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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.
Verifying haskell programs by combining testing and proving
 In Proceedings of the Third International Conference on Quality Software
"... We propose a method for improving confidence in the correctness of Haskell programs by combining testing and proving. Testing is used for debugging programs and specification before a costly proof attempt. During a proof development, testing also quickly eliminates wrong conjectures. Proving helps u ..."
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We propose a method for improving confidence in the correctness of Haskell programs by combining testing and proving. Testing is used for debugging programs and specification before a costly proof attempt. During a proof development, testing also quickly eliminates wrong conjectures. Proving helps us to decompose a testing task in a way that is guaranteed to be correct. To demonstrate the method we have extended the Agda/Alfa proof assistant for dependent type theory with a tool for random testing. As an example we show how the correctness of a BDDalgorithm written in Haskell is verified by testing properties of component functions. We also discuss faithful translations from Haskell to type theory.
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.
Testing and Proving in Dependent Type Theory (Part II: Verifying Haskell Programs by Combining Testing and Proving)
 CHALMERS UNIVERSITY OF TECHNOLOGY AND GOTEBORG UNIVERSITY
, 2003
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