SOS Rules Research Articles

Back to the format: A survey on SOS for probabilistic processes

In probabilistic process algebras the classic qualitative description of process behaviour is enriched with quantitative information on it, usually modelled in terms of probabilistic weights and/or distributions over the qualitative behaviour. In this setting, we use behavioural equivalences to check whether two processes show exactly the same behaviour, and, if this is not the case, we can use behavioural metrics to measure the distance between them. Compositional reasoning requires that equivalence, or closeness, of behaviour of two processes are not destroyed when language operators are applied on top of them in order to build larger processes. Formally, the equivalence must be a congruence, and the metric must be uniformly continuous, with respect to language operators. Instead of verifying these compositional properties by hand, operator-by-operator, it is much more convenient to prove them for a class of operators once for all, and to check that the operators one is dealing with are in that class. This is achieved by means of SOS specification formats: they consist in a set of syntactical constraints characterising a class of operators on the patterns of SOS rules, that define the operational semantics of languages. With this survey, we aim to collect and describe the specification formats that have been proposed in the literature to guarantee the compositional properties of (variants of) bisimulation equivalences and bisimulation metrics in the probabilistic setting.

SOS specifications for uniformly continuous operators

Behavioral metric semantics provide formal notions to compare probabilistic systems, giving a notion of behavioral distance characterizing how far the behavior of two systems is apart. Compositional reasoning over probabilistic systems with respect to behavioral metric semantics requires the language operators to be uniformly continuous, which ensures that a limited change in the behavior of a subsystem implies a smooth and limited change in the behavior of the whole system. We consider a hierarchy of uniform continuity properties for process algebra operators and, for each of these properties, we propose a Structural Operational Semantics specification format (namely a set of syntactical constraints on the form of the SOS rules) ensuring that the property is satisfied by construction by all operators captured by the format.

PTRebeca: Modeling and analysis of distributed and asynchronous systems

Distributed systems exhibit probabilistic and non-deterministic behaviors and may have time constraints. Probabilistic Timed Rebeca (PTRebeca) is introduced as a timed and probabilistic actor-based language for modeling distributed real-time systems with asynchronous message passing. The semantics of PTRebeca is a Timed Markov Decision Process. In this paper, we provide SOS rules for PTRebeca, introduce a new tool-set and describe the corresponding mappings. The tool-set automatically generates a Markov Automaton from a PTRebeca model in the form of the input language of the Interactive Markov Chain Analyzer (IMCA). The IMCA can be used as a back-end model checker for performance analysis of PTRebeca models against expected reachability and probabilistic reachability properties. Comparing to the existing tool-set, proposed in the conference paper, we now have the ability of analyzing significantly larger models, and we also can add different rewards to the model. We show the applicability of our approach and efficiency of our tool by analyzing a Network on Chip architecture as a real-world case study.

A Specification Format for Rooted Branching Bisimulation

Rule formats are sets of syntactical constraints over SOS rules ensuring semantical properties of the derived LTS. Given a rule format, our proposal is to relax the constraints imposed on each single rule and to introduce some constraints on the form of the whole set of rules, thus obtaining a new format ensuring the same semantical property and being less demanding than the original one. We apply our idea to a well known rule format for rooted branching bisimulation equivalence.

Linearization of CIF Through SOS

Linearization is the procedure of rewriting a process term into a linear form, which consist only of basic operators of the process language. This procedure is interesting both from a theoretical and a practical point of view. In particular, a linearization algorithm is needed for the Compositional Interchange Format (CIF), an automaton based modeling language. The problem of devising efficient linearization algorithms is not trivial, and has been already addressed in literature. However, the linearization algorithms obtained are the result of an inventive process, and the proof of correctness comes as an afterthought. Furthermore, the semantic specification of the language does not play an important role on the design of the algorithm. In this work we present a method for obtaining an efficient linearization algorithm, through a step-wise refinement of the SOS rules of CIF. As a result, we show how the semantic specification of the language can guide the implementation of such a procedure, yielding a simple proof of correctness.

