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28
Objects and Classes, Coalgebraically
 ObjectOrientation with Parallelism and Persistence
, 1995
"... The coalgebraic perspective on objects and classes in objectoriented programming is elaborated: objects consist of a (unique) identifier, a local state, and a collection of methods described as a coalgebra; classes are coalgebraic (behavioural) specifications of objects. The creation of a "new" o ..."
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Cited by 68 (17 self)
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The coalgebraic perspective on objects and classes in objectoriented programming is elaborated: objects consist of a (unique) identifier, a local state, and a collection of methods described as a coalgebra; classes are coalgebraic (behavioural) specifications of objects. The creation of a "new" object of a class is described in terms of the terminal coalgebra satisfying the specification. We present a notion of "totally specified" class, which leads to particularly simple terminal coalgebras. We further describe local and global operational semantics for objects. Associated with the local operational semantics is a notion of bisimulation (for objects belonging to the same class), expressing observational indistinguishability. AMS Subject Classification (1991): 18C10, 03G30 CR Subject Classification (1991): D.1.5, D.2.1, E.1, F.1.1, F.3.0 Keywords & Phrases: object, class, (terminal) coalgebra, coalgebraic specification, bisimulation 1. Introduction Within the objectoriente...
Automata and coinduction (an exercise in coalgebra
 LNCS
, 1998
"... The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which ..."
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Cited by 62 (16 self)
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The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the present treatment of automata theory may serve as an introduction to coalgebra.
Behavioural Differential Equations: A Coinductive Calculus of Streams, Automata, and Power Series
, 2000
"... Streams, (automata and) languages, and formal power series are viewed coalgebraically. In summary, this amounts to supplying these sets with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduct ..."
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Cited by 52 (17 self)
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Streams, (automata and) languages, and formal power series are viewed coalgebraically. In summary, this amounts to supplying these sets with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductive definitions take the shape of what we have called behavioural differential equations, after Brzozowski's notion of input derivative. A calculus is developed for coinductive reasoning about all of the afore mentioned structures, closely resembling (and at times generalising) calculus from classical analysis. 2000 Mathematics Subject Classification: 68Q10, 68Q55, 68Q85 1998 ACM Computing Classification System: F.1, F.3 Keywords & Phrases: Coalgebra, automaton, finality, coinduction, stream, formal language, formal power series, differential equation, input derivative, behaviour, semiring, maxplus algebra 1 Contents 1 Introductio...
System Identification, Approximation and Complexity
 International Journal of General Systems
, 1977
"... This paper is concerned with establishing broadlybased systemtheoretic foundations and practical techniques for the problem of system identification that are rigorous, intuitively clear and conceptually powerful. A general formulation is first given in which two order relations are postulated on a ..."
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Cited by 34 (23 self)
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This paper is concerned with establishing broadlybased systemtheoretic foundations and practical techniques for the problem of system identification that are rigorous, intuitively clear and conceptually powerful. A general formulation is first given in which two order relations are postulated on a class of models: a constant one of complexity; and a variable one of approximation induced by an observed behaviour. An admissible model is such that any less complex model is a worse approximation. The general problem of identification is that of finding the admissible subspace of models induced by a given behaviour. It is proved under very general assumptions that, if deterministic models are required then nearly all behaviours require models of nearly maximum complexity. A general theory of approximation between models and behaviour is then developed based on subjective probability concepts and semantic information theory The role of structural constraints such as causality, locality, finite memory, etc., are then discussed as rules of the game. These concepts and results are applied to the specific problem or stochastic automaton, or grammar, inference. Computational results are given to demonstrate that the theory is complete and fully operational. Finally the formulation of identification proposed in this paper is analysed in terms of Klir’s epistemological hierarchy and both are discussed in terms of the rich philosophical literature on the acquisition of knowledge. 1
Inheritance and Cofree Constructions
 European Conference on ObjectOriented Programming, number 1098 in Lect. Notes Comp. Sci
, 1995
"... The coalgebraic view on classes and objects is elaborated to include inheritance. Inheritance in coalgebraic specification (of classes) will be understood dually to parametrization in algebraic specification. That is, inheritance involves restriction (specialization), where parametrization involves ..."
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Cited by 26 (7 self)
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The coalgebraic view on classes and objects is elaborated to include inheritance. Inheritance in coalgebraic specification (of classes) will be understood dually to parametrization in algebraic specification. That is, inheritance involves restriction (specialization), where parametrization involves extension. And cofree constructions are "best" restrictions, like free constructions are "best" extensions. To make this view on inheritance precise we need a suitable notion of behaviour preserving morphism between classes, which will be defined as a "coalgebra map uptobisimulation". AMS Subject Classification (1991): 18C10, 03G30 CR Subject Classification (1991): D.1.5, D.2.1, E.1, F.1.1, F.3.0 Keywords & Phrases: object, class, inheritance, coalgebraic specification, bisimulation 1. Introduction Two basic relations in objectoriented languages are: object o belongs to class C, and: class C inherits from class C 0 (see e.g. [20]). Class membership yields what is sometimes called a...
