The Theory of Traces for Systems with Nondeterminism and Probability

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories.

[1]  Damien Pous,et al.  Complete Lattices and Up-To Techniques , 2007, APLAS.

[2]  Jurriaan Rot,et al.  A general account of coinduction up-to , 2016, Acta Informatica.

[3]  Zbigniew Semadeni,et al.  Monads and their Eilenberg-Moore algebras in functional analysis , 1973 .

[4]  Lijun Zhang,et al.  Probabilistic Logical Characterization , 2011, Inf. Comput..

[5]  Joël Ouaknine,et al.  Convex Language Semantics for Nondeterministic Probabilistic Automata , 2018, ICTAC.

[6]  Gordon D. Plotkin,et al.  Combining Computational Effects: commutativity & sum , 2002, IFIP TCS.

[7]  Rocco De Nicola,et al.  Revisiting Trace and Testing Equivalences for Nondeterministic and Probabilistic Processes , 2012, Log. Methods Comput. Sci..

[8]  Erik P. de Vink,et al.  A hierarchy of probabilistic system types , 2003, CMCS.

[9]  R. V. Glabbeek.  The Linear Time-Branching Time Spectrum I The Semantics of Concrete , Sequential ProcessesR , 2007 .

[10]  Bart Jacobs,et al.  Convexity, Duality and Effects , 2010, IFIP TCS.

[11]  Carroll Morgan,et al.  Characterising Testing Preorders for Finite Probabilistic Processes , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[12]  Ana Sokolova,et al.  Generic Trace Semantics via Coinduction , 2007, Log. Methods Comput. Sci..

[13]  Ernst-Erich Doberkat,et al.  Eilenberg-Moore algebras for stochastic relations , 2006, Inf. Comput..

[14]  Marta Z. Kwiatkowska,et al.  PRISM: Probabilistic Symbolic Model Checker , 2002, Computer Performance Evaluation / TOOLS.

[15]  Davide Sangiorgi,et al.  Enhancements of the bisimulation proof method , 2012, Advanced Topics in Bisimulation and Coinduction.

[16]  Roberto Segala,et al.  A Compositional Trace-Based Semantics for Probabilistic Automata , 1995, CONCUR.

[17]  Bart Jacobs,et al.  Introduction to Coalgebra: Towards Mathematics of States and Observation , 2016, Cambridge Tracts in Theoretical Computer Science.

[18]  F. Bartels.  On generalised coinduction and probabilistic specification formats : Distributive laws in coalgebraic modelling , 2004 .

[19]  Bart Jacobs,et al.  Coalgebraic Trace Semantics for Combined Possibilitistic and Probabilistic Systems , 2008, CMCS.

[20]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[21]  Nancy A. Lynch,et al.  Probabilistic Simulations for Probabilistic Processes , 1994, Nord. J. Comput..

[22]  Doina Precup,et al.  Equivalence Relations in Fully and Partially Observable Markov Decision Processes , 2009, IJCAI.

[23]  Klaus Keimel,et al.  Mixed powerdomains for probability and nondeterminism , 2016, Log. Methods Comput. Sci..

[24]  Martin Wirsing,et al.  Theoretical Aspects of Computing - ICTAC 2005, Second International Colloquium, Hanoi, Vietnam, October 17-21, 2005, Proceedings , 2005, ICTAC.

[25]  Michael W. Mislove.  Nondeterminism and Probabilistic Choice: Obeying the Laws , 2000, CONCUR.

[26]  Sebastian Junges,et al.  A Storm is Coming: A Modern Probabilistic Model Checker , 2017, CAV.

[27]  Jan J. M. M. Rutten,et al.  Universal coalgebra: a theory of systems , 2000, Theor. Comput. Sci..

[28]  Ohad Kammar,et al.  Semantics for probabilistic programming: higher-order functions, continuous distributions, and soft constraints , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[29]  Wang Yi,et al.  Testing Probabilistic and Nondeterministic Processes , 1992, PSTV.

[30]  Valentina Castiglioni,et al.  Trace and Testing Metrics on Nondeterministic Probabilistic Processes , 2018, EXPRESS/SOS.

[31]  Lijun Zhang,et al.  Distribution-Based Bisimulation for Labelled Markov Processes , 2017, FORMATS.

