Questions tagged [automata-theory]

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2 answers
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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 ...
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A particular generalization of free partially commutative monoids

A trace monoid, or free partially commutative monoid, is one with the presentation $\langle \Sigma \mid a_1b_1 = b_1a_1, \dots, a_nb_n = b_na_n\rangle$. The theory of trace monoids has been well ...
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1 vote
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Does Frobenius number increase if bound on input increases?

The Frobenius number F is the largest number not expressible as a non-negative linear combination of some set of positive integers $\{a_i\}$, where, $a_i$ has gcd 1. Denote $maxF(n)$ as the maximum of ...
0 votes
0 answers
51 views

First-order logics expressively equivalent to the computable languages

There is a really nice theorem that the subsets of $(\Sigma^*, =_{el}, \preceq, (S_a)_{a \in \Sigma})$ definable in first-order logic are exactly the regular sets. Where: $\Sigma^*$ is the set of ...
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4 votes
0 answers
134 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 ...
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6 votes
0 answers
107 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 ...
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2 votes
2 answers
129 views

A question on regular sets

In the end of the Abstract of the paper Minsky and Papert - Unrecognizable Sets of Numbers, the authors write "…for every infinite regular set $A$ there is a nonregular set $A'$ for which $$ \...
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4 votes
1 answer
107 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?...
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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 ...
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1 vote
1 answer
126 views

Shortest word accepted by a PDA

Given a pushdown automaton (PDA), we seek a shortest word accepted by it. A standard approach is to map the problem in the corresponding context-free grammar. Can we analyze and solve this problem ...
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1 vote
0 answers
53 views

Effect on finite transformation semigroup under a particular modification of the generators

The following question arises in connection with problems in automata theory related to the road problem. Let $f_1, f_2: [N] \to [N]$ be maps such that the transformation semigroup $S = \langle f_1, ...
2 votes
0 answers
90 views

Name for the theory of words with equal length, prefix, successors

I've worked with this theory for a while, but I've never been quite sure what to call it: $$(\Sigma^*, =_{el}, \preceq, (S_a)_{a \in \Sigma})$$ Where $\Sigma^*$ is the set of finite words on finite ...
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2 votes
1 answer
129 views

Busy beaver sequence for a simple tag-like system

This question arose in the context of tag-like systems, specifically Bitwise Cyclic Tag (BCT). Consider the following discrete dynamical system: Let $\mathbb{B} = \{\mathtt{0}, \mathtt{1}\}$. Let our ...
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4 votes
1 answer
122 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 ...
1 vote
1 answer
158 views

decidability of regularity of a language depending on representation

It is well known that many decision problems for regular languages are decidable. However, the proofs seem to rely on a witness of the regularity of said language, be it an automaton, a grammar, a ...
1 vote
0 answers
70 views

What a generating function for a language tells us about the language [closed]

What a generating function for a language tells us about the language .I need its answer in base of automata?
18 votes
2 answers
630 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)...
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6 votes
1 answer
167 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 ...
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0 answers
26 views

Probabilistic timed automata transition

I am kind of new to timed automata and I have a question related to their correctness and synchronisation. Assume that I have three states, A, B and C. I have also two clocks, $x$ and $y$ that are ...
1 vote
1 answer
146 views

Errors in Waksman's Solution to Cellular Automaton Firing Squad Problem?

Recently, a student and I have been working through Waksman's paper ``An Optimum Solution to the Firing Squad Synchronization Problem.'' The paper claims that for any value of $n$, the proposed ...
2 votes
1 answer
63 views

For synchronizing eulerian finite state machines every proper subset of states has some larger state set leads to this subset

Suppose we have a deterministic complete finite automaton which is synchronized, meaning we have a reset word, i.e. a word which resets the automaton to a definite state, regardless from which state ...
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2 votes
0 answers
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Why can a least fixed point operator only be expanded finitely many times?

If we expand modal logic with least and greatest fix point operators $\mu$ and $\nu$, respectively, we obtain the logic $L_\mu$. An alternating automaton on infinite trees has a state space that is ...
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1 vote
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258 views

Characterization of non-Zeno functions $f:\mathbb{R}\rightarrow \{0,1\}$

[Edit: I tried to integrate Nate's comments (see below).] In the context of automata over continuous time, consider Boolean-valued functions $f:\mathbb{R}\rightarrow \{0,1\}$. There are uncountably ...
1 vote
1 answer
94 views

Minimal DFA of L* [closed]

I'm learning how to minimize DFAs. Are the number of states in the minimal DFA of L, is equal to the number of states in the minimal DFA of L*? I'm trying for hours to think of examples but couldn't ...
3 votes
0 answers
157 views

Intersection of cone types

Let $G$ be a finitely generated hyperbolic group with the word metric; fix a symmetric generating set $S$ and let $\mathcal{G}$ be the Cayley graph of $G$ w.r.t. $S$. Define the cone of an element $x\...
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2 votes
0 answers
63 views

If a timed automaton always terminates, does there exist a trace with a maximum length?

