Abstract
We develop a topological theory of continuous-time automata which replaces finiteness assumptions in the classical theory of finite automata by compactness assumptions. The theory is designed to be as mathematically simple as possible while still being relevant to the question of physical feasibility. We include a discussion of which behaviors are and are not permitted by the framework, and the physical significance of these questions. To illustrate the mathematical tractability of the theory, we give basic existence results and a Myhill-Nerode theorem. A major attraction of the theory is that it covers finite automata and continuous automata in the same abstract framework.
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- 1.
While we are interested in continuous computation generally, the theory of deterministic automata provides a nice starting point because of its simplicity. In addition, the necessity of constant, instantaneous updates is a feature of many real-world continuous-time problems.
- 2.
The delay is interesting here because it connects the compactness principles for input and state space. Since the delay needs to store the last \(\tau \) part of the input in its state, it has compact state space only if the space of \(\tau \)-length inputs is compact.
- 3.
A full account of all of these results will appear in my forthcoming dissertation.
References
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: in Metric Spaces and in the Space of Probability Measures. Springer, Heidelberg (2006)
Bishop, M.: Foundations of Constructive Analysis. McGraw-Hill Book Company, New York (1967)
Bournez, O., Campagnolo, M.L.: A survey on continuous time computations. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms, pp. 383–423. Springer, New York (2008)
Henzinger, T.A.: The theory of hybrid automata. In: Inan, M.K., Kurshan, R.P. (eds.) Verification of Digital and Hybrid Systems. NATO ASI Series, vol. 170, pp. 265–292. Springer, Heidelberg (2000)
Jeandel, E.: Topological automata. Theor. Comput. Syst. 40(4), 397–407 (2007)
Lynch, N., Segala, R., Vaandrager, F.: Hybrid i/o automata. Inf. Comput. 185(1), 105–157 (2003)
Rabinovich, A.: Automata over continuous time. Theor. Comput. Sci. 300(1), 331–363 (2003)
Weihrauch, K.: Computable Analysis: An Introduction. Springer Science & Business Media, Heidelberg (2012)
Young, L.C.: Lectures on the Calculus of Variations and Optimal Control Theory, vol. 304. American Mathematical Society, London (1980)
Acknowledgements
I am grateful to my friends, instructors, and others who have helped me develop this research into its current form. There are a few I want to thank by name. Iian Smythe gave me an invaluable mathematical pointer early on. Bob Constable taught me the foundations of constructive thought, which subtly permeate the ideas presented here. Anonymous reviewers of earlier versions of this paper provided extremely useful feedback on wide-ranging matters, including among many other things the writing style and how to explain what the Young metric actually does. Lastly, I thank my advisor Anil Nerode deeply for his support and insight.
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Messick, S. (2016). Compactness in the Theory of Continuous Automata. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_18
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DOI: https://doi.org/10.1007/978-3-319-27683-0_18
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