Qualitative Logics and Equivalences for Probabilistic Systems

Zenodo (CERN European Organization for Nuclear Research), 2006

Bisimulation can be defined in a simple way using coinductive methods, and has rather pleasant properties. Ready similarity was proposed by Meyer et al. as a way to weakening the bisimulation equivalence thus getting a semantics defined in a similar way, but supported for more reasonable (weaker) observational properties. Global bisimulations were introduced by Frutos et al. in order to study different variants of non-determinism getting, in particular, a semantics under which the internal choice operator becomes associative. Global bisimulations are defined as plain bisimulations but allowing the use of new moves, called global transitions, that can change the processes not only locally in its head, but anywhere. Now we are continuing the study of global bisimulation but focusing on the way different semantics can be characterised as global bisimulation semantics. In particular, we have studied ready similarity, on the one hand because it was proposed as the strongest reasonable semantics weaker than bisimulation; on the other hand, because ready similarity was not directly defined as an equivalence relation but as the nucleus of an order relation, and this open the question whether it is also possible to define it as a symmetric bisimulation-like semantics. We have got a simple and elegant characterisation of ready similarity as a global bisimulation semantics, that provides a direct symmetric characterisation of it as an equivalence relation, without using any order as intermediate concept. Besides, we have found that it is not necessary to start from a simulation based semantics to get an equivalent global bisimulation. What has proved to be very useful is the axiomatic characterisation of the semantics. Following these ideas we have got also global bisimulation for several semantics, including refusals and traces. That provides a general framework that allows to relate both intensional and extensional semantics.

Lecture Notes in Computer Science, 2005

Continuous stochastic logic (CSL) deals with the verification of systems operating in continuous time, it may be traced to the well known tree logic CTL. The present logic contains a next-as well as an until-operator, and it permits expressing steady-state probabilities. We propose in this paper a probabilistic interpretation of this logic that is based on stochastic relations without making specific assumptions on the underlying distribution: usually some kind of exponential distribution based on a rate function is imposed, but these assumptions are shown to be unnecessarily restrictive. We study the problem of bisimulations in this fairly general context from the viewpoint of congruences for stochastic relations with the goal of finding minimal sets of formulas that permit efficient checking of models. The main contribution is to propose a general context for the interpretation of the logic including the steady-state operator, and to shed some light on the intricate nature of bisimulations. Continuous-time stochastic logic Proof This follows immediately from Proposition 7.2 in conjunction with Proposition 7.4. Specializing to the set of atomic formulas, we obtain at once: Corollary 7.2 Two states are dp(AP)-bisimilar iff they satisfy exactly the same formulas in L AP. This is not yet satisfying for practical purposes, because one has to construct the closure dp(AP) of the set of all atomic propositions, which may be done iteratively through the computation of a fixed point, as the discussion leading to Lemma 7.3 shows. Nevertheless it leads to an infinite process, handling a countable set of objects. But suppose we are in the

Electronic Notes in Theoretical Computer Science, 2014

In standard Propositional Dynamic Logic (PDL) literature [5,16,4] the semantics is given by Labeled Transition Systems, where for each program π we a associate a binary relation Rπ. Process Algebras [1,8,10,2] also give semantics to process (terms) by means of Labeled Transition Systems. In both formalisms, PDL and Process Algebra, the key notion to compare processes is bisimulation. In PDL, we also have the notion of logic equivalence, that can be used to prove that two programs π 1 and π 2 are logically equivalent π 1 ϕ ↔ π 2 ϕ. Unfortunately, logic equivalence and bisimulation do not match in PDL. Bisimilar programs are logic equivalent but the converse does not hold. This paper proposes a semantics and an axiomatization for PDL that makes logically equivalent programs also bisimilar. We prove soundness, completeness and the finite model property.

