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Push down automata, Theory of Computation

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Construct a PDA that accepts { x#y | x, y in {a, b}* such that x ? y and xi = yi for
some i, 1 = i = min(|x|, |y|) }.
For your PDA to work correctly it will need to be non-deterministic. You can
assume that you will always be given a valid string – that is, the input will always
contain one # and x and y will be strings over {a, b}. My PDA has 31 states and
and is broken into two major sections, one for |x| = |y| and one for |x| ? |y|.
For the case where we assume that |x| = |y|, you need to find a symbol that
matches at the same index of x and y (xi = yi for some i) and a symbol that does
not match at the same index of x and y (xj ? yj for some j). One way that this can
be accomplished is by finding an index i such that xi = yi and xi+1 ? yi+1 or xi+1 =
yi+1 and xi ? yi. As in programming assignment 3, you can store the index in the
stack and the values of xi and xi+1 in the state.
For the case where we assume that |x| ? |y|, you need to find an index i where
xi = yi. Since the lengths are different, we get that x ? y without finding an index j
in which xj ? yj. For this case, you can simple check that x1 = y1. If x1 ? y1, then
the other portion of the code (where we assume that |x| = |y|) will accept the
string.

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