On Some Equivalence Relations for Probabilistic Processes1

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.

A possible approach to studying behavioural equivalences in labelled transition systems is that of characterizing them in terms of homomorphic transformations. This characterization permits relying on algebraic techniques for proving systems properties and reduces equivalence checking of two systems to studying the relationships among the elements of their structures. Different algebraic characterizations of bisimulation-based equivalences in terms of particular transition systems homomorphisms have been proposed in the literature. Here we show, by an example, that trace-based equivalences are not locally characterizable and thus that the above results cannot be extended to these equivalences. However, similar results can be obtained if we confine ourselves to restricted classes of transition systems. Here, the algebraic characterizations of three well known decorated-trace equivalences (ready trace, ready and failure equivalence) for tree-like structures are presented.

Transition systems are introduced as a suitable semantic model of reactive systems. Some notions of behavioral equivalence are discussed, such as isomorphism equivalence, trace equivalence, simulation equivalence and bisimulation equivalence. Internal, unobservable actions are also considered and many behavioral equivalences are suitably adapted for this case.

An approach allowing one to simplify assertional reasoning on concurrent programs is presented. In the adopted assertional framework, to verify such properties as partial correctness, mutual exclusion, or deadlock freedom, the inductive assertions method is applied to a labelled transition systems representing a program, where concurrency is modelled by action interleavings. In order to tackle the problem of state explosion a notion of reduction of the transition system representing a verified program is introduced, where some transitions and configurations that arise from nonessential interleavings of actions are ignored. To isolate nonessential interleavings, the trace equivalence, in the sense of Mazurkiewicz, is exploited. Decidability of verifying whether a given labelled transition system is a reduction is investigated.

Contextual equivalences for cryptographic process calculi can be used to reason about correctness of protocols, but their definition suffers from quantification over all possible contexts. Here, we focus on two such equivalences, may-testing and barbed equivalence, and investigate tractable proof methods for them. To this aim, we develop an 'environment-sensitive' labelled transition system, where transitions are constrained by the knowledge the environment has of names and keys. On top of the new transition system, a trace equivalence and a co-inductive weak bisimulation equivalence are defined, both of which avoid quantification over contexts. Our main results are soundness of trace semantics and of weak bisimulation with respect to may-testing and barbed equivalence, respectively. This leads to more direct proof methods for equivalence checking. The use of such methods is illustrated via a few examples concerning implementation of secure channels by means of encrypted public channels. We also consider a variant of the labelled transition system that gives completeness, but is less handy to use.

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.

A behavioural pseudometric for probabilistic transition systems

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 system and matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministic automata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved by modelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the run time 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 such as (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the best known algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concrete system 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 of transitions.

Probabilistic models of computation have been studied for over three decades now, from foundational, logical, coalgebraic, categorical, as well as more practical verification-motivated point of view. In my work, and in this talk, we focus on the foundational, semantics side of probabilistic automata and transition systems, and in particular the relevant monads, and their algebras. The interplay of probability and non-determinism has been particularly challenging from a semantics point of view for some decades, as it does not just yield a monad. There are by now well-studied solutions to this and several monads for probability and non-determinism: some enriching the structure with convexity to obtain a monad, albeit a non-commutative one, others deliberately simplifying it, by imposing commutativity. It is well known that monads have two faces: a computational one - we think here of the powerset monad modelling non-determinism, or the probability distribution monad modelling probabilistic choices, and a universal-algebraic one - where we think of the algebraic presentation of the monads, like semilattices for the powerset monad and (variants of) convex algebras for (variants of) the probability distribution monad. Combining non-determinism and probability yields other combined monads, among which probably the most studied is the convex-subsets-of-distributions monad. This monad is presented by so-called convex semilattices, algebraic structures that are both a semilattice and a convex algebra, with suitable distributivity connecting the operations. From a semantics point of view, the algebraic presentations give us a nice way to define (and sometimes compute) language (aka trace) equivalence of the corresponding automata. Moreover, the presentations are useful for axiomatizations of language equivalence. From an algebraic point of view, these algebras are interesting and many questions about them are still open. We will discuss language semantics, its axiomatization, as well as some obtained and open algebraic problems for convex algebras and convex semilattices - and their computational consequences. In particular, we have a full characterization of congruences of (variants of) convex algebras, which for example yields decidability of distribution semantics; other results on congruences, e.g. a result on a congruence being finitely generated as a subalgebra, surprisingly accellerated the proof of completeness of infinite trace semanitcs, etc. We will review some existing results: all congruences are described and fp = fg for convex algebras, termination is a black hole, useful functors are proper on convex algebras, cancellativity of convex semi-lattices; as well as mention ongoing work on: mid-point-cancellativity, subalgebras, and homomorphisms for convex semi-lattices, as well as topological convex semilattices. This talk is based on previous joint works with Filippo Bonchi, Alexandra Silva, Valeria Vignudelli, and Harald Woracek, as well as on ongoing work and discussions with Matteo Mio, Alex Simpson, and Harald Woracek.

