Questions tagged [context-free]
The context-free tag has no usage guidance.
108
questions
4
votes
0
answers
155
views
Context Free Grammar For Complement Of { wwwww | ... } With Minimal Locality?
Definitions
Let $G$ be a context free grammar over an alphabet $\Sigma$ with non-terminals $V$.
Define the locality $l(G)$ as the length of the longest word in $(V \cup \Sigma)^*$ that has a ...
- 29
10
votes
0
answers
219
views
Smallest context-free grammar for powers of two summing to $2^k$
Let $k > 0$ and define an alphabet $\Sigma_k = \{ 2^0, 2^1, \ldots, 2^k \}$. Define:
$$P_k = \{ a_1a_2\cdots a_t \in \Sigma_k^* \mid \sum_i a_i = 2^k \}.$$
This is a finite language.
Question: Is ...
- 3,916
1
vote
1
answer
49
views
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 ...
5
votes
1
answer
161
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 $\...
- 151
9
votes
1
answer
200
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 $n \in N(L) \iff n+p \in N(L)$...
- 661
1
vote
1
answer
92
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 ...
- 11
9
votes
0
answers
89
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,...
- 7,485
2
votes
1
answer
146
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 ...
- 414
6
votes
1
answer
197
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 ...
- 29
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 $\...
- 41
0
votes
1
answer
104
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?
- 11
2
votes
0
answers
87
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 ...
- 193
5
votes
1
answer
134
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 ...
- 411
5
votes
1
answer
93
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 ...
- 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, ...
- 907
3
votes
1
answer
146
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 ...
- 907
3
votes
0
answers
107
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 ...
- 131
7
votes
1
answer
131
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 ...
- 22.3k
10
votes
3
answers
908
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 ...
- 907
5
votes
1
answer
461
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 ...
- 93
5
votes
1
answer
163
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 ...
- 13.7k
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, ...
- 107
2
votes
0
answers
62
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 ...
- 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 ...
- 22.3k
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 ...
- 10.4k
1
vote
0
answers
59
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 ...
- 81
13
votes
1
answer
295
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|...
- 22.3k
9
votes
3
answers
477
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 ...
- 91
5
votes
1
answer
191
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 ...
- 13.7k
12
votes
3
answers
470
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 $\{ www \mid w\in \Sigma^*\}$?
- 13.7k
6
votes
1
answer
287
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 ...
- 13.7k
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 ...
- 13.7k
4
votes
1
answer
106
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 ...
- 175
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 ...
- 13.7k
6
votes
1
answer
163
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 (...
- 635
1
vote
0
answers
63
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-...
- 309
5
votes
2
answers
191
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 ...
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 ...
- 4,910
9
votes
0
answers
685
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 $\mathrm{DCFL}$?
- 4,910
6
votes
1
answer
151
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 \...
- 29
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 $A\in N$, $t\in T(N\cup T)$, $t_x\in T(N\cup T\cup \{x_1,\dots,x_n\})$, $T(Z)$ ...
- 635
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 ...
- 395
2
votes
0
answers
116
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, ...
- 3,013
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 $\...
- 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 ...
- 465
19
votes
0
answers
745
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 ...
- 319
5
votes
1
answer
143
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\...
- 635
3
votes
1
answer
240
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 ...
- 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 ...
- 464
17
votes
2
answers
290
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 ...
- 465