Questions tagged [automata-theory]
The automata-theory tag has no usage guidance.
105
questions
31
votes
2
answers
1k
views
Group theory with grep?
While reading Bill Thurston's obituary in the Notices of the AMS I came across the following fascinating anecdote (pg. 32):
Bill’s enthusiasm during the early stages of mathematical discovery was ...
27
votes
1
answer
1k
views
Automatic groups - recent progress
Epstein's (et al.) "Word Processing in Groups" is a quite comprehensive monograph on automatic groups, finite automata in geometric group theory, specific examples like braid groups, fundamental ...
18
votes
2
answers
694
views
Is Post's tag system solved?
Has the 3-tag system investigated by Emil Post $(0\to00, 1\to1101)$ been solved? Is there a decision algorithm to determine which starting strings terminate, which end up in a cycle, and which (if any)...
13
votes
3
answers
856
views
Complementation of $\omega$-regular languages in reverse mathematics
Does anyone know where Büchi's theorem that $\omega$-regular languages are closed under complementation fits into the reverse-mathematics classification scheme? That is, is it equivalent over $\...
12
votes
5
answers
3k
views
Generating function of a regular language
It is well known that the generating function of a regular language $L$, i.e. $\sum n_kz^k$ where $n_k$ is the number of words of length $k$ in $L$, is rational, i.e. a quotient of two polynomials $P(...
12
votes
0
answers
268
views
Eilenberg's rational hiererchy of nonrational automata & languages — where is it now?
In the preface to his very influential books Automata, Languages and Machines (Volumes A, B), Samuel Eilenberg tantalizingly promised Volumes C and D dealing with "a hierarchy (called the rational ...
11
votes
6
answers
3k
views
Regular languages and the pumping lemma
Let's say that I want to prove that a language is not regular.
The only general technique I know for doing this is the so-called "pumping lemma", which says that if $L$ is a regular language, then ...
11
votes
1
answer
323
views
Unique words in dihedral groups
Suppose $x$ is a word over the alphabet $\{0,1\}$.
Let $a$, $b$ be elements of the group Dih$_k$ for some $k$.
Let $\varphi=\varphi_{a,b,k}$ be the map from words over $\{0,1\}$ to elements of the ...
8
votes
2
answers
2k
views
Isomorphism in category of finite automata
What does meanthat two finite automata is equivalent? I think that we must define category of finite automata, i.e. we must define $\mathrm{Hom}(A,B)$, where $A,B$ be an arbitrary finite automata. ...
8
votes
3
answers
2k
views
Wolfram's 2-state 3-symbol Turing machine
A few years ago it was announced that a 2-state symbol Turing machine was proven to be universal. However, Vaughn Pratt disputed the proof, and I gather he still disputes it. Wolfram's prize committee ...
8
votes
2
answers
919
views
A special class of regular languages: "circular" languages. Is it known?
We can define a subclass of the regular languages. Fix an alphabet $\Sigma$. Define the "circular" languages (actually, the name already exists to denote a different thing it seems, used in the field ...
7
votes
3
answers
3k
views
Is there an algorithm that can "reverse engineer" a Regular Expression?
Given a Regular language (represented as a black box to which one can apply inputs and get 0/1) Is there an algorithm that can find a finite deterministic automaton that produces that language?
7
votes
4
answers
923
views
Origin of tropical mathematics
On Wikipedia, it is claimed without a source that Imre Simon founded tropical mathematics.
The first work of his I was able to find on the subject is Limited subsets of a free monoid which uses the ...
7
votes
2
answers
617
views
Can you hide a letter without losing information?
Consider the following game between Alice and Bob.
$\Sigma$ is a finite nonempty alphabet, $\Delta \notin \Sigma$ denotes
a special symbol, and $k > 0$ is a positive integer constant representing
...
7
votes
1
answer
129
views
Generalising the adherence operator and its closure properties with regard to regular (rational) languages
Let $X$ be an alphabet and denote by $X^{\omega}$ the set of all infinite sequences (i.e. words) in $X$. A subset $L \subseteq X^{\omega}$ is called $\omega$-regular if it is acceptable by some Büchi-...
6
votes
2
answers
1k
views
Rabin's Tree Theorem
I've been reading Rabin's article on decidability in Barwise's text, and I came across Rabin's discussion of the decidability proof of his tree theory: the second-order theory with two successor ...
6
votes
1
answer
504
views
Growth zeta-functions of regular languages
Dear All,
my following question may be known and ought to be known, so in case it is folklore please could you give me the references.
To start, it is obvious that growth of rational languages are ...
