By Masami Ito

The idea of formal languages and the speculation of automata have been either initiated within the past due Nineteen Fifties, explains Ito (Kyoto Sangyo U., Japan), and the 2 fields have when you consider that constructed into very important theoretical foundations of desktop technological know-how. He seems on the from the algebraic viewpoint, starting with the algebraic constitution of automata, and in part ordered units of automata as one of those international conception. Then he delves into grammars, languages, and operations on languages. To finish, he introduces directable automata as a distinct case.

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**Example text**

Rado, who was an exceptionally tactful man, called the Christian in his problem a 'man', so the problem ran under the title 'Lion and Man'. Years after Besicovitch had found the solution above, Croft investigated several variants of the problem. For example, he proved that if the man has to run along a curve of uniformly bounded curvature then he will be eaten by a clever lion, but not by one who stays on the radius 0 M. Also, n lions can catch a man in an n-dimensional ball, but n - 1 cannot. I believe that the following problem mentioned by Littlewood is still unsolved.

Figure 11 shows eight unit squares touching another. Can we have nine such squares? 65. Infection with Three Neighbours. As in Problem 34, there is an epidemic on the n x n grid, but this time the sites are more resistant: a site becomes infected if at least three (rather than two) of its neighbours are infected. For n = 6k + 2, at least how many sites were infected at the start of an epidemic if eventually all n 2 sites are infected? Fig. 11. A square touching eight others. 16 1. The Problems 66.

7 : c;, 105. This is immediate from Bollobas's Lemma in Problem 104. 106. Assuming, as we may, that the Xk are positive reals, rewrite the bound Lk af = Lk(Ak/ k)P using telescopic sums. 108. A square cannot be cut into an odd number of triangles of equal area. Rather surprisingly, this can be shown with the aid of 2-adic norms. 109. The answer is the minimal t such that t ) n _:: : ( Lt /2J · In particular, the minimum is about log 2 n. 110. Recall from Problem 104 that a collection {(A j, B j) : 1 _:::: j _:::: n} of pairs of sets is cross-intersecting if A j n B j = 0 for every j and A j n Bh "I= 0 whenever j "I= h.