Grammar for all words other than $wq,qw$

Question

I want to generate a grammar that can't generate the words $qw$ and $wq$ but can generate the word $qwwq$. In other words, $L(G)=\{m ∈ \{q,w\}^* \mid m \neq wq,qw \}$.

My grammar:

\begin{align} &S \to qSw \mid wSq \mid qXq \mid wXw\\ &S \to qYw \mid wYq \mid q \mid w\\ &X \to qX \mid wX \mid qXw \mid wXq \mid ε \\ &Y \to qw \mid wq \\ \end{align}

@D.W. I don't a finite number of words, what I want is the whole words in {q,w}* except the "qw" and the "wq" — Nicolette Lorenzo, Mar 26 '21 at 23:38
Your language is regular as it's the complement of the finite (and hence regular) language ${wq, qw}$. — Steven, Mar 26 '21 at 23:53
@Steven yupp, do you have any idea how we can write it i just wrote CFG because i want to convert it to CNF after — Nicolette Lorenzo, Mar 27 '21 at 00:46

awillia91 · Accepted Answer · 2021-03-27T00:50:32.330

1

How about \begin{align*} S&\to q\mid w\mid qqB \mid wwB \mid qwA\mid wqA\mid \varepsilon \\ A&\to qB\mid wB \\ B&\to qB\mid wB\mid \varepsilon \end{align*}

We can just explicitly include strings of length 0, 1 or 2 that are allowed, but add another non-terminal after the length two strings which are not allowed to force us to add at least one more terminal, then make sure after that we can add any letters we want in any order.

edited Mar 27 '21 at 00:50

answered Mar 27 '21 at 00:31

awillia91

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the word "aabb" is not included for example – Nicolette Lorenzo Mar 27 '21 at 00:47
@MandiJoseph You're right I forgot to include the $B$ after the $qq$ and $ww$. I've edited it. By the way you will want to add a rule $S\to \varepsilon$ to the grammar you've written. – awillia91 Mar 27 '21 at 00:55
yes i agree with you. – Nicolette Lorenzo Mar 27 '21 at 01:07

Grammar for all words other than $wq,qw$

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