ArXiv, 2020

We investigate how various forms of bisimulation can be characterised using the technology of logical relations. The approach taken is that each form of bisimulation corresponds to an algebraic structure derived from a transition system, and the general result is that a relation $R$ between two transition systems on state spaces $S$ and $T$ is a bisimulation if and only if the derived algebraic structures are in the logical relation automatically generated from $R$. We show that this approach works for the original Park-Milner bisimulation and that it extends to weak bisimulation, and branching and semi-branching bisimulation. The paper concludes with a discussion of probabilistic bisimulation, where the situation is slightly more complex, partly owing to the need to encompass bisimulations that are not just relations.

Information and Computation, 1989

I. Castellani (1987, J. Comput. System Sri. 34, 210-235) has shown that observation equivalence of transition systems could be characterized by particular reductions: systems are equivalent if, and only if, they can be reduced to the same form. Moreover, every transition system has a minimal reduced form. We extend these results to logical equivalence, by an algebraic interpretation of temporal logics: we characterize logical equivalence of transition systems by particular reductions (saturating quasi-homomorphisms) of their power algebras of sets of states and paths and prove that every power algebra has a minimal reduced form. We then offer alternative proofs for logical characterizations of observation equivalence in particular we apply our method to prove M. Hennessy and C. Stirling's (1984, "Lecture Notes in Comput. Sci. Vol. 176," pp. 301-311, Springer-Verlag, New York/Berlin) result that "Future Perfect" logic characterizes observation equivalence of generalized transition systems, i.e., systems whose infinite behaviours are restricted by arbitrary fairness constraints.

Electronic Notes in Theoretical Computer Science, 2008

Markov transition systems for interpreting a simple negation free Hennessy-Milner logic are called distributionally equivalent iff for each formula the probability for its extension in one model is matched probabilistically in the other one. This extends in a natural way the notion of logical equivalence which is defined on the states of a transition system to its subprobability distributions. It is known that logical equivalence is equivalent to bisimilarity, i.e., the existence of a span of Borel maps that act as morphisms. We show that distributional equivalence is equivalent to bisimilarity as well, using a characterization of distributional equivalent transition systems through ergodic morphisms. As an aside, we relate bisimilar transition systems to those systems, for which cospans-taken in the category of measurable maps resp. in the Kleisli category associated with the Giry monad-exist.

2005

This paper presents various semantics in the branching-time spectrum of discrete-time and continuous-time Markov chains (DTMCs and CTMCs). Strong and weak bisimulation equivalence and simulation pre-orders are covered and are logically characterized in terms of the temporal logics Probabilistic Computation Tree Logic (PCTL) and Continuous Stochastic Logic (CSL).

2007

This paper introduces strong bisimulation for continuous-time Markov decision processes (CTMDPs), a stochastic model which allows for a nondeterministic choice between exponential distributions, and shows that bisimulation preserves the validity of CSL. To that end, we interpret the semantics of CSL—a stochastic variant of CSL for continuous-time Markov chains—on CTMDPs and show its measure-theoretic soundness.

2005

The question of when two systems are behaviourally equal has occupied a large part of the literature on verification and has yielded various equivalences (and congruences). These equivalence relations are most useful in comparing systems whose executions are not necessarily finite. An axiomatization of these equivalences gives us both, a nice algebraic handle on processes, and a proof system for checking the equality of two processes. Comparison of efficiency of non-terminating processes like an operating system has been largely untackled. We have presented here, an axiomatization for a certain subset of ordering induced bisimilarities. This axiomatization yields the axiomatization for equivalences like observational equivalence and inefficiency bisimulation as special cases. The axiomatization has been proven to be complete for finite state processes, and can be used as a proof system for checking the equality of systems.

2012

We present a format for the specification of probabilistic transition systems that guarantees that bisimulation equivalence is a congruence for any operator defined in this format. In this sense, the format is somehow comparable to the ntyft/ntyxt format in a non-probabilistic setting. We also study the modular construction of probabilistic transition systems specifications and prove that some standard conservative extension theorems also hold in our setting. Finally, we show that the trace congruence for image-finite processes induced by our format is precisely bisimulation on probabilistic systems.