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...sorts and behavioral on hidden sorts, in the sense of "indistinguishability under experiments;" special cases include hidden algebra [7, 8, 10, 9], coherent hidden algebra [5, 6], observatio=-=nal logic [15, 2]-=-, and our recent generalizations of hidden algebra [19, 18]. A related approach is coalgebra [16, 1]. Because of space constraints, we omit all proofs and most definitions, instead giving explanations...

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...s, as At : A → (A (var(t)−{⋆}) → A) with the obvious meaning. 2.2 Behavioral Specification and Satisfaction We generalize the hidden algebra of [9, 11] to include variants such as observational logic =-=[1, 3, 14]-=- and coherent hidden algebra [6, 7]. See [19] for a detailed presentation of variants, history, many other concepts, and proofs for some results mentioned here. Two important variants of behavioral lo...

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...be used in combination with coinduction. 1 Introduction Behavioral abstraction in algebraic specification appears under various names in the literature such as hidden algebra [5], observational logic =-=[8]-=-, swinging types [11], coherent hidden algebra [2], hidden logic [12], and so on. Most of these approaches appeared as a need to extend algebraic specifications to ease the process of specifying and v...

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...empotence, and combinations between all these. This feature is reflected at the execution level by term rewriting modulo such equational theories. Behavioural Specification. Behavioural specification =-=[21, 22, 13, 29]-=- provides a novel generalisation of ordinary algebraic specification. Behavioural specification characterises how objects (and systems) behave, not how they are implemented. This new form of abstracti...

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...e research area containing many approaches, including the original hidden algebra of [9, 14, 19, 18], the coherent hidden algebra of Diaconescu [6, 7], the observational logic of Bidoit and Hennicker =-=[1, 2, 24]-=-, and a new generalization of hidden algebra that tries to treat all these variants in a uniform way [35, 21]. These approaches fall into two broad categories, depending on whether or not a fixed data...

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...nd datatype constructors, observational algebras and constructor-based algebras, and observational and inductive properties of specications. Our study is based on the observational logic institution [=-=11-=-] and on a novel treatment of reachability which introduces the constructor-based logic institution. Both institutions are tailored to capture the semantically correct realizations of a specication fr...

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...always compatible, but it is easy to check that the family of observational equalities is not compatible in general. One may restrict the introduction of new observers to avoid this problem, see e.g. =-=[BH99]-=-. 3 Formal Testing from Algebraic Specifications Testing from algebraic specifications boils down to checking if axioms are satisfied by programs [Gau95]. Oracles are usually active procedures which d...