M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, 18(1):47--64, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....supporting notions of subtype (subsort) so that partial functions then appear as appropriate restrictions to given subtypes of corresponding total functions. Various explicitly partial speci cation formalisms were explored from the early days of algebraic speci cation by di erent authors, e.g. [13, 3] (see [1, 9] for surveys and bibliographies on equational speci cation, including partial approaches) Reichel s book [20] gives a systematic study of algebraic speci cation in an explicitly partial way using existence equations. The comparison between explicitly partial formalisms has been ....

M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, 18:47-64, 1982.

.... and on the experience gained in the project CIP [BBB 85] However, in contrast to most algebraic specification languages (like e.g. OBJ [JKKM88] LARCH [GHW85] ACT ONE [EM85] OBSCURE [LL88] and ACT TWO [EM90] it contains explicit support for partial functions (as a generalization of [BW82] Moreover, Spectrum is not restricted to equational or conditional equational axioms, since it does not primarily aim at executable specifications. Spectrum does not feature any built in notion of a program state, and the whole specification style is oriented towards functional programming. As ....

M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, 18:47--64, 1982.

.... the implicit extensionality axiom f, g ( x: apply(f,x) apply(g,x) f = g) The extensionality axiom differs from the usual positive conditional axioms (the most frequently used in algebraic approaches, being the widest class guaranteeing the existence of initial (free) models, see e.g. [5,6,17,18,19]) because of the internal quantification, which only involves the premises, and since the equality f(x) g(x) in the premises is strong, ie it holds iff either both sides are undefined or both denote the same element of the algebra, and not existential, ie holding iff both sides are defined and ....

....functions. We recall the notions of signature, partial algebra and evaluation of terms in appendix (see anyway [1,5,17] Partial conditional specifications. Let S = S,F) be a signature; the homomorphisms, as quite standard in the initial approach (see e.g. 5] and total S homomorphisms in [6]) are chosen in a way that the initial model, if any, satisfies the conditions of no junk and no confusion; let us recall the definition. Let A and B be two S algebras and p be a family of total functions p = p s : s A s B sS . Then p is a homomorphism from A into B iff for any opF (s 1 ....

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M. Broy and M. Wirsing, Partial abstract types, Acta Informatica 18 (1982) 47-64. 24

....with the system. Finally, applying our general results to the case of higher order specifications with positive conditional axioms, we obtain necessary and sufficient conditions for the existence of term generated initial models in that case. 0 An overview Partial algebraic specifications ([BW1, R, WB]) are one of the most interesting specification paradigms. Originally proposed as a support to the stepwise refinement procedure, they have found more recently interesting applications to the specification of concurrency and of formal semantics of languages (see eg [BW2, AR1, AR2] Higher order ....

....requirement. The most striking feature is that the existence of initial model is not guaranteed, even if the axioms are in a form which looks a natural extension of the one that guarantees the existence in the first order case. Let us consider this problem in some more detail. A classical result [BW1] states that the initial partial model exists (and is term generated) in the class of all partial models, if the specification is positive conditional, i.e. the axioms have the form i=1, n t i = e t i t = t where = e and = denote respectively existential (the sides are both defined ....

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Broy, M.; Wirsing, M. "Partial abstract types", Acta Informatica 18 (1982), 47-64.

....(b) b(2) if either b(1) t or b(2) t then B (b) t, if b(2) f then B (b) b(1) otherwise B (b) is undefined. Depending on the partiality of the functions, there are several possibilities to define homomorphisms, each one of them is useful for a different purpose (see e.g. [BW1, B, R]) in a way that the algebraic structure is preserved. Our choice follows the tradition of partial algebras (see e.g. B, BW1, T, AC1] where they are used in order to get a no junk no confusion initial object (see [MG] Def. 1.1.3. Let S = S,F) be a signature, A and B be non strict algebras ....

....Depending on the partiality of the functions, there are several possibilities to define homomorphisms, each one of them is useful for a different purpose (see e.g. BW1, B, R] in a way that the algebraic structure is preserved. Our choice follows the tradition of partial algebras (see e.g. [B, BW1, T, AC1]) where they are used in order to get a no junk no confusion initial object (see [MG] Def. 1.1.3. Let S = S,F) be a signature, A and B be non strict algebras over S. Then a homomorphism h: A B is a family h s : s A s B sS of total functions s.t. f A (a)s A implies h s (f A ....

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Broy M.; Wirsing M. "Partial abstract types", Acta Informatica 18 (1982).

....misbeliefs. Semantic Level. First we analyse the relationship between partial and total frames from a semantic point of view, i.e. disregarding their logics. Formally this means that we are working on institutions without sentences. Let us recall the basic ingredients of the partial frame (see [6, 7, 20]) Def. 2.4 A partial algebra A on a signature Sigma = S; F ) consists of a family fs A g s2S of sets and of a family ff A g f2Fw;s of partial functions s.t. if w = then either f A is undefined or f A 2 s A , else w = s 1 . s n and f A : s A 1 Theta . Theta s A n ....

