Abstract
We present general ideas about the complexity of objects and how complexity can be used to define the information in objects. In essence, the idea is that while complexity is relative to a given class of processes, information is process independent: information is complexity relative to the class of all conceivable processes. We test these ideas on the complexity of classical states. A domain is used to specify the class of processes, and both qualitative and quantitative notions of complexity for classical states emerge. The resulting theory can be used to give new proofs of fundamental results from classical information theory, to give a new characterization of entropy, to derive lower bounds on algorithmic complexity and even to establish new connections between physics and computation. All of this is a consequence of the setting which gives rise to the fixed point theorem: The least fixed point of the copying operator above complexity is information.
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
Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. III, Oxford University Press, Oxford (1994)
Alberti, P.M., Uhlmann, A.: Stochasticity and partial order: doubly stochastic maps and unitary mixing, Dordrecht, Boston (1982)
Kraft, L.G.: A device for quantizing, grouping and coding amplitude modulated pulses. M.S. Thesis, Electrical Engineering Department, MIT (1949)
Marshall, A.W., Olkin, I.: Inequalities: Theory of majorization and its applications. Academic Press Inc., London (1979)
Muirhead, R.F.: Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proc. Edinburgh Math. Soc. 21, 144–157 (1903)
Martin, K.: A foundation for computation. Ph.D. Thesis, Department of Mathematics, Tulane University (2000)
Martin, K.: Entropy as a fixed point. Oxford University Computing Laboratory, Research Report PRG-RR-03-05 (February 2003), http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-03-05.html
Martin, K.: A triangle inequality for measurement. Applied Categorical Structures 11(1) (2003)
Martin, K.: The measurement process in domain theory. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, p. 116. Springer, Heidelberg (2000)
Schrödinger, E.: Proceedings of the Cambridge Philosophical Society 32, 446 (1936)
Scott, D.: Outline of a mathematical theory of computation. Technical Monograph PRG-2, Oxford University Computing Laboratory (November 1970)
Shannon, C.E.: A mathematical theory of communication. Bell Systems Technical Journal 27, 379–423, 623–656 (1948)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Martin, K. (2004). Entropy as a Fixed Point. In: DÃaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_79
Download citation
DOI: https://doi.org/10.1007/978-3-540-27836-8_79
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22849-3
Online ISBN: 978-3-540-27836-8
eBook Packages: Springer Book Archive