Automata, Power Series, and Coinduction: Taking Input Derivatives Seriously (Extended Abstract)

“…Both these final coalgebras AN and L(A) = P(A*) are studied extensively by Rutten, see [29,32,30]. One of the things that he emphasises is the use of bisimulation as a reasoning principle.…”

Section: S* A* \ S * (X )(0 )= X X ------^X a Defined As < (1) [ S*(xmentioning

confidence: 99%

“…The coalgebraic perspective on this topic was introduced by Rutten [29,32,30], who demonstrated its fruitfulness especially for proving equalities via coinduction (using bisimulations). Rutten's work exploits the automaton structure on regular expressions introduced by Brzozowski [8,9].…”

Section: Introductionmentioning

confidence: 99%

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

Jacobs

2006

Algebra, Meaning, and Computation

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To Joseph Goguen on the occasion of his 65th birthday1.Abstract. This papers reviews the classical theory of deterministic automata and regular languages from a categorical perspective. The basis is formed by Rutten's description of the Brzozowski automaton structure in a coalgebraic framework. We enlarge the framework to a so-called bialgebraic one, by including algebras together with suitable distributive laws connecting the algebraic and coalgebraic structure of regular expressions and languages. This culminates in a reformulated proof via finality of Kozen's completeness result. It yields a complete axioma tisation of observational equivalence (bisimilarity) on regular expressions. We suggest that this situation is paradigmatic for (theoretical) computer science as the study of "generated behaviour".

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“…Coalgebraic methods have been used for dynamical systems, automata and formal languages, modal logic, transition systems, hybrid systems, infinite data types, the control of discrete event systems, formal power series, etc. (see for instance [53], [41], [42], [50], [51], [52], [22], [27]). Coalgebras have also been used as models for various programming paradigms, notably for objects and classes (see, e.g., [47], [28], and [26]).…”

Section: Abstract Behavior Typesmentioning

confidence: 99%

Abstract Behavior Types: A Foundation Model for Components and Their Composition

Arbab

2003

Formal Methods for Components and Objects

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a Software ENgineeringAbstract behavior types: a foundation model for components and their composition The notion of Abstract Data Type (ADT) has served as a foundation model for structured and object oriented programming for some thirty years. The current trend in software engineering toward component based systems requires a foundation model as well. The most basic inherent property of an ADT, i.e., that it provides a set of operations, subverts some highly desirable properties in emerging formal models for components that are based on the object oriented paradigm.We introduce the notion of Abstract Behavior Type (ABT) as a higher-level alternative to ADT and propose it as a proper foundation model for both components and their composition. An ABT defines an abstract behavior as a relation among a set of timed-data-streams, without specifying any detail about the operations that may be used to implement such behavior or the data types it may manipulate for its realization. The ABT model supports a much looser coupling than is possible with the ADT's operational interface, and is inherently amenable to exogenous coordination. We propose that both of these are highly desirable, if not essential, properties for models of components and their composition.To demonstrate the utility of the ABT model, we describe Reo: an exogenous coordination language for compositional construction of component connectors based on a calculus of channels. We show the expressive power of Reo, and the applicability of ABT, through a number of examples. IntroductionAn Abstract Data Type (ADT) defines an algebra of operations with mathematically well-defined semantics, without specifying any detail about the implementation of those operations or the data structures they operate on to realize them. As such, ADT is a powerful abstraction and encapsulation mechanism that groups data together with their related operations into logically coherent and loosely-dependent entities, such as objects, yielding better structured programs. ADT has served as a foundation model for structured and object oriented programming for some thirty years.The immense success of object oriented techniques has distracted proper attention away from critical evaluation of some of its underpinning concepts from the perspective of their utility for components. We propose that the most basic inherent property of an ADT, i.e., that it provides a set of operations in its interface, subverts some highly desirable properties in emerging models for component based systems. This is already evident in the current attempts at extending the object oriented models into the realm of components (see, e.g., Sections 3 and 5).We introduce the notion of Abstract Behavior Type (ABT) as a higher-level alternative to ADT and propose it as a proper foundation model for both components and their composition. An ABT defines an abstract behavior as a relation among a set of timed-data-streams, without specifying any detail about ...

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Automata, Power Series, and Coinduction: Taking Input Derivatives Seriously (Extended Abstract)

Cited by 16 publications

References 10 publications

Rational and Recognisable Power Series

Rational and Recognisable Power Series

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

Abstract Behavior Types: A Foundation Model for Components and Their Composition

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