44 votes

Real computers have only a finite number of states, so what is the relevance of Turing machines to real computers?

To complete the other answers: I think that Turing Machine are a better abstraction of what computers do than finite automata. Indeed, the main difference between the two models is that with finite ...
  • 7,757
32 votes
Accepted

Real computers have only a finite number of states, so what is the relevance of Turing machines to real computers?

There are two approaches when considering this question: historical that pertains to how concepts were discovered and technical which explains why certain concepts were adopted and others abandoned or ...
  • 3,741
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 ...
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 ...
  • 10.1k
19 votes
Accepted

Quadratic relationship between nondeterministic and deterministic space?

In my paper with Domaratzki and Kisman, "On the number of distinct languages accepted by finite automata with n states" published in J. Automata, Languages, and Combinatorics 7 (2002) we proved that ...
18 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 ...
18 votes
Accepted

Is Hartmanis-Stearns conjecture settled by this article?

First, the name of the conjecture is "Hartmanis-Stearns", not "Hartmanis-Stearn". Second, the Hartmanis-Stearns conjecture concerns those real numbers computable by a multi-tape Turing machine in ...
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}...
  • 992
17 votes
Accepted

For which regular expressions $\alpha$ is $\{ \beta \mid L(\alpha) = L(\beta) \}$ PSPACE-complete?

This question is addressed in Section 2 of [1], which shows (Theorem 2.6) that the problem is in P if $L(\alpha)$ is finite; coNP-complete if $L(\alpha)$ is infinite but bounded (i.e. $L(\alpha)\...
  • 188
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 $...
  • 4,731
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 ...
15 votes
Accepted

Regular language that discriminates between two deterministic CFGs

Eryk Kopczyński[1] showed in 2015 that separability (that's the name of your problem) of visibly pushdown languages by regular languages is undecidable. The class of visibly pushdown languages is a ...
14 votes
Accepted

Are regular languages closed under addition?

Yes, they are. First, consider the alphabet $\Sigma_i^3$ whose symbols are triples of digits (stacked one above each other into a pile of three digits). Over this alphabet, we can define a regular ...
14 votes
Accepted

Novel proof of pumping lemma for regular languages

Essentially the same argument is made by Andries P.J. van der Walt (1976, Lemma 2.3 and Example 2.9) for the variant of the pumping lemma where $N$ letters are marked and all three of $x$, $y$, $z$ ...
  • 3,324
13 votes

Minimizing Automata accepting $\omega$-words (i.e. infinite words)

This question generated a lot of literature in the 80's, partly due to a bad approach to the problem. This is a rather long story that I will try to summarize in this answer. 1. The case of finite ...
  • 4,731
13 votes
Accepted

Complexity of the problem of words with fewest distinct letters accepted by a finite automaton

It is NP-hard for $k=3$. The reduction is from 3-SAT-(2,2), which means that every clause contains $3$ literals and every literal occurs in at most $2$ clauses. First of all, for simplicity, let's ...
  • 13.7k
13 votes

Determinising unambiguous automata without exponential blowup

No, the exponential lower bound for determinization holds already for unambiguous NFAs. This is obtained as follows: Consider the alphabet $\{a,b\}$, and the language: $$L_k=\{w\in \{a,b\}^*:\text{the ...
  • 5,316
12 votes
Accepted

Minimizing Automata accepting $\omega$-words (i.e. infinite words)

In general, $\omega$-regular languages may not have a unique minimal DBW. For example, the language "infinitely many a's and infinitely many b's" has two 3-state DBWs (in the picture replace $\neg a$ ...
  • 5,316
12 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 ...
  • 4,731
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 = \{...
11 votes

Conjecture about two counters automata

So people keep nagging me to post this even though it only solves a simplified version of the problem. Okay then :) At the end of this, I will put some of what I learned from the paper of Ibarra and ...
11 votes
Accepted

Can we approximate the number of words accepted by an NFA?

There exists a FPRAS (Fully Polynomial Randomized Approximation Scheme) for the problem of counting the words of length $n$ accepted by a NFA in the general case (without restricting to the acyclic ...
  • 521
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 ...
11 votes
Accepted

Finding the smallest DFA that separates two words without using brute force search?

If I had to do this in practice, I would use a SAT solver. The question of whether there is a DFA with $k$ states that accepts $x$ and rejects $y$ can be easily expressed as a SAT instance. For ...
  • 10.5k
11 votes
Accepted

Is there a survey of the field of quantum automata?

You can check the recent survey by Ambainis and Yakaryilmaz: Automata and Quantum Computing. It is comprehensive and points the essential literature with some open questions. Moreover, here is a list ...
11 votes
Accepted

Simplest Machine Model Accepting $L = \{ww^Rw\;|\; w\in \Sigma^*\}$

You don't need nondeterminism or multiple heads. Even a 2DPDA can accept this language: push 2 counters per symbol while scanning from left endmarker to right; then pop 3 per symbol while scanning ...
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 ...
  • 7,757
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 ...
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 ...
  • 4,731
11 votes

NP-complete decision problems on deterministic automata

The decision version of the DFA identification problem (find a possibly non-unique smallest DFA that is consistent with a set of given labeled examples) is NP-complete: Input: Integer $k$ and sets $...

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