20
votes
Automata learning without counterexamples
Consider password automata: for each $w\in\{0,1\}^n$, the DFA $M_w$ accepts the language $\{w\}$. In this case, a membership query is the same as an equivalence query --- and clearly, you'll need ...
- 10.6k
17
votes
Accepted
Parameterized complexity of inclusion of regular languages
The particular case of language universality (are all words accepted ?) is PSPACE-complete for regular expressions or NFAs. It answers your question: in general the problem stays PSPACE-complete even ...
- 8,923
16
votes
Hierarchies in regular languages
Here is a list of several hierarchies of interest, some of which were already mentioned in other answers.
Concatenation hierarchies
A language $L$ is a marked product of $L_0, L_1, \ldots, L_n$ if
$...
- 5,101
12
votes
Hierarchies in regular languages
Expanding the comment: a natural hierarchy is the one induced by the number of states of the DFA.
We can define $\mathcal{L}_n = \{ L \mid \text{ exists an n-states DFA D s.t. } L(D) = L \}$
($D = \{...
- 23.1k
12
votes
Obscure characterizations of the regular languages
I know it is frowned upon to promote one's own results, but it turns out that I wrote an article precisely on this topic. So let me add a few characterizations of regular languages not already ...
- 5,101
11
votes
Existence of an algorithm
A language $L$ is said to be commutative if the following property holds:
for every word $a_1 \dotsm a_n \in L$ and any permutation $\sigma$ on
$\{1, \ldots, n\}$, the word $a_{\sigma(1)} \dotsm ...
- 5,101
10
votes
Accepted
Converting 2-ambiguous NFA to unambiguous NFA
I think that this is in fact not possible, thanks to the (recent and difficult!) ICALP'22 results of Göös et al..
They show that there are UFAs $A_1$ and $A_2$ with $n$ states such that the language $...
- 9,707
9
votes
Transition monoid membership for DFAs
Decidability
It's decidable. There are only finitely many possible functions $f:Q \to Q$, so you can model this as a graph reachability problem, with one vertex per function and an edge $g \to h$ if ...
- 12.3k
9
votes
In the context of regular languages, must the alphabet be finite?
It makes sense in some contexts in mathematics to consider strings or languages over infinite alphabets. For instance, this concept is used in the strong version of Higman's lemma. But a finite ...
- 51.1k
9
votes
Counting words accepted by a regular grammar
I think this is a hard counting problem, see this paper:
Counting the size of regular sequences of given length is #P-complete:
S. Kannan, Z. Sweedyk, and S. R. Mahaney. Counting
and random generation ...
- 161
9
votes
Accepted
Finding a minimal DFA whose language has a desired intersection with another
$M_C$ must accept every word of $S^+ = B$ and reject every word of $S^- = A \setminus B$.
Let $A$ and $B$ be finite and such that both $S^+$ and $S^-$ are non-empty. Then exact computation of $M_C$ ...
9
votes
Planarity of planar finite automata intersection
As mentioned in my comment, the usual product construction does not preserve planarity. In fact, there is an intersection of regular languages that can be described by a nonplanar NFA with $n$ states, ...
- 6,665
8
votes
Accepted
(N)DFA with same initial/accepting state(s)
This question is solved for deterministic automata and for unambiguous automata in the book [1]
[1] J. Berstel, D. Perrin, C, Reutenauer, Codes and automata, Vol. 129 of Encyclopedia of Mathematics ...
- 5,101
8
votes
In the context of regular languages, must the alphabet be finite?
The usual convention in formal languages and automata theory is that an alphabet is finite.
However, there are certainly some cases where it's useful to think of an alphabet being infinite. For ...
- 6,985
8
votes
Hierarchies in regular languages
I recently came across this paper which may give another relevant example (cf. the last sentence of the abstract):
Guillaume Bonfante, Florian Deloup: The genus of regular languages.
From the ...
- 5,598
8
votes
Accepted
The complexity of conversion from a regular expression to a nondeterminsitic automata and back after changing initial and final states
As observed in the proof of Theorem 6 (later dubbed the "Star Height Lemma") of Gruber/Holzer ICALP 2008, when converting a regular expression into an $\varepsilon$-NFA, then the underlying ...
