Exercise 22:

Regular description for {intercal(w1,w3)w2:(w1,w2,w3{0,1}    w1=w2=w3    value2(w1)>value2(w2)>value2(w3))}\{ \mathtt{intercal}(w_1,w_3) \mid \exists w_2: (w_1,w_2,w_3\in\{0,1\}^*\;\wedge\;|w_1|=|w_2|=|w_3|\;\wedge\;\mathtt{value}_2(w_1)>\mathtt{value}_2(w_2)>\mathtt{value}_2(w_3)) \},
where intercal(a1w1,,anwn)=a1anintercal(w1,,wn)\mathtt{intercal}(a_1w_1,\ldots,a_nw_n)=a_1\ldots a_n\mathtt{intercal}(w_1,\ldots,w_n) and intercal(λ,,λ)=λ\mathtt{intercal}(\lambda,\ldots,\lambda)=\lambda
Give a regular description for the set of words obtained by intercaling two words w1,w3w_1,w_3 for which there exists another word w2w_2 such that all three w1,w2,w3w_1,w_2,w_3 satisfy that are words over {0,1}\{0,1\} with the same length and such that the natural value obtained by interpreting w1w_1 as a binary number, that is value2(w1)\mathtt{value}_2(w_1), is bigger than value2(w2)\mathtt{value}_2(w_2), that in addition is bigger than value2(w3)\mathtt{value}_2(w_3).

Intercaling nn words w1,,wnw_1,\ldots,w_n over {0,1}\{0,1\} and with the same length gives as result a word whose sequence of symbols is: the first symbol of w1w_1, the first symbol of w2w_2, …, the first symbol of wnw_n, the second symbol of w1w_1, the second symbol of w2w_2, …, the second symbol of wnw_n, the third symbol of w1w_1, and so on.
Authors: Guillem Godoy / Documentation:
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