Questions tagged [finite-automata]
235 questions
10
votes
1 answer
Maximum number of edges in a directed graph with the following "at most one path" condition
Let $G$ be a directed graph with $n$ vertices, and let there be vertices $s$ (source) and $t$ (sink) such that
Every vertex is reachable from $s$, and can reach $t$. (I.e., for every $v$, there is a path from $s$ to $t$ going through $v$.)
For…
Caleb Stanford
- 47,093
9
votes
5 answers
Number of functions $f:[4]\times[4]\rightarrow[4]$
Let $[k] = \{0,\dots,k-1\}$.
Consider the set $F(n,m)$ of functions $f:[n]\times[m]\rightarrow[m]$.
The cardinality of $F(n,m)$ is $|F(n,m)| = m^{nm}$.
Consider the equivalence relation $f \simeq g$ between functions $f,g \in F(n,m)$ iff
there are…
Hans-Peter Stricker
- 18,719
9
votes
1 answer
How many partial derivatives does a multivariate polynomial have?
Definition: My motivation for this question stems from the following definition: Define the deterministic finite automata generated by the nonconstant* polynomial $f(x_0 , \dots , x_n) \in \mathbb{Z} [x_0 , x_1 , \dots , x_n]$, denoted…
gxcc95
- 91
7
votes
2 answers
Enumeration of finite automata
There is a nice paper Enumeration of Finite Automata by Frank Harary and Ed Palmer which presents a formula $a(n,k,m)$ for the number of finite automata with $n$ states, $k$ input symbols and $m$ output symbols. It is stated in Corollary $3$ as
…
Markus Scheuer
- 112,413
6
votes
1 answer
Proving the set of all strings of an alphabet is countably infinite
Let say that you have the alphabet $\sum$ and you wanna prove that $\sum^*$ is countably infinite. If it's countable then we can list all the strings in a well defined manner, but I think that somehow we can always come up with a string that is…
TheMathNoob
- 2,101
5
votes
2 answers
Deterministic finite automata (DFA) (have odd length or end with aaa)
Is my attempt is true or where am I wrong?
DFA : The set of strings over $\{a, b\} $ that have odd length or end with $aaa$.
zxccxz
- 89
5
votes
1 answer
Is the empty string always in a finite alphabet?
Is the empty string always an element of an aribitrary finite alphabet?
I understand that the empty string is part of the Kleene-Star of any alphabet, but is it intrinsically part of any finite alphabet where I don't explicitly mention it?
For…
Jeremy Jeffrey James
- 2,625
4
votes
0 answers
NFA to DFA conversion
I am trying to convert the following NFA to an equivalent DFA:
My steps:
There is an $\varepsilon$-transition from $q_0$ to $q_1$, hence the set of initial states is $\{q_0,q_1\}$.
From $\{q_0,q_1\}$ we can go to any state if the input is $a$, but…
Galc127
- 4,491
4
votes
1 answer
Is C* regular if C is a language with strings of prime length?
Let $C = \{a^p \ | \ p \ \text{is prime}\}$ be a language. I was able to show that $C$ is not regular using the pumping lemma.
However, I am having some trouble showing that $C^*$ is regular. Intuitively I believe it should be regular but I do not…
TheSalamander
- 401
4
votes
2 answers
Finite automata as dynamical systems
In abstract (deterministic finite) automata theory the set of states of an automaton is an arbitrary set Q, and the transistion function is a specific set δ ⊆ Q × Σ × Q (with alphabet Σ, i.e. another set).
In a "real" automaton - seen as a…
Hans-Peter Stricker
- 18,719
4
votes
3 answers
Number of states in a finite automaton
How many states are required by a deterministic finite automaton to store $m$ words each of length $n$?
I came across $2^{mn}$ as the solution but there was no explanation.
Rishi kesh
- 53
4
votes
1 answer
Is the Champernowne constant an automatic number?
The Champernowne constant in base $b \geq 2$ is obtained by concatenating the $b$-ary expansion of every integer. For example, in base $10$ this is
$$
0.123456789101112131415\dotsc
$$
Question: Is the sequence of $b$-ary digits of the base $b$…
A.P.
- 9,906
4
votes
3 answers
Epsilon and phi in automata
If $L_1 = Ø$, I read that $L_1^\ast$ is $\{\varepsilon\}$. Why is this so? I am unable to understand this. Can someone please explain? I understand that $Ø$ is an empty string while $\varepsilon$ allows us to make transitions without symbols. But,…
3
votes
1 answer
Can a NFA reach two final states at the same time?
I'm studying nondeterministic finite automata, and I understand them in principle. Compared to deterministic FA, you can have more than one transition function with the same character starting from a single state and you can have "empty"…
Paul
- 221
3
votes
2 answers
DFA that accepts strings where there are odd number of 1's, and any number of 0's
DFA that accepts strings where there are odd number of 1's, and any number of 0's.
The alphabet $\Sigma=\{0,1\}$
Well since it's odd $1$'s, then there must be at least one 1.
So I think the regex is the following:
$$R=0^*10^*$$
This is correct…
K Split X
- 6,675