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Questions tagged [context-free]

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CFG - How can I describe a language that dictates a word and its opposite?

I have this question from my Automata class and I am unsure if there's a way to do this. Assuming u,v ∈ {0,1}* and at every character in the word u, the character at the same position in the word v is ...
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5 votes
1 answer
158 views

Establishing competing memory limits for pushdown automata

Let $L$ be the language of all even-length strings whose first half is a palindrome. Let $L$ be the language of all even length strings whose first half is imbalanced—with an unequal number of $\...
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9 votes
1 answer
197 views

Words of the form $(a^n b)^n$ in a context-free language

Question For a language $L \subset \{a,b\}^*$, denote $N(L) = \{ n \geq 0 \mid (a^n b)^n \in L \}$. If $L$ is context-free, is $N(L)$ necessarily semilinear, meaning that $n \in N(L) \iff n+p \in N(L)$...
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1 vote
1 answer
87 views

Bounded non-emptiness intersection of deterministic context-free grammars

Let A and B be two determinstic context-free grammar, and let N be an integer: What's the complexity of deciding if the intersection of the languages accepted by A and B over all strings of length ...
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9 votes
0 answers
86 views

Expressiveness of pushdown automata whose stack height sequence is unambiguous

I consider pushdown automata on an alphabet $\Sigma$, which are intuitively finite automata with a stack. Formally, a pushdown automaton $A = (Q, q_0, F, \Gamma, \Delta)$ is a finite set $Q$ of states,...
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2 votes
1 answer
145 views

Does a finite, polynomially-bounded CFG translate into a polynomially-bounded DFA?

We are given a family of context-free grammars $\{ G_1, G_2, G_3, ..., G_n, ...\}$ where each $G_n$ generates strings only of length $n$ and obeys other constraints specified below. We want to study ...
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  • 414
6 votes
1 answer
187 views

Context Free Grammar For Complement Of { www | ... } with minimal pumping length?

Let $L := \{ w^3 | w \in \{0,1\}\}^C$ be the complement of the language of words that are not the 3rd power of a word over $\Sigma = \{0,1\}$. Let's define the largest minimal pumping length of a ...
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  • 389
1 vote
0 answers
82 views

Take a natural quotient of context-free grammars

Fix a finite alphabet. Let $\mathrm{CFG}$ be the set of context-free grammars on this alphabet, $\mathrm{CFL}$ the set of context-free languages, $\mathrm{UG}$ the set of unrestricted grammars and $\...
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  • 41
0 votes
1 answer
103 views

Reference for context-free grammar for Martin-Löf type theory

Are the terms and the types of Martin-Löf type theory described by context-free grammars? Have such grammars been written down somewhere?
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  • 11
2 votes
0 answers
82 views

Necessary and sufficient condition for an infinite tree to be context-free

A Buchi automaton is non-empty iff it accepts an infinite word of the form $uv^\omega$ (here $u,v$ are finite words). In other words, if $\{w\}$ is an $\omega$-regular language, then it is of that ...
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  • 193
5 votes
1 answer
132 views

Different definitions of grammar complexity

It's known that there are different "kinds" of grammar complexity of language $L$ --- nonterminal complexity (minimal possible $|N|$ for grammar $(N, \Sigma, P, S)$ generating $L$), covering ...
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  • 411
5 votes
1 answer
91 views

Nonterminal descriptional complexity of regular languages

Recently I became interested in grammar complexity of regular language. Prior to searching for literature, I tried to investigate it on my own, proving two lemmas from comment below. I am aware of an ...
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  • 411
7 votes
0 answers
100 views

Deciding whether DCFG is visibly pushdown

Is the following problem decidable? If so, what's the best algorithm known? Instance: a deterministic pushdown automaton $A$ Question: Does there exist (i) some partition of the alphabet into push, ...
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  • 907
3 votes
1 answer
141 views

Generalizations of Dyck languages?

The "narrowest" generalization of Dyck languages that I am aware of is Visibly Pushdown languages. Are there any useful classes of languages that are intermediate between Dyck languages and ...
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  • 907
3 votes
0 answers
99 views

Deciding whether an arbitrary context-free grammar generates a deterministic push-down automata?

I know that it's undecidable whether an arbitrary context-free grammar is ambiguous, but is it decidable whether that grammar is deterministic? I can't find the answer to this question anywhere on the ...
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7 votes
1 answer
130 views

Lengths of "all-accepted" words in Context Free languages

If $L$ is a Context Free language, it can happen that for some $n$, all words of length $n$ are in $L$. If we consider the set $A_L$ of such lengths represented in unary, we may guess that such set is ...
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10 votes
3 answers
883 views

Maximum shortest word accepted by pushdown automata

Given a fixed alphabet, consider all deterministic pushdown automata with $n$ states that accept a nonempty language. What is the maximum length of the shortest word accepted by a deterministic ...
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  • 907
4 votes
1 answer
423 views

Are endmarkers necessary for Deterministic Pushdown Automata?

