Axiomatising the Logic of Computer Programming by R. Goldblatt

By R. Goldblatt

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If Vs(a) = CO vs(e ~ 8,~) = vs(6) if VS(a) = I vs (~) if VS (~) = 0 to the 40 3. If either v~e 6) Vs{e) or L Us(h) is ~), then I if V8(£) = V8(6) 0 if VS(e) ~ Vs(£ = otherwise 6) is 09; U8(6). Writing DE as an abbreviation for (e = e), from clause 3 we have 7 ~0 if VS(e) = 03 Vs and hence v s(true) = v s(Df~g~e) = I. The interpretation of the connective m is the (see McCarthy (1963), Manna and McCarthy of McCarthy (1970)), in which we imagine the computer first evaluating e and then proceeding to 6 or 2.

What is traditionally known as "free logic" or " l o g i c w i t h o u t (cf. g. Scott (1967)). (1970)), but It is similar to existence assumptions" The basic idea is that each structure A is expanded to A + = A U {to}, where ~ is some entity not in A, and variables of the type of A are taken as ranging over A +. undefined in A. A variable is assigned to when it is intended that it be In this way every expression acquires a "formal value", and this allows an inductive definition of the relation "~ holds in state 8", which is written K s ~, for every formula %0.

Let 70, ~I' .... be a denumerable list of symbols, called program letters. We define the set 6]nd of commands by the f o l l o w i n g inductive closure conditions. 38 a. skip and abort b. Each p r o g r a m letter N is in Cmd ; c. If s, ~ are in C~d, then so is d. If a, 8 are in Cmd, and e is in Bxp, then (g ~ a,B) e. If ~ is in Cmd, and s is in Bxp, then The c o m m a n d are in Cmd ; (c ~ a,B) denotes the iterative command is the c o n d i t i o n a l (w~e ab02L~ is described by D i j k s t r a (~) e do s) .

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