M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, (18):47--64, 1982.

....equalities and e is an equality too, and their models classes are closed under subobjects and isomorphisms. Like the higher order ones, in general partial conditional specifications do not admit initial models, contrary to the classical case ( considered, e.g. in the work of Broy and Wirsing [6]; see also Burmeister [5] Reichel [12] of positive conditional specifications, ie those whose axioms have only existential equalities in the premises. In [1] we investigated the problem and gave necessary and sufficient conditions for the existence of initial models, related them to logical ....

....[7, 14, 3, 4] Hence we briefly collect in appendix some basic notions about partial specifications just in order to fix the notation; thus the reader can look at the appendix whenever some notations are not clear. A more ample presentation of the partial algebraic framework can be found e.g. in [5,6,13]. 1 Sound logical deduction for conditional specifications In the following when referring to generic formulas and inference systems we consider formulas and inference systems within an infinitary logic which extends first order logic by admitting countable conjunctions ( disjunctions) and ....

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Broy, M.; Wirsing, M. "Partial abstract types", Acta Informatica 18 (1982), 47-64.

....2 The condition that the axioms in SP2 have observable premises is essential here; there are well known examples of conditional equational specifications with non observable premises which do not have a terminal model as used in the above proof. The same assumption appears in [GM 82] and [BW 82] we can see no way to avoid this either. An important special case of the situation when SP2 is given as a list of conditional equations with observable premises is when there are no premises at all, i.e. when SP2 is given as a list of equations. Corollary 3.6 Consider two specifications SP1 ....

Broy, M. and Wirsing, M. Partial abstract types. Acta Informatica 18 pp. 47--64 (1982).

....which preserve the abstraction equivalence (extended to specifications) This turns out to be quite satisfactory in practice and an example shows that a stronger result cannot really be expected. We present the above ideas in the framework of partial algebras (with first order formulae) BrW 82] This is mainly to take advantage of the reader s intuition, since all of the main definitions and results as well as methodological remarks may be directly restated in the framework of an arbitrary institution [GB 84a] This means that they can be used to develop programs from specifications ....

....initial object, pushout, pullback, functor etc. which we omit here. 2 Algebraic preliminaries Most of the following definitions are more or less standard so we give them without comment or motivation; for a more detailed presentation see [GTW 76] BG 82] EM 85] for total algebras and [BrW 82] Bur 86] for partial algebras. Notation Throughout this paper we deal with many sorted sets, functions, relations, etc. for any set S, an S sorted set is just a family X = fX s g s2S of sets indexed by S, and similarly for functions, relations, etc. We will feel free to use standard ....

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Broy, M. and Wirsing, M. Partial abstract types. Acta Informatica 18 pp. 47-64 (1982).

....Dom( E ) described by some authors [Ger78, CH79, Bli81] However, they have a different purpose, since an operator, such as choose on nonempty sets, may be underspecified for a reason other than being partial. They also resemble the definedness predicate (D) used in studies of partial algebras [BW82] and in COLD [FJ92] however D is defined model theoretically, not syntactically. The definition of Exact( Delta ) is based on the exact clauses given in the trait s implications and those of included traits. This definition is lifted to arbitrary terms by requiring terms substituted for the ....

Manfred Broy and Martin Wirsing. Partial abstract types. Acta Informatica, 18(1):47--64, November 1982.

....predicates (see Theorem 10.8) in other words, for SP to be behaviourally consistent in the sense of [BHW95] In particular, 4) implies that Horn axiom premises do not contain hidden equations. This condition is already present in previous approaches to behavioural specifications (see, e.g. BW82] Corollary 4; Wir90] Theorem 5.4.5; BHW95] Example 3.24) Since we will reason about SP with respect to the final model, which is the quotient by bisimilarity of the initial model (see Section 10.3) behavioural consistency is crucial. 4) and (5) also ensure that liveness predicates can be ....

M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, 18(1):47--64, 1982.

.... could be allowed by changing the notion of an algebra (so that functions associated with operation names are not required to be total) and extending axioms and the definition of the satisfaction of an axiom by an algebra so as to provide some way of specifying the domains of functions (see [BW 82] In fact, there is no need to choose a particular fixed logical system. It is possible to parameterise Extended ML by an arbitrary logical system, or institution [GB 84] An institution comprises definitions of signature, model (algebra) sentence (axiom) and a satisfaction relation subject to ....

....types and assignment. By providing an institutionindependent semantics for Extended ML here, we make it possible in principle to remove these restrictions by plugging in an appropriate institution with syntax. For example, an institution permitting the use of partial functions based on [BW 82] is described in [ST 85c] and it is obvious how to extend this to an institution with syntax. Moreover, it is more or less apparent that Extended ML instantiated to this institution with syntax would allow one to treat partial functions in a satisfactory way. However, the situation with ....

