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.

scholarly journals State Complexity of Suffix Distance

2019 ◽  
Vol 30 (06n07) ◽  
pp. 1197-1216
Author(s):  
Timothy Ng ◽  
David Rappaport ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Finite Automaton ◽  
Regular Language ◽  
Upper Bounds ◽  
The State ◽  
Deterministic Finite Automaton ◽  
Tight Bound ◽  
State Complexity

The neighbourhood of a regular language with respect to the prefix, suffix and subword distance is always regular and a tight bound for the state complexity of prefix distance neighbourhoods is known. We give upper bounds for the state complexity of the neighbourhood of radius [Formula: see text] of an [Formula: see text]-state deterministic finite automaton language with respect to the suffix distance and the subword distance, respectively. For restricted values of [Formula: see text] and [Formula: see text] we give a matching lower bound for the state complexity of suffix distance neighbourhoods.

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.

scholarly journals ON A SUPERCLASS OF A-GRAMMARS

2017 ◽  
Vol 20 (10) ◽  
pp. 102-108
Author(s):  
V.P. Tsvetov
Keyword(s):  
Finite Automaton ◽  
Formal Languages ◽  
Regular Languages ◽  
Finite Alphabet ◽  
Graph Structure ◽  
Graph Grammars ◽  
Deterministic Finite Automaton

In this paper we consider a superclass of automaton grammars that can be represented in terms of paths on graphs. With this approach, we assume that vertices of graph are labeled by symbols of finite alphabet A . We will call such grammars graph-generated grammars or G-grammars. In contrast to the graph grammars that are used to describe graph structure transformations, G-grammars using a graphs as a means of representing formal languages. We will give an algorithm for constructing G-grammar which generate the language recognized by deterministic finite automaton. Moreover, we will show that the class of languages generated by G-grammars is a proper superset of regular languages.

Square on Deterministic, Alternating, and Boolean Finite Automata

2019 ◽  
Vol 30 (06n07) ◽  
pp. 1117-1134
Author(s):  
Galina Jirásková ◽  
Ivana Krajňáková
Keyword(s):  
Upper Bound ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper Bounds ◽  
The State ◽  
Final States ◽  
Deterministic Finite Automaton ◽  
State Complexity

We investigate the state complexity of the square operation on languages represented by deterministic, alternating, and Boolean automata. For each [Formula: see text] such that [Formula: see text], we describe a binary language accepted by an [Formula: see text]-state deterministic finite automaton with [Formula: see text] final states meeting the upper bound [Formula: see text] on the state complexity of its square. We show that in the case of [Formula: see text], the corresponding upper bound cannot be met. Using the binary deterministic witness for square with [Formula: see text] states where half of them are final, we get the tight upper bounds [Formula: see text] and [Formula: see text] on the complexity of the square operation on alternating and Boolean automata, respectively.

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 The Černý Conjecture and 1-Contracting Automata

10.37236/5616 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Henk Don
Keyword(s):  
Finite Automaton ◽  
The State ◽  
Sufficient Condition ◽  
Deterministic Finite Automaton ◽  
Special Case ◽  
Synchronizing Automaton ◽  
Synchronizing Word

A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. Černý conjectured in 1964 that a synchronizing automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$. We introduce the notion of aperiodically 1-contracting automata and prove that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Furthermore, we give a sufficient condition under which the Černý conjecture holds for aperiodically 1-contracting automata. As a special case, we prove some results for circular automata.

scholarly journals IN SEARCH OF MOST COMPLEX REGULAR LANGUAGES

2013 ◽  
Vol 24 (06) ◽  
pp. 691-708 ◽  
Author(s):  
JANUSZ BRZOZOWSKI
Keyword(s):  
Positive Integer ◽  
Finite Automaton ◽  
Cyclic Permutation ◽  
The State ◽  
Regular Languages ◽  
Final States ◽  
Deterministic Finite Automaton ◽  
Small Positive Integer ◽  
Initial State ◽  
State Complexity

Sequences (Ln| n ≥ k), called streams, of regular languages Lnare considered, where k is some small positive integer, n is the state complexity of Ln, and the languages in a stream differ only in the parameter n, but otherwise, have the same properties. The following measures of complexity are proposed for any stream: (1) the state complexity n of Ln, that is, the number of left quotients of Ln(used as a reference); (2) the state complexities of the left quotients of Ln; (3) the number of atoms of Ln; (4) the state complexities of the atoms of Ln; (5) the size of the syntactic semigroup of Ln; and the state complexities of the following operations: (6) the reverse of Ln; (7) the star of Ln; (8) union, intersection, difference and symmetric difference of Lmand Ln; and (9) the concatenation of Lmand Ln. A stream that has the highest possible complexity with respect to these measures is then viewed as a most complex stream. The language stream (Un(a, b, c) | n ≥ 3) is defined by the deterministic finite automaton with state set {0, 1, … , n−1}, initial state 0, set {n−1} of final states, and input alphabet {a, b, c}, where a performs a cyclic permutation of the n states, b transposes states 0 and 1, and c maps state n − 1 to state 0. This stream achieves the highest possible complexities with the exception of boolean operations where m = n. In the latter case, one can use Un(a, b, c) and Un(b, a, c), where the roles of a and b are interchanged in the second language. In this sense, Un(a, b, c) is a universal witness. This witness and its extensions also apply to a large number of combined regular operations.

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
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