Exercise 4:

{u,v,RΣ={a,b}    uRv}{u,v,RuRv    R2˙ (as a set, i.e., not counting repetitions)}\{\langle u,v,R\rangle\mid\Sigma=\{a,b\}\;\wedge\;u\to_R^*v\}\quad\leq\quad\{\langle u,v,R\rangle \mid u\to_R^*v\;\wedge\;|R|\notin\dot{2}\text{ (as a set, i.e., not counting repetitions)}\} (morphism not allowed in the reduction)
Reduce the word reachability problem restricted to the alphabet {a,b}\{a,b\} to the set of tuples u,v,R\langle u,v,R\rangle where u,vu,v are words, RR is a word rewrite system, uu reaches vv using RR, and the number of rules of RR is odd (by considering RR as a set, i.e., each different rule is counted only once), 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 / Documentation:
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