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BON+ is an applicative theory and closely related to the first order parts of the standard systems of explicit mathematics. As such it is also a natural framework for abstract computations. In this article we analyze this aspect of BON+ more closely. First a point is made for introducing a new operation τN, called truncation, to obtain a natural formalization of partial recursive functions in our applicative framework. Then we introduce the operational versions of a series of notions that are all equivalent to semi-decidability in ordinary recursion theory on the natural numbers, and study their mutual relationships over BON+ with τN.



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[1]Barendregt, H. P., The Lambda Calculus: Its Syntax and Semantics, North-Holland, Amsterdam, 1985.
[2]Beeson, M. J., Foundations of Constructive Mathematics: Metamathematical Studies, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3/6, Springer, Berlin, 1985.
[3]Feferman, S., A language and axioms for explicit mathematics, Algebra and Logic (Crossley, J. N., editor), Lecture Notes in Mathematics, vol. 450, Springer, Berlin, 1975, pp. 87139.
[4]Feferman, S., Constructive theories of functions and classes, Logic Colloquium ’78 (Boffa, M., van Dalen, D., and McAloon, K., editors), Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam, 1979, pp. 159224.
[5]Feferman, S. and Jäger, G., Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic, vol. 65 (1993), no. 3, pp. 243263.
[6]Feferman, S., Jäger, G., and Strahm, T., Foundations of explicit mathematics, in preparation.
[7]Hinman, P. G., Recursion-Theoretic Hierarchies, Perspectives in Mathematical Logic, vol. 9, Springer, Berlin, 1978.
[8]Kahle, R., Applikative Theorien und Frege-Strukturen, Ph.D. thesis, Institut für Informatik und angewandte Mathematik, Universität Bern, 1997.
[9]Kahle, R., N-strictness in applicative theories, Archive for Mathematical Logic, vol. 39 (2000), no. 2, pp. 125144.
[10]Kanamori, A., The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer, Berlin, 2003.
[11]Longley, J. and Normann, D., Higher-Order Computability, Springer, Berlin, 2015.
[12]Louvau, A., Some results in the Wadge hierarchy of Borel sets, Cabal Seminar 79–81 (Kechris, A. S., Martin, D. A., and Moschovakis, Y. N., editors), Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 2855.
[13]Montalbán, A. and Shore, R. A., The limits of determinacy in second order arithmetic, Proceedings of the London Mathematical Society, vol. 104 (2012), no. 2, pp. 223252.
[14]Nemoto, T., Determinacy of Wadge classes and subsystems of second order arithmetic, Mathematical Logic Quarterly, vol. 55 (2009), no. 2, pp. 154176.
[15]Nemoto, T. and Sato, K., A marriage of Brouwer’s Intuitionism and Hilbert’s Finitism I: Arithmetic, this JOURNAL, accepted for publication.
[16]Rin, B. G. and Walsh, S., Realizability semantics for quantified modal logic: Generalizing Flagg’s 1985 construction, The Review of Symbolic Logic, vol. 9 (2016), no. 4, pp. 752809.
[17]Rosebrock, T., Some models and semi-decidability notions of applicative theories, Ph.D. thesis, in preparation.
[18]Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, I, Studies in Logic and the Foundations of Mathematics, vol. 121, North-Holland, Amsterdam, 1988.
[19]Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, II, Studies in Logic and the Foundations of Mathematics, vol. 123, North-Holland, Amsterdam, 1988.



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