Hierarchical states in the Compositional Interchange Format

CIF is a language designed for two purposes, namely as a specification language for hybrid systems and as an interchange format for allowing model transformations between other languages for hybrid systems. To facilitate the top-down development of a hybrid system and also to be able to express models more succinctly in the CIF formalism, we need a mechanism for stepwise refinement. In this paper, we add the notion of hierarchy to a subset of the CIF language, which we call hCIF. The semantic domain of the CIF formalism is a hybrid transition system, constructed using structural operational semantics. The goal of this paper is to present a semantics for hierarchy in such a way that only the SOS rules for atomic entities in hCIF are redesigned in comparison to CIF. Furthermore, to be able to reuse existing tools like simulators of the CIF language, a procedure to eliminate hierarchy from an automaton is given.

A non-interleaving process calculus for multi-party synchronisation

We introduce the wire calculus. Its dynamic features are inspired by Milner's CCS: a unary prefix operation, binary choice and a standard recursion construct. Instead of an interleaving parallel composition operator there are operators for synchronisation along a common boundary and non-communicating parallel composition. The (operational) semantics is a labelled transition system obtained with SOS rules. Bisimilarity is a congruence with respect to the operators of the language. Quotienting terms by bisimilarity results in a compact closed category.

Generating priority rewrite systems for OSOS process languages

We propose an algorithm for generating a Priority Rewrite System (PRS) for an arbitrary process language in the OSOS format such that rewriting of process terms is sound for bisimulation and head normalising. The algorithm is inspired by a procedure which was developed by Aceto, Bloom and Vaandrager and presented in Turning SOS rules into equations [L. Aceto, B. Bloom, F.W. Vaandrager, Turning SOS rules into equations, Information and Computation 111 (1994) 1–52]. For a subclass of OSOS process languages representing finite behaviours the PRSs that are generated by our algorithm are strongly normalising (terminating) and confluent, where termination is proved using the dependency pair and dependency graph techniques. Additionally, such PRSs are complete for bisimulation on closed process terms modulo associativity and commutativity of the choice operator of CCS. We illustrate the usefulness of our results, and the benefits of rewriting with priorities in general, with several examples.

A deterministic logical semantics for pure Esterel

Esterel is a synchronous design language for the specification of reactive systems. There exist two main semantics for Esterel. On the one hand, the logical behavioral semantics provides a simple and compact formalization of the behavior of programs using SOS rules. But it does not ensure deterministic deadlock-free executions, as it may define zero, one, or many possible behaviors for a given program and input sequence. Since nondeterministic programs have to be rejected by compilers, this means that it defines behaviors for incorrect programs, which is awkward. On the other hand, the constructive semantics is deterministic (amongst other properties) but at the expense of a much more complex formalism. In this work, we build and thoroughly analyze a new deterministic semantics for Esterel that retains the simplicity of the logical behavioral semantics from which it derives. It defines, at most, one behavior per program and input sequence. We further extend this semantics with the ability to deal with errors so that incorrect programs are no longer (negatively) characterized by a lack of behavior, but (positively) by the existence of an incorrect behavior. In our view, this new semantics, with or without explicit errors, provides a better framework for formal and automated reasoning about Esterel programs.

Operational Semantics of Reversibility in Process Algebra

Reversible computation has a growing number of promising application areas such as the modelling of biochemical systems, program debugging and testing, and even programming languages for quantum computing. We discuss reversibility in major process algebras from the point of view of operational semantics. The main difficulty seems to be with the definitions of forward and reverse computation for the dynamic operators, and we confine ourselves to these, leaving the static operators for further work. We consider a solution where predicates in SOS rules play a vital role.

