Algorithm 8 - RealYannakakis($q$, $D$): There is a correctness issue here. Right now, the output is not $q(D)$ since step 27 returns $Or(D)$, which is a relation instance that has attributes which are not exactly given by $\bar{x}$. That is, step 25 does a projection $\pi{s\cup\bar{x}}$ which means that it also contains all of the (body) attributes given by the current node $s$ in the join tree. If you replace the return at line 27 with $\pi_{\bar{x}}(O_r(D))$, it would be correct.
Hi, I'd like to report an issue: