Abstract not found

. Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intended that the state should be "hidden" with only certain features accessible through attributes and methods. States should become equal, if no external observation may distinguish them. It has recently been discovered that state based systems such as transition systems, automata, lazy data structures and objects give rise to structures dual to universal algebra, which are called coalgebras. Equality is replaced by indistinguishability and co-induction replaces induction as proof principle. However, as it turns out, one has to look at universal algebra from a more general perspective (using elementary category theoretic notions) before the dual concept is able to capture the relevant ...

We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

Ordinary software engineers and programmers can easily understand regular patterns, as shown by the immense interest in and the success of scripting languages like Perl, based essentially on regular expression pattern matching. We believe that regular expressions provide an elegant and powerful specification language also for monitoring requirements, because an execution trace of a program is in fact a string of states. Extended regular expressions (EREs) add complementation to regular expressions, which brings additional benefits by allowing one to specify patterns that must not occur during an execution. Complementation gives one the power to express patterns on strings more compactly. In this paper we present a technique to generate optimal monitors from EREs. Our monitors are deterministic finite automata (DFA) and our novel contribution is to generate them using a modern coalgebraic technique called coinduction. Based on experiments with our implementation, which can be publicly tested and used over the web, we believe that our technique is more efficient than the simplistic method based on complementation of automata which can quickly lead to a highly-exponential state explosion.

We introduce the lambda-coiteration schema for a distributive law lambda of a functor T over a functor F. Under certain conditions it can be shown to uniquely characterise functions into the carrier of a final F-coalgebra, generalising the basic coiteration schema as given by finality. The duals of primitive recursion and course-of-value iteration, which are known extensions of coiteration, arise as instances of our framework. One can furthermore obtain schemata justifying recursive specifications that involve operators such as addition of power series, regular operators on languages, or parallel and sequential composition of processes. Next...

This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of F-automata, devices that operate on pointed F-coalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite two-player graph game. We also introduce a language of coalgebraic fixed point logic for F-coalgebras, and we provide a game semantics for this language. Finally we show that any formula p of the language can be transformed into an F-automaton Ap which is equivalent to p in the sense that Ap accepts precisely those pointed F-coalgebras in which p holds.

The work in this thesis has been carried out under the auspices of the research school IPA (Institute for Programming research and Algorithmics).vrije universiteit Böhm-Like Trees for Rewriting academisch proefschrift ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. T. Sminia, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der Exacte Wetenschappen op maandag 20 maart 2006 om 15.45 uur in de aula van de universiteit, De Boelelaan 1105 door

In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines.

We develop a coalgebraic theory of Kleene algebra with tests (KAT) along the lines of Rutten (1998) for Kleene algebra (KA) and Chen and Pucella (2003) for a limited version of KAT, resolving some technical issues raised by Chen and Pucella. Our treatment includes a simple definition of the Brzozowski derivative for KAT expressions and an automata-theoretic interpretation involving automata on guarded strings. We also give a complexity analysis, showing that an efficient implementation of coinductive equivalence proofs in this setting is tantamount to a standard automatatheoretic construction. It follows that coinductive equivalence proofs can be generated automatically in PSPACE. This matches the bound of Worthington (2008) for the automatic generation of equational proofs in KAT. 1

Abstract. Event-pattern reactive programs are front-end programs for distributed reactive components that preprocess an incoming stream of event stimuli. Their purpose is to recognize temporal patterns of events that are relevant to the serviced program and ignore all other events, outsourcing some of the component’s complexity and shielding it from event overload. Correctness of event-pattern reactive programs is essential, because bugs may result in loss of relevant events and hence failure to react appropriately. We introduce PAR, a specification language for event-pattern reactive programs. We propose a new approach for defining such languages in terms of observations and actions. This approach applies standard techniques from coalgebra to obtain instances of the corecursion and coinduction principles. Corecursion is used to formally define the operational semantics of PAR, and coinduction allows to prove general equivalences between (ground and parameterized) PAR programs. This is the first of a series of papers in which we study questions of expressive completeness, complexity, and formal verification techniques for specification languages of event-pattern reactive programs. 1