J. Bergstra, J. Heering, and P. Klint. Module algebra. JACM, 37(2):335--372, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....presented 0 theories for some 0 . Diaconescu, Goguen, Stefaneas [14] and Ro su [41] present logic paradigm independent (or institutional [18] approaches to information hiding and integration of it with other operations on modules. Work on module algebra by Bergstra, Heering and Klint [1] also investigates information hiding formally. Behavioral abstraction is another development in algebraic speci cation which appears under various names in the literature such as hidden algebra in works by Goguen, Diaconescu and many others [17, 19, 23, 22, 42, 26] observational logic in works ....

J. Bergstra, J. Heering, and P. Klint. Module algebra. JACM, 37(2):335-372, 1990.

....(d 0 ) 2 ) then there exists 2 SenINS ( Sigma ) such that: 1 j= INS Sigma 1 SenINS (d) and SenINS (t) j= INS Sigma 2 2 : 2 Of course, the interpolation property does not hold in every institution. For instance, it does not hold for institution EQ of equational logic (see [BHK 90] However, it does hold for some richer institutions like e.g. institution FOLEQ of the first order logic with equality (see [BM 86] Definition 8 (D; T ) amalgamation property. A (D; T ) institution INS satisfies (D; T ) amalgamation property iff for any d; d 0 2 D and t; t 0 2 T that ....

....rules presented above. The above entailment relation is sound wrt the semantical consequence relation. The proof follows directly from the proof of soundness presented in [ST 88] Now, to prove its completeness we need some more notions. Similar definitions were presented in [Cen 94] cf. also [BHK 90] Definition 12 Normal form. We say that the specification SP is in the normal form if it has a form: derive h Sigma ; Phii by d; where (d : Sig[SP] Sigma ) 2 D and Phi SenINS ( Sigma ) 2 Definition 13. Specifications SP 1 and SP 2 are equivalent (written SP 1 = SP 2 ) if Sig[SP 1 ] ....

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J. A. Bergstra, J. Heering, P. Klint. Module algebra. Journal of the ACM, 37(2):335-372, April 1990.

....in an algebraic fashion. 1. 1 Relation with Other Work Operators on modules and their semantics have been studied in e.g. 1] in the context of CLEAR) 5] in the context of PLUSS) 3] and [4] using category theory) 16] using model class semantics) Our main source of inspiration has been [2], where the approach is similar to Wirsing s in [16] extended to theory semantics and countable model semantics. The role of the interpolation theorem for the theory semantics of import and export has been pointed out in [7] Besides giving a survey of logical aspects of modularisation, this ....

.... of a renamed signature element is the renaming of the type of that signature element) Observe that we are liberal in the de nition of renamings in the sense that they need not be bijective, so e.g. P : R; Q : R] is a correct renaming (if P , Q and R have the same type) this is in contrast to [2], where renamings are bijective and even involutive (i.e. if (P ) Q then (Q) P ) We de ne domain and range of a renaming by dom( def fI j (I) 6= Ig rg( def f (I) j (I) 6= Ig ( dom( All renamings are nitely generated, so domain and range of a renaming are nite and hence ....

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J.A. Bergstra, J. Heering, P. Klint, Module algebra, Journal of the ACM 37 (1990) 335-372

.... i 2 Sen I ( i ) for i = 1; 2, if Sen I (t 0 ) 1 ) j= I 0 Sen I (d 0 ) 2 ) then there exists Sen I ( such that: 1 j= I 1 Sen I (d) and Sen I (t) j= I 2 2 : 2 A characterization of above interpolation properties in terms of a module algebra can be found in [BHK 90] and also in [DGS 93] Lemma 2.17 If the (D; T ) institution I has in nite conjunction and satis es the weak (D; T ) interpolation property then it also satis es (D; T ) interpolation property. 2 Example 2.18 The (D; T ) institution EQ where DEQ is the class of signature inclusions and TEQ ....

....(see [RG 88] but not the (D; T ) interpolation property, whereas the (D; T ) institution FOEQ where DFOEQ is the class of signature inclusions and TFOEQ is the class of signature injections satis es both the interpolation properties. The above facts follow from the arguments presented in [BHK 90] 2 Remark 2.19 It follows from Example 2.18 and Lemma 2.17 that the (D; T ) institution EQ from the above example does not have in nite conjunction in the sense of De nition 2.6 (which is obvious anyway) 2 De nition 2.20 (Compactness) The institution I is compact i for any sentence 2 ....

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J. A. Bergstra, J. Heering, P. Klint. Module algebra. Journal of the ACM, 37(2):335-372, April 1990.

....what we propose here could be applied to most existing algebraic specification languages (e.g. OBJ3 [13] or Larch [9] 1 We only present here the generalization and instantiation operators. Other operators have been defined in [17] and are similar to many others found in the literature (e.g. [19, 20, 3]) The syntax of our specifications is based on multi sorted signatures ( Sigma = S; Op) Sigma terms (noted by small s; t) and Sigma equations (noted by small e) Equations may be simple (t 1 = t 2 ) or conditional (t 1 = t 2 if e 1 : e n ) The semantics of our specifications are ....

