scholarly journals A QUADRATIC UPPER BOUND ON THE SIZE OF A SYNCHRONIZING WORD IN ONE-CLUSTER AUTOMATA

2011 ◽  
Vol 22 (02) ◽  
pp. 277-288 ◽  
Author(s):  
MARIE-PIERRE BÉAL ◽  
MIKHAIL V. BERLINKOV ◽  
DOMINIQUE PERRIN
Keyword(s):  
Upper Bound ◽  
The State ◽  
Additional Property ◽  
Prefix Code ◽  
Deterministic Automaton ◽  
Huffman Codes ◽  
Synchronizing Word ◽  
Černý’S Conjecture

Černý's conjecture asserts the existence of a synchronizing word of length at most (n - 1)2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p·ar = q·as for some integers r, s (for a state p and a word w, we denote by p·w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n2). This applies in particular to Huffman codes.

scholarly journals Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

2016 ◽  
Vol 27 (02) ◽  
pp. 127-145 ◽  
Author(s):  
Jorge Almeida ◽  
Emanuele Rodaro
Keyword(s):  
Upper Bound ◽  
Broad Class ◽  
Regular Languages ◽  
Theoretic Approach ◽  
Synchronizing Automata ◽  
Natural Notion ◽  
Strongly Connected ◽  
Synchronizing Automaton ◽  
Synchronizing Word ◽  
Černý’S Conjecture

We present a ring theoretic approach to Černý's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those in Černý's series. Semisimplicity gives also the advantage of “factorizing” the problem of finding a synchronizing word into the sub-problems of finding “short” words that are zeros into the projection of the simple components in the Wedderburn-Artin decomposition. In the general case this last problem is related to the search of radical words of length at most [Formula: see text] where n is the number of states of the automaton. We show that the solution of this “Radical Conjecture” would give an upper bound [Formula: see text] for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases and Černý's conjecture for the class of strongly semisimple synchronizing automata. These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots.

STATE COMPLEXITY OF ADDITIVE WEIGHTED FINITE AUTOMATA

2007 ◽  
Vol 18 (06) ◽  
pp. 1407-1416 ◽  
Author(s):  
KAI SALOMAA ◽  
PAUL SCHOFIELD
Keyword(s):  
Upper Bound ◽  
Finite Automaton ◽  
Regular Language ◽  
Finite Automata ◽  
The State ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Weighted Finite Automata

It is known that the neighborhood of a regular language with respect to an additive distance is regular. We introduce an additive weighted finite automaton model that provides a conceptually simple way to reprove this result. We consider the state complexity of converting additive weighted finite automata to deterministic finite automata. As our main result we establish a tight upper bound for the state complexity of the conversion.

On the length of uncompletable words in unambiguous automata

2019 ◽  
Vol 53 (3-4) ◽  
pp. 115-123
Author(s):  
Antonio Boccuto ◽  
Arturo Carpi
Keyword(s):  
Upper Bound ◽  
Minimal Length ◽  
Deterministic Automaton ◽  
Transformation Monoid

This paper deals with uncomplete unambiguous automata. In this setting, we investigate the minimal length of uncompletable words. This problem is connected with a well-known conjecture formulated by A. Restivo. We introduce the notion of relatively maximal row for a suitable monoid of matrices. We show that, if M is a monoid of {0, 1}-matrices of dimension n generated by a set S, then there is a matrix of M containing a relatively maximal row which can be expressed as a product of O(n3) matrices of S. As an application, we derive some upper bound to the minimal length of an uncompletable word of an uncomplete unambiguous automaton, in the case that its transformation monoid contains a relatively maximal row which is not maximal. Finally we introduce the maximal row automaton associated with an unambiguous automaton A. It is a deterministic automaton, which is complete if and only if A is. We prove that the minimal length of the uncompletable words of A is polynomially bounded by the number of states of A and the minimal length of the uncompletable words of the associated maximal row automaton.

