UTP2: Higher-Order Equational Reasoning by Pointing

We describe a prototype theorem prover, UTP2, developed to match the style of hand-written proof work in the Unifying Theories of Programming semantical framework. This is based on alphabetised predicates in a 2nd-order logic, with a strong emphasis on equational reasoning. We present here an overview of the user-interface of this prover, which was developed from the outset using a point-and-click approach. We contrast this with the command-line paradigm that continues to dominate the mainstream theorem provers, and raises the question: can we have the best of both worlds?Comment: In Proceedings UITP 2014, arXiv:1410.785

Heterogeneous Semantics and Unifying Theories

Model-driven development is being used increasingly in the development of modern computer-based systems. In the case of cyber-physical systems (including robotics and autonomous systems) no single modelling solution is adequate to cover all aspects of a system, such as discrete control, continuous dynamics, and communication networking. Instead, a heterogeneous modelling solution must be adopted. We propose a theory engineering technique involving Isabelle/HOL and Hoare & He’s Unifying Theories of Programming. We illustrate this approach with mechanised theories for building a contractual theory of sequential programming, a theory of pointer-based programs, and the reactive theory underpinning CSP’s process algebra. Galois connections provide the mechanism for linking these theories

Angelic Processes for CSP via the UTP

Demonic and angelic nondeterminism play fundamental roles as abstraction mechanisms for formal modelling. In contrast with its demonic counterpart, in an angelic choice failure is avoided whenever possible. Although it has been extensively studied in refinement calculi, in the context of process algebras, and of the Communicating Sequential Processes (CSP) algebra for refinement, in particular, it has been elusive. We show here that a semantics for an extended version of CSP that includes both demonic and angelic choice can be provided using Hoare and He's Unifying Theories of Programming (UTP). Since CSP is given semantics in the UTP via reactive designs (pre and postcondition pairs) we have developed a theory of angelic designs and a conservative extension of the CSP theory using reactive angelic designs. To characterise angelic nondeterminism appropriately in an algebra of processes, however, a notion of divergence that can undo the history of events needs to be considered. Taking this view, we present a model for CSP where angelic choice completely avoids divergence just like in the refinement calculi for sequential programs

Angelic Processes

In the formal modelling of systems, demonic and angelic nondeterminism play fundamental roles as abstraction mechanisms. The angelic nature of a choice pertains to the property of avoiding failure whenever possible. As a concept, angelic choice first appeared in automata theory and Turing machines, where it can be implemented via backtracking. It has traditionally been studied in the refinement calculus, and has proved to be useful in a variety of applications and refinement techniques. Recently it has been studied within relational, multirelational and higher-order models. It has been employed for modelling user interactions, game-like scenarios, theorem proving tactics, constraint satisfaction problems and control systems. When the formal modelling of state-rich reactive systems is considered, it only seems natural that both types of nondeterministic choice should be considered. However, despite several treatments of angelic nondeterminism in the context of process algebras, namely Communicating Sequential Processes, the counterpart to the angelic choice of the refinement calculus has been elusive. In this thesis, we develop a semantics in the relational setting of Hoare and He's Unifying Theories of Programming that enables the characterisation of angelic nondeterminism in CSP. Since CSP processes are given semantics in the UTP via designs, that is, pre and postcondition pairs, we first introduce a theory of angelic designs, and an isomorphic multirelational model, that is suitable for characterising processes. We then develop a theory of reactive angelic designs by enforcing the healthiness conditions of CSP. Finally, by introducing a notion of divergence that can undo the history of events, we obtain a model where angelic choice avoids divergence. This lays the foundation for a process algebra with both nondeterministic constructs, where existing and novel abstract modelling approaches can be considered. The UTP basis of our work makes it applicable in the wider context of reactive systems

Model Checking of State-Rich Formalisms (By Linking to Combination of State-based Formalism and Process Algebra)

