J. Meseguer and J. Goguen. Initiality, induction and computability. In Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....names in the context Gamma is arbitrary. It might seem surprising that the definition of jOBS does not make use of the higher order features of the language, except as a result of the way that equality is expressed via quantification over predicates. So jOBS is just the same as in e.g. SW83] [MG85], NO88] The reason for this choice is that the natural modification of the definition of jOBS to make use of higher order formulae (Definition 5.19) gives exactly the same relation, see Corollary 5.22. The following definition is the key to understanding the relationship between ....

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

....development and modern programming, but also in algebraic speci cation. Majster [32] suggested that algebraic speci cations are practically limited because certain algebras cannot be speci ed as an initial algebra of a nite set of equations, but later, Bergstra and Tucker [2] see also [33]) showed that in fact any computable algebra can be speci ed as the restriction of an initial 0 algebra of a nite set of 0 equations, for some nite 0 larger than . Therefore, there are some theories of interest that do not admit nite speci cations but are restrictions ....

J. Meseguer and J. Goguen. Initiality, induction and computability. In Algebraic Methods in Semantics, pages 459-541. Cambridge, 1985.

....for software is natural. Although complicated, some general results about correctness and term rewriting properties for importing are known. What makes this approach important is the strong theoretical foundation, laid using algebra and equational term rewriting (Goguen et al. 1978] Meseguer and Goguen [1985], Wirsing [1990] Loeckx et al. 1996] and the slowly maturing software tools such as OBJ (Goguen et al. 2000] ASF SDF (Bergstra et al. 1989] van Deursen et al. 1996] Maude (Clavel et al. 1996] The algebraic theory of data type specifications use signatures as interfaces. 2.1.2 ....

Meseguer and Goguen [1985] J Meseguer and J A Goguen. Initiality, induction and computability. In M Nivat, editor, Algebraic Methods in Semantics, chapter 14, pages 459--541. Cambridge University Press, 1985.

....et al. 1996; Didrich et al. 1994; Kamperman and Walters 1996] ffl Equational semantics is very well understood. A rich body of notions and results can be brought to bear to prove useful properties of equational systems, such as completeness, soundness, and relations between various models [Meseguer and Goguen 1985; Wirsing 1990] ffl Implementations and formal results used for one application of equational semantics (e.g. interpreters) can often be reused or extended with ease to other settings (e.g. partial evaluators) 2. Equational Interpreters 2 program do x : a; w : x; w : b; if w = x ....

....The idea is simple: if the result of inserting two distinct closed program fragments in all result yielding closed program contexts is the same, then the two fragments should be viewed as equivalent. The algebraic equivalent of observational equivalence is the notion of a final algebra [Meseguer and Goguen 1985, Sec. 5] Wirsing 1990, Sec. 5.4] One pleasant property of equational semantics is that observable equivalence can often be captured simply by enriching the equations that define an operational, or initial semantics, with additional equations that axiomatize observational equivalence. For P such ....

MESEGUER, J. AND GOGUEN, J. A. 1985. Initiality, induction and computability. In Algebraic Methods in Semantics, M. Nivat and J. C. Reynolds, Eds. Cambridge University Press, 459--541.

....is such that its theories have initial models, then an appropriate form of inductive reasoning is always valid when proving sentences with respect to the initial models of its theories. This method is very general; for example, for equational logic, induction and initiality are equivalent concepts [31]. 1 1 Since the notion of initiality is very general, the corresponding inductive reasoning principles may in each case take di erent forms. For example, in an equational logic allowing 5 We now explain why the requirements listed above are sucient for turning a logical framework into a ....

J. Meseguer and J. A. Goguen. Initiality, induction and computability. In M. Nivat and J. C. Reynolds, editors, Algebraic Methods in Semantics, pages 459-541. Cambridge University Press, 1985.

....are emphasized in our coalgebraic description of objects. But there is more to say, see the last section. The suitability of coalgebras for the description of object oriented features was recognized before, see e.g. 21, 20, 14, 15] Elements may be traced back to earlier sources, like [16] or [18], where the word coalgebra is not yet used (in [18] one finds the phrase abtract machine instead) In [11] the 2. Preliminaries 2 two level structure of specifications in the language COLD are explained: first there is a specification of one s application domain using algebraic data types, and ....

....(But there is more to say, see the last section. The suitability of coalgebras for the description of object oriented features was recognized before, see e.g. 21, 20, 14, 15] Elements may be traced back to earlier sources, like [16] or [18] where the word coalgebra is not yet used (in [18] one finds the phrase abtract machine instead) In [11] the 2. Preliminaries 2 two level structure of specifications in the language COLD are explained: first there is a specification of one s application domain using algebraic data types, and then there is the system description in terms of ....

