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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.
Can proofs be animated by games?
 . URZYCZYN ED., TLCA 2005, LNCS 3461
, 2005
"... Proof animation is a way of executing proofs to find errors in the formalization of proofs. It is intended to be "testing in proof engineering". Although the realizability interpretation as well as the functional interpretation based on limitcomputations were introduced as means for pro ..."
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Proof animation is a way of executing proofs to find errors in the formalization of proofs. It is intended to be "testing in proof engineering". Although the realizability interpretation as well as the functional interpretation based on limitcomputations were introduced as means for proof animation, they were unrealistic as an architectural basis for actual proof animation tools. We have found game theoretical semantics corresponding to these interpretations, which is likely to be the right architectural basis for proof animation.
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.
unknown title
"... Automatabased representations have recently been investigated as a tool for representing and manipulating sets of integer vectors. In this paper, we study some structural properties of automata accepting the encodings (most significant digit first) of the natural solutions of systems of linear Dio ..."
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Automatabased representations have recently been investigated as a tool for representing and manipulating sets of integer vectors. In this paper, we study some structural properties of automata accepting the encodings (most significant digit first) of the natural solutions of systems of linear Diophantine inequations, i.e., convex polyhedra in ¡£¢ Based on those structural properties, we develop an algorithm that takes as input an automaton and generates a quantifierfree formula that represents exactly the set of integer vectors accepted by the automaton. In addition, our algorithm generates the minimal Hilbert basis of the linear system. In experiments made with a prototype implementation, we have been able to synthesize in seconds formulas and Hilbert bases from automata with more than 10,000 states. 1.