J.A. Bergstra, J. Heering, and R. Klint. Module algebra. JACM, 37(2):335--372, 1990.

....Roughly speaking, this property states that models of given signatures can be combined to yield a uniquely determined model of a compound signature, provided that the original models are mutually compatible. The amalgamation property allows the computation of normal forms for 2 speci cations [5, 7]; it is a prerequisite for good behaviour w.r.t. parametrization [14] and conservative extensions [12] The combination of implementations in the semantics of architectural speci cations crucially depends on amalgamation [26] The proof system for development graphs with hiding [21] which allow a ....

J.A. Bergstra, J. Heering, and P. Klint, Module algebra, J. ACM 37 (1990), no. 2, 335-372.

....the simple notion of importing in the current model. The extensive literature on abstract data types and wide spectrum algebraic specification languages provides a rich source of motivation for other concrete models of IDLs and extensions to our general framework. Examples of sources include Bergstra et al. 1990], Ehrig and Mahr [1990] Goguen and Tracz [2000] Loeckx et al. 1996] Bidoit et al. 1999] Mosses [1999] A prototype of the specification has been developed in Prolog which has been used to test the axiomatisation. A next step is to use the general model of Section 3 to design a metalanguage ....

Bergstra et al. [1990] J Bergstra, P Heering, and P Klint. Module algebra. Journal of the ACM, 37(2):335--372, 1990.

....designers of PROLOG systems for a reject from PROLOG developpers of such techniques. Nevertheless, we think that it is essential for PROLOG today to endow it with reliable and well proven development techniques for large software. Among these, we are dealing here with modularity and genericity ([4], 5] 11] These techniques contribute to significantly improve decomposition, abstraction, reusability and maintenance for large software, therefore their quality. For some years, the concept of modularity has been introduced in PROLOG to come up to a proposal of standardization from ISO. ....

....using the mapping f. Using the predicate = 2 we write map(Mapping, map(Mapping, E1 L1] E2 L2] Goal = Mapping, E1, E2] call(Goal) map(Mapping, L1, L2) We can use map to increment all elements of a list : inc(X,Y) Y is X 1. map(inc, 1, 2, 3] L2) L2 = [2, 3, 4] In fact, to use genericity in logic programming in a strict way, at the present time, it is better to use languages based on a higher order logic than the one of PROLOG (see for example HiLog [7] Godel [10] Lambda Prolog [14] but this leads to a completely different class of languages that ....

J.A. Bergstra, J. Heering, and P. Klint. Module Algebra. JACM, 37(2):335--372, April 1990.

....skeleton of a general construction and indicates that a potentially large class of entailments can be extended so that a uniform presentation of the interpolants is available. 1 Introduction There is a well established relation between interpolation [8] and modularity properties of refinements [23, 24, 29, 5, 32, 40, 39, 12, 11] and databases [25] On the other hand, many logics that have been used in refinement or databases lack the desirable interpolation properties. To compensate for this inadequacy, several groups of researchers have proposed techniques to restrict these logics to fragments that have the desirable ....

J.A. Bergstra, J. Heering, and P. Klint. Module algebra. ACM, 37(2):335--372, 1990.

....available and the manipulation of uniform schemata is supported. 1 Introduction There is a well established relation between interpolation [7, 25, 40, 2] and modularity properties of refinement [26, 27, 53, 50, 52, 51, 15, 14, 13] and databases [29] Also, some operations of module algebras [5] are linked directly with a splitting version of interpolation ( 37] discussing an earlier version of [5] The behaviour of the language restriction operator [14, 13] also information hiding operator in [9] is associated with an interpolation property of the specification formalism. In ....

.... relation between interpolation [7, 25, 40, 2] and modularity properties of refinement [26, 27, 53, 50, 52, 51, 15, 14, 13] and databases [29] Also, some operations of module algebras [5] are linked directly with a splitting version of interpolation ( 37] discussing an earlier version of [5]) The behaviour of the language restriction operator [14, 13] also information hiding operator in [9] is associated with an interpolation property of the specification formalism. In addition, the presence of uniform interpolants facilitates formal reasoning and syntactic manipulations in all ....

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J.A. Bergstra, J. Heering, and P. Klint. Module algebra. ACM, 37(2):335--372, 1990.

....a Development Workspace on Espec . 1 Introduction There is a well established relation between interpolation [6, 21, 31, 2] and modularity properties of refinement via implementations (e.g. 12, 11, 10] and [22, 23, 41, 38, 40, 39] and databases [24] Also, some operations of module algebras [4] have been linked directly with a splitting version of interpolation ( 28] discussing an earlier version of [4] and the behaviour of the language restriction (also information hiding in [8] is associated in [11, 13, 10] with an interpolation property of the specification formalism. ....

