Refining Nodes and Edges of State Machines

State machines are hierarchical automata that are widely used to structure complex behavioural specifications. We develop two notions of refinement of state machines, node refinement and edge refinement. We compare the two notions by means of examples and argue that, by adopting simple conventions, they can be combined into one method of refinement. In the combined method, node refinement can be used to develop architectural aspects of a model and edge refinement to develop algorithmic aspects. The two notions of refinement are grounded in previous work. Event-B is used as the foundation for our refinement theory and UML-B state machine refinement influences the style of node refinement. Hence we propose a method with direct proof of state machine refinement avoiding the detour via Event-B that is needed by UML-B

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

Feature refinement

Development by formal stepwise refinement offers a guarantee that an implementation satisfies a specification. But refinement is frequently defined in such a restrictive way as to disallow some useful development steps. Here we de- fine feature refinement to overcome some limitations of re- finement and show its usefulness by applying it to examples taken from the literature. Using partial relations as a canonical state-based semantics and labelled transition systems as a canonical event-based semantics, we degine functions formally linking the state- and event-based operational semantics. We can then use this link to move notions of refinement between the event- and state-based worlds. An advantage of this abstract approach is that it is not restricted to a specific syntax or even a specific interpretation of the operational semantic

Flexible refinement

To help make refinement more usable in practice we introduce a general, flexible model of refinement. This is defined in terms of what contexts an entity can appear in, and what observations can be made of it in those contexts. Our general model is expressed in terms of an operational semantics, and by exploiting the well-known isomorphism between state-based relational semantics and event-based labelled transition semantics we were able to take particular models from both the state- and event-based literature, reflect on them and gradually evolve our general model. We are also able to view our general model both as a testing semantics and as a logical theory with refinement as implication. Our general model can used as a bridge between different particular special models and using this bridge we compare the definition of determinism found in different special models. We do this because the reduction of nondeterminism underpins many definitions of refinement found in a variety of special models. To our surprise we find that the definition of determinism commonly used in the process algebra literature to be at odds with determinism as defined in other special models. In order to rectify this situation we return to the intuitions expressed by Milner in CCS and by formalising these intuitions we are able to define determinism in process algebra in such a way that it no longer at odds with the definitions we have taken from other special models. Using our abstract definition of determinism we are able to construct a new model, interactive branching programs, that is an implementable subset of process algebra. Later in the chapter we show explicitly how five special models, taken from the literature, are instances of our general model. This is done simply by fixing the sets of contexts and observations involved. Next we define vertical refinement on our general model. Vertical refinement can be seen both as a generalisation of what, in the literature, has been called action refinement or non-atomic refinement. Alternatively, by viewing a layer as a logical theory, vertical refinement is a theory morphism, formalised as a Galois connection. By constructing a vertical refinement between broadcast processes and interactive branching programs we can see how interactive branching programs can be implemented on a platform providing broadcast communication. But we have been unable to extend this theory morphism to implement all of process algebra using broadcast communication. Upon investigation we show the problem arises with the examples that caused the problem with the definition of determinism on process algebra. Finally we illustrate the usefulness of our flexible general model by formally developing a single entity that contains events that use handshake communication and events that use broadcast communication

A refinement of Rasmussen's s-invariant

In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's slice genus bound s for each stable cohomology operation. We show that in the case of the Steenrod square Sq^2 our refinement is strictly stronger than s.Comment: 26 pages, 2 figure

Managing LTL properties in Event-B refinement

Refinement in Event-B supports the development of systems via proof based step-wise refinement of events. This refinement approach ensures safety properties are preserved, but additional reasoning is required in order to establish liveness and fairness properties. In this paper we present results which allow a closer integration of two formal methods, Event-B and linear temporal logic. In particular we show how a class of temporal logic properties can carry through a refinement chain of machines. Refinement steps can include introduction of new events, event renaming and event splitting. We also identify a general liveness property that holds for the events of the initial system of a refinement chain. The approach will aid developers in enabling them to verify linear temporal logic properties at early stages of a development, knowing they will be preserved at later stages. We illustrate the results via a simple case study