Omega-rational expressions with bounded synchronization delay

© 2013, Springer Science+Business Media New York. In 1965 Sch ̈utzenberger published his famous result that star-free languages (SF) and aperiodic languages (AP) coincide over finite words, often written as SF = AP. Perrin generalized SF = AP to infinite words in the mid 1980s. In 1973 Sch ̈utzenberger presented another (and less known) characteri- zation of aperiodic languages in terms of rational expressions where the use of the star operation is restricted to prefix codes with bounded synchronization delay and no complementation is used. We denote this class of languages by SD. In this paper, we present a generalization of SD = AP to infinite words. This became possible via a substantial simplification of the proof for the cor- responding result for finite words. Moreover, we show that SD = AP can be viewed as more fundamental than SF = AP in the sense that the classical 1965 result of Sch ̈utzenberger and its 1980s extension to infinite words by Perrin are immediate consequences of SD = AP

Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions

What is the common link, if there is any, between Church-Rosser systems, prefix codes with bounded synchronization delay, and local Rees extensions? The first obvious answer is that each of these notions relates to topics of interest for WORDS: Church-Rosser systems are certain rewriting systems over words, codes are given by sets of words which form a basis of a free submonoid in the free monoid of all words (over a given alphabet) and local Rees extensions provide structural insight into regular languages over words. So, it seems to be a legitimate title for an extended abstract presented at the conference WORDS 2017. However, this work is more ambitious, it outlines some less obvious but much more interesting link between these topics. This link is based on a structure theory of finite monoids with varieties of groups and the concept of local divisors playing a prominent role. Parts of this work appeared in a similar form in conference proceedings where proofs and further material can be found.Comment: Extended abstract of an invited talk given at WORDS 201

A Survey on the Local Divisor Technique

Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. We also introduce the notion of localizable language class as a new abstract concept which unifies some of the proofs for the results above

A survey on the local divisor technique

© 2015 Elsevier B.V. Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. We also introduce the notion of a localizable language class as a new abstract concept which unifies some of the proofs for the results above

Characterizing classes of regular languages using prefix codes of bounded synchronization delay

In this paper we continue a classical work of Sch\"utzenberger on codes with bounded synchronization delay. He was interested to characterize those regular languages where the groups in the syntactic monoid belong to a variety $H$. He allowed operations on the language side which are union, intersection, concatenation and modified Kleene-star involving a mapping of a prefix code of bounded synchronization delay to a group $G\in H$, but no complementation. In our notation this leads to the language classes $SD_G(A^\infty)$ and $SD_H(A^\infty$). Our main result shows that $SD_H(A^\infty)$ always corresponds to the languages having syntactic monoids where all subgroups are in $H$. Sch\"utzenberger showed this for a variety $H$ if $H$ contains Abelian groups, only. Our method shows the general result for all $H$ directly on finite and infinite words. Furthermore, we introduce the notion of local Rees products which refers to a simple type of classical Rees extensions. We give a decomposition of a monoid in terms of its groups and local Rees products. This gives a somewhat similar, but simpler decomposition than in Rhodes' synthesis theorem. Moreover, we need a singly exponential number of operations, only. Finally, our decomposition yields an answer to a question in a recent paper of Almeida and Kl\'ima about varieties that are closed under Rees products

Omega VLF timing revision 1

The report specifically discusses time dissemination techniques, including epoch determination, frequency determination, and ambiguity resolution. It also discusses operational considerations including equipment, path selection, and adjustment procedure. epoch (the actual location or timing of periodic events) is shown to be both maintainable and calibratable by the techniques described to better than 3-microsecond accuracy; and frequency (the uniformity of the time scale) to about one part in 10 to the 12th power

An Introduction to Digital Signal Processing

An Introduction to Digital Signal Processing aims at undergraduate students who have basic knowledge in C programming, Circuit Theory, Systems and Simulations, and Spectral Analysis. The book is focused on basic concepts of digital signal processing, MATLAB simulation and implementation on selected DSP hardware in which the candidate is introduced to the basic concepts first before embarking to the practical part which comes in the later chapters. Initially Digital Signal Processing evolved as a postgraduate course which slowly filtered into the undergraduate curriculum as a simplified version of the latter. The goal was to study DSP concepts and to provide a foundation for further research where new and more efficient concepts and algorithms can be developed. Though this was very useful it did not arm the student with all the necessary tools that many industries using DSP technology would require to develop applications. This book is an attempt to bridge the gap. It is focused on basic concepts of digital signal processing, MATLAB simulation and implementation on selected DSP hardware. The objective is to win the student to use a variety of development tools to develop applications. Contents• Introduction to Digital Signal processing.• The transform domain analysis: the Discrete-Time Fourier Transform• The transform domain analysis: the Discrete Fourier Transform• The transform domain analysis: the z-transform• Review of Analogue Filter• Digital filter design.• Digital Signal Processing Implementation Issues• Digital Signal Processing Hardware and Software• Examples of DSK Filter Implementatio