47 research outputs found
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
Star-Free Languages are Church-Rosser Congruential
The class of Church-Rosser congruential languages has been introduced by
McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser
congruential (belongs to CRCL), if there is a finite, confluent, and
length-reducing semi-Thue system S such that L is a finite union of congruence
classes modulo S. To date, it is still open whether every regular language is
in CRCL. In this paper, we show that every star-free language is in CRCL. In
fact, we prove a stronger statement: For every star-free language L there
exists a finite, confluent, and subword-reducing semi-Thue system S such that
the total number of congruence classes modulo S is finite and such that L is a
union of congruence classes modulo S. The construction turns out to be
effective
Formalized proof, computation, and the construction problem in algebraic geometry
An informal discussion of how the construction problem in algebraic geometry
motivates the search for formal proof methods. Also includes a brief discussion
of my own progress up to now, which concerns the formalization of category
theory within a ZFC-like environment
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
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
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