Equational Axioms for Probabilistic Bisimilarity (Preliminary Report)

This paper gives an equational axiomatization of probabilistic bisimulation equivalence for a class of finite-state agents previously studied by Stark and Smolka ((2000) Proof, Language, and Interaction: Essays in Honour of Robin Milner, pp. 571-595). The axiomatization is obtained by extending the general axioms of iteration theories (or iteration algebras), which characterize the equational properties of the fixed point operator on (omega-)continuous or monotonic functions, with three axiom schemas that express laws that are specific to probabilistic bisimilarity. Hence probabilistic bisimilarity (over finite-state agents) has an equational axiomatization relative to iteration algebras

A connection between concurrency and language theory

We show that three fixed point structures equipped with (sequential) composition, a sum operation, and a fixed point operation share the same valid equations. These are the theories of (context-free) languages, (regular) tree languages, and simulation equivalence classes of (regular) synchronization trees (or processes). The results reveal a close relationship between classical language theory and process algebra

Complete axiomatization for the total variation distance of Markov chains

We propose a complete axiomatization for the total variation distance of finite labelled Markov chains. Our axiomatization is given in the form of a quantitative deduction system, a framework recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) to extend classical equational deduction systems by means of inferences of equality relations t≡εs indexed by rationals, expressing that “t is approximately equal to s up to an error ε”. Notably, the quantitative equational system is obtained by extending our previous axiomatization (CONCUR 2016) for the probabilistic bisimilarity distance with a distributivity axiom for the prefix operator over the probabilistic choice inspired by Rabinovich's (MFPS 1983). Finally, we propose a metric extension to the Kleene-style representation theorem for finite labelled Markov chains w.r.t. trace equivalence due to Silva and Sokolova (MFPS 2011)

A uniform framework for modelling nondeterministic, probabilistic, stochastic, or mixed processes and their behavioral equivalences

Labeled transition systems are typically used as behavioral models of concurrent processes, and the labeled transitions define the a one-step state-to-state reachability relation. This model can be made generalized by modifying the transition relation to associate a state reachability distribution, rather than a single target state, with any pair of source state and transition label. The state reachability distribution becomes a function mapping each possible target state to a value that expresses the degree of one-step reachability of that state. Values are taken from a preordered set equipped with a minimum that denotes unreachability. By selecting suitable preordered sets, the resulting model, called ULTraS from Uniform Labeled Transition System, can be specialized to capture well-known models of fully nondeterministic processes (LTS), fully probabilistic processes (ADTMC), fully stochastic processes (ACTMC), and of nondeterministic and probabilistic (MDP) or nondeterministic and stochastic (CTMDP) processes. This uniform treatment of different behavioral models extends to behavioral equivalences. These can be defined on ULTraS by relying on appropriate measure functions that expresses the degree of reachability of a set of states when performing single-step or multi-step computations. It is shown that the specializations of bisimulation, trace, and testing equivalences for the different classes of ULTraS coincide with the behavioral equivalences defined in the literature over traditional models

Equational axioms for probabilistic bisimilarity

This paper gives an equational axiomatization of probabilistic bisimulation equivalence for a class of finite-state agents previously studied by Stark and Smolka ((2000) Proof, Language, and Interaction: Essays in Honour of Robin Milner, pp. 571-595). The axiomatization is obtained by extending the general axioms of iteration theories (or iteration algebras), which characterize the equational properties of the fixed point operator on (#-)continuous or monotonic functions, with three axiom schemas that express laws that are specific to probabilistic bisimilarity

Probabilistic Process Algebra

Every day we witness the fast development of the hardware and software technology. This, of course, is the reason that new and more complex systems controlled by some kind of computational-based devices become an unseparated part of our daily life. As more as the system complexity increases, as more the reasoning about its correct behaviour becomes dif??cult. A variety of consequences may occur as a result of a failure, ranging from simple annoying to life threatening ones. Thus for some systems it is crucial that they exhibit a correct functioning. However, for systems with an extremely complex construction it is almost impossible to give an absolute guarantee for their correctness. In this case, it is still satisfactory to know that the possibility for a system to fail is low enough. Formal methods have been developed for establishing correctness of computer systems. They provide rigorous methods with which one can formally specify properties of a systems's intended behaviour, and also can check if the system conforms to that speci??cation. In case of complex systems we need a formal method that allows us to reason in compositional way, it provides us with techniques that can be used to build larger systems from the composition of smaller ones. Process algebra carries exactly this idea; it provides operators that allow to compose processes in order to obtain a more complex process. Besides, every process algebra contains a set of axioms. Every axiom is an algebraic equation that carries our intuition and insight in process behaviour, it expresses which two processes behaviour we consider equal. In such a way, manipulation with processes becomes manipulation with equations in the algebraic sense. But, equations and operators do not have any meaning unless we place them in a certain real ¿world¿ and match the terms of the process algebra with the entities of the real world. This step is traditionally called ¿giving a semantic of the syntax¿. The structure constructed in this way is called a model of the considered process algebra. For every given process algebra we can construct an in??nite number of models, but only several of them are interesting for the purpose process algebra was developed as a formal method. However, there is a tendency always to use so-called a bisimulation model. In this thesis we propose several process algebras and construct their models based on the notion of bisimulation

Equational axioms for probabilistic bisimilarity (preliminary report

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Equational Axioms for Probabilistic Bisimilarity (Preliminary Report)

This paper gives an equational axiomatization of probabilistic bisimulation equivalence for a class of finite-state agents previously studied by Stark and Smolka ((2000) Proof, Language, and Interaction: Essays in Honour of Robin Milner, pp. 571-595). The axiomatization is obtained by extending the general axioms of iteration theories (or iteration algebras), which characterize the equational properties of the fixed point operator on (omega-)continuous or monotonic functions, with three axiom schemas that express laws that are specific to probabilistic bisimilarity. Hence probabilistic bisimilarity (over finite-state agents) has an equational axiomatization relative to iteration algebras