The process of data flow exchange in complex concurrent systems is often redundant and uncertain, leading to resource inefficiency and 'state explosion'. Equivalence relationships identify processes of the same behaviour, remove redundancy to simplify system verification and analysis, and effectively mitigate the 'state explosion' problem. In Glabbeek's equivalence spectrum, trace equivalence (systems that produce the same sequence of actions are considered equivalent) and bisimulation equivalence (requiring bidirectional behaviour correspondence between systems) are widely accepted. In this paper, we discuss the method of trace equivalence judgement on the polynomial algebraic event structure, propose the judgement of polynomial event equivalence, and use Gröbner basis for calculation. Gröbner basis, as an algebraic tool to determine the equality of polynomial systems, has the advantages of accuracy and efficiency, and provides a strict mathematical basis for the equivalence determination. Finally, a practical case is presented to show the optimisation effect of this method.

When a Petri net is synthesised from a labelled transition system, it is frequently desirable that certain additional constraints are fulfilled. For example, in circuit design, one is often interested in constructing safe Petri nets. Targeting such subclasses of Petri nets is not necessarily computationally more efficient than targeting the whole class. For example, targeting safe nets is known to be NP-complete while targeting the full class of place/transition nets is polynomial, in the size of the transition system. In this paper, several classes of Petri nets are examined, and their suitability for being targeted through efficient synthesis from labelled transition systems is studied and assessed. The focus is on choice-free Petri nets and some of their subclasses. It is described how they can be synthesised efficiently from persistent transition systems, summarising and streamlining in tutorial style some of the authors’ and their groups’ work over the past few years.

In our previous paper [1], a new model of a Labeled Transition System (LTS)-type implementation was proposed. In ordinary LTSs, transitions are labeled by actions; therefore, they can be called LTSs of actions. The new model is an LTS of observations; in this model, observations and test actions (buttons) are used instead of actions. This model generalizes many testing semantics that are based on the LTS of actions but use additional observations (refusals, ready sets, etc.). Moreover, systems with priority, which are not described by the LTS of actions, are simulated uniformly. In the present paper, we develop this approach by focusing on the composition of systems. The point is that, on observation traces, one cannot define a composition with respect to which a composition of LTSs would possess the property of additivity: the set of traces of a composition of LTSs coincides with the set of all pairwise compositions of traces of LTS operands. This is explained by the fact that an observation in a composition state is not calculated based on observations in states-operands. In this paper, we propose an approach that eliminates this drawback. To this end, we label the transitions of LTSs by symbols (events) that, on the one hand, can be composed to guarantee the property of additivity, and, on the other hand, can be used to generate observations under testing: a transition by an event gives rise to an observation related to this event. This model is called an LTS of events. In this paper, we define (1) a transformation of an LTS of events into an LTS of observations to conform with the principles of our previous paper [1]; (2) a composition of LTSs of events; (3) a composition of specifications that preserves conformance: a composition of conformal implementations is conformal to a composition of specifications; and (4) a uniform simulation of LTSs of actions in terms of the LTSs of events, which allows one to consider an implementation in any interaction semantics admissible for LTSs of actions. In this case, a composition of LTSs of events obtained as a result of simulation of the original LTSs of actions is equivalent to the LTS of events obtained as a result of simulation of the composition of these LTSs of actions.