In the book by Kozen (Automata and Computability), the transition function of deterministic pushdown automata (DPDAs) is supposed, in contrast with non-deterministic pushdown automata (NPDAs), to ...
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5 votes
1 answer
160 views

Are PDAs without ϵ moves and with bounded stack operation as powerful as PDAs with them?

It is known that PDAs without $\epsilon$ moves are as powerful as PDAs with them. However, it seems to me that the proof allows several stack operations in one move. What happens if we allow at most ...
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  • 13.5k
1 vote
0 answers
71 views

Sentences in what kinds of grammar in the Chomsky hierarchy can be parsed by an LSTM of a given size?

Given an LSTM $N$ of a given size $A$, a sentence $S$ with a given number of words $B$, a Chomsky grammar hierarchy level $C$ in 0-3, a Chomsky grammar $G$ of level $C$ of size $D$, A given fixed, ...
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2 votes
0 answers
61 views

A Context-Sensitive Grammar which cannot be recognised by a Parsing Expression Grammar

It is (currently) an open question of whether every context-free grammar can be recognised by some parsing expression grammar. [1] However, has it been proven that there exists an example of a ...
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  • 121
21 votes
2 answers
2k views

Languages that we cannot (dis)prove to be Context-Free

I'm looking for languages which are "probably not Context-Free" but we are not able to (dis)prove it using known standard techniques. Is there a recent survey on the subject or an open problem ...
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3 votes
1 answer
149 views

Compressing grammars by introducing ambiguity and left-recursion

This is a reference request. What is known about the following questions? Problem: Given a grammar $G$ (for example context-free) with language $L$ we can introduce a new grammar $G'$ which also ...
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1 vote
0 answers
58 views

Is there an unambiguous grammar that has no left recursion or left factors, but is not in $LL(1)$?

I know that, for a grammar $G$ to belong to $LL(1)$, it is necessary that $G$ is not ambiguous; that is, every sentence has a unique parse tree in $G$. $G$ has no left recursion; that is, we can't ...
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13 votes
1 answer
292 views

Is { ww' | HamDist(w,w')>1 } context-free?

After reading the recent question "Is the complement of $\{ www \mid ...\}$ context-free?"; I remembered a similar problem I wasn't able to disprove: Is $L = \{ ww' \mid w,w' \in \{0,1\}^* \land |w|...
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9 votes
3 answers
471 views

Continuous mathematics and formal language theory

Whether there are some results on solving formal languages problems using mathematical analysis, continuous mathematics. For example, solving the intersection non-emptiness problem for a context-free ...
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  • 91
5 votes
1 answer
189 views

For which $R$ is $\{0^a10^b10^c\mid R(a,b,c)\}$ context-free?

Unless I'm mistaken, a language of the form $\{0^a10^b\mid R(a,b)\}$ is context-free if and only if $R$ is a finite union of linear (in)equalities involving integer constants and the variables $a$ and ...
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  • 13.5k
12 votes
3 answers
468 views

Is the complement of { www | … } context-free?

It is well-known that the complement of $\{ ww \mid w\in \Sigma^*\}$ is context-free. But what about the complement of $\{ www \mid w\in \Sigma^*\}$?
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  • 13.5k
6 votes
1 answer
286 views

Size of complement of context-free language

Let $L$ be a context-free language, $\bar L$ be its complement and $\bar L_n$ be the length $n$ words in $\bar L_n$. What is known about $|\bar L_n|$? Note that it is known that $|L_n|$ is either ...
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  • 13.5k
8 votes
0 answers
101 views

Does ${\bf CFLPAD}={\bf PPAD}$?

What happens if we define ${\bf PPAD}$ such that instead of a polytime Turing-machine/polysize circuit, a (non-)deterministic finite/push-down automaton encodes the problem? I asked a similar ...
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  • 13.5k
4 votes
1 answer
102 views

What is the current state of the art in black-box grammar induction?

Grammar induction of Context Free Languages seems to be a very well researched field. I would like to know the current state of the art in inducing a Context Free Grammar (I am reading up Higuera's ...
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20 votes
1 answer
2k views

Number of words of length n in a context-free language

Denote by $w_n$ the number of words of length $n$ in a (possibly ambiguous) context-free language. What is known about $w_n$? I'm sure this has been studied a lot, but I couldn't find anything at ...
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  • 13.5k
6 votes
1 answer
161 views

Example of context-free tree language which can not be generated by monadic CFTG

Assuming that a context-free tree language (CFTL) is that which is generated by a context-free tree grammar (CFTG), I am looking for an example of CFTL which can not be generated by a monadic CFTG (...
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1 vote
0 answers
62 views

What is the interpretation of an infinite formal context-free grammar?

