By Joseph A. Goguen, Grant Malcolm
Algebraic Semantics of principal courses offers a self-contained and novel "executable" advent to formal reasoning approximately principal courses. The authors' fundamental target is to enhance programming skill through bettering instinct approximately what courses suggest and the way they run.The semantics of crucial courses is laid out in a proper, carried out notation, the language OBJ; this makes the semantics hugely rigorous but uncomplicated, and offers help for the mechanical verification of software properties.OBJ used to be designed for algebraic semantics; its declarations introduce symbols for kinds and features, its statements are equations, and its computations are equational proofs. therefore, an OBJ "program" is an equational idea, and each OBJ computation proves a few theorem approximately this kind of conception. which means an OBJ application used for outlining the semantics of a software already has an actual mathematical that means. furthermore, average strategies for mechanizing equational reasoning can be utilized for verifying axioms that describe the impression of important courses on summary machines. those axioms can then be utilized in mechanical proofs of houses of programs.Intended for complicated undergraduates or starting graduate scholars, Algebraic Semantics of principal courses comprises many examples and routines in application verification, all of that are performed in OBJ.
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Extra info for Algebraic Semantics of Imperative Programs
Let us now consider lists of natural numbers using so-called error supersoris: obj NATLISTE is s ort s Nat List EL ist . subsort s Nat < List < EL ist op 0 : - > Nat op s - : Nat -> Nat - > List op nil op - * List List - > List op - * ELi s t EList -> EList op head_ : EL ist -> Nat op t ail : EList - > ELi s t o p errorL i s t : - > ELi s t var s X Y Z : List var N : Nat . eq X * nil ::: X eq nil * X ::: X eq (X * Y)* Z ::: X *(Y * Z) . eq head(N * X) ::: N eq tail(N * X) ::: X . eq head N N . nil .
E . , they are strings pr efix ed with a quote ( ' ) . ural number expressions which c o nt ai n quoted identifiers . obj EXP i s pr QIO . sort Exp . subsort Id < Exp . op 0 : - > Exp . op s - : Exp - > Exp [pree 1 J Exp Exp -> Exp [pr ee 5J op - +op - * - : Exp Exp -> Exp [pr ee 3] endo (a) The following mo dule defines a n operation subst : Exp Id Exp -> E xp which takes an exp res si on E 1 , a quoted identifier X an d an exp ress io n E2, an d r e t u r ns the result of replacing all O Ccurrences of X in E2 by E 1 ; i .
Our next object uses subsorts in an interesting way; indeed, this kind of example cannot b e done adequately with just many sorted algebra (this is proved in ). The constructors 0 and s b uild natural numbers, w hile nil and * build lists. The subsort declaration Nat < NeList s ays that every number is a (singleton) list. 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.
Algebraic Semantics of Imperative Programs by Joseph A. Goguen, Grant Malcolm