Abstract
This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories, stemming from an initial algebra in the underlying category. This approach requires some non-trivial assumptions, like dcpo enrichment, which do not always hold, even in cases where one can reasonably speak of traces (like for weighted automata). More recently, it has been noticed that trace semantics can also arise by first performing a determinization construction. In this paper, we develop a systematic approach, in which the two approaches correspond to different orders of composing a functor and a monad, and accordingly, to different distributive laws. The relevant final coalgebra that gives rise to trace semantics does not live in a Kleisli category, but more generally, in a category of Eilenberg-Moore algebras. In order to exploit its finality, we identify an extension operation, that changes the state space of a coalgebra into a free algebra, which abstractly captures determinization of automata. Notably, we show that the two different views on trace semantics are equivalent, in the examples where both approaches are applicable.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Balan, A., Kurz, A.: On coalgebras over algebras. Theor. Comp. Sci. 412(38), 4989–5005 (2011)
Barr, M., Wells, C.: Toposes, Triples and Theories. Springer, Berlin (1985) (revized and corrected version), www.cwru.edu/artsci/math/wells/pub/ttt.html
Bonsangue, M.M., Milius, S., Silva, A.: Sound and complete axiomatizations of coalgebraic language equivalence, arxiv.org/abs/1104.2803 (2011)
Borceux, F.: Handbook of Categorical Algebra. Encyclopedia of Mathematics, vol. 50, 51 and 52. Cambridge Univ. Press (1994)
Boreale, M.: Weighted Bisimulation in Linear Algebraic Form. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 163–177. Springer, Heidelberg (2009)
Castro, P., Panangaden, P., Precup, D.: Equivalence relations in fully and partially observable Markov decision processes. In: Proc. IJCAI 2009, pp. 1653–1658 (2009)
Cîrstea, C.: Maximal traces and path-based coalgebraic temporal logics. Theor. Comput. Sci. 412(38), 5025–5042 (2011)
Coumans, D., Jacobs, B.: Scalars, monads and categories. In: Heunen, C., Sadrzadeh, M. (eds.) Compositional Methods in Physics and Linguistics, Oxford Univ. Press (2012)
Crafa, S., Ranzato, F.: A Spectrum of Behavioral Relations over LTSs on Probability Distributions. In: Katoen, J.-P., König, B. (eds.) CONCUR 2011 – Concurrency Theory. LNCS, vol. 6901, pp. 124–139. Springer, Heidelberg (2011)
Deng, Y., van Glabbeek, R., Hennessy, M., Morgan, C.: Characterising testing preorders for finite probabilistic processes. Logical Methods in Computer Science 4(4) (2008)
Doberkat, E.: Eilenberg-moore algebras for stochastic relations. Information and Computation 204(12), 1756–1781 (2006)
Hasuo, I., Jacobs, B., Sokolova, A.: Generic trace theory via coinduction. Logical Methods in Computer Science 3(4:11) (2007)
Hermanns, H., Parma, A., Segala, R., Wachter, B., Zhang, L.: Probabilistic logical characterization. Information and Computation 209(2), 154–172 (2011)
Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, 3rd edn. Wesley (2006)
Jacobs, B.: Coalgebraic Walks, in Quantum and Turing Computation. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 12–26. Springer, Heidelberg (2011)
Johnstone, P.T.: Adjoint lifting theorems for categories of algebras. Bull. London Math. Soc. 7, 294–297 (1975)
Kissig, C., Kurz, A.: Generic trace logics (2011), http://arxiv.org/abs/1103.3239
Manes, E.G.: Algebraic Theories. Springer, Berlin (1974)
Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin (1971)
Moggi, E.: Notions of computation and monads. Information and Computation 93(1), 55–92 (1991)
Parma, A., Segala, R.: Logical Characterizations of Bisimulations for Discrete Probabilistic Systems. In: Seidl, H. (ed.) FOSSACS 2007. LNCS, vol. 4423, pp. 287–301. Springer, Heidelberg (2007)
Pavlovic, D., Mislove, M.W., Worrell, J.B.: Testing Semantics: Connecting Processes and Process Logics. In: Johnson, M., Vene, V. (eds.) AMAST 2006. LNCS, vol. 4019, pp. 308–322. Springer, Heidelberg (2006)
Segala, R., Lynch, N.A.: Probabilistic Simulations for Probabilistic Processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR 1994. LNCS, vol. 836, pp. 481–496. Springer, Heidelberg (1994)
Silva, A.: Kleene coalgebra. PhD thesis, Radboud University Nijmegen (2010)
Silva, A., Bonchi, F., Bonsangue, M., Rutten, J.: Generalizing the powerset construction, coalgebraically. In: Proc. FSTTCS 2010. LIPIcs, vol. 8, pp. 272–283 (2010)
Silva, A., Sokolova, A.: Sound and complete axiomatization of trace semantics for probabilistic systems. In: Proc. MFPS 2011. ENTCS, vol. 276, pp. 291–311 (2011)
Varacca, D., Winskel, G.: Distributing probability over non-determinism. Mathematical Structures in Computer Science 16(1), 87–113 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 IFIP International Federation for Information Processing
About this paper
Cite this paper
Jacobs, B., Silva, A., Sokolova, A. (2012). Trace Semantics via Determinization. In: Pattinson, D., Schröder, L. (eds) Coalgebraic Methods in Computer Science. CMCS 2012. Lecture Notes in Computer Science, vol 7399. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32784-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-32784-1_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32783-4
Online ISBN: 978-3-642-32784-1
eBook Packages: Computer ScienceComputer Science (R0)