By Marie-Claude Gaudel, Gilles Bernot (auth.), Prof. Dr. Egidio Astesiano, Prof. Dr. Hans-Jörg Kreowski, Prof. Dr. Bernd Krieg-Brückner (eds.)

ISBN-10: 364259851X

ISBN-13: 9783642598517

ISBN-10: 3642641512

ISBN-13: 9783642641510

The goal of software program engineering is the availability and research of tools for the improvement of software program structures of top of the range with correctness as a key factor. A method is named right if it does what one desires, if it meets the necessities. to accomplish and to assure right platforms, the necessity of formal tools with rigorous semantics and the opportunity of verification is commonly authorized. Algebraic specification is a software program engineering technique of this angle. whilst Liskov and Zilles, Guttag and the ADJ-group with Goguen, Thatch er, Wagner and Wright brought the fundamental rules of algebraic specification within the mid seventies within the U. S. A. and Canada, they initiated a really winning and nonetheless flourishing new sector. within the overdue seventies, algebraic specification turned an incredible examine subject additionally in lots of ecu international locations. initially, the algebraic framework used to be meant for the mathematical starting place of ab stract facts kinds and the formal improvement of first-order applicative seasoned grams. in the meantime, the variety of purposes has been prolonged to the appropriate specification of whole software program platforms, the uniform definition of syntax and semantics of programming languages, and to the stepwise improvement of right platforms from the requirement definitions to the working courses. The actions within the final 25 years have resulted in an abundance of strategies, equipment, ways, theories, languages and instruments, that are mathemati cally based in common algebra, type idea and logic.

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2a. 2b. 3a. 3b. Donald Sannella and Andrzej Tarlecki If ~ ~ I[! ) ~ ModE(~)' If A ~ B then ThE(B) ~ ThE(A). ~ ~ ThE(ModE(~))' ~ ModE(ThE(A)). ModE(~) ModE(ThE(ModE(~)))' A ThE(A) = = ThE(ModE(ThE(A))). A E-equation cp is a semantic (or model-theoretic) consequence of a set ~ of E-equations, written ~ ~E cp, if cp E CIE(~) (equivalently, if ModE(~) ~E cp). We will write ~ ~ cp instead of ~ ~ E cp where the signature 17 is obvious. 16. Semantic consequence is preserved by translation along signature morphisms: for any signature morphism u: 17 --+ 17', set ~ of 17equations, and 17 -equation cp, if~ ~E cp then u(~) ~E' u(cp).

The most straightforward is to require each algebra A to incorporate a (single-sorted) algebra of carriers, Carr (A) , having sets which interpret types as values and an operation to interpret each type constructor. Then, for each operation f E (}tl ... tn,t and for each instantiation of type variables i: V -t ICarr (A) I, A has to provide a function fA,i: i#(td x ... x i#(t n ) -t i#(t). Various conditions may be imposed to ensure that the interpretation of polymorphic operations is parametric, by requiring f A,i and f A,i' to be appropriately related for different type variable instantiations i, il.

3 Error algebras To model operations that may produce erroneous or exceptional results, we can partition each of the carrier sets of an algebra into an error part and an OK part. Operations in signatures are classed as safe or unsafe, where the former are required to yield OK values when applied to OK arguments. Homomorphisms are required to preserve OK-ness. Like operations, variables in equations are classed as safe or unsafe; the former range over OK values only, while the latter range over all values.

### Algebraic Foundations of Systems Specification by Marie-Claude Gaudel, Gilles Bernot (auth.), Prof. Dr. Egidio Astesiano, Prof. Dr. Hans-Jörg Kreowski, Prof. Dr. Bernd Krieg-Brückner (eds.)

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