Quantitative Relations and Approximate Process Equivalences

Coalgebraic Methods in Computer Science, 2012

We consider the problem of modeling syntax and semantics of probabilistic processes with continuous states (e.g. with continuous data). Syntax and semantics of these systems can be defined as algebras and coalgebras of suitable endofunctors over Meas, the category of measurable spaces. In order to give a more concrete representation for these coalgebras, we present an SOS-like rule format which induces an abstract GSOS over Meas; this format is proved to yield a fully abstract universal semantics, for which behavioural equivalence is a congruence. To this end, we solve several problems. In particular, the format has to specify how to compose the semantics of processes (which basically are continuous state Markov processes). This is achieved by defining a language of measure terms, i.e., expressions specifically designed for describing probabilistic measures. Thus, the transition relation associates processes with measure terms. As an example application, we model a CCS-like calculus of processes placed in an Euclidean space. The approach we follow in this case can be readily adapted to other quantitative aspects, e.g. Quality of Service, physical and chemical parameters in biological systems, etc.

We tackle the problem of non robustness of simulation and bisimulation when dealing with probabilistic processes. It is important to ignore tiny deviations in probabilities because these often come from experience or estimations. A few approaches have been proposed to treat this issue, for example metrics to quantify the non bisimilarity (or closeness) of processes. Relaxing the definition of simulation and bisimulation is another avenue which we follow. We define a new semantics to a known simple logic for probabilistic processes and show that it characterises a notion of -simulation. We also define two-players games that correspond to these notions: the existence of a winning strategy for one of the players determines -(bi)simulation. Of course, for all the notions defined, letting = 0 gives back the usual notions of logical equivalence, simulation and bisimulation. However, in contrast to what happens when = 0, two-way -simulation for > 0 is not equal tobisimulation. Next we give a polynomial time algorithm to compute a naturally derived metric: distance between states s and t is defined as the smallest such that s and t are equivalent. Finally we show that most of these notions can be extended to deal with probabilistic systems that allow non-determinism as well.

Lecture Notes in Computer Science, 2004

In order to describe probabilistic processes by means of a formal model, some considerations have to be taken into account. In this paper we present some of the ideas appeared in the literature that could help to define appropriate formal frameworks for the specification of probabilistic processes. First, we will explain the different interpretations of the probabilistic information included in this kind of models. After that, the different choice operators used in the most common probabilistic languages are enumerated. Once we have an appropriate language, we have to give its semantics. Thus, we will review some of the theories based on bisimulation and testing semantics. We will conclude by studying the extensions of the chosen languages with other operators such as parallel composition and hiding.

2009

In this paper we show how Rate Transition Systems (RT S s) can be used as a unifying framework for the definition of the semantics of stochastic process algebras. RT S s facilitate the compositional definition of such semantics exploiting operators on the next state functions which are the functional counterpart of classical process algebra operators. We apply this framework to representative fragments of major stochastic process calculi namely T IP P , P EP A and IML and show how they solve the issue of transition multiplicity in a simple and elegant way. We, moreover, show how RT S s help describing different languages, their differences and their similarities. For each calculus, we also show the formal correspondence between the RT S s semantics and the standard SOS one. Research partially funded by EU IP SENSORIA (contract n. 016004), CNR-RSTL project XXL, FIRB-MUR project TOCAI.IT and by PRIN-MIUR PACO. 1 Sometimes actions are assumed to have zero duration; then the associated r.v. is interpreted as a delay, before the action takes place.

Data & Knowledge Engineering, 2008

In various application domains there is a desire to compare process models, e.g., to relate an organization-specific process model to a reference model, to find a web service matching some desired service description, or to compare some normative process model with a process model discovered using process mining techniques. Although many researchers have worked on different notions of equivalence (e.g., trace equivalence, bisimulation, branching bisimulation, etc.), most of the existing notions are not very useful in this context. First of all, most equivalence notions result in a binary answer (i.e., two processes are equivalent or not). This is not very helpful because, in real-life applications, one needs to differentiate between slightly different models and completely different models. Second, not all parts of a process model are equally important. There may be parts of the process model that are rarely activated (i.e., ''process veins'') while other parts are executed for most process instances (i.e., the ''process arteries''). Clearly, differences in some veins of a process are less important than differences in the main arteries of a process. To address the problem, this paper proposes a completely new way of comparing process models. Rather than directly comparing two models, the process models are compared with respect to some typical behavior. This way, we are able to avoid the two problems just mentioned. The approach has been implemented and has been used in the context of genetic process mining. Although the results are presented in the context of Petri nets, the approach can be applied to any process modeling language with executable semantics.

Journal of Computer and System Sciences, 2000

This paper deals with probabilistic and nondeterministic processes represented by a variant of labelled transition systems where any outgoing transition of a state s is augmented with probabilities for the possible successor states. Our main contribution are algorithms for computing the bisimulation equivalence classes as introduced by Larsen & Skou 44] and the simulation preorder a la Segala & Lynch 57]. The algorithm for deciding bisimilarity is based on a variant of the traditional partitioning technique 43, 51] and runs in time O(mn(log m + log n)) where m is the number of transitions and n the number of states. The main idea for computing the simulation preorder is the reduction to maximum ow problems in suitable networks. Using the method of Cheriyan, Hagerup & Mehlhorn 15] for computing the maximum ow, the algorithm runs in time O((mn 6 + m 2 n 3 )= log n). Moreover, we show that the network-based technique is also applicable to compute the simulation-like relation of 40] in fully probabilistic systems (a variant of ordinary labelled transition systems where the non-determinism is totally resolved by probabilistic choices).

2009 24th Annual IEEE Symposium on Logic In Computer Science, 2009

We associate a statistical vector to a trace and a geometrical embedding to a Markov Decision Process, based on a distance on words, and study basic Membership and Equivalence problems. The Membership problem for a trace w and a Markov Decision Process S decides if there exists a strategy on S which generates with high probability traces close to w. We prove that Membership of a trace is testable and Equivalence of MDPs is polynomial time approximable. For Probabilistic Automata, Membership is not testable, and approximate Equivalence is undecidable. We give a class of properties, based on results concerning the structure of the tail sigma-field of a finite Markov chain, which characterizes equivalent Markov Decision Processes in this context.

Information and Computation, 2003

Labelled Markov processes are probabilistic versions of labelled transition systems. In general, the state space of a labelled Markov process may be a continuum. In this paper, we study approximation techniques for continuousstate labelled Markov processes.

Logical Methods in Computer Science, 2006

We develop a pseudo-metric analogue of bisimulation for generalized semi-Markov processes. The kernel of this pseudo-metric corresponds to bisimulation; thus we have extended bisimulation for continuous-time probabilistic processes to a much broader class of distributions than exponential distributions. This pseudo-metric gives a useful handle on approximate reasoning in the presence of numerical information -such as probabilities and time -in the model.

2013

A syntactic framework called SGSOS, for defining well-behaved Markovian stochastic transition systems, is introduced by analogy to the GSOS congruence format for nondeterministic processes. Stochastic bisimilarity is guaranteed a congruence for systems defined by SGSOS rules. Associativity of parallel composition in stochastic process algebras is also studied within the framework.