J. Meseguer and J. Goguen. Initiality, induction and computability. In M. Nivat and J.C. Reynolds, editors, Algebraic Methods in Semantics, pages 459--541. Cambridge Univ. Press, 1985.

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Jose Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459-541. Cambridge, 1985.

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Jose Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.

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Jose Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985. 19

.... method and inheritance) with overloaded many sorted algebra, and with the notation of OBJ (for which see [25] In order to x notation, and also to set the stage for later developments, we brie y review some basics of overloaded many sorted algebra; further details may be found in [16] and [30]. We emphasise overloading because of its importance for the object paradigm. Many readers may wish to skip directly to Section 3. 2.1 Overloaded Many Sorted Algebra Given a sort set S, an S indexed (or sorted) set A is a family fA s j s 2 Sg of sets indexed by the elements of S. In this ....

....) for all appropriate contexts c whenever, for j = 1; m, for all appropriate contexts c j . As with unconditional equations, we write A j e. 2 The algebraic approach to the object paradigm is pre gured in work of Goguen and Meseguer on (what they then called) abstract machines [20, 30]; the hidden sorted algebra approach di ers from this mainly in its use of behavioural satisfaction for equations, an idea introduced by Reichel [32] in the context of partial algebras. Reichel [33] later introduced the related idea of behavioural equivalence for states, which we also use here ....

Jose Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459-541. Cambridge, 1985. 26

....or Hoare logics are not needed. The denotational semantics of OBJ is algebraic, as in the algebraic theory of abstract data types [84, 83, 153, 90] and in particular, the denotation of an OBJ object is an algebra, a collection of sets with functions among them . The initial algebra approach [83, 116] takes the unique (up to isomorphism) initial algebra as the standard, or most representative model of a set of equations (there may of course be many other models) i.e. as the representation independent standard of comparison for correctness. It is shown in [11] see also [116] that an ....

....approach [83, 116] takes the unique (up to isomorphism) initial algebra as the standard, or most representative model of a set of equations (there may of course be many other models) i.e. as the representation independent standard of comparison for correctness. It is shown in [11] see also [116]) that an algebra is initial if and only if it satisfies the following properties: 1. no junk: every element can be named using the given constant and operator symbols; and 2. no confusion: all ground equations true of the algebra can be proved from the given equations. For canonical systems ....

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Jos'e Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.

....that other modules might have used details of the implementation. Second, a classic result of Bergstra and Tucker [2] says that every computable algebra has a nite equational speci cation with some hidden operations, and examples show that the hidden operations are sometimes necessary (see [21] for a survey of this area) Third, 15] shows that every [ nite] behavioral (also called hidden) algebraic speci cation [12, 14, 26, 15, 24] has an equivalent [ nite] information hiding speci cation with the same models, but using ordinary satisfaction. Category theory and institutions are ....

Jose Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459-541. Cambridge, 1985.

....from ours, we do not survey it here; see [6] and its references. Space precludes many details, including proofs, some de nitions, and further examples; see the full technical report [19] Parts of the paper assume familiarity with basics of category theory and algebraic speci cation (e.g. [17, 21]) 2 Examples This section introduces parameterized programming through examples, beginning with the rst order case, and then higher order examples. Example 1 The simplest non trivial module has only a single sort: th TRIV is sort Elt . end The keyword th introduces theories, which ....

Jose Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459-541. Cambridge, 1985.

....worry that other modules might have used details of the implementation. Second, a classic result of Bergstra and Tucker [2] says that every computable algebra has a nite equational speci cation with some hidden operations, and examples show that the hidden operations are sometimes necessary (see [20] for a survey of this area) Third, 15] shows that every [ nite] behavioral (also called observational, or hidden) algebraic speci cation [14, 25, 15, 23] has an equivalent [ nite] information hiding speci cation with the same models, but using ordinary satisfaction. Category theory and ....

Jose Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459-541. Cambridge, 1985.

.... it still has initial models for all theories, as shown in work on the semantics of the language Eqlog, which combines functional and logic programming [21, 20] Yet another extension considers machines that may have internal states, identifying two such machines iff their behaviour is equivalent [19, 46]. This gives a semantics for abstract objects which generalises that of abstract data types, as in the semantics of FOOPS [23] An early attempt to handle state appears in Guttag s thesis [31] later formalised by Wand [60] using final algebras. However, approaches which admit a class of models, ....

....= T 0 [M] and v ; Phi = Phi 0 . 2 3 Object Oriented Concepts This section applies the types as theories viewpoint to object oriented specification and programming. Although algebra has already been applied to the object oriented paradigm in a variety of different ways (e.g. see [19, 46, 53, 23, 33, 8, 56]) the present approach is based on institutions. See [47] for a general discussion of object oriented programming. 3.1 Inheritance of Modules Many basic issues concerning inheritance are orthogonal to issues concerning object and state, because they have to do with importing one module into ....

