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.

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.

Trim Strongly Connected Synchronizing Automata and Ideal Languages

2018 ◽  
Vol 162 (2-3) ◽  
pp. 183-203
Author(s):  
Marina Maslennikova ◽  
Emanuele Rodaro
Keyword(s):  
Synchronizing Automata ◽  
Strongly Connected

scholarly journals Randomized Construction of Complexes with Large Diameter

2020 ◽  
Author(s):  
Francisco Criado ◽  
Andrew Newman
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Large Diameter ◽  
Simplicial Complexes ◽  
Similar Improvement ◽  
Strongly Connected ◽  
Probabilistic Construction ◽  
Combinatorial Diameter

Abstract We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected $$(d-1)$$ ( d - 1 ) -dimensional simplicial complexes on n vertices, denoted $$H_s(n, d)$$ H s ( n , d ) . Using a probabilistic construction we give a new lower bound on $$H_s(n, d)$$ H s ( n , d ) that is within an $$O(d^2)$$ O ( d 2 ) factor of the upper bound. This improves on the previously best known lower bound which was within a factor of $$e^{\varTheta (d)}$$ e Θ ( d ) of the upper bound. We also make a similar improvement in the case of pseudomanifolds.

scholarly journals COMPLEXITY OF ATOMS OF REGULAR LANGUAGES

2013 ◽  
Vol 24 (07) ◽  
pp. 1009-1027 ◽  
Author(s):  
JANUSZ BRZOZOWSKI ◽  
HELLIS TAMM
Keyword(s):  
Upper Bound ◽  
Regular Language ◽  
Regular Languages ◽  
State Complexity ◽  
Empty Intersection ◽  
Language L ◽  
The Right

The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs of a given language, i.e., those NFAs that have no equivalent states. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2n − 1 if r = 0 or r = n; for 1 ≤ r ≤ n − 1 the bound is[Formula: see text] For each n ≥ 2, we exhibit a language with 2n atoms which meet these bounds.

scholarly journals Towards a Framework for Assessing Cybersecurity Risks in Internet of Things (IOT) Devices

2021 ◽  
Vol 19 (8) ◽  
pp. 148-163
Author(s):  
Zhilei Qiao ◽  
Julio C. Rivera ◽  
Mi Zhou
Keyword(s):  
Internet Of Things ◽  
Broad Class ◽  
Mobile App ◽  
The Internet ◽  
Theoretic Approach ◽  
Iot Security ◽  
Graph Theoretic ◽  
Security Issues ◽  
Technical Requirements ◽  
Business Entities

The term Internet of Things (IoT) refers to a broad class of devices used by business entities as well as consumers to provide or consume a broad array of services. All these devices share their need to connect to the internet to deliver their native functionality. This connection requirement exposes the devices to the cybersecurity threats found on the internet. Existing literature on IoT cybersecurity solution models has shown that different technologies, such as communication technologies, mobile-app based authorization framework, graph-theoretic approach or blockchain technologies, have been majorly proposed to solve IoT security issues. However, these studies only focus on some specific IoT security issues like data theft or security issues on some specific layer across the whole IoT architecture. Therefore, there is a lack of systematic framework to solve IoT cybersecurity issues. This paper presents a framework for assessing such risks. In the qualitative analysis results, the device threats seem more severe than data confidentiality and privacy issues. This surprising finding highlights the significances of security taxonomy because both issues are based on different technical requirements. Our study has important managerial and practical implications for users, managers, and policymakers.

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