Probabilistic Transition Systems Research Articles

Quotients of Probabilistic Boolean Networks

A probabilistic Boolean network (PBN) is a discrete-time system composed of a collection of Boolean networks between which the PBN switches in a stochastic manner. This article focuses on the study of quotients of PBNs. Given a PBN and an equivalence relation on its state set, we consider a probabilistic transition system that is generated by the PBN; the resulting quotient transition system then automatically captures the quotient behavior of this PBN. We therefore describe a method for obtaining a probabilistic Boolean system that generates the transitions of the quotient transition system. Applications of this quotient description are discussed, and it is shown that for PBNs, controller synthesis can be performed easily by first controlling a quotient system and then lifting the control law back to the original network. A biological example is given to show the usefulness of the developed results.

Distribution Bisimilarity via the Power of Convex Algebras

Probabilistic automata (PA), also known as probabilistic nondeterministic labelled transition systems, combine probability and nondeterminism. They can be given different semantics, like strong bisimilarity, convex bisimilarity, or (more recently) distribution bisimilarity. The latter is based on the view of PA as transformers of probability distributions, also called belief states, and promotes distributions to first-class citizens. We give a coalgebraic account of distribution bisimilarity, and explain the genesis of the belief-state transformer from a PA. To do so, we make explicit the convex algebraic structure present in PA and identify belief-state transformers as transition systems with state space that carries a convex algebra. As a consequence of our abstract approach, we can give a sound proof technique which we call bisimulation up-to convex hull.

Model independent approach to probabilistic models

There is a lot of research on probabilistic transition systems. There are not many studies in probabilistic process models. The lack of investigation into the interactive aspect of probabilistic processes is mainly due to the difficulty caused by the discrepancy between probabilistic choices and nondeterministic actions. The paper proposes a uniform approach to probabilistic process models and a bisimulation theory for probabilistic concurrency.

From generic partition refinement to weighted tree automata minimization

Partition refinement is a method for minimizing automata and transition systems of various types. Recently,we have developed a partition refinement algorithm that is generic in the transition type of the given systemand matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministicautomata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved bymodelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the runtime analysis of our algorithm to cover additional instances, notably weighted automata and, more generally,weighted tree automata. For weights in a cancellative monoid we match, and for non-cancellative monoids suchas (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the bestknown algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concretesystem types by implementing a simple refinement interface. Moreover, the algorithm and the tool are modular,and partition refiners for new types of systems are obtained easily by composing pre-implemented basic functors.Experiments show that even for complex system types, the tool is able to handle systems with millions oftransitions.

Logical characterization of branching metrics for nondeterministic probabilistic transition systems

In this paper we propose a logical characterization of (bi)simulation metrics obtained by a probabilistic variant of Hennessy-Milner logic enriched with variables, whose semantics is defined following the equational μ-calculus approach. Our characterization is based on the novel notions of mimicking formulae and syntactical distance on formulae. The former ones are the quantitative analogous to characteristic formulae. The latter is a 1-bounded pseudometric on formulae measuring their syntactical disparities. The characterization is obtained by showing that the (bi)simulation distance between processes corresponds to the syntactical distance between their mimicking formulae.We also discuss the expressive power of mimicking formulae with respect to probabilistic (bi)simulations. We show that two processes are bisimilar if and only if their mimicking formulae are syntactically equivalent. Moreover, we obtain that mimicking formulae of processes coincide with their characteristic formulae for ready simulation and that negation free mimicking formulae coincide with the characteristic formulae for simulation.

Stochastic equivalence for performance analysis of concurrent systems in dtsiPBC

We propose an extension with immediate multiactions of discrete time stochastic Petri box calculus (dtsPBC), pre- sented by I.V. Tarasyuk. The resulting algebra dtsiPBC is a discrete time analogue of stochastic Petri box calculus (sPBC) with immediate multiactions, proposed by H. MaciV. Valero and others within a continuous time domain. The step operational semantics is constructed via labeled probabilistic transition systems. The denotational seman- tics is defined on the basis of a subclass of labeled discrete time stochastic Petri nets with immediate transitions. A consistency of the both semantics is demonstrated. In order to evaluate performance, the corresponding semi-Markov chains and (reduced) discrete time Markov chains are analyzed. We define step stochastic bisimulation equivalence of expressions and prove that it can be applied to reduce their transition systems and underlying semi-Markov chains while preserving the functionality and performance characteristics. We explain how this equivalence may help to sim- plify performance analysis of the algebraic processes. In a case study, a method of modeling, performance evaluation and behaviour preserving reduction of concurrent systems is outlined and applied to the shared memory system.

