# Questions tagged [regular-languages]

Questions about properties of the class of regular languages and individual languages.

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### Is there a regular tree language in which the average height of a tree of size $n$ is neither $\Theta(n)$ nor $\Theta(\sqrt{n})$?

We define a regular tree language as in the book TATA: It is the set of trees accepted by a non-deterministic finite tree automaton (Chapter 1) or, equivalently, the set of trees generated by a ...
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### Using logic to prove non-regularity of a language

A language $L$ is regular if and only if it is definiable by a sentence in monadic second order logic (MSO) over strings (J.R. Buchi, Weak second-order arithmetic and Finite automata; Z. Math. Logik ...
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### Test whether two languages are equal, when give in algebraic form

This sub-problem is motivated by Algorithm to test whether a language is regular. Suppose we have two languages $L_1,L_2$ that are expressed in "algebraic" form, as formalized below. I want to ...
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### Regularity profiles

A standard exercise in formal language theory uses Lagrange's four-square theorem to construct a language $L$ such that $L$ isn't regular but $L^2$ is regular. (Let $A = \{ a^{n^2} : n \geq 0 \}$. ...
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### Growth function for non-regular languages

For language $L$ over an alphabet $\Sigma$ denote $\gamma_L(n)$ as the number of words of length $n$ in the language $L$. It is known that for regular languages this function represents a sequence ...
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### How to disambiguate symbolic regular expressions

What I mean by a "symbolic regular expression" (if there already is a different name for this I'm not aware of it) is a regular expression that may include exponents that are symbolic arithmetic ...
1answer
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### NP completeness of deciding whether a set of examples, consisting of strings and states, has a corresponding DFA?

I'm working on a textbook problem, 7.36 in Sipser 3rd edition. It claims that if we are given an integer $N$ and set of pairs $(s_i, q_i)$, where $s_i$ are binary strings and $q_i$ are states (we are ...
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### Subexponential size of string to prove $\{xy : x,y \in \{0,1\}^\star, |x| = |y|, x \ne y\}$ is not regular?

In the standard proof of this language not being regular using the Pumping Lemma for Regular languages, one picks $0^p 1^p 0^{p+p!} 1^p$ where $p$ is the pumping constant and using that can derive the ...
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### BNF rule to regular expression

I'm looking for a way to find out whether a specific rule in a BNF grammar can be converted to a regular expression. (With "regular expression" (RE), I mean the simple mathematical kind. I'm ...
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### Regular language is closed given transposition of rightmost character to leftmost

It would appear straightforward to show that a regular language is closed given the transposition of the rightmost character to the front. However after drawing a few sample DFA for the phenomenon, I'...
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### How to prove that a language created from a context-free gramar's left side is regular(or left-linear)?

Given a context-free grammar $G$, let $\longrightarrow_G$ be the (one-step) rightmost derivation relation, and $\longrightarrow^*_G$ its reflexive and transitive closure. Let $S$ be the start symbol ...
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### concatenation of regular and DCFL

$S -> AB$ $A -> aA / epsilon$ $B ->aBb / epsilon$ What is the class of language generated by the above grammar ? I think that it generates $a^* a^n b^n | n>=0$ so it should be regular ...
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### What kind of structural features of strings can be described by regular grammars?

Context-free grammars, as well as other types of grammars, can naturally associate structure with the strings of the defined language, for example tree structures in the case of context-free language. ...
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### Intuition for irregular languages

I'm struggling in understanding how to recognize irregular languages. I know what the meaning of irregular language but still find it hard to recognize. Are there any tips to recognize better and to ...
0answers
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### Over every non-empty alphabet there exist languages which are non-regular

I am not sure about the answer. Intuitivly I would say that there are alphabets for which there are no non-regular languages. In particular I am thinking of languages with only one element. But I am ...
2answers
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### Are regular grammar languages defined from “accepting” states?

In a transition diagram, the language L(D) where D is the diagram is defined as all the words that are formed from following an "accepting" walk. Does the same apply for languages of regular ...
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### Is there a direct way to obtain the RE for the handle-finding DFA of a grammar?

LR parser for a (CFG) grammar uses a handle-find automaton (which is a DFA) to find the handles. Such automata can be constructed by computing the canonical collection of sets of LR(0)/LR(1) items. Is ...
3answers
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### PDA for a language where the second part is not the reverse of the first part

I came across an exercise for constructing a PDA for the following language: $$L = \{ncm \mid n,m\in\{a,b\}^* \text{ and } n \ne m^R\}.$$ Where $L \subseteq ({a,b,c})^*$ So $n$ and $m$ are both a ...
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### Is every language described by a grammar?

I read the following argument showing that not every language is described by a grammar: For a fixed alphabet $\Sigma$ and variables $V$ there are uncountable many languages over $\Sigma$ since the ...
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### Language of rationals is regular, what is the number set equivalent to PDA?

Consider rational numbers given as their decimal expansions, then for every number we can build a finite automaton able to accept it. To simplify the argument, assume that finite rational expansions ...