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.

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 Subnomality in soluble minimax groups

1974 ◽  
Vol 17 (1) ◽  
pp. 113-128 ◽  
Author(s):  
D. J. McCaughan
Keyword(s):  
Large Class ◽  
Upper Bound ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Wreath Products ◽  
Minimal Length ◽  
Intersection Property ◽  
Finite Chain ◽  
Subnormal Subgroups

A subgroup H of a group G is said to be subnormal in G if there is a finite chain of subgroups, each normal in its successor, connecting H to G. If such chains exist there is one of minimal length; the number of strict inclusions in this chain is called the subnormal index, or defect, of H in G. The rather large class of groups which have an upper bound for the subnormal indices of their subnormal subgroups has been inverstigated to same extent, mainly with a restriction to solublegroups — for instance, in [10] McDougall considered soluble p-groups in this class. Robinson, in [14], restricted his attention to wreath products of nilpotent groups but extended his investigations to the strictly larger class of groups in which the intersection of any family of subnormal subgroups is a subnormal subgroup. These groups are said to have the subnormal intersection property.

The Price of Bounded Preemption

2021 ◽  
Vol 8 (1) ◽  
pp. 1-21
Author(s):  
Noga Alon ◽  
Yossi Azar ◽  
Mark Berlin
Keyword(s):  
Single Machine ◽  
Upper Bound ◽  
Loss Factor ◽  
Processing Time ◽  
Minimal Length ◽  
Release Time ◽  
Preemptive Scheduling ◽  
Tight Bound ◽  
Bounded Degree ◽  
Specific Scenario

In this article we provide a tight bound for the price of preemption for scheduling jobs on a single machine (or multiple machines). The input consists of a set of jobs to be scheduled and of an integer parameter k ≥ 1. Each job has a release time, deadline, length (also called processing time), and value associated with it. The goal is to feasibly schedule a subset of the jobs so that their total value is maximal; while preemption of a job is permitted, a job may be preempted no more than k times. The price of preemption is the worst possible (i.e., largest) ratio of the optimal non-bounded-preemptive scheduling to the optimal k -bounded-preemptive scheduling. Our results show that allowing at most k preemptions suffices to guarantee a Θ(min {log k +1 n , log k +1 P }) fraction of the total value achieved when the number of preemptions is unrestricted (where n is the number of the jobs and P the ratio of the maximal length to the minimal length), giving us an upper bound for the price; a specific scenario serves to prove the tightness of this bound. We further show that when no preemptions are permitted at all (i.e., k =0), the price is Θ (min { n , log P }). As part of the proof, we introduce the notion of the Bounded-Degree Ancestor-Free Sub-Forest (BAS) . We investigate the problem of computing the maximal-value BAS of a given forest and give a tight bound for the loss factor, which is Θ(log k +1 n ) as well, where n is the size of the original forest and k is the bound on the degree of the sub-forest.

scholarly journals Upper bound on minimal length from deuteron

2014 ◽  
Vol 4 ◽  
pp. 52-53 ◽  
Author(s):  
S.B. Faruque ◽  
Md. Afjalur Rahman ◽  
Md. Moniruzzaman
Keyword(s):  
Upper Bound ◽  
Minimal Length

scholarly journals Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

2017 ◽  
Vol 16 (03) ◽  
pp. 1750043
Author(s):  
Martino Garonzi ◽  
Dan Levy ◽  
Attila Maróti ◽  
Iulian I. Simion
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Minimal Length ◽  
Real Constant ◽  
Carter Subgroup ◽  
Positive Real ◽  
Solvable Length ◽  
A Finite Group ◽  
Positive Real Constant

We consider factorizations of a finite group [Formula: see text] into conjugate subgroups, [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is nilpotent or solvable. We derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group which, for a large class of groups, is linear in the non-solvable length of [Formula: see text]. We also show that every solvable group [Formula: see text] is a product of at most [Formula: see text] conjugates of a Carter subgroup [Formula: see text] of [Formula: see text], where [Formula: see text] is a positive real constant. Finally, using these results we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.

scholarly journals A Short Note on Minimal Length

2019 ◽  
Vol 11 (2) ◽  
pp. 151-155
Author(s):  
M. Moniruzzaman ◽  
S. B. Faruque
Keyword(s):  
Hydrogen Atom ◽  
Upper Bound ◽  
Short Note ◽  
Interaction Strength ◽  
Minimal Length ◽  
Fundamental Interaction ◽  
Compton Wavelength ◽  
Nuclear Systems

After revival of the concept of minimal length, many investigations have been devoted, in literature, to estimate upper bound on minimal length for systems like hydrogen atom, deuteron etc. We report here a possible origin of minimal length for atomic and nuclear systems which is connected with the fundamental interaction strength and the Compton wavelength. The formula we appear at is numerically close to the upperbounds found in literature.

scholarly journals Lorentz force and ponderomotive force in the presence of a minimal length

2016 ◽  
Vol 14 (01) ◽  
pp. 1750005 ◽  
Author(s):  
Behrooz Khosropour
Keyword(s):  
Electromagnetic Field ◽  
Lorentz Force ◽  
Uncertainty Principle ◽  
Upper Bound ◽  
Ponderomotive Force ◽  
Generalized Uncertainty Principle ◽  
Minimal Length ◽  
Field Tensor ◽  
Faraday’S Law ◽  
Electromagnetic Field Tensor

In this work, according to the electromagnetic field tensor in the framework of generalized uncertainty principle (GUP), we obtain the Lorentz force and Faraday’s law of induction in the presence of a minimal length. Also, the ponderomotive force and ponderomotive pressure in the presence of a measurable minimal length are found. It is shown that in the limit [Formula: see text], the generalized Lorentz force and ponderomotive force become the usual forms. The upper bound on the isotropic minimal length is estimated.

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