Abstract
We seek universal categorical conditions ensuring the representability of all partial recursive functions. In the category Pfn of sets and partial functions, the natural numbers provide both an initial algebra and a final coalgebra for the functor 1 + −. We recount how finality yields closure of the partial functions on natural numbers under Kleene’s μ-recursion scheme. Noting that Pfn is not cartesian, we then build on work of Paré and Román, obtaining weak initiality and finality conditions on natural numbers algebras in monoidal categories that ensure the (weak) representability of all partial recursive functions. We further obtain some positive results on strong representability. All these results adapt to Kleisli categories of cartesian categories with natural numbers algebras. However, in general, not all partial recursive functions need be strongly representable.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alves, S., Fernández, M., Florido, M., Mackie, I.: Linear recursive functions. In: Comon-Lundh, H., Kirchner, C., Kirchner, H. (eds.) Rewriting, Computation and Proof. LNCS, vol. 4600, pp. 182–195. Springer, Heidelberg (2007)
Barr, M., Wells, C.: Category Theory for Computing Science. Prentice Hall (1998); Also available as Reprints in Theory and Applications of Categories, vol. 22, pp. 1–538 (2012), www.tac.mta.ca/tac/reprints/
Bucalo, A., Führmann, C., Simpson, A.K.: An equational notion of lifting monad. Theor. Comput. Sci. 294(1/2), 31–60 (2003)
Cockett, J.R.B., Lack, S.: Restriction categories II: partial map classification. Theor. Comput. Sci. 294(1/2), 61–102 (2003)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications, vol. 93. CUP (2003)
Gladstone, M.: Simplification of the recursion scheme. J. Symb. Logic 36(4), 653–665 (1971)
Lambek, J., Scott, P.J.: Introduction to Higher-Order Categorical Logic. Cambridge Studies in Advanced Mathematics, vol. 7. CUP (1988)
Jacobs, B.P.F.: Semantics of weakening and contraction. Annals of Pure and Applied Logic 69, 73–106 (1994)
Jockusch Jr., C.G., Soare, R.I.: \(\mathrm{\Pi}^0_1\) classes and degrees of theories. Trans. Amer. Math. Soc. 173(2), 33–56 (1972)
Johnstone, P.T.: Sketches of an Elephant: a Topos Theory Compendium, vol. 1. OUP (2002)
Mackie, I., Román, L., Abramsky, S.: An internal language for autonomous categories. Journal of Applied Categorical Structures 1, 311–343 (1993)
Paré, R., Román, L.: Monoidal categories with natural numbers object. Studia Logica 48(3), 361–376 (1989)
Power, A.J., Robinson, E.: Premonoidal categories and notions of computation. Mathematical Structures in Computer Science 7(5), 453–468 (1997)
Smoryński, C.: Logical Number Theory I. Springer (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Plotkin, G. (2013). Partial Recursive Functions and Finality. In: Coecke, B., Ong, L., Panangaden, P. (eds) Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky. Lecture Notes in Computer Science, vol 7860. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38164-5_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-38164-5_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38163-8
Online ISBN: 978-3-642-38164-5
eBook Packages: Computer ScienceComputer Science (R0)