Computer-based systems are becoming more and more complex. It is really a grand challenge to assure the dependability of these systems with the growing complexity, especially for high integrity and safety critical systems that require extremely high dependability. Circus, as a formal language, is designed to tackle this problem by providing precision preservation and correctness assurance. It is a combination of Z, CSP, refinement calculus and Dijkstra's guarded commands. A main objective of Circus is to provide calculational style refinement that differentiates itself from other integrated formal methods. Looseness, which is introduced from constants and uninitialised state space in Circus, and nondeterminism, which is introduced from disjunctive operations and CSP operators, make model checking of Circus more difficult than that of sole CSP or Z. Current approaches have a number of disadvantages like nondeterminism and divergence information loss, abstraction deterioration, and no appropriate tools to support automation. In this thesis, we present a new approach to model-check state-rich formalisms by linking them to a combination of a state-based formalism and a process algebra. Specifically, the approach illustrated in this thesis is to model-check Circus by linking to CSP || B. Eventually, we can use ProB, a model checker for B, Event-B, and CSP || B etc., to check the resultant CSP || B model. A formal link from Circus to CSP || B is defined in our work. Our link solution is to rewrite Circus models first to make all interactions between the state part and the behavioural part of Circus only through schema expressions, then translate the state part and the behavioural part to B and CSP respectively. In addition, since the semantics of Circus is based on Hoare and He's Unifying Theories of Programming (UTP), in order to prove the soundness of our link, we also give UTP semantics to CSP || B. Finally, because both ends of the link have their semantics defined in UTP, they are comparable. Furthermore, in order to support an automatic translation process, a translator is developed. It has supported almost all constructs defined in the link though with some limitations. Finally, three case studies are illustrated to show the usability of our model checking solution as well as limitations. The bounded reactive buffer is a typical Circus example. By our model checking approach, basic properties like deadlock freedom and divergence freedom for both the specification and the implementation with a small buffer size have been verified. In addition, the implementation has been verified to be a refinement of the specification in terms of traces and failures. Afterwards, in the Electronic Shelf Edge Label (ESEL) case study, we demonstrate how to use Circus to model different development stages of systems from the specification to two more specific systems. We have verified basic properties and sequential refinements of three models as well as three application related properties. Similarly, only the systems with a limited number of ESELs are verified. Finally, we present the steam boiler case study. It is a real and industrial control system problem. Though our solution cannot model check the steam boiler model completely due to its large state space, our solution still proves its benefits. Through our model checking approach, we have found a substantial number of errors from the original Circus solution. Then with counterexamples during animation and model checking, we have corrected all these found errors

Isabelle/UTP: Mechanised Theory Engineering for the UTP

Isabelle/UTP is a mechanised theory engineering toolkit based on Hoare and He’s Unifying Theories of Programming (UTP). UTP enables the creation of denotational, algebraic, and operational semantics for different programming languages using an alphabetised relational calculus. We provide a semantic embedding of the alphabetised relational calculus in Isabelle/HOL, including new type definitions, relational constructors, automated proof tactics, and accompanying algebraic laws. Isabelle/UTP can be used to both capture laws of programming for different languages, and put these fundamental theorems to work in the creation of associated verification tools, using calculi like Hoare logics. This document describes the relational core of the UTP in Isabelle/HOL

Isabelle/UTP : Mechanised Theory Engineering for Unifying Theories of Programming

Isabelle/UTP is a mechanised theory engineering toolkit based on Hoare and He’s Unifying Theories of Programming (UTP). UTP enables the creation of denotational, algebraic, and operational semantics for different programming languages using an alphabetised relational calculus. We provide a semantic embedding of the alphabetised relational calculus in Isabelle/HOL, including new type definitions, relational constructors, automated proof tactics, and accompanying algebraic laws. Isabelle/UTP can be used to both capture laws of programming for different languages, and put these fundamental theorems to work in the creation of associated verification tools, using calculi like Hoare logics. This document describes the relational core of the UTP in Isabelle/HOL