By Joseph A. Goguen, Grant Malcolm
Algebraic Semantics of critical courses offers a self-contained and novel "executable" advent to formal reasoning approximately crucial courses. The authors' basic objective is to enhance programming skill by way of bettering instinct approximately what courses suggest and the way they run.The semantics of valuable courses is laid out in a proper, carried out notation, the language OBJ; this makes the semantics hugely rigorous but uncomplicated, and offers aid for the mechanical verification of software properties.OBJ used to be designed for algebraic semantics; its declarations introduce symbols for types and features, its statements are equations, and its computations are equational proofs. therefore, an OBJ "program" is an equational conception, and each OBJ computation proves a few theorem approximately this sort of thought. which means an OBJ application used for outlining the semantics of a application already has an exact mathematical that means. additionally, common recommendations for mechanizing equational reasoning can be utilized for verifying axioms that describe the impact of crucial courses on summary machines. those axioms can then be utilized in mechanical proofs of homes of programs.Intended for complicated undergraduates or starting graduate scholars, Algebraic Semantics of significant courses comprises many examples and routines in software verification, all of that are performed in OBJ.
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Extra info for Algebraic Semantics of Imperative Programs
E . , it serves as a representation indep en d ent standard of comparison for correctness . Thus , it ma ke s sense to let the denotation of a given OBJ obj ect with signature � and equa t ions E be the class of all i n i t ial (I:, E ) -algebras . ative" denot ation for a set E, the operational re wr i t i ng semantics agrees with in the sense t h a t the reduced terms form an initial algebra (this res u l t was shown in  , and is explained in detail in [ 1 5] ) . To make th is remark precise , we fi rst define some of the most fund ament al concepts Under certain conditions on the denotational initial a lgeb ra semantics, in term r ew riting : Definition 20 A term rewri ti n g system is Church-Rosser iff whenever t � i t and t � t z , t he re i s some term t o such that h � t o and t 2 � t o .
A) Show that the identity function id precisely, we : BT -? B T is a homomorphism . ( M ore should say that the family of identity functions id BTr e e B T BTr e e B T Nat id Nat -? -? B T BTr e e B TNat is a E-homomorphism. ) (b) Further examples o f r;-homomorphisms are given by the fol lowing OBJ module, which introduces operations to sum and count the tips o f a tree : obj HOMS is pr BTREE . ops h l h2 : Btree -> Nat . v ar N : Nat . vars X Y : BTr e e . eq h 1 ( t ip N ) = N eq eq eq endo ( c) hl (X ++ Y) h2 ( t i p N ) h2 ( X + + Y ) = h1 (X) + h1 (y ) + h2 ( Y ) 1 .
We can only proceed by applying one of the two equations of STORE: which one can be applied depends on whether or not 'Y is equal to z.
Algebraic Semantics of Imperative Programs by Joseph A. Goguen, Grant Malcolm