J. Meseguer, J. A. Goguen. Initialtrot, induction and computabilitt. Algebraic methods in semantics, Cambridge University Press, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....relation is constructed and the quotient of the term algebra is taken. For details we refer to [4] A reader convinced of the existence right away, can skip section 8, and does not need to worry about the shape of the construction. In terms of the initial algebra approach as discussed in [3,4], one can take a signature P with one sort, where the operations consist of a constant for each element of the domain, and one binaxy operation. Then our binaxy structure is the initial algebra in the vaxiety of P algebras of the given set of laws, that is the class consisting of all P algebras ....

J. Meseguer, J. A. Goguen. Initialtrot, induction and computabilitt. Algebraic methods in semantics, Cambridge University Press, 1985.

....to use are even more open. How does one develop axiomatisations and proof systems for stream algebras Initial algebra speci cations of discrete data types are important theoretically and practically [37] Equational de nability and computability are known to be equivalent for countable algebras [17, 25]. But there are problems with continuous data types such as streams. In short, there is no shortage of either open problems or potential applications for topological data types in general, or stream data types in particular. In this note we will study the construction of an algebra A of ....

J. Meseguer and J.A. Goguen, Initiality, induction and computability, Algebraic methods in semantics (M. Nivat and J. Reynolds, editors), Cambridge University Press, 1985, pp. 459-541.

....some specification, and to define the semantics of such observations in order to obtain a framework that will capture the essence of program correctness. Up to now, various notions of observability have been introduced, involving observation techniques based on sorts [5] 21] 10] 4] 18] [11], 19] 14] 13] operations [1] terms [17] 7] or formulae [16] 17] It is unfortunately difficult to compare these various notions of observability and to decide which one is better suited to 3 solve the problem described above. The aim of this paper is to provide grounds for such a ....

Meseguer J., Goguen J.A. Initiality, Induction and Computability (Nivat M., Reynolds J.C. eds.) Algebraic Methods in Semantics, Cambridge Univ. Press, 459-540 (1985)

....all the axioms given there. An algebra is an acceptable realisation of this specification as long as the membership relation behaves properly (i.e. gives the right answers for every choice of argument) This theme has been discussed in [GGM 76] BM 81] Rei 81] GM 82] Sch 86] Kam 83] MG 83] ST 85a,87a] and elsewhere; the idea goes back (at least) to work on automata theory in the 1950 s [Moo 56] To formalize these ideas we will consider another class of specification building operations called abstractors. Intuitively, any equivalence relation on Sigma algebras determines a ....

Meseguer, J. and Goguen, J.A. Initiality, induction and computability. Algebraic Methods in Semantics (M. Nivat and J. Reynolds, eds.), Cambridge Univ. Press, pp. 459-541 (1983).

....empty. Together with an appropriate facility (see 3. 3) for hiding and renaming, subclass inheritance directly provides a simple notion of implementation [ST 88] but one may in practice require a more sophisticated notion of implementation up to observational equivalence ( Rei 81] SW 83] MG 85] ST 87] NO 88] In any case, the notion of subclass refinement should support the development process of both requirement and design specifications, in which additional design decisions restrict the set of admissible models, eventually leading towards a specification that includes all the ....

....is correct up to behavioural equivalence , meaning that all observable consequences of SETS X (equations between terms of type bool or X.T, in this case) are satisfied by SETSasLISTS X L. Thus we regard SETSasLISTS as a correct implementation of its specification. See e.g. Rei 81] SW 83] MG 85] Sch 87] ST 87] NO 88] and [ST 89] for more motivation and technical details. The possibilities for currying and successive parameterisation are even better in the following version (at the expense of an extra local instantiation) since we can partially instantiate, for example, as ....

J. Meseguer and J.A. Goguen: Initiality, induction and computability. Algebraic Methods in Semantics (M. Nivat and J. Reynolds, eds.), 459-540. Cambridge Univ. Press (1985).

....the correctness of a candidate implementation, one must first choose a paradigm and then make the comparison. This paper concentrates, therefore, on how to compare an implementation algebra to a paradigm, once a paradigm is selected. Several authors have studied such notions previously (including [4, 7, 17, 18] see Section 8 of [23] for a survey) For our purposes the most important technique is that of Schoett [18] Schoett casts the problem as one of showing that a partial algebra A can be used in place of the paradigm, a partial algebra B, without exhibiting surprising behavior. He argues that ....

....the more distantly related, but nevertheless influential, work of Goguen and Meseguer. The decision to formulate our results as a behavior realization adjunction was inspired by Goguen s [5] categorical theory of automata [5] and its subsequent extension to general modules by Goguen and Meseguer [6, 7]. But the relationship between our theory of behavior, realization, and the associated adjunction and the corresponding theory of Goguen and Meseguer is not straightforward. Roughly speaking, in [6, 7] the behavior of an algebra A is defined pretty much the way Schoett does, namely, as the ....

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J. Goguen and J. Meseguer, Initiality, Induction and Computability, Algebraic Methods in Semantics (M. Nivat and J. Reynolds, eds.), Cambridge University Press, Cambridge, 1985, pp. 459--541.

....related work, future work, and offer some conclusions. Related work. Our decision to formulate our results as a behavior realization adjunction was considerably influenced by Goguen s categorical theory of automata [6] and its subsequent extension to (discrete) algebras by Goguen and Meseguer [8, 9]. Our notion of dual realization can be viewed in a loose sense as a generalization of Goguen and Meseguer s realization of the behavior of an automaton or algebra, although strictly speaking the two notions are incomparable. In their work they speak of the behavior of an entire algebra, which ....

Joseph Goguen and Jos'e Meseguer, Initiality, Induction and Computability, Algebraic Methods in Semantics (M. Nivat and J. Reynolds, eds.), Cambridge University Press, Cambridge, 1985, pp. 459--541.

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