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$\{\langle u,v,R\rangle\mid\Sigma=\{a,b\}\;\wedge\;u\to_R^*v\}\quad\leq\quad\{\langle u,v,R\rangle \mid |\Sigma|\leq 2\;\wedge\;u\to_R^*v\text{ with an odd number of steps}\}$ (morphism not allowed in the reduction)
Reduce the word reachability problem restricted to the alphabet
$\{a,b\}$ to
the set of tuples
$\langle u,v,R\rangle$ where
$u,v$ are words,
$R$ is a word
rewrite system, the used alphabet has at most
$2$ symbols, and
$u$ reaches
$v$
using
$R$ with an odd number of steps, in order to prove that such set is
undecidable (not recursive).
The use of morphism is not allowed in the description of this reduction.
Authors: Guillem Godoy
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