27
votes
Accepted
Number of words of length n in a context-free language
Every context-free language has either polynomial growth or exponential growth. In the notation of the question poser:
Either there is a polynomial $p$ so that $w_n\le p(n)$ for all $n$
Or there ...
- 5,762
19
votes
Accepted
Are DPDAs without a $\epsilon$ moves as powerful as DPDAs with them?
Perhaps I found some relevant information in:
Jean-Michel Autebert, Jean Berstel, Luc Boasson; Context-Free Languages and Pushdown Automata; Handbook of Formal Languages; 1997, pp 111-174
DPDAs ...
- 22.6k
15
votes
Languages that we cannot (dis)prove to be Context-Free
Another good one is the complement of the set $S$ of contiguous subwords (aka "factors") of the Thue-Morse sequence ${\bf t} = 0110100110010110 \cdots $. To give some context, Jean Berstel proved ...
- 6,947
12
votes
Languages that we cannot (dis)prove to be Context-Free
How about the language $L_{TP}$ of twin primes? I.e., all pairs of natural numbers $(p,p')$ (represented, say, in unary), such that $p,p'$ are both prime and $p'=p+2$? If twin primes conjecture is ...
- 10.2k
11
votes
Accepted
Size of complement of context-free language
From the proof that determining if a CFL ${L}$ = $\Sigma^*$ is undecidable, the set of strings $ID_0\#ID_1^R\#ID_2\#ID_3^R\#\ldots\#ID_t$ where $ID_0,ID_1,\ldots,ID_t$ is a list of the configurations ...
- 8,566
11
votes
Accepted
Is the complement of { www | … } context-free?
Still CFL I believe, with an adaptation of the classical proof. Here's a sketch.
Consider $L = \{xyz : |x|=|y|=|z| \land (x \neq y \lor y \neq z)\}$, which is the complement of $\{www\}$, with the ...
- 3,936
10
votes
Accepted
What is the state complexity of the copy language?
The technique described by Yuval:
Do there exists polynomial size CFG that describe this finite language?
(
you may also read:
Lower bounds on the size of CFGs for specific finite languages
)
...
- 1,085
10
votes
Accepted
Known and described subclasses of Context-Free Grammars class
Density might be interesting concept for you. The density function is defined as
$$\delta_L(n) := |L\cap \Sigma^n|,$$
where $\Sigma^n$ denotes the set of all strings of length $n$ over $\Sigma$.
...
- 452
10
votes
Accepted
Is equivalence of unambiguous context-free languages decidable?
This is currently an open problem.
As correctly pointed out, if it is decidable, then one expects the proof to be hard since it generalises the famous DPDA equivalence problem.
On the other hand, the ...
- 118
10
votes
Is the complement of { www | … } context-free?
Here is the way I think about solving this problem, with a PDA. In my opinion, it's intuitively clearer.
A word $x$ is not of the form $www$ iff either (i) $|x| \not\equiv 0$ (mod 3), which is easy ...
- 6,947
9
votes
Accepted
A reference for a "more algebraic" approach to pushdown automata and CFLs?
Sakarovitch's PhD thesis from 1976, titled Monoïdes syntactiques et languages algébriques (syntactic monoids and algebraic languages), revolves around this topic. At the time, this led to the ...
- 3,936
9
votes
How is proving a context free language to be ambiguous undecidable?
The answer by apolge presents the proof that it is undecidable whether an arbitrary context free grammar is ambiguous. The question of whether a context free language is inherently ambiguous is a ...
9
votes
Maximum shortest word accepted by pushdown automata
The precise answer depends on your model of PDA (models differ among different authors; compare Sipser to Hopcroft &Ullman). And number of states alone is not a good measure for PDA's, because ...
- 6,947
8
votes
Does there exist a hardest DCFL?
The paper
J.-M. Autebert, Une note sur le cylindre des langages déterministes,
Theoretical Computer Science 8 (1979), 395-399
gives a short proof of the following result (credited to Greibach) ...
- 4,761
8
votes
Does there exist a hardest DCFL?
