Download e-book for iPad: Axiomatising the Logic of Computer Programming by R. Goldblatt

By R. Goldblatt

ISBN-10: 3540112103

ISBN-13: 9783540112105

ISBN-10: 3540390383

ISBN-13: 9783540390381

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Example text

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. or non-termination of evaluation) sequ¢ntiaZ one If e is undefined (meaning "error", then the whole expression is undefined. If evalu- ation terminates with ~ found to have value I (meaning e is true) then 6 is evaluated. Otherwise £ is found to be false and ~ evaluated. The connective m provides similar sequential interpretations of the standard logical connectives if we define not-E (hence (~ = as (~ = 6,£Aue) as (c m £rue,6) 6 as (E D 6,faZse) e iff 6 as (e = 6 , ( 6 s imp£i~ 6 or 6 and AS operations on B + 1 false) as = f~e,Z~e)).

1 (note also that ~0t-e could have b e e n d e f i n e d as (e D f~66e,t2L~e)} , and thus we could have used m to define =. However w e p r e f e r to t r e a t the latter as a p r i m i t i v e symbol, to emphasise b o t h its f u n d a m e n t a l importance to the theory and the fact that its i n t e r p r e t a t i o n as the e q u a l i t y r e l a t i o n in B is e x a c t l y the same as its i n t e r p r e t a t i o n as this relation in any o t h e r data type. In general a function f of the form A 1 × ....

We introduce, for each natural number n, a premiss making an assertion about what happens after n loops. e. "do 0 loops"). S~p always termin- ates, leaving everything just as it was, hence the Dummy Rule m{sk/p}m. NOW f o r a given command ~ , we i n d u c t i v e l y a ° n+l = define skip = 0(; O(n SO that ~n means "do ~ n times". Next, consider the assertion (19n) q0{~n} ( ~ e ~ ~). This states that if q) is true beforehand, then after n executions of G if a situation is reached at which ~ is false, ~ will at that point be true.

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Axiomatising the Logic of Computer Programming by R. Goldblatt


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