[32]  Ana Sokolova,et al.  Probabilistic systems coalgebraically: A survey , 2011, Theor. Comput. Sci..

[33]  Ernst-Erich Doberkat.  Erratum and Addendum: Eilenberg-Moore algebras for stochastic relations , 2008, Inf. Comput..

[34]  Sam Staton.  Relating Coalgebraic Notions of Bisimulation , 2009, CALCO.

[35]  Carroll Morgan,et al.  Testing Finitary Probabilistic Processes , 2009, CONCUR.

[36]  Hans A. Hansson.  Time and probability in formal design of distributed systems , 1991, DoCS.

[37]  Alexandra Silva,et al.  Layer by layer - Combining Monads , 2017, ICTAC.

[38]  Leslie Pack Kaelbling,et al.  Planning and Acting in Partially Observable Stochastic Domains , 1998, Artif. Intell..

[39]  Matteo Mio,et al.  Upper-Expectation Bisimilarity and Łukasiewicz μ-Calculus , 2014, FoSSaCS.

[40]  Roberto Segala,et al.  Modeling and verification of randomized distributed real-time systems , 1996 .

[41]  Daniele Varacca,et al.  Probability, Nondeterminism and Concurrency: Two Denotational Models for Probabilistic Computation , 2003 .

[42]  Helmut Röhrl,et al.  Convexity theories IV. Klein-Hilbert parts in convex modules , 1995, Appl. Categorical Struct..

[43]  Jean Goubault-Larrecq.  Prevision Domains and Convex Powercones , 2008, FoSSaCS.

[44]  Christel Baier,et al.  Probabilistic ω-automata , 2012, JACM.

[45]  Davide Sangiorgi,et al.  On the bisimulation proof method , 1998, Mathematical Structures in Computer Science.

[46]  Holger Hermanns,et al.  Probabilistic Bisimulation: Naturally on Distributions , 2014, CONCUR.

[47]  Azaria Paz,et al.  Probabilistic automata , 2003 .

[48]  Ana Sokolova,et al.  Termination in Convex Sets of Distributions , 2017, CALCO.

[49]  Uwe Wolter,et al.  On Corelations, Cokernels, and Coequations , 2000, CMCS.

[50]  Alexandra Silva,et al.  The Power of Convex Algebras , 2017, CONCUR.

[51]  Wang Yi,et al.  Testing and Refinement for Nondeterministic and Probabilistic Processes , 1994, FTRTFT.

[52]  John Power,et al.  Combining computational effects: commutativity and sum , 2001 .

[53]  Ohad Kammar,et al.  A convenient category for higher-order probability theory , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[54]  Dominic R. Verity,et al.  ∞-Categories for the Working Mathematician , 2018 .

[55]  M. H. Stone.  Postulates for the barycentric calculus , 1949 .

[56]  Alexandra Silva,et al.  Trace semantics via determinization , 2012, J. Comput. Syst. Sci..

[57]  Alexandra Silva,et al.  A coalgebraic view on decorated traces , 2016, Math. Struct. Comput. Sci..

[58]  Moshe Y. Vardi.  Automatic verification of probabilistic concurrent finite state programs , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[59]  Klaus Keimel,et al.  Semantic Domains for Combining Probability and Non-Determinism , 2005, Electronic Notes in Theoretical Computer Science.

[60]  Lijun Zhang,et al.  When Equivalence and Bisimulation Join Forces in Probabilistic Automata , 2014, FM.

[61]  Bartek Klin,et al.  Bialgebras for structural operational semantics: An introduction , 2011, Theor. Comput. Sci..

[62]  Alexandra Silva,et al.  Generalizing the powerset construction, coalgebraically , 2010, FSTTCS.

[63]  Peter Norvig,et al.  Artificial Intelligence: A Modern Approach , 1995 .

[64]  Rocco De Nicola,et al.  Relating strong behavioral equivalences for processes with nondeterminism and probabilities , 2013, Theor. Comput. Sci..

[65]  Gordon D. Plotkin,et al.  Towards a mathematical operational semantics , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[66]  Rob J. van Glabbeek,et al.  The Linear Time - Branching Time Spectrum I , 2001, Handbook of Process Algebra.

[67]  Christel Baier,et al.  Principles of model checking , 2008 .