Show whether the language with almost as many 0 as 1 in every prefix is regular [closed]

This is the exercise:

Let A be a language defined over the alphabet Σ = {0, 1} composed by the strings with the property that in every prefix, the number of 0s and the number of 1s differ by at most 2. Determine if A is a regular language and prove it.

Ok, as @roi-divon commented, A is a regular language. I've tried to obtain de minimal DFA accepting A but I haven't got it... Someone knows?

closed as unclear what you're asking by David Richerby, Rick Decker, Juho, Luke Mathieson, ShaullNov 21 '14 at 15:01

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The language is regular.

Denote

$L^k = \{ x \ | \ \forall \alpha \in Prefix(x): |\#_0(\alpha)-\#_1(\alpha)| \leq k \}$

Your language is $L = L^1$.

We shall prove, using Myhill-Nerode theorem that each $L^1$ is regular (actually for any $k,\ L^k$ is regular).

Define equivalence classes of $R_L$ as follow:

• for $-k \leq i \leq k: \ S^i = \{ x \ | \ \#_0(\alpha)-\#_1(\alpha) = i, \forall \alpha \in Prefix(x): |\#_0(\alpha)-\#_1(\alpha)| \leq k \}$
• $S_{out} = \{ w | \exists \alpha\in Prefix(x): |\#_0(\alpha)-\#_1(\alpha)| \gt k\}$

We can see that $L^k = \bigcup_{-k \leq i \leq k}S^i$

Clearly, the equivalence classes are not empty, they don't intersect and the union is $\Sigma^*$

Assume $x,y \in S_{out}$, we show that $x R_L y$: clearly from every $z \in \Sigma^*: \ xz,yz \notin L$ (have "bad" prefixes")

Assume $x,y \in S^i$, and assume there is a $z \in \Sigma^*$ such that $xz \in L$ but $yz \notin L$. Then $yz$ has a prefix $y \beta$ such that the difference between 0's and 1's is larger than k ($\beta$ is prefix for z). By counting the number of 0's and 1's of $x \beta$ we get that it's larger then k, in contradiction that $xz \in L$

Now we show that if $x,y$ in different classes, then $(x,y) \notin R_L$.

Assume $x \in S^i, \ y \in S_{out}$: then for $z = \epsilon: \ xz = x \in L, yz = y \notin L$. Therefore $(x,y) \notin R_L$

Assume $x \in S^i, \ y \in S^j$, and $j \neq i$. w.l.o.g, $i > j$. Examine $z = 0^{k-i+1}$. Counting 0's and 1's, we get that $xz \notin L$, and $yz \in L$. Therefore $(x,y) \notin R_L$

By that We've proved that $L = L^1$ is regular

• And would you know how to obtain a minimal DFA accepting A? I've tried it, but... – eyhqtl Nov 18 '14 at 9:08
• Also, rather than this lengthy proof via Myhill-Nerode, it's much easier to just produce a DFA for this: six states should suffice. – David Richerby Nov 18 '14 at 9:13
• Please consider not to encourage undesirable posting behaviour. – Raphael Nov 18 '14 at 9:42
• @DavidRicherby May i submit the DFA? – muradin Nov 18 '14 at 14:18