Abstract
Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces and measurable functions. Our main contribution is that the existence of a final coalgebra in the Kleisli category of these monads is closely connected to the measure-theoretic extension theorem for sigma-finite pre-measures. In fact, we obtain a practical definition of the trace measure for both finite and infinite traces of PTS that subsumes a well-known result for discrete probabilistic transition systems.
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.
References
Adámek, J., Bonchi, F., Hülsbusch, M., König, B., Milius, S., Silva, A.: A Coalgebraic Perspective on Minimization and Determinization. In: Birkedal, L. (ed.) FOSSACS 2012. LNCS, vol. 7213, pp. 58–73. Springer, Heidelberg (2012)
Ash, R.B.: Real Analysis and Probability. Probability and Mathematical Statistics – A Series of Monographs and Textbooks. Academic Press, New York (1972)
Chen, D., van Breugel, F., Worrell, J.: On the Complexity of Computing Probabilistic Bisimilarity. In: Birkedal, L. (ed.) FOSSACS 2012. LNCS, vol. 7213, pp. 437–451. Springer, Heidelberg (2012)
Cîrstea, C.: Generic infinite traces and path-based coalgebraic temporal logics. Electronic Notes in Theoretical Computer Science 264(2), 83–103 (2010)
Doberkat, E.: Stochastic relations: foundations for Markov transition systems. Chapman & Hall/CRC studies in informatics series. Chapman & Hall/CRC (2007)
Elstrodt, J.: Maß- und Integrationstheorie, 5th edn. Springer (2007)
Giry, M.: A categorical approach to probability theory. In: Categorical Aspects of Topology and Analysis. Lecture Notes in Mathematics, vol. 915, pp. 68–86. Springer (1981)
van Glabbeek, R., Smolka, S.A., Steffen, B., Tofts, C.M.N.: Reactive, generative and stratified models of probabilistic processes. Information and Computation 121, 59–80 (1995)
Hasuo, I., Jacobs, B., Sokolova, A.: Generic trace theory. In: International Workshop on Coalgebraic Methods in Computer Science. Electronic Notes in Theoretical Computer Science, vol. 164, pp. 47–65. Elsevier (2006)
Hasuo, I., Jacobs, B., Sokolova, A.: Generic trace semantics via coinduction. Logical Methods in Computer Science 3(4:11), 1–36 (2007)
Jacobs, B., Rutten, J.: A tutorial on (co)algebras and (co)induction. Bulletin of the European Association for Theoretical Computer Science 62, 222–259 (1997)
Kerstan, H.: Trace Semantics for Probabilistic Transition Systems - A Coalgebraic Approach. Diploma thesis, Universität Duisburg-Essen (September 2011), http://jordan.inf.uni-due.de/publications/kerstan/kerstan_diplomathesis.pdf
Kerstan, H., König, B.: Coalgebraic trace semantics for probabilistic transition systems based on measure theory. Tech. rep., Abteilung für Informatik und Angewandte Kognitionswissenschaft, Universität Duisburg-Essen (2012), http://jordan.inf.uni-due.de/publications/kerstan/coalgpts_concur12_long.pdf
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer (1998)
Panangaden, P.: Labelled Markov Processes. Imperial College Press (2009)
Rutten, J.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249, 3–80 (2000)
Sokolova, A.: Coalgebraic Analysis of Probabilistic Systems. Ph.D. thesis, Technische Universiteit Eindhoven (2005)
Sokolova, A.: Probabilistic systems coalgebraically: A survey. Theoretical Computer Science 412(38), 5095–5110 (2011); cMCS Tenth Anniversary Meeting
van Breugel, F., Worrell, J.: Approximating and computing behavioural distances in probabilistic transition systems. Theoretical Computer Science 360, 373–385 (2005)
van Breugel, F., Worrell, J.: A behavioural pseudometric for probabilistic transition systems. Theoretical Computer Science 331, 115–142 (2005)
Viglizzo, I.D.: Final Sequences and Final Coalgebras for Measurable Spaces. In: Fiadeiro, J.L., Harman, N.A., Roggenbach, M., Rutten, J. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 395–407. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kerstan, H., König, B. (2012). Coalgebraic Trace Semantics for Probabilistic Transition Systems Based on Measure Theory. In: Koutny, M., Ulidowski, I. (eds) CONCUR 2012 – Concurrency Theory. CONCUR 2012. Lecture Notes in Computer Science, vol 7454. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32940-1_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-32940-1_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32939-5
Online ISBN: 978-3-642-32940-1
eBook Packages: Computer ScienceComputer Science (R0)