10
votes
Accepted
Can we efficiently enumerate the words accepted by a DFA by order of increasing weight?
EDIT: Added Lemma 2 which covers all cases asked about.
Lemma 1. Given a DFA with alphabet $\{0,1\}$ and an integer $n$, it is possible to enumerate all length-$n$ words in the language of the DFA, ...
- 10.9k
9
votes
Accepted
Is DFA language inclusion decidable in quasi-linear time?
EDIT: the question has an answer by Michael Wehar. A better than quadratic running time contradicts the strong exponential time hypothesis.
https://cstheory.stackexchange.com/a/29166/2367
ORIGINAL ...
- 6,665
8
votes
Accepted
Error in Robson's proof about separating strings?
Converting my comment to an answer:
This is taken care of in the paragraph just before Theorem 1. If the accepting state is reached at some $v_j$, then we necessarily have $j>i$, and we can compose ...
- 881
8
votes
Accepted
Number of minimal DFAs of size at most $m$?
According to Ishigami Y., Tani S. (1993) The VC-dimension of finite automata with $n$ states,
http://link.springer.com/chapter/10.1007/3-540-57370-4_58
,
the VC-dimension of the concept class of $n$-...
- 10.6k
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
6
votes
Accepted
Finite Automata with succinct representation of chains of states
This is a very partial answer, but I have some ideas:
Clearly the union of NSAs can be taken without any blowup - just use the nondeterministic union of the initial states.
As for determinization, ...
- 5,616
6
votes
Accepted
Separating words with random DFAs
It appears, via code, that if you take a random string $x$ and then form $y$ by flipping only the first bit of $x$, then a random DFA on $n/5$ states fails to separate $x,y$ with high probability. So, ...
- 321
6
votes
NFA to 2DFA: what are the upper and lower bounds?
The recent survey Two-Way Finite Automata: Old and Recent Results by Pighizzini states in the introduction:
The costs of the simulations of 1NFAs by 2DFAs and of 2NFAs by 2DFAs
are still unknown. ...
- 6,665
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
Bounds on this Strategy for Separating Words
The second section of Robson's "Separating strings with small automata" proves $F(n) = O((n \log n)^{1/2})$. The string sequence $(10^n)^n$ gives a lower bound of $\Omega(n^{1/2})$. If the ...
- 176
5
votes
Relationship between size of Boolean functions and DFAs
Complementing the other answers, here are a few research papers that explicitly study the size of (one-way) DFAs that represent Boolean functions in the way the OP describes.
Maximum and average state ...
- 6,665
5
votes
Relationship between size of Boolean functions and DFAs
Regarding question 3:
There are $S^{2S} \cdot 2^S$ different DFAs on $S$ states (fixing the initial state), and so most Boolean functions require $\Omega(2^n/n)$ states. This is the same calculation ...
- 14.5k
5
votes
Relationship between size of Boolean functions and DFAs
Here are are my attempts to answer. I'm not an expert on this subject. Please check all details for yourself.
No. Consider $f$ defined by $f(x)=1$ iff $x_1 \ne x_{n/2+1}$ or $x_2 \ne x_{n/2+2}$ or ...
- 12.3k
5
votes
Random Cerny Conjecture
I think this problem has little to do with Cerny's conjecture. There the problem is to find a word that works for every pair of states. Here it is enough to show that the word will work whp. for any ...
- 14.1k
5
votes
2DFA to 1DFA - Converting two way deterministic finite automata to one way deterministic finite automata
Kozen presents a constructive proof of the equivalence of 1dfa and 2dfa in a chapter of his book titled "Automata and Computability". If I recall correctly, it is a standard argument and the ...
- 361
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
Accepted
Oncina-Garcia RPNI algorithm for learning DFAs
The algorithm is named RPNI, not RNPI.
Given that the language generating the inputs is regular and that enough examples are given (the characteristic set), the algorithm returns the canonical (i.e., ...
4
votes
Accepted
2DFA to 1DFA - Converting two way deterministic finite automata to one way deterministic finite automata
First, a warning: this will involve exponential $(2^{n \log n})$ blowup in the number of states (see here). However, if your application is fine with computing the states of the DFA "on the fly" then ...
- 892
4
votes
Number of minimal DFAs of size at most $m$?
(NB: the upper bound given in the accepted answer is better or equal to the one given here)
An upper bound is proposed in this paper given in one of the previous comments: “On the number of distinct ...
- 427
3
votes
Finite Automata with succinct representation of chains of states
A model that seems somewhat relevant is capacitated automata:
http://www.cs.huji.ac.il/~ornak/publications/fsttcs14b.pdf
- 1,225
3
votes
Accepted
Is Bayes optimal RL of a finite set of DFAs feasible?
Even determining whether there is a policy which always succeeds is NP-complete, by a reduction from constructing optimal decision trees (Hyafil and Rivest, Constructing Optimal Binary Decision Trees ...
- 376
2
votes
Separating words and graph isomorphism
I hadn't thought about the Separating Words problem before, but based on looking at the Robson paper referenced in the linked answer, I don't immediately see how techniques from GI would apply.
At a ...
- 37.8k
2
votes
Can we efficiently enumerate the words accepted by a DFA by order of increasing weight?
After discussing it further with a3nm, I propose an algorithm that is
different than Neal's algorithm and works in a more general
setting. The approach gives polynomial delay algorithm but it uses
...
- 2,174
2
votes
Accepted
What is the current state of the art on exact identification of DFAs with a maximum N states
The last step of the proposed reasoning can be done as described in https://cs.stackexchange.com/questions/48136/testing-two-dfas-generate-the-same-language-by-trying-all-strings-upto-a-certain and in ...
- 408
1
vote
I have to make a dfa over the alphabet Σ = { 0, 1, 2 } of strings that end with the same digit twice; e.g., strings that end in 00, 11, 22
this forum is meant for research level questions. Your question seems to be an exercise to a first course on formal languages and would fit more in the following forum https://cs.stackexchange.com/
...
- 775
1
vote
Accepted
Counting the different subsets of nodes seen when iterating a subset through a directed graph
The argument in Chrobak’s paper can be applied to this problem as well, with the same bounds.
Let $\{D_i:i<k\}$ be the set of strongly connected components of $G$ that contain a cycle (i.e., other ...
- 18.2k
1
vote
A conjecture related to the Cerny conjecture - counterexample/reference request
I found a partial answer to question 2. The same idea is discussed in the last 10 minutes of this lecture.
- 91
1
vote
When designing a DFA, am I allowed to design two separate Machines and perform an Intersection on them?
Thanks to the hint given by @Micheal Wehar, I looked into Cartesian Product constructions and Intersections and I came across this answer:
https://stackoverflow.com/questions/7780521/how-to-use-the-...
- 17
1
vote
Non equivalent states shortest word
Use the pigeonhole principle. Let $w$ be the shortest differentiating word. Suppose its length is larger than the number of states of the DFA. Consider the sequence of states traversed on input $w$....
- 12.3k
1
vote
Finding self-similar homomorphisms of a FSM transducer
I guess I'm a few years late with this answer but here goes:
Every homomorphism $h:\Sigma\rightarrow\Sigma^{*}$ has a corresponding deterministic transducer with a single state $q$ (which is ...
- 361
Only top scored, non community-wiki answers of a minimum length are eligible