Analyzing a variant of Clobber: The game of San Jego

ICGA Journal ◽  
2020 ◽  
pp. 1-15
Author(s):  
Raphael Thiele ◽  
Ingo Althöfer
Keyword(s):  
State Space ◽  
Upper Bound ◽  
The State ◽  
Perfect Information ◽  
Space Complexity ◽  
Game Tree ◽  
Black And White ◽  
Game Theoretic ◽  
Player Game

San Jego is a two-player game with perfect information. It is a variation of the games Clobber (2001) and Cannibal Clobber. Given is a rectangular board with black and white pieces on the cells. Multiple pieces which are stacked on each other are called towers. The piece on top of a tower indicates her owner. A move consists of picking an own tower and placing it completely on top of an adjacent tower. If both players can not move anymore, the game ends. Winner is the player with the highest tower. An upper bound for the state-space complexity of San Jego is determined. Furthermore, the game-tree complexity is approximated theoretically and numerically. For small board sizes, the optimal game-theoretic values are calculated and the advantage of the first move is determined.

State Complexity of Insertion

2016 ◽  
Vol 27 (07) ◽  
pp. 863-878 ◽  
Author(s):  
Yo-Sub Han ◽  
Sang-Ki Ko ◽  
Timothy Ng ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Regular Language ◽  
Finite Automata ◽  
The State ◽  
Tight Bound ◽  
State Complexity ◽  
Nondeterministic State

It is well known that the resulting language obtained by inserting a regular language to a regular language is regular. We study the nondeterministic and deterministic state complexity of the insertion operation. Given two incomplete DFAs of sizes m and n, we give an upper bound (m+2)·2mn−m−1·3m and find a lower bound for an asymp-totically tight bound. We also present the tight nondeterministic state complexity by a fooling set technique. The deterministic state complexity of insertion is 2Θ(mn) and the nondeterministic state complexity of insertion is precisely mn+2m, where m and n are the size of input finite automata. We also consider the state complexity of insertion in the case where the inserted language is bifix-free or non-returning.

scholarly journals State Complexity of Catenation Combined with a Boolean Operation: A Unified Approach

2016 ◽  
Vol 27 (06) ◽  
pp. 675-703 ◽  
Author(s):  
Pascal Caron ◽  
Jean-Gabriel Luque ◽  
Ludovic Mignot ◽  
Bruno Patrou
Keyword(s):  
Upper Bound ◽  
The State ◽  
Symmetric Difference ◽  
Boolean Operation ◽  
Unified Approach ◽  
State Complexity

We study the state complexity of catenation combined with symmetric difference. First, an upper bound is computed using some combinatoric tools. Then, this bound is shown to be tight by giving a witness for it. Moreover, we relate this work with the study of state complexity for two other combinations: catenation with union and catenation with intersection. We extract a unified approach which allows to obtain the state complexity of any combination involving catenation and a binary boolean operation.

LINEAR-TIME PRIME DECOMPOSITION OF REGULAR PREFIX CODES

2003 ◽  
Vol 14 (06) ◽  
pp. 1019-1031 ◽  
Author(s):  
JUREK CZYZOWICZ ◽  
WOJCIECH FRACZAK ◽  
ANDRZEJ PELC ◽  
WOJCIECH RYTTER
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Prefix Code ◽  
Prime Decomposition ◽  
Deterministic Automaton ◽  
Prefix Codes ◽  
Efficient Storage ◽  
Finite State ◽  
Infinite Set

One of the new approaches to data classification uses prefix codes and finite state automata as representations of prefix codes. A prefix code is a (possibly infinite) set of strings such that no string is a prefix of another one. An important task driven by the need for the efficient storage of such automata in memory is the decomposition (in the sense of formal languages concatenation) of prefix codes into prime factors. We investigate properties of such prefix code decompositions. A prime decomposition is a decomposition of a prefix code into a concatenation of nontrivial prime prefix codes. A prefix code is prime if it cannot be decomposed into at least two nontrivial prefix codes. In the paper a linear time algorithm is designed which finds the prime decomposition F1F2…Fk of a regular prefix code F given by its minimal deterministic automaton. Our results are especially interesting for infinite regular prefix codes.

PREFIX-FREE ŁUKASIEWICZ LANGUAGES

2007 ◽  
Vol 18 (06) ◽  
pp. 1417-1423
Author(s):  
LUDWIG STAIGER
Keyword(s):  
Upper Bound ◽  
Desirable Property ◽  
Prefix Code ◽  
Good Information ◽  
Information Theoretic ◽  
The One

Generalised Łukasiewicz languages are simply described languages having good information-theoretic properties. An especially desirable property is the one of being a prefix code. This paper addresses the question under which conditions a generalised Łukasiewicz language is a prefix code. Moreover, an upper bound on the delay of decipherability of a generalised Łukasiewicz language is derived.

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.

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