Download Lectures on constructive functional programming by Richard Bird PDF

By Richard Bird

The topic of those lectures is a calculus of services for deriving courses from their requirements. This calculus includes a number of ideas and notations for outlining features over a variety of info kinds
(including lists, timber and arrays), including their algebraic and different houses. each one lecture starts off with a particular challenge, and the idea essential to clear up it's theu built. during this approach we are hoping to teach sensible method of the matter of systematically cal­culating courses from their requirements can take its position along different methodologies.

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3 Loops In the functional approach to program derivation, the final product of a calculation is an expression denoting a mathematical function. This expres­ sion still has to be translated into a specific programming language in order for it to be executable by computer. One obvious candidate is a functional programming language, such as ML or Miranda l . However, there is no rea­ son why the final expression should not be translated into a conventional imperative language. For example, a left reduction can easily be translated into a loop.

It is instructive to determine the conditions under which a general set of equations h[] h[a] h(x * y) = = e fa H(x, y,f x,f y) determines a unique function h, not necessarHy a homomorphism. After all, such sets of equations constitute the basic means for specifying functions on lists. v) ~ (x If h' is a well-defined function, then so is h. We have = 11"2 h • h' where ",(a, b) ~ b. In order to determine the conditions under which the above equations determine h', let {J be the smallest set of values such that 1.

Suppose h(x*y)=hxEllhy for some operator Ell. Since h[2, 1] = 2,h[4] = [4], and h[2] = [2], we have h[2, 1, 4] = h[2, 11 Ell h[4] [2] Ell [4] = h[2] Ell h[4] h[2,4] This is a contradiction, since h[2, 1,4] = [2] and h[2,4] = [2,4]. 2 Left reductions We defined the directed reductions in the first lecture, but the pattern of the equations does not follow those laid down in the previous section. mple, mpe is a well-defined function. 29 In the monoid view of lists, the formal definition of fBfe is fe [] fe [a] $ fe (x $ e $ e$a $ fe' y * y) where e':::;: $ fe x Equivalently, setting f e :::;: ffJfe, we have f e[] f era] e :::;: eE»a f(Jex)y fe(x*y) ~Ne leave to the reader to check that the above equations satisfy the consis­ tency conditions of the previous section.

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