- 6,665
7
votes
Accepted
Testing whether letters can be scheduled to achieve a word in a regular language
The problem is NP-hard for $L = A^*$ where $A$ is the finite language containing the following words:
$x111$, $x000$,
$y100$, $y010$, $y001$,
$00c11$, $01c10$, $10c01$, and $11c00$
The reduction is ...
- 2,883
7
votes
Complexity of checking if two words have an interleaving in a language
For a word $w=w_1\ldots w_{\ell}$ and for two integers $i,j$ with $1\le i\le j\le \ell$ we denote by $w(i,j)$ the subword $w_iw_{i+1}\ldots w_j$ of $w$.
Furthermore we let $w(0,0)$ denote the empty ...
- 5,792
7
votes
Accepted
Complexity of DFA intersection in this specific case?
The precise bound is $2^n$. The lower bound was given in the comments: the state complexity of $A^*a_1A^* \cap \dotsm \cap A^*a_nA^*$ is $2^n$.
For the upper bound, it suffices to observe that if $B$ ...
- 5,101
7
votes
Accepted
Kleene Algebra for star-free regular expressions
You might be interested in bounded synchronization delay expressions.
See [1] for details on these expressions.
To sum up, they are equivalent to star-free expressions, but instead of using complement,...
- 8,923
7
votes
Accepted
Is the function $f(a_1 \dotsm a_n) = a_1(a_1a_2)(a_1a_2a_3)\ \dotsm\ (a_1 \dotsm a_n)$ regularity-preserving?
Here is a proposition for an elementary proof:
Let $\mathcal A=(A,Q,q_0,F,\delta)$ be a DFA for $L$, we want to build a DFA $\mathcal A'=(A,Q',q_0',F',\delta')$ for $f^{-1}(L)$. Intuitively, when ...
- 8,923
7
votes
Accepted
Regular Expressions that converts into unambiguous automata
The paper Ambiguity in Graphs and Expressions (Book et al., 1971) discusses constructing regular expressions that preserve the ambiguity of the input NFA and vice versa.
That is, they give a ...
- 399
7
votes
Accepted
star height of star-free languages
The examples of arbitrary star-height given on the wikipedia page on the star-height problem are star-free:
On arbitrary alphabet:
:\begin{alignat}{2}
e_1 &= a_1^* \\
e_2 &= \left(a_1^*a_2^*...
- 8,923
6
votes
minimizing size of regular expression
It is PSPACE-complete to decide whether an expression accepts all words, i.e. is equivalent to $(a|b|c|...)^*$. It is not hard to get convinced that in this proof of PSPACE-completeness (see e.g. ...
- 8,923
6
votes
Accepted
Pumping Lemma over 2 way Automaton
Claim: If $M$ is a 2-way NFA (without endmarkers) with $n$ states
which acceptes a unary language $L\not=\emptyset$ then there is a word $w \in L$ of
length at most $3n+5$.
Proof sketch:
Let $v\in L$ ...
- 1,417
6
votes
Accepted
NFA to DFA Powerset Construction : A Partial determinization algorithm with trade-off between running time and size for the resulting automata?
The paper [HP06] is in the spirit of your idea, although in a different direction, in the context of infinite words.
It can be adapted more easily to finite words.
In the powerset construction, we ...
- 8,923
6
votes
Accepted
Existence of injective length-preserving rational function to a smaller alphabet
It turns out the answer is "no": in [2] the authors show that a regular language $L$ exists with density $\rho_L(n) \leq |B|^n$ such that no regular language over $B$ has the same density; ...
- 1,234
6
votes
Accepted
Is there a simple characterization of regular languages closed under circular shifts?
We can propose an automaton model characterizing regular circular languages: a C-automaton is an NFA where all states are initial. A run must see an accepting state somewhere, and must start and end ...
- 8,923
5
votes
Accepted
Automata : Language Containment, Minimality & Graph Homomorphism
For inclusion, using your condition that non-final states can be mapped to final states does not work. Consider for instance that $A$ is a rejecting sink $p_0$, and $B$ is the minimal automaton for ...
- 8,923
5
votes
Hierarchies in regular languages
There are several natural hierarchies for regular languages of infinite words, that convey a notion of "complexity of the language", for instance:
Number of ranks needed in a deterministic parity ...
- 8,923
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