23 votes
Accepted

Deciding emptiness of intersection of regular languages in subquadratic time

Simple answer: If there does exist a more efficient algorithm that runs in $O(n^{\delta})$ time for some $\delta < 2$, then the strong exponential time hypothesis would be refuted. We will prove a ...
user avatar
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 ...
user avatar
  • 10k
17 votes
Accepted

Regular versus TC0

Take $S_5$ as alphabet and $$L= \{ \sigma_1\cdots \sigma_n \in S_5^*\mid \sigma_1\circ\cdots\circ\sigma_n = \text{Id}\}$$ Barrington proved in [2] that $L$ is $\textrm{NC}^1$-complete for $\textrm{AC}...
user avatar
  • 992
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 $...
user avatar
  • 4,721
15 votes

Regular versus TC0

Regular languages with unsolvable syntactic monoids are $\mathrm{NC}^1$-complete (due to Barrington; this is the underlying reason behind the more commonly quoted result that $\mathrm{NC}^1$ equals ...
user avatar
14 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 ...
user avatar
  • 7,653
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 = \{...
user avatar
11 votes
Accepted

computing maximal bit density over a FSM

First, you mean "sup" rather than "max", because it is easy to construct examples of regular languages, such as 00(011)*00 where there is no max. (The sup may not be attained.) Second, by "FSM" I ...
user avatar
11 votes
Accepted

On the realisation of monoids as syntactic monoids of languages

It seems there is a paper answering this exact question, and even in the more general case of $\omega$-regular languages, but I cannot find an open-access version. If somebody finds a link without ...
user avatar
  • 7,653
11 votes

On the realisation of monoids as syntactic monoids of languages

In a more elementary way than Denis's answer, the following is extracted from Pippenger's "Theories of Computability", p.87, and immediate to check. Definition: Let $M$ be a monoid, and $Y \subseteq ...
user avatar
11 votes

On the realisation of monoids as syntactic monoids of languages

The terminology rigid seems to be relatively new compared to the term disjunctive used in the late 70's (and probably before, I didn't check for earlier references). A subset $P$ of a monoid $M$ is ...
user avatar
  • 4,721
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 ...
user avatar
  • 4,721
10 votes
Accepted

Who introduced nondeterministic M-automata, and proved that the finite ones recognize the rational sets over M?

According to J. Sakarovich, Elements of Automata Theory, p. 372, It seems that the result appears for the first time in [1]; hidden in a footnote of this seminal but difficult article, it was then ...
user avatar
  • 4,721
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 ...
user avatar
9 votes

Deciding whether a context-free language is regular

Regularity is decidable for DCFL, but it is undecidable for general Context-Free Languages. Regarding DCFL, I have two references (from Hopcroft+Ullman 79): A regularity test for pushdown machines, ...
user avatar
  • 1,502
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 ...
user avatar
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 ...
user avatar
  • 10.4k
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 $M_C$ ...
user avatar
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, ...
user avatar
8 votes
Accepted

Is it decidable whether the output length of a transducer is bounded by the input length?

The other contributor deleted his answer, maybe to let me extend my above comment, so here it is. Let $T$ be a possibly nondeterministic transducer, and $L$ be a regular language. Modify $T$ into a ...
user avatar
8 votes
Accepted

minimal finite automata given in-words and out-words

If your FSM is a DFA, then this is the Minimum Consistent DFA Problem, which is well known in the machine learning community. This problem is NP complete Gold1978 - Complexity of Automaton ...
user avatar
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 ...
user avatar
  • 4,721
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 ...
user avatar
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 ...
user avatar
7 votes

Complexity of intersection of regular languages as context-free grammars

This is a great question and it really lies within my interests. I'm glad that you asked it Max. Let $n$ DFA's with at most $O(n)$ states each be given. It would be nice if there existed a PDA with ...
user avatar
7 votes
Accepted

Do bounded-visit nondeterministic linear bounded automata recognize only regular languages?

A bit overkill, but: this paper shows (among other nice things) that non-deterministic Hennie transducers realize exactly the class of non-deterministic MSO-definable transductions. The latter have ...
user avatar
  • 550
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 ...
user avatar
  • 5,712
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 ...
user avatar
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 $B$ ...
user avatar
  • 4,721
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,...
user avatar
  • 7,653

Only top scored, non community-wiki answers of a minimum length are eligible