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Questions related to formal languages, grammars, and automata theory

8 votes
1 answer
1k views

If L is context-free, must FH(L) be context-free?

Define $FH(L) = \{x \in \Sigma^* : \exists y \in \Sigma^* \text{ with } |x| = |y| \text{ such that } xy \in L\}$. In other words, $FH(L)$ is the set of first halves of even length strings in $L$. …
David Smith's user avatar
7 votes
2 answers
868 views

Parikh's Theorem: CFL's "contain" regular languages?

The first sentence of the Wikipedia article for Parikh's Theorem states: "Parikh's theorem in theoretical computer science says that if one looks only at the relative number of occurrences of ter …
David Smith's user avatar
4 votes
1 answer
589 views

Show that every grammar for an inherently ambiguous CFL has infinitely many ambiguities

Prove that if a CFL $L$ is inherently ambiguous, then for any grammar $G$ with $L(G) = L$, there are infinitely many strings in $L$ that have (at least) 2 different derivations in $G$. Here's a s …
David Smith's user avatar
3 votes
3 answers
728 views

Give an example of a language whose Myhill-Nerode equivalence relation is such that if $x,y ...

Suppose $\Sigma = \{0,1\}$. Provide an example of a language $L \subseteq \Sigma^*$ with the property that its associated Myhill-Nerode equivalence relation, $R_L$, is such that every one of its equi …
David Smith's user avatar
3 votes
2 answers
780 views

If $L$ is regular, must the language $L_1 = \{w : w^Rw \in L\}$ be regular, or may it be non...

The reverse, $w^{R}$, of a string $w = w_1w_2...w_n$ is the string $w_n...w_2w_1$. Suppose that L is a regular language. Must the language $L_1 = \{w : w^Rw \in L\}$ be regular, or may it be non-reg …
David Smith's user avatar
2 votes
2 answers
136 views

Prove that REG is closed against removing all but lexicographicaly largest words (per length)

Let $\Sigma_n = \{0, 1, ... , n-1\}$. Suppose $L \subseteq$ $\Sigma^*_n$, and let $\qquad\displaystyle\mathcal{B}(L) = \{ x \in L : x = \textrm{lex}\max L_m, m \in \mathbb{N}_0 \}$, where $L_m = L \ …
David Smith's user avatar
1 vote
1 answer
618 views

Show that a regular language L contains only palindromes if and only if all words of length ...

This is an extension of a previous question asked by a different user earlier: Let $x, u, v, w, y, x', u', v', w', y'$ be words satisfying $y'x' = xy$. $y'u'x' = xuy$. $y'v'x' = xvy$. $y'w'x' = xwy$ …
David Smith's user avatar
1 vote
2 answers
95 views

Is the language of strings with an integer ratio of the number of a's to the number of b's c...

Consider the language $L \subseteq \{a,b,c\}^*$, where $w \in L$ if and only if the ratio of the number of $a$'s in $w$ to the number of $b$'s in $w$ is an integer. I've been unable to find a countere …
David Smith's user avatar
0 votes
1 answer
141 views

For two regular languages, why is the set of words from one that don't have a subsequence in...

In general, a string $x$ is a subsequence of $w = w_1\dots w_n$ if there are integers $i_1<\dots< i_k$ such that $x = w_{i_1}\dots w_{i_k}$. The subsequence is proper if $k < n$ and $k > 0$. With th …
David Smith's user avatar