Abstract
We explore recursion theory on the reals, the analog counterpart of recursive function theory. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions, such as composition and various forms of differential equations. We define classes of real recursive functions, in a manner similar to the classical approach in recursion theory, and we study their complexity. In particular, we prove both upper and lower bounds for several classes of real recursive functions, which lie inside the primitive recursive functions and, therefore, can be characterized in terms of standard computational complexity.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V. I. Arnold. Equations Différentielles Ordinaires. Editions Mir, 5 ème edition, 1996.
A. Babakhanian. Exponentials in differentially algebraic extension fields. Duke Math. J., 40:455–458, 1973.
M.S. Branicky. Universal computation and other capabilities of hybrid and continuous dynamical systems. Theoretical Computer Science, 138(1):67–100, 1995.
L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over the real numbers: NP-completness, recursive functions and universal machines. Bull. Amer. Math. Soc., 21:1–46, 1989.
M.L. Campagnolo. Computational complexity of real recursive functions and analog circuits. PhD thesis, Instituto Superior Técnico, 2001.
P. Clote. Computational models and function algebras. In E.R. Griffor, editor, Handbook of Computability Theory, pages 589–681. Elsevier, 1999.
M.L. Campagnolo, C. Moore, and J.F. Costa. An analog characterization of the Grzegorczyk hierarchy. To appear in the Journal of Complexity.
N. J. Cutland. Computability: An Introduction to Recursive Function Theory. Cambridge University Press, 1980.
D. Graça. The general purpose analog computer and recursive functions over the reals. Master’s thesis, Instituto Superior Técnico, 2002.
A. Grzegorczyk. Computable functionals. Fund. Math., 42:168–202, 1955.
P. Hartman. Ordinary Differential Equations. Birkhauser, 2nd edition, 1982.
P. Henrici. Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York, 1962.
J. E. Hopcroft and J. D. Ullman. Introduction to Automata Theory, Languages and Computation. Addisson-Wesley, 1979.
P. Koiran and C. Moore. Closed-form analytic maps in one or two dimensions can simulate Turing machines. Theoretical Computer Science, 210:217–223, 1999.
K.-I. Ko. Complexity Theory of Real Functions. Birkhauser, 1991.
C. Moore. Recursion theory on the reals and continuous-time computation. Theoretical Computer Science, 162:23–44, 1996.
C. Moore. Finite-dimensional analog computers: flows, maps, and recurrent neural networks. In C.S. Calude, J. Casti, and M.J. Dinneen, editors, Unconventional Models of Computation, DMTCS. Springer-Verlag, 1998.
P. Odifreddi. Classical Recursion Theory. Elsevier, 1989.
P. Odifreddi. Classical Recursion Theory II. Elsevier, 2000.
R.W. Ritchie. Classes of predictably computable functions. Transactions Amer. Math. Soc., 106:139–173, 1963.
H.E. Rose. Subrecursion: Functions and Hierarchies. Clarendon Press, 1984.
J. Traub and A.G. Werschulz. Complexity and Information. Cambridge University Press, 1998.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Campagnolo, M.L. (2002). The Complexity of Real Recursive Functions. In: Unconventional Models of Computation. UMC 2002. Lecture Notes in Computer Science, vol 2509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45833-6_1
Download citation
DOI: https://doi.org/10.1007/3-540-45833-6_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44311-7
Online ISBN: 978-3-540-45833-3
eBook Packages: Springer Book Archive