Abstract
We consider spaces of linear signals equipped with the prefix relation and a suitably defined generalized ultrametric distance function. We introduce a new class of abstract structures, which we call generalized ultrametric semilattices, and prove a representation theorem stating that generalized ultrametric semilattices with totally ordered distance sets are isomorphic to such spaces of linear signals. It follows that the definition of generalized ultrametric semilattices with totally ordered distance sets captures all formal properties of such spaces.
This work was supported in part by the Center for Hybrid and Embedded Software Systems (CHESS) at UC Berkeley, which receives support from the National Science Foundation (NSF awards #0720882 (CSR-EHS: PRET), #0931843 (CPS: Large: ActionWebs), and #1035672 (CPS: Medium: Ptides)), the Naval Research Laboratory (NRL #N0013-12-1-G015), and the following companies: Bosch, National Instruments, and Toyota.
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References
Hitzler, P., Seda, A.K.: Generalized metrics and uniquely determined logic programs. Theoretical Computer Science 305(1-3), 187–219 (2003)
Priess-Crampe, S., Ribenboim, P.: Ultrametric spaces and logic programming. The Journal of Logic Programming 42(2), 59–70 (2000)
Naundorf, H.: Strictly causal functions have a unique fixed point. Theoretical Computer Science 238(1-2), 483–488 (2000)
Liu, X., Matsikoudis, E., Lee, E.A.: Modeling timed concurrent systems. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 1–15. Springer, Heidelberg (2006)
Matsikoudis, E., Lee, E.A.: The fixed-point theory of strictly causal functions. Technical Report UCB/EECS-2013-122, EECS Department, University of California, Berkeley (June 2013)
Priess-Crampe, S., Ribenboim, P.: Fixed points, combs and generalized power series. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 63(1), 227–244 (1993)
Smyth, M.B.: Quasi-uniformities: Reconciling domains with metric spaces. In: Main, M.G., Mislove, M.W., Melton, A.C., Schmidt, D. (eds.) MFPS 1987. LNCS, vol. 298, pp. 236–253. Springer, Heidelberg (1988)
Flagg, R.C., Kopperman, R.: Continuity spaces: Reconciling domains and metric spaces. Theoretical Computer Science 177(1), 111–138 (1997)
Edalat, A., Heckmann, R.: A computational model for metric spaces. Theoretical Computer Science 193(1-2), 53–73 (1998)
Krötzsch, M.: Generalized ultrametric spaces in quantitative domain theory. Theoretical Computer Science 368(1-2), 30–49 (2006)
Lee, E.A., Varaiya, P.: Structure and Interpretation of Signals and Systems, 2nd edn. (2011), http://LeeVariaya.org
Lee, E.A., Sangiovanni-Vincentelli, A.: A framework for comparing models of computation. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 17(12), 1217–1229 (1998)
Lee, E.A.: Modeling concurrent real-time processes using discrete events. Annals of Software Engineering 7(1), 25–45 (1999)
Hodges, W.: Model Theory. Encyclopedia of Mathematics and its Applications, vol. 42. Cambridge University Press (1993)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002)
Enderton, H.B.: A Mathematical Introduction to Logic, 2nd edn. Harcourt/Academic Press (2001)
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Matsikoudis, E., Lee, E.A. (2013). An Axiomatization of the Theory of Generalized Ultrametric Semilattices of Linear Signals. In: GÄ…sieniec, L., Wolter, F. (eds) Fundamentals of Computation Theory. FCT 2013. Lecture Notes in Computer Science, vol 8070. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40164-0_24
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DOI: https://doi.org/10.1007/978-3-642-40164-0_24
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