By Joseph A. Goguen, Grant Malcolm

Algebraic Semantics of significant courses provides a self-contained and novel "executable" creation to formal reasoning approximately primary courses. The authors' fundamental aim is to enhance programming skill through bettering instinct approximately what courses suggest and the way they run.The semantics of relevant courses is laid out in a proper, applied notation, the language OBJ; this makes the semantics hugely rigorous but easy, and offers aid for the mechanical verification of software properties.OBJ was once designed for algebraic semantics; its declarations introduce symbols for types and features, its statements are equations, and its computations are equational proofs. hence, an OBJ "program" is an equational thought, and each OBJ computation proves a few theorem approximately this sort of concept. which means an OBJ software used for outlining the semantics of a software already has an exact mathematical that means. in addition, common innovations for mechanizing equational reasoning can be utilized for verifying axioms that describe the impression of principal courses on summary machines. those axioms can then be utilized in mechanical proofs of homes of programs.Intended for complex undergraduates or starting graduate scholars, Algebraic Semantics of significant courses includes many examples and workouts in application verification, all of that are performed in OBJ.

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**Example text**

A th o r o u gh treatment of both many sorted general al gebr a and OBJ 3 is give n in [ 1 5] . A c om p re hen s ive but sophisticated t re at m e nt of o r de r sorted algebra is g ive n in [22] . J 3 Ve r s ion 2 is [26] . G e n e r al algeb r a has m any ap p l i c atio ns i n C om p ut i ng Science beyond those t reat e d in this book. Some of t h ese are c on c u rr e nt processing [36] , program d eri vat io n [4) , h ar dwar e verification [ 1 5] , program refinement [37] , and computer security [49] .

Then nil is just a list, and * builds non-empty lists ( of sort N eLi s t , with of course NeList < L i s t ) . A key point is that the selectors head and t ail are defined only on the non-empty lists. In this ap pr oac h , head ( nil) is not "undefined" or some such (potentially) vag u e thing, but is simply an ill formed ter m . Restricting the domains of o p e ratio n s to sub sorts can be a much simpler way to to handle errors than many others that are common in functional languages, such as raising exceptions.

Thus we conclude that swap(S) [[ 'X]] swap(S) [['Y]] for all S of sort Store, ::: S [[ 'Y]] ::: S[[ 'X] ] as desired. When you reduce a term which contains a variable, OBJ Warning: in a reduction the term contains This war n i n g can be ignored, as responds with a variable. OBJ performs the reduction nevertheless. An alter native way to prove a statement with a universal quantification involves declaring a new constant operator. For example, the following OBJ score, in which constant operator s : -> Store a new is declared, is an alternative way of proving the correctness of swap: Copyrighted Material 55 Stores, Variables, Values, and Assignment open STORE .