578 research outputs found
Classes of tree languages and DR tree languages given by classes of semigroups
In the first section of the paper we give general conditions under which a class of recognizable tree languages with a given property can be defined by a class of monoids or semigroups defining the class of string languages having the same property. In the second part similar questions are studied for classes of (DR) tree languages recognized by deterministic root-to-frontier tree recognizers
Large Aperiodic Semigroups
The syntactic complexity of a regular language is the size of its syntactic
semigroup. This semigroup is isomorphic to the transition semigroup of the
minimal deterministic finite automaton accepting the language, that is, to the
semigroup generated by transformations induced by non-empty words on the set of
states of the automaton. In this paper we search for the largest syntactic
semigroup of a star-free language having left quotients; equivalently, we
look for the largest transition semigroup of an aperiodic finite automaton with
states.
We introduce two new aperiodic transition semigroups. The first is generated
by transformations that change only one state; we call such transformations and
resulting semigroups unitary. In particular, we study complete unitary
semigroups which have a special structure, and we show that each maximal
unitary semigroup is complete. For there exists a complete unitary
semigroup that is larger than any aperiodic semigroup known to date.
We then present even larger aperiodic semigroups, generated by
transformations that map a non-empty subset of states to a single state; we
call such transformations and semigroups semiconstant. In particular, we
examine semiconstant tree semigroups which have a structure based on full
binary trees. The semiconstant tree semigroups are at present the best
candidates for largest aperiodic semigroups.
We also prove that is an upper bound on the state complexity of
reversal of star-free languages, and resolve an open problem about a special
case of state complexity of concatenation of star-free languages.Comment: 22 pages, 1 figure, 2 table
Maximal subgroups of free idempotent generated semigroups over the full linear monoid
We show that the rank r component of the free idempotent generated semigroup
of the biordered set of the full linear monoid of n x n matrices over a
division ring Q has maximal subgroup isomorphic to the general linear group
GL_r(Q), where n and r are positive integers with r < n/3.Comment: 37 pages; Transactions of the American Mathematical Society (to
appear). arXiv admin note: text overlap with arXiv:1009.5683 by other author
On the closedness of nilpotent DR tree languages under Boolean operations
This note deals with the closedness of nilpotent deterministic root-to-frontier tree languages with respect to the Boolean operations union, intersection and complementation. Necessary and sufficient conditions are given under which the union of two deterministic tree languages is also deterministic. The paper ends with a characterization of the largest subclass of the
class of nilpotent deterministic root-to-frontier tree languages closed under the formation of complements
Monoid automata for displacement context-free languages
In 2007 Kambites presented an algebraic interpretation of
Chomsky-Schutzenberger theorem for context-free languages. We give an
interpretation of the corresponding theorem for the class of displacement
context-free languages which are equivalent to well-nested multiple
context-free languages. We also obtain a characterization of k-displacement
context-free languages in terms of monoid automata and show how such automata
can be simulated on two stacks. We introduce the simultaneous two-stack
automata and compare different variants of its definition. All the definitions
considered are shown to be equivalent basing on the geometric interpretation of
memory operations of these automata.Comment: Revised version for ESSLLI Student Session 2013 selected paper
On DR tree automata, unary algebras and syntactic path monoids
We consider deterministic root-to-frontier (DR) tree recognizers and the tree languages recognized by them from an algebraic point of view. We make use of a correspondence between DR algebras and unary algebras shown by Z. Esik (1986). We also study a question raised by F. GĂ©cseg (2007) that concerns the definability of families of DR-recognizable tree languages by syntactic path monoids. We show how the families of DR-recognizable tree languages path-definable by a variety of finite monoids (or semigroups) can be derived from varieties of string languages. In particular, the three pathdefinable families of GĂ©cseg and B. Imreh (2002, 2004) are obtained this way
- âŠ