On-the-fly model checking for extended action-based probabilistic operators

The quantitative analysis of concurrent systems requires expressive and user-friendly property languages combining temporal, data handling, and quantitative aspects. In this paper, we aim at facilitating the quantitative analysis of systems modeled as PTSs (Probabilistic Transition Systems) labeled by actions containing data values and probabilities. We propose a new regular probabilistic operator that specifies the probability measure of a path described by a generalized regular formula involving arbitrary computations on data values. This operator, which subsumes the Until operators of PCTL and their action-based counterparts, can provide useful quantitative information about paths having certain (e.g., peak) cost values. We integrated the regular probabilistic operator into MCL (Model Checking Language) and we devised an associated on-the-fly model checking method, based on a combined local resolution of linear and Boolean equation systems. We implemented the method in the EVALUATOR model checker of the CADP toolbox and experimented it on realistic PTSs modeling concurrent systems.

Łukasiewicz μ-calculus

The paper explores properties of the Łukasiewicz μ-calculus, or Łμ for short, an extension of Łukasiewicz logic with scalar multiplication and least and greatest fixed-point operators (for monotone formulas). We observe that Łμ terms, with n variables, define monotone piece-wise linear functions fr om [0, 1]n to [0, 1]. Two effective procedures for calculating the output of Łμ terms on rational inputs are presented. We then consider the Łukasiewicz modal μ-calculus, which is obtained by adding box and diamond modalities to Łμ. Alternatively, it can be viewed as a generalization of Kozen’s modal μ-calculus adapted to probabilistic nondeterministic transition systems (PNTS’s). We show how properties expressible in the well-known logic PCTL can be encoded as Łukasiewicz modal μ-calculus formulas. We also show that the algorithms for computing values of Łukasiewicz μ-calculus terms provide automatic (albeit impractical) methods for verifying Łukasiewicz modal μ-calculus properties of finite rational PNTS’s.

A general SOS theory for the specification of probabilistic transition systems

This article focuses on the formalization of the structured operational semantics approach for languages with primitives that introduce probabilistic and non-deterministic behavior. We define a general theoretic framework and present the ▪ rule format that guarantees that bisimulation equivalence (in the probabilistic setting) is a congruence for any operator defined in this format. We show that the bisimulation is fully abstract w.r.t. the ▪ format and (possibilistic) trace equivalence in the sense that bisimulation is the coarsest congruence included in trace equivalence for any operator definable within the ▪ format (in other words, bisimulation is the smallest congruence relation guaranteed by the format). We also provide a conservative extension theorem and show that languages that include primitives for exponentially distributed time behavior (such as IMC and Markov automata based language) fit naturally within our framework.

Minimization of Probabilistic Models of Programs

In this paper, we consider a problem of reduction of probabilistic transition systems (PTS) in order to reduce the complexity of model checking of such systems. The problem of model checking of a PTS is to calculate truth values of formulas of the probabilistic temporal logic PCTL in an initial state of the PTS. We introduce the concept of equivalence of states of a PTS and present an algorithm for removing equivalent states. A result of this algorithm is a PTS such that all of its properties expressed by formulas of PCTL coincide with those of the original PTS.

Rooted branching bisimulation as a congruence for probabilistic transition systems

We propose a probabilistic transition system specification format, referred to as probabilistic RBB safe, for which rooted branching bisimulation is a congruence. The congruence theorem is based on the approach of Fokkink for the qualitative case. For this to work, the theory of transition system specifications in the setting of labeled transition systems needs to be extended to deal with probability distributions, both syntactically and semantically. We provide a scheduler-free characterization of probabilistic branching bisimulation as adapted from work of Andova et al. for the alternating model. Counter examples are given to justify the various conditions required by the format.

SOS rule formats for convex and abstract probabilistic bisimulations

Probabilistic transition system specifications (PTSSs) in the $nt \mu f\theta / nt\mu x\theta$ format provide structural operational semantics for Segala-type systems that exhibit both probabilistic and nondeterministic behavior and guarantee that bisimilarity is a congruence for all operator defined in such format. Starting from the $nt \mu f\theta / nt\mu x\theta$ format, we obtain restricted formats that guarantee that three coarser bisimulation equivalences are congruences. We focus on (i) Segala's variant of bisimulation that considers combined transitions, which we call here "convex bisimulation"; (ii) the bisimulation equivalence resulting from considering Park & Milner's bisimulation on the usual stripped probabilistic transition system (translated into a labelled transition system), which we call here "probability obliterated bisimulation"; and (iii) a "probability abstracted bisimulation", which, like bisimulation, preserves the structure of the distributions but instead, it ignores the probability values. In addition, we compare these bisimulation equivalences and provide a logic characterization for each of them.

Identifying All Preorders on the Subdistribution Monad

The countable valuation monad, the countable distribution monad, and the countable subdistribution monad are often used in the coalgebraic treatment of discrete probabilistic transition systems. We identify preorders on them using a technique based on the preorder ⊤⊤-lifting and elementary facts about preorders on real intervals preserved by convex combinations. We show that there are exactly 15, 5, and 41 preorders on the countable valuation monad, the countable distribution monad, and the countable subdistribution monad respectively. We also give concrete definitions of these preorders. By applying Hesselink and Thijs's/ Hughes and Jacobs's construction to some preorder on the countable subdistribution monad, we obtain probabilistic bisimulation between Markov chains ignoring states with deadlocks.