Broy, M. and Wirsing, M. Partial abstract types. Acta Informatica 18 pp. 47-64.

....earlier precedents, such as [16] 21] and [5] quite many recent papers (as e.g. 2] 24] 17] 14] 19] 13] have proposed partial versions of first order predicate logic aiming at this field of application. Partiality has been also investigated within the field of data type specification [3], 20] These works show quite different views of partiality, semantics and proof calculi. Some of them adopt a third truth value (undefined) which has been argued to be useful in certain cases for reasoning about refinements and implementations of specifications (see [24] 13] 3 . In a ....

M. Broy, M. Wirsing. Partial Abstract Types. Acta Informatica 18, 47-64, 1982.

....because well definedness of terms cannot be statically checked. The problem of partiality has been tackled pursuing different principles and, probably, different goals. A considerable effort has been made on partial models, where operations are partial and errors are represented by undefinedness ([7, 46, 1] among many others) Another research direction focused on many sorted total models with error elements (see [14, 12, 4, 3] among others) sometimes without success. Direction which evolved towards order sorted techniques (see section 5 and 6) which are more elegant and more efficient. 3 ....

Manfred Broy and Martin Wirsing. Partial abstract types. Acta Informatics, 18:47--64, 1982.

....predicates, Dom( E ) described by some authors [12, 9, 5] However, they have a different purpose, since an operator, such as choose on nonempty sets, may be underspecified for a reason other than being partial. They also resemble the definedness predicate (D) used in studies of partial algebras [7] and in COLD [10] however D is defined model theoretically, not syntactically. The definition of Exact( Delta ) is based on the exact clauses given in the trait s implications and those of included traits. This definition is lifted to arbitrary terms by requiring terms substituted for the ....

M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, 18(1):47--64, Nov. 1982.

....which still have to be worked out, FOOPS has the potential of expressing all object oriented concepts and constructions as outlined in section 2. 3.2 Entity Algebras Entity algebras [Re91] provide models for data types, processes, and objects. They are special partial algebras with predicates [BW82] including the following features: ffl The elements of certain sorts represent dynamic elements, called entities. Their dynamics is realized by allowing to perform labelled transitions. ffl Entities may have subcomponents. The structure of these subcomponent entities is not fixed but can be ....

Broy,M.;Wirsing,M.: Partial abstract types. Acta Informatica 18, 1982, 47--64

....: T : E ) 8 x : T : Exact( E ) Exact( if E 1 then E 2 else E 3 ) Exact( E 1 ) Exact( E 2 ) Exact( E 3 ) Exact( E ) false, otherwise Figure 10: Definition of Exact. other than being partial. They also resemble the definedness predicate (D) used in studies of partial algebras [7] and in COLD [10] however D is defined model theoretically, not syntactically. The definition of Exact( Delta ) is based on the exact clauses given in the trait s implications (and those of included traits) This definition is lifted to arbitrary terms by requiring terms substituted for the ....

M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, 18(1):47--64, Nov. 1982.

....predicates, Dom( E ) described by some authors [11, 8, 5] However, they have a different purpose, since an operator, such as choose on nonempty sets, may be underspecified for a reason other than being partial. They also resemble the definedness predicate (D) used in studies of partial algebras [6] and in COLD [9] however D is defined model theoretically, not syntactically, and is not incompletely extended to boolean terms. The definition of Exact( Delta ) is based on the exact clauses given in the trait s implications (and those of included traits) This definition is lifted to ....

Manfred Broy and Martin Wirsing. Partial abstract types. Acta Informatica, 18(1):47--64, November 1982.

....any, s.t. ji A;B j = i jAj;jBj . If Sigma , Sigma 0 and A is a Sigma 0 algebra, then we write A j Sigma instead of A ji Sigma; Sigma 0 . Assume in what follows a fixed static framework ST = Sig; Sorts; Alg; j Gammaj . In the examples, ST is the static framework of partial algebras (see [7]) In the following, if S is a set, then [S] denotes S [ fg and [s] stands for either an element of S or for the empty string. Definition10. If S is a set of sorts, then a method signature M Sigma over S is a (S Theta[S] family of symbols called method symbols; if m 2 M Sigma w; s] then ....

M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, 18, 1982.

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M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, 18(1):47--64, 1982.

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M. Broy and M. Wirsing. Partial abstract types. Acta Informatica, 18, 1982.

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M. Broy and M. Wirsing. Partial Abstract Types. Acta Informatica, 18:47--64, 1982.

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M. Broy, M. Wirsing, Partial Abstract Types, Acta Informatica 18 (1982) 47-64

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BW Broy, M.; Wirsing, M. "Partial abstract types", Acta Informatica 18, 1982.

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M. Broy, M. Wirsing. Partial abstract types. Acta Informatica 18, pp.47-64, 1982.

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