6
votes
1
answer
523
views
K-fellow traveler property and automatic structure
I have been reading several articles about automatic groups and metric spaces of negative curvature. However it is not clear for me the relationship between automatic groups, hyperbolcity and the k-...
6
votes
1
answer
181
views
Embedding Turing machine [closed]
I have some questions about Turing machines. Is there an embedding method where you embed Turing machines, finite automata into continuous space or graphs? Or are there geometrical approaches to ...
6
votes
2
answers
586
views
Deterministic finite-state automaton driven by a Markov chain
I've stumbled on some problem, and I have the feeling that this is closed to something well-studied in dynamical systems. The problem is the following. Consider a finite-state automaton with state ...
6
votes
1
answer
161
views
Separating infinite words sharing factors by automata
Two infinite words $\xi, \eta \in X^{\omega}$ are separated by an (Büchi-)automaton if it accepts one but not the other.
Denote by $F_n(\xi)$ the factors of length $n$ of an infinite word $\xi$ and ...
6
votes
0
answers
111
views
Are "germ" automata studied?
I've been exploring the idea of a nondeterministic continuous automaton based on germs:
Two functions $f,g: \mathbb{R} \to S$ have the same right germ at $x$ if there is some interval $[x,a)$ on which ...
6
votes
0
answers
554
views
Computing the pro-solvable closure of a finitely generated subgroup of a free group
The pro-solvable topology on a group $G$ is the unique group topology such that the set of normal subgroups $N\lhd G$ with $G/N$ a finite solvable group is a fundamental system of neighborhoods of the ...
6
votes
0
answers
249
views
"Question-answer" bisimulation
I often come across relations that would be defined as a bisimulation, except that the label match can be "inexact", that is, in the bisimulation game, a move labelled with "a" can be replied to with "...
5
votes
1
answer
143
views
Computations with conetypes of hyperbolic groups
I'd like to know if there exists (and, in this case, where I can find it) some computer program/programming language/any kind of software that can find explicitly the conetypes of a hyperbolic group ...
5
votes
1
answer
304
views
Is it decidable whether the support of a rational $\mathbb{Z}$-series is a regular language?
Let $S \in \mathbb{Z}\langle\langle A\rangle\rangle$ be a rational series in noncommutative variables. The support of $S$ is the set of all words $u \in A^*$ such that $(S, u) \not= 0$. It is ...
5
votes
2
answers
531
views
Neighbourhood of a word and Levenshtein distance
The Levenshtein distance or Edit distance $$ lev(U,V) $$ between two strings $U$ and $V$ over a finite alphabet $\Sigma$ of size $ \left| \Sigma \right| = \sigma ,$ is the minimal number of insertions,...
5
votes
1
answer
381
views
Rabin's proofs of emptiness and complementation problems for automata on infinite trees
I have originally asked this question on Math.SE, but I think it is more suitable here.
I have been reading M. Rabin's 1969 article Decidability of Second-Order Theories and Automata on Infinite ...
5
votes
0
answers
237
views
A problem on automatic groups and geodesic paths on the Cayley graph
Let $\Gamma = \langle S \mid R \rangle$ be a finitely generated group, with the neutral element $e \not \in S= S^{-1}$.
Let $\ell : \Gamma \to \mathbb{N}$ be the world length related to $S$.
For ...
5
votes
0
answers
319
views
Büchi automata with acceptance strategy [closed]
I have already asked this question on cstheory.stackexchange, but without success. Maybe it is too close to an "open problem", although it is not a famous one. Anyway I try here, I can ...
4
votes
2
answers
2k
views
Are context-free languages with context-free complements necessarily deterministic context-free?
Let $L \subseteq A^\star$ be a formal language over $A$ generated by a context-free grammar, and $L' = A^\star - L$ be the relative complement in $A^\star$.
If $L$ and $L'$ are both context-free, are ...
4
votes
1
answer
121
views
Multi-head two-way finite automata versus logarithmic space
It is known that the languages decided by logarithmic-space Turing machines are exactly those decided by finite automata with multiple, bidirectional (2-way) scanning heads. Where could I find a proof?...
4
votes
1
answer
416
views
Giving the same concept different names in the same paper
I found a seminal paper of renowned authors (Inference of Finite Automata Using Homing Sequences (1993) by Ron Rivest and Robert Schapire) in which the authors define the very same set-theoretic ...
4
votes
1
answer
165
views
Subsets of $\omega$-regular lanuages accepted by automata with special acceptance condition
Let $\mathcal A = (X, Q, \delta, q_0, F)$ be a deterministic finite automata with the following acceptance condition on infinite words:
The automata accepts $\xi \in X^{\omega}$ with respect to $F$ ...