Jos'e Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.

....This subsection shows that terms over regular signatures have a well defined least sort, and also that the standard MSA term algebra construction gives an initial order sorted algebra. We first review the inductive construction of the many sorted term algebra T Sigma using the same notation as in [57], except that we will be more pedantic, using ( and ) to denote parentheses used as formal syntactic symbols; however, this pedantry is only temporary. If Sigma is a many sorted signature with sort set S, then: ffl Sigma ;s T Sigma;s ; ffl if oe 2 Sigma w;s and if ti 2 T Sigma;si for i = ....

....w;s for w 6= It is easy to see that Sigma(X ) is regular if Sigma is. Now form T Sigma(X) and view it as an order sorted Sigma algebra just by forgetting about the constants in X; let us denote this algebra T Sigma (X) The following result and proof are entirely analogous to the MSA case [57]. Theorem 2.13 Given a regular order sorted signature (S; Sigma) let A be a Sigma algebra and let a: X A be an S sorted function; hereafter we call such a function an assignment. Then there is a unique order sorted Sigma homomorphism a : T Sigma (X) A such that a (x) a(x) for each ....

[Article contains additional citation context not shown here]

Jos'e Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.

....be observed directly, but only indirectly through the visible results of experiments, which consist of applying a sequence of methods and then examining an attribute. Hidden algebra originated in [19] extending earlier work of Goguen and Meseguer on (what they then called) abstract machines [33, 50], mainly through using behavioral satisfaction for equations, an idea introduced by Reichel [54] in the context of partial algebras. Reichel [55] later introduced behavioral equivalence for states, which is also used in hidden algebra. Goguen [19] showed that hidden algebra with some intuitive ....

....as for other comments, which have helped to improve the paper. 2 Prerequisites, Notation and Preliminaries We assume familiarity with many sorted algebra, but to establish notation, we will briefly review some main concepts and results. For compatible expositions with more detail, see [28] or [50]; this approach, based on indexed sets, originated in lectures by Joseph Goguen at the University of Chicago in 1968. Some of the examples in Sections 4.2 and 5 assume basic knowledge of term rewriting, such as confluence and termination. Introductions to term rewriting may be found in [9, 21] ....

Jos'e Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.

....1 Given a poset (S; let = denote the transitive and symmetric closure of . Then = is an equivalence relation whose equivalence classes are called the connected components of (S; 2 However, the reader should be aware that satisfaction of an equation depends crucially on its variable set [23]. 5 For conciseness, sometimes we use variations on the notation for equations: X; l; r; C) stands for (8X) l = r if C; l; r; C) when X = and simply l = r, if X = and C = 2.5 Order sorted Equational Deduction This section gives rules of deduction for OSA with conditional equations. ....

Jos'e Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.

....course, much of the interesting work could be automated, but this is certainly impossible in general. Some logical foundations for this approach are summarized in Appendix A. We assume familiarity with basic many sorted algebra and with OBJ. The relevant background appears in [44,37,58,61] and [78], among many other places. We try to avoid category theory, but in some cases (e.g, Section 3.9) its greater elegance and power are so compelling that we could not resist. Section 2 introduces hidden algebra, with Section 2.3 giving necessary and suf cient conditions for a speci cation to be ....

....of a xed universe of data values, and in its use of behavioral satisfaction. The founding hidden paper is [32] which builds on earlier algebraic work on abstract data types [26,58,59] and is a natural extension of prior work by Goguen and Meseguer on (what they then called) abstract machines [48,78]. Hidden algebra also generalizes automaton theory, which again has a long tradition in computing science 4 ; we would particularly mention the pioneering work of Eilenberg and Wright [23] which took a categorical approach to the tree automata generalization of state transition automata; tree ....

[Article contains additional citation context not shown here]

Jose Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459-541. Cambridge, 1985.

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J. Meseguer and J. Goguen. Initiality, induction and computability. In Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.

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J. Meseguer and J. Goguen. Initiality, induction and computability. In Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.

No context found.

J. Meseguer and J. Goguen. Initiality, induction and computability. In M. Nivat and J. Reynolds, editors, Algebraic Methods in Semantics, pages 459--540. Cambridge, 1985.

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

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Jose Meseguer and Joseph Goguen. Initiality, induction and computability. In Maurice Nivat and John Reynolds, editors, Algebraic Methods in Semantics, pages 459--541. Cambridge, 1985.

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MESEGUER, J., AND GOGUEN, J. Initiality, induction and computability. In Algebraic Methods in Semantics, M. Nivat and J. Reynolds, Eds. Cambridge University Press, 1985, pp. 459--541.

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