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

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 supply and research of equipment for the advance of software program platforms of top of the range with correctness as a key factor. A approach is termed 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 opportunity of verification is greatly permitted. Algebraic specification is a software program engineering technique of this angle. while 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 truly profitable and nonetheless flourishing new quarter. within the overdue seventies, algebraic specification turned a huge examine subject additionally in lots of eu international locations. initially, the algebraic framework was once meant for the mathematical beginning of ab­ stract information varieties and the formal improvement of first-order applicative seasoned­ grams. in the meantime, the variety of functions has been prolonged to the fitting specification of entire 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 working courses. The actions within the final 25 years have resulted in an abundance of suggestions, equipment, methods, theories, languages and instruments, that are mathemati­ cally based in common algebra, classification thought and logic.

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Extra info for Algebraic Foundations of Systems Specification

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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.

Tn) I p: S1 x ... dX)ls,i = 1, ... ,n} 1 When trying to specify transitive closure in a purely functional style, we either get additional truth values in the free extension, or equate true and false. Adding a first-order axiom stating that there are exactly two truth values destroys the existence of free extensions. These problems arise because homomorphisms are not so flexible: they have to preserve both truth and falsehood. t'l 1\ ... 1\ t'n => t'n+! I t'i E At(E, X), i = 1, ... ,n + 1} <> In other words, conditional axioms are positive Horn-Clauses, built using the predicates in II and the equality symbol.

6 Higher-order functions Higher-order functions (which take functions as parameters and/or return functions as results) can be accommodated by interpreting certain sort names as (subsets of) function spaces. Given a set S of (base) sorts, let S-+ be the closure of S under formation of function types: S-+ is the smallest set such that S ~ S-+ and for all S1, ... ,Sn, S E S-+, S1 X •.. X Sn -t s E S-+. Then a higher-order signature E is a pair (S, n) where n is an S-+ -indexed set of operation names.

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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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