On the Duality between Observability and Reachability
 PROC. 4TH INT. CONF. FOUNDATIONS OF SOFTWARE SCIENCE AND COMPUTATION STRUCTURES (FOSSACS'01
, 2001
"... Observability and reachability are important concepts in formal software development. While observability concepts allow to specify the required observable behavior of a program or system, reachability concepts are used to describe the underlying data in terms of datatype constructors. In this paper ..."
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Cited by 12 (4 self)
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Observability and reachability are important concepts in formal software development. While observability concepts allow to specify the required observable behavior of a program or system, reachability concepts are used to describe the underlying data in terms of datatype constructors. In this paper, we show that there is a duality between observability and reachability, both from a methodological and from a formal point of view. In particular, we establish a correspondence between observer operations and datatype constructors, observational algebras and constructorbased algebras, and observational and inductive properties of specifications. Our study is based on the observational logic institution [11] and on a novel treatment of reachability which introduces the constructorbased logic institution. Both institutions are tailored to capture the semantically correct realizations of a specification from the observational and reachability points of view. The duality between the observability and reachability concepts is then formalized in a categorytheoretic setting.
The BehaviorRealization Adjunction and Generalized Homomorphic Relations
 COMP. SCI
, 1996
"... A model theory for proving correctness of abstract data types is developed within the framework of the behaviorrealization adjunction. To allow for incomplete specifications, proofofcorrectness ... ..."
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Cited by 9 (5 self)
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A model theory for proving correctness of abstract data types is developed within the framework of the behaviorrealization adjunction. To allow for incomplete specifications, proofofcorrectness ...
Precise past  fuzzy future
 International Journal of ManMachine Studies
, 1983
"... This paper examines the motivation and foundations of fuzzy sets theory, now some 20 years old, particularly possible misconceptions about possible operators and relations to probability theory. It presents a standard uncertainty logic (SUI.) that subsumes standard propositional, fuzzy and probabili ..."
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Cited by 6 (4 self)
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This paper examines the motivation and foundations of fuzzy sets theory, now some 20 years old, particularly possible misconceptions about possible operators and relations to probability theory. It presents a standard uncertainty logic (SUI.) that subsumes standard propositional, fuzzy and probability logics, and shows how many key results may be derived within SUL without further constraints. These include resolutions of standard paradoxes such as those of the bald man and of the barber, decision rules used in pattern recognition and control, the derivation of numeric truth values from the axiomatic form of the SUL, and the derivation of operators such as the arithmetic mean. The addition of the constraint of truthfunctionality to a SUL is shown to give fuzzy, or Lukasiewicz infinitelyvalued, logic. The addition of the constraint of the law of the excluded middle to a SUL is shown to give probability, or modal $5, logic. An example is given of the use of the two logics in combination to give a possibility vector when modelling sequential behaviour with uncertain observations.
ObjectOriented Hybrid Systems of Coalgebras plus Monoid Actions
 Algebraic Methodology and Software Technology (AMAST
, 1996
"... . Hybrid systems combine discrete and continuous dynamics. We introduce a semantics for such systems consisting of a coalgebra together with a monoid action. The coalgebra captures the (discrete) operations on a state space that can be used by a client (like in the semantics of ordinary (nontempora ..."
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Cited by 5 (1 self)
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. Hybrid systems combine discrete and continuous dynamics. We introduce a semantics for such systems consisting of a coalgebra together with a monoid action. The coalgebra captures the (discrete) operations on a state space that can be used by a client (like in the semantics of ordinary (nontemporal) objectoriented systems). The monoid action captures the influence of time on the state space, where the monoids that we consider are the natural numbers monoid (N; 0; +) of discrete time, and the positive reals monoid (R0 ; 0; +) of real time. Based on this semantics we develop a hybrid specification formalism with timed method applications: it involves expressions like s:meth@ff, with the following meaning: in state s let the state evolve for ff units of time (according to the monoid action), and then apply the (coalgebraic) method meth. In this formalism we specify various (elementary) hybrid systems, investigate their correctness, and display their behaviour in simulations. We furthe...
Analogy categories, virtual machines and , ,structured programming
 GI2 Jahrestagung, Lecture Notes in Computer Science
, 1975
"... Abstract This paper arises from a number of studies of machine/problem relationships, software development techniques, language and machine design. It develops a categorytheoretic framework for the analysis of the relationships between programmer, virtual machine, and problem that are inherent in d ..."
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Cited by 5 (5 self)
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Abstract This paper arises from a number of studies of machine/problem relationships, software development techniques, language and machine design. It develops a categorytheoretic framework for the analysis of the relationships between programmer, virtual machine, and problem that are inherent in discussions of "ease of programming", "good programming techniques", "structured programming", and so ono The concept of "analogy " is introduced as an expllcatum of the comprehensibility of the relationship between two systems. Analogy is given a formal definition in terms of a partially ordered structure of analogy categories whose minimal element is a "truth " c ~ "proof " category ° The theory is constructive and analogy relationships are computable between defined systems, c ~ classes of system. Thus the structures developed may be used to study the relationships between programmer, problem, and virtual machine in practical situations. io