Generating regular expressions not containing abc

Question

Let sigma = {a,b,c}. How do I generate a language L that does not containg abc? Any guidance is appreciated!

Should $L$ contain all words except abc? I.e., are you looking for a regular expression for $L = \Sigma^* \setminus { abc }$? — Steven, Mar 18 '20 at 22:45
Yes! So my L for now contains all words where aaa appears exactly once. But it also generates abc. And I need the L without abc. — TanaM, Mar 18 '20 at 22:47
I added an answer that shows how to obtain a regular expression for $L$. Anyway, the language associated to your regular expression does contain the word "abc"... — Steven, Mar 18 '20 at 22:53
all words except $abc$, or all words that do not contain $abc$ [as a subword] ?? — Hendrik Jan, Mar 19 '20 at 01:30

Steven · Answer 1 · 2020-03-18T23:00:59.200

1

Design a DFA that recognizes "abc" (make sure to include all transitions). Complement it (make accepting states non-accepting, and make non-accepting states accepting) in order to get a DFA for $L = \Sigma^* \setminus \{ abc \}$. Finally, write down the regular expression of the complemented DFA.

As a brute force solution that does not use DFAs: write a regular expression for all words of lengths 0, 1, 2, and more than 3. This should be easy. Then write, a regular expression for all words of length 3 except abc. For example: $(b+c)(a+b+c)^2 + ab(a+b) + a(a+c)(a+b+c)$

Take the union of all the above REs.

edited Mar 18 '20 at 23:00

answered Mar 18 '20 at 22:49

Steven

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Thank you. But it is only the beginning of the course and we haven't learned, therefore, I am not familiar with DFA :| – TanaM Mar 18 '20 at 22:52
I edited my answer to add an alternative solution. – Steven Mar 18 '20 at 23:04

score 1 · Answer 2 · answered Mar 19 '20 at 13:10

The language of all words not containing $abc$ as a subword can be handled by focussing on the letter $b$. Whenever it occurs in the string we should check that one of the following conditions hold:

it is the first letter of the string
it is directly after another $b$ or after a $c$, or symmetrically
it is directly before ...
it is the last letter of the string

If that is true, I think it can be handled efficiently with a regular expression.

Be safe, all!

Generating regular expressions not containing abc

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