Let $L$ be a language as follows: $$ \begin{align*} L &::= a\ |\ L^{*}\\ \end{align*} $$ Now, suppose I apply some sort of transformation $T : N \rightarrow N$ where $N$ is the set of non-...
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  • 309
5 votes
2 answers
187 views

Reducing the Height of Context-Free Grammars

Let $G$ be a context free grammar in Chomsky normal form (CNF) with language $L(G)\subseteq \Sigma^n$. In other words, all strings generate by $G$ have size $n$. Say that a string $w\in L(G)$ has ...
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5 votes
1 answer
1k views

What is the complexity of counting parse trees?

A Counting Problem Given a CFG $G$ and a string $s$, how many distinct parse trees are there for the string $s$? An Example Instance Let's consider an example instance consisting of a CFG $G$ with ...
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9 votes
0 answers
677 views

Is it known if $\mathrm{CFL} \subseteq\mathrm{ NSPACE}(o(log^2(n)))$?

$\mathrm{CFL}$ is the class of context-free languages. Question Is $\mathrm{CFL}$ known to be solvable in $o(log^{2}(n))$ non-deterministic space? What about $\mathrm{DCFL}$?
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6 votes
1 answer
146 views

Is this generalization of context free grammars known and strict?

Let $\Sigma$ be an finite alphabet and $(N, \circ)$ a semigroup. The semigroup operation on $N$ can be extended to $\mathscr{P}(N)$: $N_1 \circ N_2 := \{ \; n_1 \circ n_2 \; | \; n_1 \in N_1, \; n_2 \...
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  • 389
5 votes
0 answers
161 views

"Context" understanding in tree grammars

The Context-Free tree grammar has rules of the form: $A\rightarrow t$ or $A(x_1,\dots,x_n)\rightarrow t_x$, where $A\in N$, $t\in T(N\cup T)$, $t_x\in T(N\cup T\cup \{x_1,\dots,x_n\})$, $T(Z)$ ...
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6 votes
1 answer
1k views

Show that minimal CFG is undecidable via mapping reduction

Actually the problem below is an exercise in Sipser's textbook (Problem 5.36). However, from my perspective, it is not so trivial so that I post this question on this site instead of CS.SE. The ...
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  • 395
2 votes
0 answers
112 views

Is scalable hardware support for LogCFL (= sAC^1) possible?

The (uniform) circuit classes $TC^0$, $NC^1$ and $sAC^1$ seem to lend themselves to efficient hardware implementation. But using an FPGA approach to create the circuits on the fly seems problematic, ...
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2 votes
1 answer
228 views

Deformation of finite regular languages [closed]

Let $L \subseteq \{0,1\}^n$ be any finite regular language s.t it has an acyclic DFA. Let $C$ be some class of acyclic DFAs. Let $\sigma \in S_n$ be a permutation on $n$ symbols. We can apply $\...
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  • 31
19 votes
1 answer
1k views

Is equivalence of unambiguous context-free languages decidable?

It is well known that the equivalence problem is undecidable for general context-free languages. However, all proofs of this fact that I am aware of seem to involve some ambiguous context-free ...
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19 votes
0 answers
734 views

Why is the Pumping Lemma sometimes called Bar-Hillel's Lemma?

There are several papers in the literature that refer to the Pumping Lemma for context free languages as Bar-Hillel's Lemma (for example, here, here, and on the Wikipedia page). However, the first ...
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5 votes
1 answer
142 views

Regarding proper form of production rules of Context-free tree grammars

Is it possible to describe Context-free Tree Grammar $G_t$ such that set of yields of its trees will coincide with Context-sensitive word language $a^nb^nc^n$? $\{a^nb^nc^n | n>0\}=\{Yield(t)|t\...
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3 votes
1 answer
239 views

Compression algorithms for low-complexity strings?

Let $s$ denote a string over a finite alphabet, $n_s = |s|$ be the length of $s$, and $n_s^{*}$ denote the minimum description size of $s$ under a given computational model (TM, CFG, etc.). Are there ...
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  • 35
13 votes
2 answers
1k views

Is SAT a context-free language?

I am considering the language of all satisfiable propositional logic formulae, SAT (to ensure that this has a finite alphabet, we would encode propositional letters in some suitable way [edit: the ...
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  • 454
17 votes
2 answers
288 views

A reference for a "more algebraic" approach to pushdown automata and CFLs?

In the Sakarovitch's book on automata theory, it is written in the introduction to the section on rationals in the free group that the material presented therein lays "the foundation of a truly ...
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16 votes
1 answer
244 views

Characterization of $L$ for which $\{ww : w \in L\}$ is not a CFL?

It is a standard proof in automata courses that for $L = \Sigma^\star$ and $|\Sigma| \ge 2$ that $S(L) = \{ww : w \in L\}$ is not a context-free language. It is also true that for any finite $L$, $S(...
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  • 545
3 votes
0 answers
175 views

Context-Sensitive Grammar characteristic properties

This question can look like some kind of puzzle, but it is actually part of more complex applied problem. Let's consider subspace of Context-Sensitive Grammars, which contains grammars which can not ...
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