I'm working on solving the question answered at this page but with different values at the table, my alphabet is {a,b,c} Words that have the same right- and left-associative product

Currently I'm in the stage where I have drawn the DFA of the multiplication table, found its reverse which was an NFA.

Here is the NFA I got by reversing the multiplication table's DFA enter image description here

I apology for the miss draw, but I hope its readable.

Now I have taken the input "abcb" and applied it on the above NFA, and I have gone through this tree enter image description here

As you can see here that the input is all consumed at the branch "C" and I could reach the Final State. Would someone elaborate to me how can I backtrack from that branch which is "C" in this case and indicate that the "C" is the state which shall be marked as the Final State my NFA?

  • $\begingroup$ I don't understand what you mean by "backtrack" here. How does any of this relate to the titular question? Who wants to remember something -- you, some algorithm, or the automaton? $\endgroup$ – Raphael Apr 19 '16 at 6:48

Hint: Let the input be $x_1,\ldots,x_n$. As you mention, it is easy to compute the left-associative product $L_n$ "as you go": $L_{m+1} = L_m x_{m+1}$. The right-associative product recurrence has the wrong direction: $R_m = x_{m+1} R_{m+1}$. To cope with that, at each step you have to guess the value of $R_{m+1}$; it needs to be a value which conforms to your earlier guess of $R_m$ and to the value of $x_{m+1}$. At the end, compare your initial guess of $R_1$ to the value of $L_n$. Arrange things so that you only need to keep track of finitely many values of products.

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  • $\begingroup$ I appreciate your answer, but I'm a real beginner at this kind of science, I couldn't understand your answer. Can you try to put it in a simpler form for me to understand? - Also how do I can keep track of anything without having memory? $\endgroup$ – CSGuy Nov 20 '15 at 22:31
  • $\begingroup$ @CSGuy Unfortunately, I am not willing to provide further hints. $\endgroup$ – Yuval Filmus Nov 20 '15 at 22:32
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    $\begingroup$ That's fine, thank you I will study a bit more and try to decrypt the answer. $\endgroup$ – CSGuy Nov 20 '15 at 22:33

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