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Say I have a language over the alphabet {x,y,(,)} . The language's rules are: any string over that alphabet with balanced parentheses.

Clearly x(), (xy) , ()()xxx, ((xx)) are accepted. BUT

My question is, is a string with no parentheses considered as having balanced parentheses? For example are: y, xxyy, xyxyxxx accepted?

And then also what about an empty String?

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This is what formal definitions are for...

That said, it makes the most sense to allow no parentheses. Roughly speaking, a string has balanced parentheses if every left parenthesis is "matched" by a right parenthesis (of course this is far from being a formal definition). If there is no left parenthesis, then everything is OK.

Once you accept that, there is absolutely nothing wrong about the empty string – it is a string like any other. In particular, it is a string over your alphabet with balanced parentheses.

One way to formally define your language is using a context-free grammar. While there are many possible grammars, here is one that tried to capture the "matching" aspect when scanning the string from left to right:

$$ \begin{align*} &S \to \epsilon \\ &S \to xS \\ &S \to yS \\ &S \to (S)S \end{align*} $$

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  • $\begingroup$ Thank you. I wasn't given a formal definition unfortunately, but in (roughly) the wording I stated in my OP. The real question is to design the PDA from the CFG. But - that I will do on my own, just couldn't figure out the "corner case" scenario I was pondering. Thanks! $\endgroup$ – NateH06 Jan 25 '17 at 22:02

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