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Regular description for $\{ \mathtt{intercal}(w_1,w_2,w_4) \mid \exists w_3: (w_1,w_2,w_3,w_4\in\{0,1\}^*\;\wedge\;|w_1|=|w_2|=|w_3|=|w_4|\;\wedge\;\mathtt{value}_2(w_1)+\mathtt{value}_2(w_2)=\mathtt{value}_2(w_3)>\mathtt{value}_2(w_4)) \}$,
where $\mathtt{intercal}(a_1w_1,\ldots,a_nw_n)=a_1\ldots a_n\mathtt{intercal}(w_1,\ldots,w_n)$ and $\mathtt{intercal}(\lambda,\ldots,\lambda)=\lambda$
Give a regular description for the set of words obtained by intercaling three words
$w_1,w_2,w_4$
over
$\{0,1\}$ with the same length and such that there exists another word
$w_3\in\{0,1\}^*$, also
with the same length as
$w_1,w_2,w_4$, and satisfying the following property:
the sum of the natural values obtained
from
$w_1,w_2$ by interpreting them
as a binary numbers is equal to
$w_3$, that is
$\mathtt{value}_2(w_1)+\mathtt{value}_2(w_2)=\mathtt{value}_2(w_3)$,
and it is bigger than
$\mathtt{value}_2(w_4)$.
Intercaling
$n$ words
$w_1,\ldots,w_n$ over
$\{0,1\}$ and with the same length gives
as result a word whose sequence of symbols is:
the first symbol of
$w_1$, the first symbol of
$w_2$,
…, the first symbol of
$w_n$, the second symbol of
$w_1$, the second symbol
of
$w_2$, …, the second symbol of
$w_n$, the third symbol of
$w_1$, and so on.
Authors: Guillem Godoy
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