Distributing probability over non-determinism

We study the combination of probability and non-determinism from a categorical point of view. In category theory, non-determinism and probability are represented by suitable monads. However, these two monads do not combine well as they are. To overcome this problem, we introduce the notion of indexed valuations. This notion is used to define a new monad that can be combined with the usual non-deterministic monad via a categorical distributive law. We give an equational characterisation of our construction. We discuss the computational meaning of indexed valuations, and we show how they can be used by giving a denotational semantics of a simple imperative language.

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

[2]  Mariëlle Stoelinga,et al.  An Introduction to Probabilistic Automata , 2002, Bull. EATCS.

[3]  Annabelle McIver,et al.  Refinement-oriented probability for CSP , 1996, Formal Aspects of Computing.

[4]  Michael W. Mislove.  Discrete Random Variables over Domains , 2005, ICALP.

[5]  Catuscia Palamidessi,et al.  Probabilistic Asynchronous pi-Calculus , 2000, FoSSaCS.

[6]  Roberto Segala,et al.  Axiomatizations for Probabilistic Bisimulation , 2001, ICALP.

[7]  Klaus Keimel,et al.  Semantic Domains for Combining Probability and Non-Determinism , 2005, Electron. Notes Theor. Comput. Sci..

[8]  Daniele Varacca,et al.  The powerdomain of indexed valuations , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

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

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

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

[12]  Eugenio Moggi,et al.  Notions of Computation and Monads , 1991, Inf. Comput..

[13]  George Gratzer,et al.  Universal Algebra , 1979 .

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

[15]  T. Henzinger,et al.  Trading memory for randomness , 2004 .

[16]  Claire Jones,et al.  Probabilistic non-determinism , 1990 .

[17]  N. D. Gautam.  The validity of equations of complex algebras , 1957, Arch. Math. Log..

[18]  C. Jones,et al.  A probabilistic powerdomain of evaluations , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[19]  Gordon D. Plotkin,et al.  Notions of Computation Determine Monads , 2002, FoSSaCS.

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

[21]  Regina Tix,et al.  Continuous D-cones: convexity and powerdomain constructions , 1999 .

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