By D. John Cooke

Principal to Formal tools is the so-called Correctness Theorem which relates a specification to its right Implementations. This theorem is the objective of conventional application checking out and, extra lately, of software verification (in which the theory needs to be proved). Proofs are tough, even though inspite of using strong theorem provers. This quantity explains and illustrates another technique, which permits the development of (necessarily right) algorithms from a specification utilizing algebraic ameliorations and refinement thoughts which stop the advent of error. in keeping with educating fabric used largely at Loughborough collage, John Cooke introduces the fundamentals, utilizing uncomplicated examples and many distinct operating (which can usually be re-used). developing right software program will supply valuable examining for college kids and practitioners of computing device technological know-how and software program Engineering to whom correctness of software program is of top significance.

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F = g and f(x) = g(x) for (3). An important property of ‘°’, of functional composition, is that when (say) f:A — B, g:B — C, and h:C — D and f · g and g · h , so that when x ˜ f, h(g(f(x))) is properly defined. It follows that ( h ° g) ° f = h ° (g ° f). We say that functional composition is associative. ) (4). It is also convenient here to give the definition of the graph of a function. The distinction between a function and its graph is often blurred, but here goes. Given a function f of type X — Y, the graph of f is the relation (which is of type (X Ù Y) ) consisting of the pairs ”x, f(x)’ for all x for which f(x) exists.

We can now introduce the notation associated with the concept of a set, already mentioned informally in passing. Sets can be defined in several ways, and here we shall use two. Firstly, a (small) set can be defined explicitly; for example: A – {1,2,3,5,7,9} All the elements of the set are written out. ” 40 Constructing Correct Software Following on from these definitions we can write A Ù B and T explicitly: A Ù B is the set {”1,1’ ”1,2’ ”1,4’ ”1,6’ ”1,8’ , ”2,1’ , ”2,2’ , ”2,4’ , ”2,6’ , ”2,8’ , ”3,1’ , , ”3,2’ , , ”3,4’ , , ”3,6’ , , ”3,8’ , ”5,1’ , ”7,1’ , ”9,1’ , ”5,2’ , ”7,2’ , ”9,2’ , ”5,4’ , ”7,4’ , ”9,4’ , ”5,6’ , ”7,6’ , ”9,6’ , ”5,8’ , ”7,8’ , ”9,8’ } and T is the set { ”1,2’ ”1,4’ ”1,6’ ”1,8’ , , ”2,4’ , ”3,4’ , , ”2,6’ , ”3,6’ , ”5,6’ , , ”2,8’ , ”3,8’ , ”5,8’ , ”7,8’ } or, in a more usual, but perhaps less comprehensible, form5: { ”1,2’ , ”1,4’ , ”2,4’ , ”3,4’ , ”1,6’ , ”2,6’ , ”3,6’ , ”5,6’ , ”1,8’ , ”2,8’ , ”3,8’ , ”5,8’ , ”7,8’ } Notice that the relation T, and in fact any relation, is also a set.

40 Constructing Correct Software Following on from these definitions we can write A Ù B and T explicitly: A Ù B is the set {”1,1’ ”1,2’ ”1,4’ ”1,6’ ”1,8’ , ”2,1’ , ”2,2’ , ”2,4’ , ”2,6’ , ”2,8’ , ”3,1’ , , ”3,2’ , , ”3,4’ , , ”3,6’ , , ”3,8’ , ”5,1’ , ”7,1’ , ”9,1’ , ”5,2’ , ”7,2’ , ”9,2’ , ”5,4’ , ”7,4’ , ”9,4’ , ”5,6’ , ”7,6’ , ”9,6’ , ”5,8’ , ”7,8’ , ”9,8’ } and T is the set { ”1,2’ ”1,4’ ”1,6’ ”1,8’ , , ”2,4’ , ”3,4’ , , ”2,6’ , ”3,6’ , ”5,6’ , , ”2,8’ , ”3,8’ , ”5,8’ , ”7,8’ } or, in a more usual, but perhaps less comprehensible, form5: { ”1,2’ , ”1,4’ , ”2,4’ , ”3,4’ , ”1,6’ , ”2,6’ , ”3,6’ , ”5,6’ , ”1,8’ , ”2,8’ , ”3,8’ , ”5,8’ , ”7,8’ } Notice that the relation T, and in fact any relation, is also a set.