Abstract
The relation between least and diagonal fixed points is a well known and completely studied question for a large class of partially ordered models of the lambda calculus and combinatory logic. Here we consider this question in the context of algebraic recursion theory, whose close connection with combinatory logic recently become apparent. We find a comparatively simple and rather weak general condition which suffices to prove the equality of least fixed points with canonical (corresponding to those produced by the Curry combinator in lambda calculus) diagonal fixed points in a class of partially ordered algebras which covers both combinatory spaces of Skordev and operative spaces of Ivanov. Especially, this yields an essential improvement of the axiomatization of recursion theory via combinatory spaces.
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References
Barendregt, H.: The Lambda Calculus: its Syntax and Semantics. Amsterdam, 1981
Di Paola R.A., A. Heller.: Dominical categories: recursion theory without elements. The Journal of Symb. Logic 52, 594–635 (1987)
Ivanov, L.L.: Iterative operator spaces. Thesis. Sofia University, 1980 (in Bulgarian)
Ivanov, L.L.: Algebraic Recursion Theory. Ellis Horwood, Chichester, 1986
Kleene, S.C.: Introduction to metamathematics. Groningen, 1952
Skordev, D.G.: Recursion theory on iterative combinatory spaces. Bull. Acad. Polon. Sci. 24, 23–31 (1976)
Skordev, D.G.: Combinatory Spaces and recursiveness in them (in Russian) Sofia, BAN, 1980
Skordev, D.G.: Computability in Combinatory Spaces. Amsterdam, 1992
Zashev, J.: Code evaluation in operative spaces with storage operation. Annuaire de l'Universite de Sofia, Faculte de Mathematiques et Informatique 87, 307–326 (1993)
Zashev, J. On the recursion theorem in iterative operative spaces. The Journal of Symb. Logic 66, 1727–1748 (2001)
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Supported in part by the Ministry of Education and Science of Republic of Bulgaria, contract No 705
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Zashev, J. Diagonal fixed points in algebraic recursion theory. Arch. Math. Logic 44, 973–994 (2005). https://doi.org/10.1007/s00153-005-0307-x
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DOI: https://doi.org/10.1007/s00153-005-0307-x