Generalized maximum likelihood noncoherent block detection for decode-and-forward relay systems

This paper considers noncoherent cooperative decode-and-forward (DF) half-duplex multi-branch relay systems. Each relay branch is modeled as a probabilistic transition system at the last hop, and thus it can be considered as a relaying chain comprising multi-hop relays. An approximation to the generalized maximum likelihood (ML) noncoherent block detection is derived for uncoded M-ary modulation in a faded noisy environment. In particular, the derived noncoherent block detection in a noiseless case is equivalent to a multichannel reception with full diversity. Furthermore, the generalized detection is extended specifically to block coded M-ary phase shift keying (MPSK) modulation. For a DF three node relay system using block coded quadrature phase shift keying (QPSK), simulation results are provided to examine the end-to-end error performance of the noncoherent detection with considering the effects of network geometry and power allocation, respectively. It is shown that under a fixed power allocation, a proper relay placement can yield near full diversity for large signal-to-noise ratio.

Approximating Markov Processes by Averaging

Normally, one thinks of probabilistic transition systems as taking an initial probability distribution over the state space into a new probability distribution representing the system after a transition. We, however, take a dual view of Markov processes as transformers of bounded measurable functions. This is very much in the same spirit as a “predicate-transformer” view, which is dual to the state-transformer view of transition systems. We redevelop the theory of labelled Markov processes from this viewpoint; in particular, we explore approximation theory. We obtain three main results. (i) It is possible to define bisimulation on general measure spaces and show that it is an equivalence relation. The logical characterization of bisimulation can be done straightforwardly and generally. (ii) A new and flexible approach to approximation based on averaging can be given. This vastly generalizes and streamlines the idea of using conditional expectations to compute approximations. (iii) We show that there is a minimal process bisimulation-equivalent to a given process, and this minimal process is obtained as the limit of the finite approximants.

Coalgebraic Trace Semantics for Continuous Probabilistic Transition Systems

Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces and measurable functions. Our main contribution is that the existence of a final coalgebra in the Kleisli category of these monads is closely connected to the measure-theoretic extension theorem for sigma-finite pre-measures. In fact, we obtain a practical definition of the trace measure for both finite and infinite traces of PTS that subsumes a well-known result for discrete probabilistic transition systems. Finally we consider two example systems with uncountable state spaces and apply our theory to calculate their trace measures.

SLAP: Specification logic of actions with probability

A logic for specifying probabilistic transition systems is presented. Our perspective is that of agents performing actions. A procedure for deciding whether sentences in this logic are valid is provided. One of the main contributions of the paper is the formulation of the decision procedure: a tableau system which appeals to solving systems of linear equations. The tableau rules eliminate propositional connectives, then, for all open branches of the tableau tree, systems of linear equations are generated and checked for feasibility. Proofs of soundness, completeness and termination of the decision procedure are provided.

Łukasiewicz mu-Calculus

The paper explores properties of {\L}ukasiewicz mu-calculus, a version of the quantitative/probabilistic modal mu-calculus containing both weak and strong conjunctions and disjunctions from {\L}ukasiewicz (fuzzy) logic. We show that this logic encodes the well-known probabilistic temporal logic PCTL. And we give a model-checking algorithm for computing the rational denotational value of a formula at any state in a finite rational probabilistic nondeterministic transition system.

Discrete Time Stochastic Petri Box Calculus with Immediate Multiactions dtsiPBC

We propose discrete time stochastic Petri box calculus extended with immediate multiactions, called dt-siPBC. The step operational semantics is constructed via labeled probabilistic transition systems. The denotational semantics is defined via labeled discrete time stochastic Petri nets with immediate transitions (LDTSIPNs). A consistency of both semantics is demonstrated. In order to evaluate performance, the corresponding semi-Markov chains are analyzed. In a case study, performance of the shared memory system is evaluated.

Compositionality of Approximate Bisimulation for Probabilistic Systems

Probabilistic transition system specifications using the rule format ntmuft-ntmuxt provide structural operational semantics for Segala-type systems and guarantee that probabilistic bisimilarity is a congruence. Probabilistic bisimilarity is for many applications too sensitive to the exact probabilities of transitions. Approximate bisimulation provides a robust semantics that is stable with respect to implementation and measurement errors of probabilistic behavior. We provide a general method to quantify how much a process combinator expands the approximate bisimulation distance. As a direct application we derive an appropriate rule format that guarantees compositionality with respect to approximate bisimilarity. Moreover, we describe how specification formats for non-standard compositionality requirements may be derived.