I have a theoretical question regarding timed automata and I would like to know if someone has already given an answer to it, since that would be useful for my research. So my question is the ...
1 vote
0 answers
220 views

What does homomorphism between languages mean to the correspoding Turing Machines?

According to the article: every c.e.language over $\Sigma^*$can be formed by homomorphism from a Dyck language over $\Sigma^{'}$ intersection with a minimal linear language over $\Sigma^{'}$ to the ...
11 votes
1 answer
314 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 ...
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 ...
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0 votes
2 answers
219 views

Verification of Turing-equivalent automata

Correct me if I slept in my computer science studium: If an automaton is Turing-equivalent, the Halting problem shows that there are programs we can not verify (since we can't even predict their ...
1 vote
0 answers
293 views

Life. Intermediate stages

My question is pure mathematics when restricted to the cellular automata theory. John von Neumann got the grasp of and defined life. Many years later biologists supported von Neumann's definition of ...
5 votes
1 answer
323 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 ...
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1 vote
1 answer
261 views

Modal logic in combination with automata theory

I'm planning to write a paper about the possibility of describing modal logic and the multiple world aspect of it with techniques of automata theory. To not duplicate my work does anyone have more ...
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4 votes
1 answer
387 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 ...
5 votes
0 answers
231 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 ...
2 votes
1 answer
148 views

Understanding the paper: "Guarded Fixed Point Logic"

This question is specifically about the paper "Guarded Fixed Point Logic" by Gradel and Walukiewicz. Among other things they prove the decidability of the satisfiability problem for Fixpoint Loosely ...
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5 votes
2 answers
463 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,...
1 vote
1 answer
2k views

Non-regular languages fulfilling the Pumping Lemma

Some non-regular languages don't yield to the Pumping Lemma ($L_1=a^nb^mc^m$ should work). But now consider the set of non-regular languages L only over the alphabet {a}. (Like $L_2=a^{n^2}$ or ...
-2 votes
1 answer
128 views

How one can use a real math function on transaction in Hybrid Petri Net fundamental equation?

Say we have a simple HPN with 2 continuous places $A$ and $B$ and one transition. We want a transition not only add and substract $N$ marks from $A$ and add $M$ to $B$ but use mathematical function $...
-2 votes
1 answer
267 views

Deterministic Finite Automata question [closed]

I am very new to finite automata, and I came across an issue in my professors lecture slides which I think is wrong, and I'd wonder if any of you could confirm: Alphabet: {1} Automata Surely the ...
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2 votes
1 answer
258 views

How does "inhibitor arc" fit into fundamental equation of Hybrid Petri Nets?

In "ON HYBRID PETRI NETS" by DAVID AND ALLA published in 2001 on page 26 is given an example of how fundamental equation solves a HPN for given start and end time values. A system looks like And ...
5 votes
1 answer
136 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 ...
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2 votes
0 answers
45 views

Relation between indexed languages (OI-macro or context-free tree) and scattered context languages

I'm not sure about the relation between indexed languages (generated by indexed grammars--Aho) and scattered context languages (generated by scattered context grammars--J Hopcroft). I think that ...
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5 votes
1 answer
473 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-...
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1 vote
1 answer
236 views

The automorphism groups of smallest grammars of a language string are isomorphic

Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = \...
3 votes
0 answers
219 views

Estimating the growth rate of nondeterministic finite automata

Given a nondeterministic finite automaton $\mathcal{A}$ (or a regular expression, or a regular grammar), can we efficiently compute the number $|L_k(\mathcal{A})|$ of accepted words of length $k$? No,...
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6 votes
2 answers
553 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 ...
1 vote
0 answers
221 views

LTL - Büchi-automaton Translation [closed]

I need some help in Generalized Büchi automaton .. I do understand the translation of a LTL-formula ϕ into Generalized Büchi automaton A= (Q, Δ, I, F), with F= {F1,...,Fn} My problem is F .. I know ...
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3 votes
1 answer
255 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}, \...
2 votes
1 answer
192 views

QBF of exponential length?

We consider a slightly extended version of a nondeterministic finite automaton, call it a "propositional nondeterministic finite automaton". It is defined as follows. Consider a fixed propositional ...