By Richard Bird

The topic of those lectures is a calculus of services for deriving courses from their requirements. This calculus contains more than a few options and notations for outlining capabilities over numerous info forms
(including lists, timber and arrays), including their algebraic and different homes. every one lecture starts off with a selected challenge, and the idea essential to remedy it really is theu constructed. during this means we are hoping to teach sensible method of the matter of systematically cal­culating courses from their standards can take its position along different methodologies.

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Second, there is the question of how many directions one might wish to identify as being useful. We shall adopt the following solutions to these questions. Each directed reduction will involve a single operator, and we focus on two legitimate directions: top-down (top reductions), and left-right (left reductions). We shall use the notation EElt for a top-reduction, and for a left reduction. ) The effect of a top reduction is illustrated by e+ EIlt Thus e tx ( 123) = 4 5 6 7 8 9 ((1E1l4)EIl7 (2 Ell 5) Ell 8 (3 Ell 6) Ell 9 ) produces a single row as its result.

Y reductions, in the way that list accumulations were related to list reductions in the first lecture, we need to consider the array analogues'of inits and tails. This we do next. 9 Tops and bottoms There are four reasonable ways of dissecting an array: we shall call them tops, bottoms, lefts and rights. Two of them are illustrated by the following examples: lefts 0~ n = (( non 0~ n) 1 2 3 123 PS(456)= tO 789 ( 123 456 (123 456 789 The essential points here are that lefts and rights each return a single row, while tops and bottoms each return a single column.

0fe where e =~ and a0b=a$fb We omit the simple proof. I . f!. efficiently. n write minimax where 00 is the identity element of a0x Since! x Using the specialisation lemma. c) Now, a is a left-zero of EBa (and so, by the way, is 00), and -00 is a. left-zero of 0. 6 The alpha-beta algorithm \Ve now generalise the minimax problem to trees. Consider the data-type tree ::~ Tip numlFork [tree] The syntax of this declaration is that of the functional language Miranda, and it is also employed in the notation of Bird and \Vadler (reference [2] of Lecture 1).

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