# Regex for $L = \{ w \mid w \in \Sigma^* \text{ and each substring } u \text{ of } w \text{ where } |u| = 4 \text{ contains the character } 0 \}$

The question asks to write a regex to the following language $$L$$ above $$\Sigma = \left \{0,1 \right \}$$.

$$L = \{ w \mid w \in \Sigma^* \text{ and each substring } u \text{ of } w \text{ where } |u| = 4 \text{ contains the character } 0 \}$$

Note that if $$|w| \leq 3$$ then there is no substring of length 4, so there is no need for the string to contain 0.

I came up with the following regular expression:

$$r = (0 \Sigma^3 + \Sigma 0 \Sigma^2 + \Sigma^2 0 \Sigma + \Sigma^3 0)^* (\epsilon + \Sigma + \Sigma^2 + \Sigma^3) : \Sigma = (0 + 1)$$

It is a wrong answer because for example the word $$0111101$$ can be generated by $$r$$.

I tried to convert the NFA in the picture to regex, but the regex was too long and it's missing the point of the question because I suppose to generate the regex by understanding what characteristic a word in $$L$$.

I came up to the conclusion that words in $$L$$ can't have sustring $$1111$$ but I don't know how to use it in order to create a regex.

It seems I'm failing short from the solution and I would like to know how I can transform $$r$$ to the required regex.

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Basically, your language asks you to restrict the run length of 1s to at most three. Here's how you can construct the required regex:

Step 1: Construct a DFA $$M$$ that accepts all strings that have at least one run of 1s or length four (or more). It should have five states with a single final state.

Step 2: Construct the complement $$M'$$ of $$M$$ by alternating the final and non-final status of the states.

Step 3: Build the regex for $$M'$$ using Arden's theorem.

If you want to directly derive the regex, here's how you can proceed: after the consecutive three 1s, you must put a zero if it is not the end of the string. But a zero may appear anywhere; there is no restriction on it. Thus, the regex is: $$(0+10+110+1110)^*(\epsilon+1+11+111)$$

• @coderR Thank you for your answer. Unfortunately I am not allowed to use Arden's theorem because where I learn it's not on the material. I already asked a question with the same core idea - a systematic approach to tackle the problem. The answer is that their is no way to "escape" from an automaton, but I'm been expected to not use it and to come up with an appropriate regex by trying to understand the words in the given language. Can you please give me your thought process on how you will tackle the problem without using an automaton ? Commented Apr 18 at 21:11
• Since I have already seen the DFA, I'm a bit biased now. Nevertheless, I have tried to provide an intuitive way to construct the regex from scratch. Commented Apr 19 at 7:49
• Thank you very much for the help !! Commented Apr 20 at 9:53