Algorithms

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By R Cooke, K L. Lockett, J A Bellman

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44 Constructing Correct Software Even if another y exists, this one is new. In programming terms, its scope is “| .......... }” and it does not exist outside of these symbols, and within them it hides any other y which might have been introduced earlier. This is a notation from logic that corresponds exactly to this familiar concept of scope from blockstructured programming languages. Fuller explanations will be given later; for now this is simply helping to describe what the y is all about. R – {x:X | (∃y:Y)(”x,y’˜R) } where R: (X Ù Y) “The domain of R is the set of all x’s (of type X) such that, for each x, there exists a y of type Y, ‘(∃y:Y)’, where the pair ”x,y’ is in R” The syntax (the layout) used here will probably be strange to anyone who has not encountered formal logic.

This in itself is not important but what in general is important is that we can distinguish the values which can be placed at the start of an arrow (the input values) and those at the sharp end (result values). Notice that two arrows might point to the same result, and not all of the data points or result points are necessarily used. Other kinds of diagram are sometimes used, as we shall see. 3. 3 Technical Background 37 For the function f to be consistent with a specification, the specification must include all the ‘arrows’ of f but might also have many others.

Also defined is the mirror-image concept of the range of the relation R. This is written R and is defined: R – {y:Y | (∃x:X)(”x,y’˜R) } where R: (X Ù Y) This gives the set of all values in Y that are to be found at the pointed end of the arrows which comprise R. 9. In this figure, the relation R1 is used as the example. 9 Referring to the figure and definition of R1, R 1 –{”x,y’: X Ù Y | x = y + 1 } it is clear that it is fruitless to attempt to find an answer (an answer compatible with R 1 ) starting from the value 1 as input.

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Algorithms, graphs, and computers by R Cooke, K L. Lockett, J A Bellman


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