.... [6, 21, 31, 2] and modularity properties of refinement via implementations (e.g. 12, 11, 10] and [22, 23, 41, 38, 40, 39] and databases [24] Also, some operations of module algebras [4] have been linked directly with a splitting version of interpolation ( 28] discussing an earlier version of [4]) and the behaviour of the language restriction (also information hiding in [8] is associated in [11, 13, 10] with an interpolation property of the specification formalism. Furthermore, the presence of uniform interpolants facilitates formal reasoning and syntactic manipulations in all these ....

J.A. Bergstra, J. Heering, and P. Klint. Module algebra. ACM, 37(2):335--372, 1990.

.... Interpolation and Modularisation There is a well established relation between splitting versions of interpolation [7, 21, 31, 2] and modularity properties of refinement via implementations, e.g. 11, 10, 9] and [22, 41, 38, 40, 39] and databases [24] Also, some operations of module algebras [6] have been linked directly with a splitting version of interpolation ( 29] discussing an earlier version of [6] and the behaviour of language restriction ( information hiding in [8] is associated in [10, 12, 9] with an interpolation property of the specification formalism. Furthermore, the ....

.... [7, 21, 31, 2] and modularity properties of refinement via implementations, e.g. 11, 10, 9] and [22, 41, 38, 40, 39] and databases [24] Also, some operations of module algebras [6] have been linked directly with a splitting version of interpolation ( 29] discussing an earlier version of [6]) and the behaviour of language restriction ( information hiding in [8] is associated in [10, 12, 9] with an interpolation property of the specification formalism. Furthermore, the presence of uniform interpolants facilitates formal reasoning and syntactic manipulations in all these cases (as ....

J.A. Bergstra, J. Heering, and P. Klint. Module algebra. ACM, 37(2):335--372, 1990.

....to the Craig Interpolation Lemma. For a further analysis including a detailed proof (for the classical first order case) the reader is referred to [42] and [43] Some years later the important role of the Craig Interpolation property was independently observed by a group working with Bergstra [4], who attributed the lack of certain modularity properties in their formalisms to the absence of the interpolation property. Further work on this, revealed an important observation [32] For other logics, such as (conditional) equational logic, the various formulations of Craig interpolation are ....

J. Bergstra, J. Heering, and P. Klint. Module algebra. ACM, 37(2):335--372, 1990.

.... generally, faithful morphisms under pushouts (amalgamation of conservative extensions) 34, 33, 10, 9] The latter provides the formal basis for some fundamental modularity properties of refinement [21, 22, 35, 32, 34] 33, 10, 9, 8] and databases [24] while some operations of module algebras [2] have been linked directly with a similar form of interpolation ( 27] discussing an earlier version of [2] The paper is structured as follows: In section 2 we review the basic concepts involved and the relevant previous work on this subject. The generalised form of the interpolation property we ....

.... The latter provides the formal basis for some fundamental modularity properties of refinement [21, 22, 35, 32, 34] 33, 10, 9, 8] and databases [24] while some operations of module algebras [2] have been linked directly with a similar form of interpolation ( 27] discussing an earlier version of [2]) The paper is structured as follows: In section 2 we review the basic concepts involved and the relevant previous work on this subject. The generalised form of the interpolation property we are interested in, called CRI, is highlighted in section 3, where we also explain why CRI is in general ....

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J.A. Bergstra, J. Heering, and P. Klint. Module algebra. ACM, 37(2):335--372, 1990.

....6 Concluding Remarks The concept of renaming has been widely studied and employed in mathematics and computer science for di erent purposes, including program composition. For instance, adopting an algebraic approach for de ning the semantics of module and module compositions, Bergstra et al. [3] introduced a renaming operator for combining di erent signatures. However, the kind of renaming they proposed is permutative , that is, if a is replaced by b, then b is replaced by a. This can be useful for avoiding name clashes, but not for purposes such as increasing or decreasing the ....

J. A. Bergstra, J. Heering, and P. Klint. Module Algebra. Journal of the ACM, 37(2):335-372, 1990.

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J.A. Bergstra, J. Heering, and P. Klint. Module Algebra (revised version).

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J. Bergstra, J. Heering, and P. Klint. Module algebra. JACM, 37(2):335--372, 1990.

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J. Bergstra, J. Heering, and P. Klint. Module algebra. Journal of the ACM, 37(2):335--372, 1990.

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Bergstra, J., Heering, J., and Klint, P. (1990). Module algebra. Journal of the Association for Computing Machinery, 37:335--372.

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J. Bergstra, J. Heering, and P. Klint. Module algebra. J. ACM, 37(2):335-372, 1990.

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J. A. Bergstra, J. Heering, and P. Klint. Module Algebra. Journal of the ACM, 37(2):335--372, 1990.

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J.A. Bergstra, J. Heering, and P. Klint. Module algebra. volume 37, pages 335-372, 1990.

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J.A. Bergstra, J. Heering, and P. Klint. Module algebra. volume 37, pages 335--372, 1990.

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J. A. Bergstra, J. Heering, P. Klint. Module algebra. Journal of the ACM, 37(2):335-372, April 1990.

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J. A. Bergstra, J. Heering, P. Klint. Module algebra. Journal of the ACM, 37(2):335-372, April 1990.

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