Axiomatising the Logic of Computer Programming by R. Goldblatt

By R. Goldblatt

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A member of Cmd is a s y m b o l i c p r e s e n t a t i o n of the form of an actual In the n e x t chapter, the ~'s w i l l be replaced by assignments (x := 0). Members of Omd o t h e r than p r o g r a m letters w i l l be referred to as structured commands. Cmd is d e n u m e r a b l y infinite. F0rmu]ae. The class Fma of (well-formed) formulae comprises w h a t we have informally called "assertions" and ~'statements" above. It is d e f i n e d as follows. e. Bxp ~ Fma I. Each B o o l e a n e x p r e s s i o n is a formula, 2.

Nings". procedure and function declarations, array types, and The underlying philosophy here is one of "small begin- The prospect of producing a single axiom system that derives just the valid assertions about all programs in, say, Pascal is a daunting one, and is best approached by a series of individual studies of the many different notions involved. Once each of these is well understood the problem of putting them all together into one struc- 85 ture can be more realistically tackled. A start has been made here: hopefully one that points the way to further progress.

The early foundational studies of this century were concerned to cope with "paradoxes" and "crises" that resulted from the application of little understood methods to new and unfamiliar ideas about infinite sets. Today the practising mathematician understands very well the properties of infinities, which he treats the finite. and manipulates them with the same confidence with But although the use of formal logic played a vital role in bringing this circumstance about, it does not follow that its present techniques will apply to subsequent discoveries, or that new ideas and arguments can be confined to its traditional framework.

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