Konstantinos Tourlas

Hiproofs: A Hierarchical Notion of Proof Tree

Motivated by the concerns of theorem-proving, we generalise the notion of proof tree to that of hierarchical proof tree. Hierarchical trees extend ordinary trees by adding partial order structure to the set of nodes: that allows us to visualise a node as a rectangle in the plane rather than as a point, letting us use the containment relation to express structure additional to that given by a tree. A hierarchical proof tree, or hiproof for short, is a hierarchical tree with nodes labelled by tactics. We motivate the details of our definition by reference to the behaviour of tactics in tactical theorem proving. We characterise the construction of the ordinary proof tree underlying a hierarchical proof tree as a left adjoint. We then analyse the notion of proof refinement with respect to hierarchy, and we give a characterisation of hiproofs that is more directly suited to implementation.

An Assessment of the IEC 1131-3 Standard on Languages for Programmable Controllers

Programmable Logic Controllers (PLCs) are playing an increasing role in the construction of safety critical systems. The standard IEC 1131–3 defines a number of interrelated languages for the expression of PLC programs.

Justification of smart sensors for nuclear applications

This paper describes the results of a research study sponsored by the UK nuclear industry into methods of justifying smart sensors. Smart sensors are increasingly being used in the nuclear industry; they have potential benefits such as greater accuracy and better noise filtering, and in many cases their analogue counterparts are no longer manufactured. However, smart sensors (as it is the case for most COTS) are sold as black boxes despite the fact that their safety justification might require knowledge of their internal structure and development process. The study covered both management aspects of interacting with manufacturers to obtain the information needed, and the technical aspects of designing an appropriate safety justification approach and assessing feasibility of a range of technical analyses. The analyses performed include the methods we presented at Safecomp 2002 and 2003.

An Algebraic Foundation for Higraphs

Higraphs, which are structures extending graphs by permitting a hierarchy of nodes, underlie a number of diagrammatic formalisms popular in computing. We provide an algebraic account of higraphs (and of a mild extension), with our main focus being on the mathematical structures underlying common operations, such as those required for understanding the semantics of higraphs and Statecharts, and for implementing sound software tools which support them.

Design for Proof: An Approach to the Design of Domain-Specific Languages

Abstract. We propose that the domain of a Domain-Specific Language (DSL) can be characterised by:1. the class of environments in which systems developed in the language are expected to operate; and2. the class of properties which such systems are expected to possess.The design of DSLs should therefore include the development of a proof system that eases the task of proving the properties in the class identified for the anticipated operating environments.We develop these ideas in the context of industrial computing systems by presenting a semantics and proof system for a language based on IEC 1131-3, the international standard programming language for programmable controllers.Of particular significance in this example is the use of a diagrammatic representation and the development of a proof system for a class of invariance properties that requires only local knowledge of the structure of diagrams.

Formalising pragmatic features of graph-based notations

Graph-based notations form a significant subclass of visual languages. Studies of the use of such notations in practice have shown that users often employ pragmatic aspects, such as layout, to capture important domain information. Moreover this pragmatic information can support and guide reasoning over such representations. However, typical formalisations of graph-based notations often pay scant regard to such pragmatic considerations. This paper highlights an algebraic account of graph-based notations which is sensitive to relevant layout information. We illustrate, with examples taken from software engineering practice, how this algebra both captures pragmatic aspects of graphs and supports direct reasoning over their structure.

Reasoning in higraphs with loose edges

Harel (1988) introduces the notion of zooming out as a useful operation in working with higraphs. Zooming out allows one to consider less detailed versions of a higraph by dropping some detail from the description in a structured manner. Although this is a very useful operation it seems it can be misleading in some circumstances by allowing the user of the zoomed out higraph to make false inferences given the usual transition system semantics for higraphs. We consider one approach to rectifying this situation by following through Harels suggestion that, in some circumstances, it may be useful to consider higraphs with edges that have no specific origin or destination. We call these higraphs loose higraphs and show that an appropriate definition of zooming on loose higraphs avoids some of the difficulties arising from the use of zooming. We also consider a logic for connectivity in loose higraphs.

On the Geometric Modelling of Visual Languages

Abstract We present arguments for the complementary role that intrinsically geometric models of visual languages can play in many applications, alongside non-geometry based models (such those based on graph grammars, logical predicates or algebraic structures). We characterise which models are intrinsically geometric, and illustrate the concept by providing such a model for higraphs. Further, we examine the relevance of geometric models to the design and implementation of tools which effectively support users in working with visual languages.

Zooming-out on Higraph-based diagrams - Syntactic and Semantic Issues

Abstract Computing system representations based on Harels notion of hierarchical graph, or higraph, have become popular since the invention of Statecharts. Such hierarchical representations support a useful filtering operation, called “zooming-out”, which is used to manage the level of detail presented to the user designing or reasoning about a large and complex system. In the framework of (lightweight) category theory, we develop the mathematics of zooming e ut for higraphs with loose edges, formalise the transition semantics of such higraphs and conduct an analysis of the effect the operation of zooming out has on the semantic interpretations, as required for the soundness of reasoning arguments depending on zoom-out steps.

An Algebraic Foundation for Graph-based Diagrams in Computing

We develop an algebraic foundation for some of the graph-based structures underlying a variety of popular diagrammatic notations for the specification, modelling and programming of computing systems. Using hypergraphs and higraphs as leading examples, a locally ordered category Graph(C) of graphs in a locally ordered category C is defined and endowed with symmetric monoidal closed structure. Two other operations on higraphs and variants, selected for relevance to computing applications, are generalised in this setting.