There actually is a hardest DCFL, which is a deterministic version of Greibach's; it was introduced by Sudborough in 78 in On
deterministic
context-free
languages,
multihead
automata,
and
the
...
- 3,936
8
votes
Does there exist a hardest DCFL?
An identical homomorphism characterization of DCFL does not seem to be possible. The following is extracted from Greibach's original paper.
We show that every context-free language can be expressed ...
8
votes
Accepted
Are endmarkers necessary for Deterministic Pushdown Automata?
Short answer: it depends on how you set the acceptance condition of the DPDA model: final state or empty stack.
The endmarkers are not necessary for DPDAs in which the accept condition is final state ...
- 22.6k
7
votes
Accepted
Is SAT a context-free language?
Just an alternative proof using a mix of well known results.
Suppose that:
variables are expressed with the regular expression $d = (+|-)1(0|1)^*$
and that the (regular) language (over $\Sigma = \{0,...
- 22.6k
7
votes
Maximum shortest word accepted by pushdown automata
(Answer inspired by Lamine's comment)
We assume the automaton is only allowed to push one symbol per state (otherwise, you could make the stack arbitrarily large with only two states). With a stack ...
- 917
7
votes
Lengths of "all-accepted" words in Context Free languages
The shortest word in $A_L$ is not bounded by a recursive function in the size of a given context-free grammar describing $L$. See here for more results in that direction:
https://doi.org/10.4230/...
- 5,759
6
votes
Known and described subclasses of Context-Free Grammars class
Your two grammars seem very similar. They are both linear grammars in two non-terminals. (Morally one, really -- in both examples the language of S is contained in the language semiring generated by ...
- 406
6
votes
Accepted
For which $R$ is $\{0^a10^b10^c\mid R(a,b,c)\}$ context-free?
What you're looking for is an old result of Ginsburg and Spanier actually related to one of the oldest open questions of the field. See Ginsburg's book The Mathematical Theory of CFLs.
Defs. A ...
- 3,936
6
votes
Continuous mathematics and formal language theory
Lamine commented on the connection to the Chomsky-Schützenberger enumeration theorem. Recently, a few research problems in formal language theory were solved using continuous mathematics via this ...
- 5,759
5
votes
Is SAT a context-free language?
If the number of variables is finite then so is the number of satisfiable conjunctions, so your SAT language is finite (and hence regular). [Edit: this claim assumes the CNFSAT form.]
Otherwise, let'...
- 10.2k
5
votes
A reference for a "more algebraic" approach to pushdown automata and CFLs?
In addition to the references given by Michaël Cadilhac, let me add this paper
Berstel, J.; Boasson, L. Towards an algebraic theory of context-free languages. Fund. Inform. 25 (1996), no. 3-4, 217--...
- 4,761
5
votes
Is the complement of { www | … } context-free?
Just a different ("grammar oriented") perspective to prove that the complement of $\{ w^k \}$ is CF for any fixed $k$ using closure properties.
First note that in the complement of $\{ w^k \}$ there ...
- 22.6k
5
votes
Continuous mathematics and formal language theory
One of the first connections is via generating functions. The Chomsky-Schützenberger theorem states that the generating function of the number of words of a unambiguous CFL is algebraic. In his paper, ...
- 1,137
5
votes
Accepted
Maximum shortest word accepted by pushdown automata
Counter Automata
I was a co-author for a paper where we investigated this problem for counter automata. We were able to show that the length of a shortest string accepted by an $n$-state (non-empty) ...
- 5,065
5
votes
Accepted
Generalizations of Dyck languages?
I believe that a type of automata that your are looking for could be "visibly counter automata". It is easy to imagine what they should be - they are just standard automata but equipped with ...
- 1,185
5
votes
Accepted
Does a finite, polynomially-bounded CFG translate into a polynomially-bounded DFA?
The sizes $|Q_n|$ can grow exponentially even in the context of question 1 (and thus questions 2 and 3 as well).
For $n = 2 k$ even, define the grammar $G_n$ of size $O(n)$ by
$$
\begin{align*}
S \to {...
- 661
Only top scored, non community-wiki answers of a minimum length are eligible