A Deterministic Logical Semantics for Esterel

Esterel is a synchronous design language for the specification of reactive systems. There exist two main semantics for Esterel. On the one hand, the logical behavioral semantics provides a simple and compact formalization of the behavior of programs using SOS rules. But it does not ensure deterministic executions for all programs and all inputs. As non-deterministic programs have to be rejected as incorrect, this means it defines behaviors for incorrect programs, which is not convenient. On the other hand, the constructive semantics is deterministic (amongst other properties) but at the expense of a much more complex formalism. In this work, we construct and thoroughly analyze a new deterministic semantics for Esterel that retains the simplicity of the logical behavioral semantics, from which it derives. In our view, it provides a much better framework for formal reasoning about Esterel programs.

The Box Algebra=Petri Nets+Process Expressions

The paper describes a Petri net as well as a structural operational semantics for an algebra of process expressions. It specifically addresses this problem for the box algebra, a model of concurrent computation which combines Petri nets and standard process algebras. The main result is that it is possible to obtain a framework where process expressions can be given two, entirely consistent, kinds of semantics: one based on Petri nets, the other on SOS rules. This consistency can also be extended to a partial order semantics.

The Box Algebra=Petri Nets+Process Expressions

The paper describes a Petri net as well as a structural operational semantics for an algebra of process expressions. It specifically addresses this problem for the box algebra, a model of concurrent computation which combines Petri nets and standard process algebras. The main result is that it is possible to obtain a framework where process expressions can be given two, entirely consistent, kinds of semantics: one based on Petri nets, the other on SOS rules. This consistency can also be extended to a partial order semantics.

Compositional SOS and beyond: a coalgebraic view of open systems

In this paper we address the issue of providing a structured coalgebra presentation of transition systems with algebraic structure on states determined by an equational specification Γ. More precisely, we aim at representing such systems as coalgebras for an endofunctor on the category of Γ-algebras. The systems we consider are specified by using arbitrary SOS rules, which in general do not guarantee that bisimilarity is a congruence. We first show that the structured coalgebra representation works only for systems where transitions out of complex states can~be derived from transitions out of corresponding component states. This decomposition property of transitions indeed ensures that bisimilarity is a congruence. For a system not satisfying this requirement, next we propose a closure construction which adds context transitions, i.e., transitions that spontaneously embed a state into a bigger context or vice versa. The notion of bisimulation for the enriched system coincides with the notion of dynamic bisimilarity for the original one, i.e., with the coarsest bisimulation which is a congruence. This is sufficient to ensure that the structured coalgebra representation works for the systems obtained as result of the closure construction.

From SOS Specifications to Structured Coalgebras: How to Make Bisimulation a Congruence

In this paper we address the issue of providing a structured coalgebra presentation of transition systems with algebraic structure on states determined by an equational specification Γ. More precisely, we aim at representing such systems as coalgebras for an endofunctor on the category of Γ-algebras. The systems we consider are specified by using a quite general format of SOS rules, the algebraic format, which in general does not guarantee that bisimilarity is a congruence.We first show that the structured coalgebra representation works only for systems where transitions out of complex states can be derived from transitions out of corresponding component states. This decomposition property of transitions indeed ensures that bisimilarity is a congruence. For a system not satisfying this requirement, next we propose a closure construction which adds context transitions, i.e., transitions that spontaneously embed a state into a bigger context or vice-versa. The notion of bisimulation for the enriched system coincides with the notion of dynamic bisimilarity for the original one, that is, with the coarsest bisimulation which is a congruence. This is sufficient to ensure that the structured coalgebra representation works for the systems obtained as result of the closure construction.

A π-calculus with explicit substitutions

A new formulation of the π-calculus, where name instantiation is handled explicitly via the introduction of a suitable combinator, is presented. The bisimulation semantics originally developed for the π-calculus are retrieved by giving the description of the corresponding strategies for name instantiation. The explicit handling of name instantiation allows us to reduce the π-calculus transitional semantics to a standard SOS framework. Hence, π-calculus bisimulation models can take fully advantage of the SOS meta-theory developed for ‘static’ process calculi. For instance, complete axiomatic characterizations of π-calculus bisimulation equivalences can be automatically derived by turning SOS rules into equations. This formulation of the π-calculus is very promising for the development of semantic-based automatic verification tools.