scholarly journals Automaticity of primitive words and irreducible polynomials

2013 ◽  
Vol Vol. 15 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Anne Lacroix ◽  
Narad Rampersad
Keyword(s):  
Finite Field ◽  
Finite Automaton ◽  
Prime Numbers ◽  
Deterministic Finite Automaton ◽  
Irreducible Polynomials ◽  
Letter Alphabet ◽  
International Audience ◽  
Primitive Words

Automata, Logic and Semantics International audience If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.

DETERMINISTIC BLOW-UPS OF MINIMAL NONDETERMINISTIC FINITE AUTOMATA OVER A FIXED ALPHABET

2008 ◽  
Vol 19 (03) ◽  
pp. 617-631 ◽  
Author(s):  
JOZEF JIRÁSEK ◽  
GALINA JIRÁSKOVÁ ◽  
ALEXANDER SZABARI
Keyword(s):  
Finite Automaton ◽  
Finite Automata ◽  
Magic Numbers ◽  
Deterministic Finite Automaton ◽  
Nondeterministic Finite Automata ◽  
Letter Alphabet ◽  
Input Alphabet ◽  
Nondeterministic Finite Automaton

We show that for all integers n and α such that n ⩽ α ⩽ 2n, there exists a minimal nondeterministic finite automaton of n states with a four-letter input alphabet whose equivalent minimal deterministic finite automaton has exactly α states. It follows that in the case of a four-letter alphabet, there are no "magic numbers", i.e., the holes in the hierarchy. This improves a similar result obtained by Geffert for a growing alphabet of size n + 2.

scholarly journals The \v Cerný conjecture for aperiodic automata

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
A. N. Trahtman
Keyword(s):  
Finite Automaton ◽  
Deterministic Finite Automaton ◽  
International Audience ◽  
Synchronizing Word

International audience A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Cerny conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n-1)2. We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n-1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true.

scholarly journals On the length of shortest 2-collapsing words

2009 ◽  
Vol Vol. 11 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Alessandra Cherubini ◽  
Andrzej Kisielewicz ◽  
Brunetto Piochi
Keyword(s):  
Finite Automaton ◽  
The State ◽  
Finite Alphabet ◽  
Deterministic Finite Automaton ◽  
Sigma Delta ◽  
Natural Action ◽  
Deterministic Automaton ◽  
International Audience

Automata, Logic and Semantics International audience Given a word w over a finite alphabet Sigma and a finite deterministic automaton A = < Q,Sigma,delta >, the inequality vertical bar delta(Q,w)vertical bar <= vertical bar Q vertical bar - k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton ( with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Sigma there is a 2-collapsing word whose length is vertical bar Sigma vertical bar(3)+6 vertical bar Sigma vertical bar(2)+5 vertical bar Sigma vertical bar/2. Then we produce shorter 2-collapsing and 2-synchronizing words over alphabets of 4 and 5 letters.

Model reformulation for conflict-free routing problems using Petri Net and Deterministic Finite Automaton

2015 ◽  
Vol 20 (3) ◽  
pp. 262-269 ◽  
Author(s):  
Ryosuke Nakamura ◽  
Kenji Sawada ◽  
Seiichi Shin ◽  
Kenji Kumagai ◽  
Hisato Yoneda
Keyword(s):  
Petri Net ◽  
Finite Automaton ◽  
Deterministic Finite Automaton ◽  
Routing Problems ◽  
Model Reformulation

scholarly journals Short Kloosterman Sums for Polynomials over Finite Fields

2003 ◽  
Vol 55 (2) ◽  
pp. 225-246 ◽  
Author(s):  
William D. Banks ◽  
Asma Harcharras ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Small Degree ◽  
Kloosterman Sums ◽  
Degree Relative ◽  
Irreducible Polynomials ◽  
Arbitrary Polynomial ◽  
Polynomials Over Finite Fields ◽  
Titchmarsh Theorem

AbstractWe extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring [x]/M(x) for collections of polynomials either of the form f−1g−1 or of the form f−1g−1 + afg, where f and g are polynomials coprime to M and of very small degree relative to M, and a is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

Irreducible polynomials over a finite field

1990 ◽  
Vol 30 (6) ◽  
pp. 915-925 ◽  
Author(s):  
E. N. Kuz'min
Keyword(s):  
Finite Field ◽  
Irreducible Polynomials

𝔽 p 2 -maximal curves with many automorphisms are Galois-covered by the Hermitian curve

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Daniele Bartoli ◽  
Maria Montanucci ◽  
Fernando Torres
Keyword(s):  
Finite Field ◽  
Prime Numbers ◽  
Hermitian Curve ◽  
Maximal Curve ◽  
Maximal Curves

Abstract Let 𝔽 be the finite field of order q 2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curve H q + 1 : y q + 1 = x q + x ${{\mathcal{H}}_{q+1}}:{{y}^{q+1}}={{x }^{q}}+x$ is also 𝔽-maximal. For prime numbers q we show that every 𝔽-maximal curve x $\mathcal{x}$ of genus g ≥ 2 with | Aut(𝒳) | > 84(g − 1) is Galois-covered by H q + 1 . ${{\mathcal{H}}_{q+1}}.$ The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curve x $\mathcal{x}$ for q = 71 of genus g = 7 with | Aut(𝒳) | = 84(7 − 1) which is not Galois-covered by the Hermitian curve H 72 . ${{\mathcal{H}}_{72}}.$

Magic squares of squares over a finite field

2021 ◽  
pp. 111-122
Author(s):  
Stewart Hengeveld ◽  
Giancarlo Labruna ◽  
Aihua Li
Keyword(s):  
Finite Field ◽  
Prime Numbers ◽  
Similar Question ◽  
Magic Squares ◽  
Content Type ◽  
Magic Square ◽  
Twin Primes ◽  
Quadratic Residues ◽  
Perfect Squares ◽  
Open Question

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

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