Axiomatising the Logic of Computer Programming by R. Goldblatt

By R. Goldblatt

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Example text

R := g]~ will be explicitly Thus our system constitutes something of a departure even in its treatment of the standard quantifiers. EQUIVALENCE OF PROGRAMS. As an instance of r C a ++ [r := a]~ we have reflecting the fact that the command (r := r) changes nothing, and so is "equivalent" to S ~ p . But the latter satisfies ~-~ [ s k i p ] ~ and so [r is valid. e. that they halt for the same inputs and bring about the same output situations. g. that a w ~ e - c o m m a n d 29 is equivalent to the dummy command ~hen its test expression is false.

L ~ -~ ( [ w h / / e is valid. Hence ~ do a ]~ ++ q~) We shall modify this slightly below, after m a k i n g a d i s t i n c t i o n between i n t e r n a l and e x t e r n a l v e r s i o n s of negation. ASSIGNMENTS. In the new notion the Rule of A s s i g n m e n t becomes r ~ ~ [ r := ~]~, b u t the intuition that suggests that this is v a l i d does the same to [r := ~]~ ~ r , and h e n c e to ~r +~ [ r := ~]~ (this w i l l be m o d i f i e d b e l o w to allow for the p o s s i b i l i t y that ~ has no value).

V s(f0/~e) = 0 2. 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.

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