Scheduling Hard Sporadic Tasks by Means of Finite Automata and Generating Functions

For regular expressions in (strong) star normal form a large set of efficient algorithms is known, from conversions into finite automata to characterisations of unambiguity. In this paper we study the average complexity of this class of expressions using analytic combinatorics. As it is not always feasible to obtain explicit expressions for the generating functions involved, here we show how to get the required information for the asymptotic estimates with an indirect use of the existence of Puiseux expansions at singularities. We study, asymptotically and on average, the alphabetic size, the size of the [Formula: see text]-follow automaton and of the position automaton, as well as the ratio and the size of these expressions to standard regular expressions.

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Generalized Jacobsthal numbers and restricted k-ary words

Pure Mathematics and Applications ◽

10.1515/puma-2015-0034 ◽

2019 ◽

Vol 28 (1) ◽

pp. 91-108

Author(s):

José L. Ramirez ◽

Mark Shattuck

Keyword(s):

Complete Graph ◽

Generating Functions ◽

Finite Automata ◽

General Setting ◽

Polynomial Extension ◽

Combinatorial Argument ◽

Domino Tilings ◽

Basic Properties ◽

Binomial Type

Abstract We consider a generalization of the problem of counting ternary words of a given length which was recently investigated by Koshy and Grimaldi [10]. In particular, we use finite automata and ordinary generating functions in deriving a k-ary generalization. This approach allows us to obtain a general setting in which to study this problem over a k-ary language. The corresponding class of n-letter k-ary words is seen to be equinumerous with the closed walks of length n − 1 on the complete graph for k vertices as well as a restricted subset of colored square-and-domino tilings of the same length. A further polynomial extension of the k-ary case is introduced and its basic properties deduced. As a consequence, one obtains some apparently new binomial-type identities via a combinatorial argument.

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A novel cryptosystem based on abstract automata and Latin cubes

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2015.52.2.1309 ◽

2015 ◽

Vol 52 (2) ◽

pp. 221-232

Author(s):

Pál Dömösi ◽

Géza Horváth

Keyword(s):

Block Cipher ◽

Finite Automata ◽

Theoretical Background ◽

Point Of View ◽

Theoretical Point ◽

Transition Matrices ◽

Encryption And Decryption ◽

Secret Keys ◽

Automata Network ◽

Fully Functioning

In this paper we introduce a novel block cipher based on the composition of abstract finite automata and Latin cubes. For information encryption and decryption the apparatus uses the same secret keys, which consist of key-automata based on composition of abstract finite automata such that the transition matrices of the component automata form Latin cubes. The aim of the paper is to show the essence of our algorithms not only for specialists working in compositions of abstract automata but also for all researchers interested in cryptosystems. Therefore, automata theoretical background of our results is not emphasized. The introduced cryptosystem is important also from a theoretical point of view, because it is the first fully functioning block cipher based on automata network.

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Parallel Combining Different Approaches to Multi-pattern Matching for Fpga-based Security Systems

Advances in Cyber-Physical Systems ◽

10.23939/acps2020.01.008 ◽

2017 ◽

Vol 5 (1) ◽

pp. 8-15

Author(s):

Sergii Hilgurt ◽

Keyword(s):

Information Security ◽

Pattern Matching ◽

Intrusion Detection System ◽

Detection System ◽

Finite Automata ◽

Network Intrusion Detection ◽

Security Systems ◽

Analytical Technique ◽

Network Intrusion ◽

Pros And Cons

The multi-pattern matching is a fundamental technique found in applications like a network intrusion detection system, anti-virus, anti-worms and other signature- based information security tools. Due to rising traffic rates, increasing number and sophistication of attacks and the collapse of Moore’s law, traditional software solutions can no longer keep up. Therefore, hardware approaches are frequently being used by developers to accelerate pattern matching. Reconfigurable FPGA-based devices, providing the flexibility of software and the near-ASIC performance, have become increasingly popular for this purpose. Hence, increasing the efficiency of reconfigurable information security tools is a scientific issue now. Many different approaches to constructing hardware matching circuits on FPGAs are known. The most widely used of them are based on discrete comparators, hash-functions and finite automata. Each approach possesses its own pros and cons. None of them still became the leading one. In this paper, a method to combine several different approaches to enforce their advantages has been developed. An analytical technique to quickly advance estimate the resource costs of each matching scheme without need to compile FPGA project has been proposed. It allows to apply optimization procedures to near-optimally split the set of pattern between different approaches in acceptable time.

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Complexity and Universality of Iterated Finite Automata

Complex Systems ◽

10.25088/complexsystems.18.1.145 ◽

2009 ◽

Vol 18 (1) ◽

pp. 145-158

Author(s):

Jiang Zhang ◽

Keyword(s):

Finite Automata

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Short Generating Functions for some Semigroup Algebras

The Electronic Journal of Combinatorics ◽

10.37236/1729 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 23

Author(s):

Graham Denham

Keyword(s):

Rational Function ◽

Generating Functions ◽

Hilbert Series ◽

Simple Expression ◽

Arbitrary Field ◽

Graded Algebra ◽

Semigroup Algebras ◽

Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

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New family of Whitney numbers

Filomat ◽

10.2298/fil1702309e ◽

2017 ◽

Vol 31 (2) ◽

pp. 309-320 ◽

Cited By ~ 2

Author(s):

B.S. El-Desouky ◽

Nenad Cakic ◽

F.A. Shiha

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Stirling Numbers ◽

Explicit Formulas ◽

Basic Properties ◽

New Family ◽

Whitney Numbers ◽

Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

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Identities related to special polynomials and combinatorial numbers

Filomat ◽

10.2298/fil1715833y ◽

2017 ◽

Vol 31 (15) ◽

pp. 4833-4844 ◽

Cited By ~ 2

Author(s):

Eda Yuluklu ◽

Yilmaz Simsek ◽

Takao Komatsu

Keyword(s):

Generating Functions ◽

Functional Equations ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Euler Numbers ◽

Special Polynomials ◽

Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

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Symplectic capacities

10.1093/oso/9780198794899.003.0013 ◽

2017 ◽

Author(s):

Dusa McDuff ◽

Dietmar Salamon

Keyword(s):

Euclidean Space ◽

Generating Functions ◽

Weinstein Conjecture ◽

Finite Dimensional ◽

Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

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