<p>Fokkink and Zantema ((1994) Computer Journal 37:259-267) have shown that<br />bisimulation equivalence has a finite equational axiomatization over the language<br />of Basic Process Algebra with the binary Kleene star operation (BPA*). In light<br />of this positive result on the mathematical tractability of bisimulation equivalence<br />over BPA*, a natural question to ask is whether any other (pre)congruence relation<br />in van Glabbeek's linear time/branching time spectrum is finitely (in)equationally<br />axiomatizable over it. In this paper, we prove that, unlike bisimulation equivalence,<br />none of the preorders and equivalences in van Glabbeek's linear time/branching time<br />spectrum, whose discriminating power lies in between that of ready simulation and<br />that of completed traces, has a finite equational axiomatization. This we achieve by<br />exhibiting a family of (in)equivalences that holds in ready simulation semantics, the<br />finest semantics that we consider, whose instances cannot all be proven by means of<br />any finite set of (in)equations that is sound in completed trace semantics, which is<br />the coarsest semantics that is appropriate for the language BPA*. To this end, for<br />every finite collection of (in)equations that are sound in completed trace semantics, we<br />build a model in which some of the (in)equivalences of the family under consideration<br />fail. The construction of the model mimics the one used by Conway ((1971) Regular<br />Algebra and Finite Machines, page 105) in his proof of a result, originally due to<br />Redko, to the effect that infinitely many equations are needed to axiomatize equality<br />of regular expressions.</p><p>Our non-finite axiomatizability results apply to the language BPA* over an arbitrary<br />non-empty set of actions. In particular, we show that completed trace equivalence<br />is not finitely based over BPA* even when the set of actions is a singleton.<br />Our proof of this result may be easily adapted to the standard language of regular expressions to yield a solution to an open problem posed by Salomaa ((1969) Theory<br />of Automata, page 143).<br />Another semantics that is usually considered in process theory is trace semantics.<br />Trace semantics is, in general, not preserved by sequential composition, and is<br />therefore inappropriate for the language BPA*. We show that, if the set of actions<br />is a singleton, trace equivalence and preorder are preserved by all the operators in<br />the signature of BPA, and coincide with simulation equivalence and preorder, respectively.<br />In that case, unlike all the other semantics considered in this paper, trace<br />semantics have nite, complete equational axiomatizations over closed terms.</p><p> </p><p>AMS Subject Classification (1991): 08A70, 03C05, 68Q10, 68Q40, 68Q45,<br />68Q55, 68Q68, 68Q70.<br />CR Subject Classification (1991): D.3.1, F.1.1, F.1.2, F.3.2, F.3.4, F.4.1.<br />Keywords and Phrases: Concurrency, process algebra, Basic Process Algebra<br />(BPA*), Kleene star, bisimulation, ready simulation, simulation, completed trace semantics,<br />ready trace semantics, failure trace semantics, readiness semantics, failures<br />semantics, trace semantics, equational logic, complete axiomatizations.</p><p> </p>

Rate transition systems (RTS) are a special kind of transition systems introduced for defining the stochastic behavior of processes and for associating continuous-time Markov chains with process terms. The transition relation assigns to each process, for each action, the set of possible futures paired with a measure indicating the rates at which they are reached. RTS have been shown to be a uniform model for providing an operational semantics to many stochastic process algebras. In this paper, we define Uniform Labeled TRAnsition Systems (ULTraS) as a generalization of RTS that can be exploited to uniformly describe also nondeterministic and probabilistic variants of process algebras. We then present a general notion of behavioral relation for ULTraS that can be instantiated to capture bisimulation and trace equivalences for fully nondeterministic, fully probabilistic, and fully stochastic processes.

Larsen and Skou introduced a notion of bisimulation for probabilistic transition systems. They characterized probabilistic bisimilarity in terms of a probabilistic modal logic and also in terms of ‘button pressing’ tests. Desharnais et al. extended the notion of probabilistic bisimulation and the logical characterization of probabilistic bisimilarity to labelled Markov processes. These processes generalize probabilistic transition systems in that they also allow continuous state spaces. We extend the characterization of probabilistic bisimilarity in terms of testing to labelled Markov processes. One of our main technical contributions is the construction of a final object in a category of labelled Markov processes and the identification of a natural metric on the state space of the final labelled Markov process. This metric provides us with another characterization of probabilistic bisimilarity: states are probabilistic bisimilar if and only if they have distance 0.