The formal development of proofs is becoming more and more practical thanks to advancements of hardware and software. However, the task