4
votes
1
answer
2k
views
Deciding equivalence of regular languages
Given two regular expressions $R$ and $S$ on an alphabet $\Sigma$ it is possible to decide their equivalence as follows:
build two finite automata $M_R$ and $M_S$ such that $L(R) = L(M_R)$ and $L(S) =...
4
votes
1
answer
128
views
Can one reduce to 'reversing' the right multiplier finite-state automata of an automatic group to obtain a biautomatic structure?
Let $\left( G, A, W, \left\{ R_{a} \right\}_{a \in A \cup \{ 1 \}} \right)$ be a group equipped with an automatic structure, where $G$ is the group, $A$ is a finite set of generators of $G$, $W$ is ...
4
votes
1
answer
964
views
Algebraic structure generated by primitive graph operations
Let $M$ be a finite set, and
$S(M) = \{(f_0, f_1) | f_0, f_1: M → M\}$.
Each element of $S(M)$ can be considered as a finite directed graph with the set of nodes $M$, which has exactly two arrows ...
4
votes
0
answers
155
views
Corollaries of Kleene's Theorem (Regular Languages)
Kleene's theorem that finite automata (specifically, nondeterministic) are expressively equivalent to regular expressions seems to be a powerful and not immediately obvious tool for untangling the ...
4
votes
0
answers
124
views
Properties of classical automata preserved in Büchi automata
Given two NFW $A$ and $B$, we regarded $A$ and $B$ as Büchi automata.
We can show that the containment property is not preserved in Büchi automata. That is, we can construct a example: $L(A) \...
4
votes
0
answers
215
views
How should one generate a random set of mappings?
My motivation for this question comes from the study of synchronizing automata. There is a general consensus that random automata are synchronizing and have short synchronizing words. I am hoping ...
4
votes
0
answers
153
views
connectivity in automata by words of length n-1
Let $A$ be a complete strongly connected automaton with $n$ states. Does always exist a word $v$ of length at most $n-1$ such that its underlying graph is connected?
That is for any pair of distinct ...
3
votes
4
answers
1k
views
Is there a physically realizable inductive turing machine that can solve Hilbert's $10$th problem and can it overcome Church-Turing Hypothesis?
There is a claim on https://en.wikipedia.org/wiki/Super-recursive_algorithm#Inductive_Turing_machines that 'Simple inductive Turing machines are equivalent to other models of computation such as ...
3
votes
5
answers
2k
views
Theory mainly concerned with $\lambda$-calculus?
Automata theory is mainly concerned with Turing machines and all its relatives-in-spirit. $\lambda$-calculus is rather rarely mentioned in textbooks on automata theory.
What's the common name of the ...
3
votes
1
answer
1k
views
Collatz conjecture— finite state machine transducer construction, origination?
wikipedia has an entry on the Collatz conjecture with a section on As an abstract machine that computes in base two. this apparently describes a construction of a FSM transducer computing sequential ...
3
votes
3
answers
545
views
Finite variation and idempotent languages and automata
Let $L$ be a regular language over alphabet $\Sigma$ and let $A:=(Q,\Sigma,\delta, q_0, F)$ be the minimal DFA recognizing $L$. For every $w\in \Sigma^*$ define the variation of $w$ w.r.t. $L$ by
$$\...
3
votes
3
answers
901
views
'Closure' of CFLs under complementation and intersection
Consider two context-free languages $L_1, L_2$. Of course, $L_1 - L_2, L_1\cap L_2, \bar{L}_1$, etc. are not necessarily context-free, but they are context-sensitive (the second is easy, the other two ...
3
votes
1
answer
773
views
Language equivalence between deterministic and non-deterministic counter net
One-Counter Nets (OCNs) are finite-state machines equipped with an integer counter that
cannot decrease below zero and cannot be explicitly tested for zero.
An OCN $A$ over alphabet $\sum$ accepts a ...
3
votes
1
answer
517
views
Study of free monoids of the recursive S. Eilenberg.
Compared to the usual treatises on recursion (eg, Rogers H. "Computability and Undecidability." McGraw-Hill, New York) the book of Samuel Eilenberg & Calvin C. Elgot "Recursiveness" treats such ...
3
votes
1
answer
264
views
Exponential objects in a category of abstract automata
I'm working with a more or less standard definition of the category Aut(C) of automata over a category C (where C has finite products) which has tuples $$
A=\langle I_{A},O_{A},S_{A},\sigma_{A}, \...
3
votes
2
answers
783
views
Turing-complete primitive blind automata
Let $N$ be the set of natural numbers, $S$ be the set of finite binary sequences, and
$Q = [N \rightarrow N] \times [N \rightarrow N],$
where $[N \rightarrow N]$ is the set of all computable ...