By W.H.J. Feijen, A.J.M. van Gasteren, David Gries, J. Misra

More than the rest, this publication is a tribute to Edsger W. Dijkstra, at the get together of his 60th birthday, through quite a few of these lucky sufficient to be encouraged by means of him and his paintings and to be referred to as his good friend or relation, his grasp, colleague, or scholar. This ebook comprises fifty-four technical contributions in numerous components of recreation, even though lots of them care for a space of specific difficulty to Dijkstra: programming. every one contribution is comparatively brief and will be digested in a single sitting. jointly, they shape a pleasant go element of the self-discipline of programming first and foremost of the nineties. whereas many be aware of of Dijkstra's technical contributions, they won't concentrate on his final target, the mastery of complexity in arithmetic and computing technology. He has forcefully argued that good looks and style are necessary to this mastery. The name of this booklet, selected to mirror his final aim, comes from a sentence in a piece of writing of his on a few appealing arguments utilizing mathematical induction: *"... once we **recognize the conflict opposed to chaos, mess, and unmastered **complexity as one in all computing sci*- *ence's significant callings, **we needs to admit that 'Beauty Is Our Business'.*"

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**Extra resources for Beauty Is Our Business: A Birthday Salute to Edsger W. Dijkstra**

**Example text**

Let x(O :S i < N) be the given sequence. Required is the subsequence of x with indices 4>(0 :S i < n), where 4> : int ........ int satisfies the conditions: (i) (boundary conditions) 4>(0) =0 and 4>(n) =N (ii) (solution is a subsequence) (Vi, j : 0 :S i < j < n: O:S 4>( i) < 4>(j) < N) (iii) (solution is decreasing) (Vi ,j : 0 :S i < j < n: x(4)(i)) > x(4)(j))) (iv) (solution is left-minimal) (Vj: O:S j < n : (Vk: 4>(j) :S k < 4>(j + 1) : x(4)(j)) :S x(k))) The first two conditions say that what is wanted is a subsequence of x beginning with the first element of x.

But we can do better than that. In proving Theorem (15) , we have actually shown that for p mod 10 E {1,9}, pi F p - 1 and pi F p - 1 hold. Therefore, pi f(p - 1) (see (4)), and we can change (20) to (21) If p mod 10 E {I , 9}, then *(p - l,p) and s(p) Ip - 1, If p mod 10 E {3, 7}, then (2(p + 1),p) and s(p) 12(p + 1). Result (21) was obtained by D. D. Wall in 1960, using less elementary tools . We have gained other insight as well. Since Fp ±1 has an even subscript, means that either pi F(p±1) /2 or pi E(p±1) /2 ' Again, observing the patterns in Table 1 suggests some conjectures: pi F p±1 (22) Theorem If If If If p p p p For any odd prime, p =I- 5, mod mod mod mod 20 E { 1, 9}, 20 E {ll , 19}, 20 E { 3, 7}, 20 E {13, 17}, then then then then pi F (p-l) /2 , pi E(p-l) /2, pi E (p+l) /2 , pi F(p+l) /2 ' These follow dir ectly from the following Lemma (23), with i = (p-1) /2 and F 2i+ l = Fp ~ 1 in the first two cases and with i = (p+ 1)/2 and F 2i+ l = F p + 2 ~ -1 in the second two cases. *

2]. This restores commutativity, but, as we shall see in the next section, there remain important laws that fail to hold. The semantics defined in this note has th e prop erty that all laws of propositional calculus carry through. Conventions and notations We suppos e a set At of logical atom s and a valuation av E At ---. {F, U, T} given: this is meant to model the fact that formulae without logical connectives can be false, undefined or true respectively. For convenience's sake it is assumed that F and T do not themselves belong to At.