Algebraic Foundations of Systems Specification by Marie-Claude Gaudel, Gilles Bernot (auth.), Prof. Dr. Egidio

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

The target of software program engineering is the availability and research of equipment for the advance of software program platforms of top of the range with correctness as a key factor. A method is named right if it does what one wishes, if it meets the necessities. to accomplish and to assure right structures, the necessity of formal tools with rigorous semantics and the potential of verification is generally permitted. Algebraic specification is a software program engineering procedure of this angle. whilst Liskov and Zilles, Guttag and the ADJ-group with Goguen, Thatch­ er, Wagner and Wright brought the elemental principles of algebraic specification within the mid seventies within the U. S. A. and Canada, they initiated a really profitable and nonetheless flourishing new sector. within the overdue seventies, algebraic specification grew to become an incredible learn subject additionally in lots of ecu international locations. initially, the algebraic framework was once meant for the mathematical starting place of ab­ stract info varieties and the formal improvement of first-order applicative professional­ grams. in the meantime, the diversity of purposes has been prolonged to the perfect specification of whole software program platforms, the uniform definition of syntax and semantics of programming languages, and to the stepwise improvement of right structures from the requirement definitions to the operating courses. The actions within the final 25 years have resulted in an abundance of thoughts, equipment, methods, theories, languages and instruments, that are mathemati­ cally based in common algebra, class conception and logic.

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Let us look at an incremental use of conditional specifications to describe a data type as the initial model of a presentation. 15. Let us consider the problem of the definition of a very primitive dynamic data type that is a subset of the CCS language. The Calculus of Communicating Systems has been introduced to study in isolation the problems due to concurrency and to describe reactive systems. It provides primitives to express the interactions between the components of complex systems. The intuition is that there is a set of agents that can perform actions either individually or in cooperation with each other.

False}, ... }. 23. Bool f-t O} • Taking a presentation to denote the (non-empty) class of its initial models is called taking its initial semantics. The initiality property identifies a model of (17, iP) up to isomorphism: any two initial models are isomorphic, and any model isomorphic to an initial model is itself initial. We therefore refer to the initial model of a presentation. 25. 14) since it is isomorphic to TEBool/=

Tn ) = f(t~, ... ,t~) Sl X ... t=t' VY. 20. 14. The following derivation proves iPBool f-EBool Vp:bool. ,(p /\ ,false) = ,p: ,false = true Vp:bool. p = p Vp:bool. ,false = true Vp:bool. p /\ ,false = p /\ true Vp:bool. p /\ true = p Vp:bool. ,(p/\ ,false) = ,(p/\ true) Vp:bool. ,(p/\ true) = ,p Vp:bool. ,(p /\ ,false) = 'p • As mentioned above, f- E is both sound (only valid consequences may be derived) and complete (all valid consequences may be derived) for FE. 21. For any set iP of E -equations and any E -equation cp, iP f- E cp if and only if iP FE cpo Simplifying the above calculus by omitting explicit quantifiers in equations yields an unsound system because algebras may have empty carrier sets.

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