Let A,B, C be fad. Consider the equation X = AX + BX + C. Must a solution X be fad?
Could you help me solve this question? fad is a regular language
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Quick question, what is fad? – user1700890 Sep 26 '17 at 00:43
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Fad is finite automata definable or regular language – AmrutaMV Sep 26 '17 at 05:47
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Is the `+` operator indicating a union or a concatenation? I'm assuming union. It looks like X is definable as `(A|B)*C`. – Welbog Sep 26 '17 at 16:32
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Unclear what you are asking, mostly because you have not explained what you mean by your notations. – C. M. Sperberg-McQueen Sep 27 '17 at 16:59
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Assume juxtaposition (AX) means concatenation, and + means union. Then, let A = B = {e} and C = {}, the FAD language containing only the empty string and the empty language, respectively. Then let X be any non-FAD language. Clearly, the equation X = AX + BX + C is true since AX = X, BX = X, and X + X + {} = X.
Here are FAs for {e} and {} (proof, if desired, is left as an exercise):
/-\
--->[q0]-s->q1 | s
\-/
/-\
--->q0 | s
\-/
If juxtaposition and union mean something else, the answer may change. For instance, it's possibly that + means concatenation, but then I don't know what to make of juxtaposition (union? intersection?).
Patrick87
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