# I need help in this regural expression exercises [closed]

Write a regular expression for the language of words over $$\{0,1,2\}$$ satisfying the following requirements:

• The word has length at least 3.
• The last symbol is 2.
• The second to last symbol isn't 0.
• The combination of the last two symbols doesn’t appear anywhere else in the string.

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• It's really better if you solved this on your own. – Yuval Filmus Sep 28 at 22:28
• Yeah I know..I have been trying the last 3 hours but I am really stuck.I could use some guidance at least – stathis korakas Sep 28 at 22:36
• Perhaps you could show us what you did manage to do. – Yuval Filmus Sep 28 at 22:37
• (0+1+2)[(10*+11*+2*)]*(12*) I wrote this for the case where the second to last symbol is 1 but it doesn’t seem right at all – stathis korakas Sep 28 at 22:50

In order to solve this exercise, we need to come up for regular expressions for two classes of words:

• Words of the form $$x00$$, where $$x \neq \epsilon$$ and the only occurrence of $$00$$ is at the end.
• Words of the form $$x01$$, where $$x \neq \epsilon$$ and the only occurrence of $$01$$ is at the end.

Let us start with the first class. For starters, $$x$$ must end with $$1$$ or $$2$$. So such words are of one of the forms $$y100$$ or $$y200$$, where $$y$$ doesn't contain $$00$$. Presumably you already know how to describe all words not containing $$00$$ using a regular expression.

For the second class, the only constraints on $$x$$ are that it is non-empty and doesn't contain $$01$$. If you don't like the "non-empty" constraint, you can use the following case distinction:

• If $$x = y0$$ or $$x = y2$$, then the only constraint on $$y$$ is that it doesn't contain $$01$$.
• If $$x = y1$$, then $$y$$ cannot end with $$0$$. It could be empty. If it ends with $$2$$, then $$x$$ is of the form $$z21$$. If it ends with $$1$$, then we're again in the same situation: the preceding symbol (if any) cannot be $$0$$. Continuing in this way, we see that either $$x \in 1^+$$ or $$x$$ is of the form $$y2z$$, where $$y$$ doesn't contain $$01$$ and $$z = 1^n$$ for some $$n \geq 1$$.

I'll let you figure out how to describe strings $$y$$ avoiding $$01$$ – this is very similar to the case of avoiding $$00$$.

• Thank you so much for your time.One more question..do you think by drawning the DFA would help me to answer the problem because I tried it and I got even more confused – stathis korakas Sep 28 at 23:23
• I'm not sure it would be too helpful, since regular expressions are quite different from DFAs. However, there is an algorithm that converts a DFA to an equivalent regular expression. – Yuval Filmus Sep 28 at 23:33
• Alright I’ll keep that in mind. – stathis korakas Sep 28 at 23:36