Circus specifications define both data and behavioural aspects of systems using a combination of Z and CSP constructs. Previously, a denotational semantics has been given to Circus; however, a shallow embedding of Circus in Z, in which the mapping from Circus constructs to their semantic representation as a Z specification, with yet another language being used as a meta-language, was not useful for proving properties like the refinement laws that justify the distinguishing development technique associated with Circus. This work presents a final reference for the Circus denotational semantics based on Hoare and He’s Unifying Theories of Programming (UTP); as such, it allows the proof of meta-theorems about Circus including the refinement laws in which we are interested. Its correspondence with the CSP semantics is illustrated with some examples. We also discuss the library of lemmas and theorems used in the proofs of the refinement laws. Finally, we give an account of the mechanisation of the Circus semantics and of the mechanical proofs of the refinement laws.

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Design-by-contract is an important technique for model-based design in which a composite system is specified by a collection of contracts that specify the behavioural assumptions and guarantees of each component. In this paper, we describe a unifying theory for reactive design contracts that provides the basis for modelling and verification of reactive systems. We provide a language for expression and composition of contracts that is supported by a rich calculational theory. In contrast with other semantic models in the literature, our theory of contracts allow us to specify both the evolution of state variables and the permissible interactions with the environment. Moreover, our model of interaction is abstract, and supports, for instance, discrete time, continuous time, and hybrid computational models. Being based in Unifying Theories of Programming (UTP), our theory can be composed with further computational theories to support semantics for multi-paradigm languages. Practical reasoning support is provided via our proof framework, Isabelle/UTP, including a proof tactic that reduces a conjecture about a reactive program to three predicates, symbolically characterising its assumptions and guarantees about intermediate and final observations. This allows us to verify programs with a large or infinite state space. Our work advances the state-of-the-art in semantics for reactive languages, description of their contractual specifications, and compositional verification.Moreover, the empty relation false is usually used to denote either a program that fails to terminate, or else is "miraculous", having no possible behaviours. This embedding of programs into logic naturally provides great opportunities for verification by automated proof [19]. Moreover, the standard laws of programming [32] are all theorems with respect to the operator denotations. We also emphasise that these operators are all alphabet polymorphic, and can therefore be used to compose predicates of varying types, so long as the side conditions are satisfied. A selection of theorems of Definition 2.2 is shown below. Theorem 2.3 (Relational Calculus Laws). P ; I I = I I ; P = P P ; false = false ; P = false (P ; Q) ; R = P ; (Q ; R) (P b Q) ; R = P ; R b Q ; R i∈I P(i) ; Q = i∈I P(i) ; Q P ; i∈I Q(i) = i∈I P ; Q(i)

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Hoare and He's theory of reactive processes provides a unifying foundation for the formal semantics of concurrent and reactive languages. Though highly applicable, their theory is limited to models that can express event histories as discrete sequences. In this paper, we show how their theory can be generalised by using an abstract trace algebra. We show how the algebra, notably, allows us to also consider continuoustime traces and thereby facilitate models of hybrid systems. We then use this algebra to reconstruct the theory of reactive processes in our generic setting, and prove characteristic laws for sequential and parallel processes, all of which have been mechanically verified in the Isabelle/HOL proof assistant.

Robots are becoming ubiquitous: from vacuum cleaners to driverless cars, there is a wide variety of applications, many with potential safety hazards. The work presented in this paper proposes a set of constructs suitable for both modelling robotic applications and supporting verification via model checking and theorem proving. Our goal is to support roboticists in writing models and applying modern verification techniques using a language familiar to them. To that end, we present RoboChart, a domain-specific modelling language based on UML, but with a restricted set of constructs to enable a simplified semantics and automated reasoning. We present the RoboChart metamodel, its well-formedness rules, and its process-algebraic semantics. We discuss verification based on these foundations using an implementation of RoboChart and its semantics as a set of Eclipse plug-ins called RoboTool. Keywords State machines • Formal semantics • Process algebra • CSP • Model checking • Timed properties • Domain-specific language for robotics Communicated by Dr. Jeff Gray.

We present a refinement strategy for Circus , which is the combination of Z, CSP, and the refinement calculus in the setting of Hoare and He’s unifying theories of programming. The strategy unifies the theories of refinement for processes and their constituent actions, and provides a coherent technique for the stepwise refinement of concurrent and distributed programs involving rich data structures. This kind of development is carried out using Circus ’s refinement calculus, and we describe some of its laws for the simultaneous refinement of state and control behaviour, including the splitting of a process into parallel subcomponents. We illustrate the strategy and the laws using a case study that shows the complete development of a small distributed program.

We present algebraic laws for a language similar to a subset of sequential Java that includes inheritance, recursive classes, dynamic binding, access control, type tests and casts, assignment, but no sharing. These laws are proved sound with respect to a weakest precondition semantics. We also show that they are complete in the sense that they are sufficient to reduce an arbitrary program to a normal form substantially close to an imperative program; the remaining object-oriented constructs could be further eliminated if our language had recursive records. This suggests that our laws are expressive enough to formally derive behaviour preserving program transformations; we illustrate that through the derivation of provably-correct refactorings.