By Peter Gacs, Laszlo Lovasz

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Now we can ﬁx the new tiles on the edge of the bigger square. e. we have obtained a tiling of the whole plane. e. it refers only to two tiles dominoes, the tiles will be correctly matched in the ﬁnal tiling, too. 2 Let us construct a Turing machine doing the following. For a word x ∈ Σ∗0 , it ﬁrst of all decides whether it codes a kit (this is easy); if not then it goes into an inﬁnite cycle. If yes, then with this set, it tries to tile one after the other the squares 1 × 1, 2 × 2, 3 × 3, etc. For each concrete square, it is decidable in a ﬁnite number of steps, whether it is tileable, since the sides can only be numbered in ﬁnitely many ways by the numbers occurring in the kit, and it is easy to verify whether among the tilings obtained this way there is one for which every tile comes from the given kit.

For a word x ∈ Σ∗0 , it ﬁrst of all decides whether it codes a kit (this is easy); if not then it goes into an inﬁnite cycle. If yes, then with this set, it tries to tile one after the other the squares 1 × 1, 2 × 2, 3 × 3, etc. For each concrete square, it is decidable in a ﬁnite number of steps, whether it is tileable, since the sides can only be numbered in ﬁnitely many ways by the numbers occurring in the kit, and it is easy to verify whether among the tilings obtained this way there is one for which every tile comes from the given kit.

Xn . To every elementary conjunction Ei , let there correspond a vertex into wich edges run from the input points belonging to the literals occurring in Ei , and which computes the conjunction of these. Finally, edges lead from these vertices into the output point t which computes their disjunction. Note that this circuit has large fan-in and fan-out. We can consider each Boolean circuit as an algorithm serving to compute some Boolean function. g. Turing machines: a circuit can deal only with inputs and outputs of a given size.