1,361 research outputs found
Two-Way Unary Temporal Logic over Trees
We consider a temporal logic EF+F^-1 for unranked, unordered finite trees.
The logic has two operators: EF\phi, which says "in some proper descendant \phi
holds", and F^-1\phi, which says "in some proper ancestor \phi holds". We
present an algorithm for deciding if a regular language of unranked finite
trees can be expressed in EF+F^-1. The algorithm uses a characterization
expressed in terms of forest algebras.Comment: 29 pages. Journal version of a LICS 07 pape
Inverse problems of symbolic dynamics
This paper reviews some results regarding symbolic dynamics, correspondence
between languages of dynamical systems and combinatorics. Sturmian sequences
provide a pattern for investigation of one-dimensional systems, in particular
interval exchange transformation. Rauzy graphs language can express many
important combinatorial and some dynamical properties. In this case
combinatorial properties are considered as being generated by substitutional
system, and dynamical properties are considered as criteria of superword being
generated by interval exchange transformation. As a consequence, one can get a
morphic word appearing in interval exchange transformation such that
frequencies of letters are algebraic numbers of an arbitrary degree.
Concerning multydimensional systems, our main result is the following. Let
P(n) be a polynomial, having an irrational coefficient of the highest degree. A
word (w=(w_n), n\in \nit) consists of a sequence of first binary numbers
of i.e. . Denote the number of different subwords
of of length by .
\medskip {\bf Theorem.} {\it There exists a polynomial , depending only
on the power of the polynomial , such that for sufficiently
great .
On the rational subset problem for groups
We use language theory to study the rational subset problem for groups and
monoids. We show that the decidability of this problem is preserved under graph
of groups constructions with finite edge groups. In particular, it passes
through free products amalgamated over finite subgroups and HNN extensions with
finite associated subgroups. We provide a simple proof of a result of
Grunschlag showing that the decidability of this problem is a virtual property.
We prove further that the problem is decidable for a direct product of a group
G with a monoid M if and only if membership is uniformly decidable for
G-automata subsets of M. It follows that a direct product of a free group with
any abelian group or commutative monoid has decidable rational subset
membership.Comment: 19 page
A Fibrational Approach to Automata Theory
For predual categories C and D we establish isomorphisms between opfibrations
representing local varieties of languages in C, local pseudovarieties of
D-monoids, and finitely generated profinite D-monoids. The global sections of
these opfibrations are shown to correspond to varieties of languages in C,
pseudovarieties of D-monoids, and profinite equational theories of D-monoids,
respectively. As an application, we obtain a new proof of Eilenberg's variety
theorem along with several related results, covering varieties of languages and
their coalgebraic modifications, Straubing's C-varieties, fully invariant local
varieties, etc., within a single framework
Unambiguous morphic images of strings
We study a fundamental combinatorial problem on morphisms in free semigroups: With
regard to any string α over some alphabet we ask for the existence of a morphism σ such
that σ(α) is unambiguous, i.e. there is no morphism T with T(i) ≠ σ(i) for some symbol
i in α and, nevertheless, T(α) = σ(α). As a consequence of its elementary nature, this
question shows a variety of connections to those topics in discrete mathematics which
are based on finite strings and morphisms such as pattern languages, equality sets and,
thus, the Post Correspondence Problem.
Our studies demonstrate that the existence of unambiguous morphic images essen-
tially depends on the structure of α: We introduce a partition of the set of all finite
strings into those that are decomposable (referred to as prolix) in a particular manner
and those that are indecomposable (called succinct). This partition, that is also known
to be of major importance for the research on pattern languages and on finite fixed
points of morphisms, allows to formulate our main result according to which a string α
can be mapped by an injective morphism onto an unambiguous image if and only if α is
succinct
Unambiguous morphic images of strings
Motivated by the research on pattern languages, we study a fundamental combinatorial question on morphisms in free semigroups: With regard to any string α over some alphabet we ask for the existence
of a morphism σ such that σ(α) is unambiguous, i.e. there is no morphism ρ with ρ ≠ σ and ρ(α) = σ(α). Our main result shows that a rich and natural class of strings is provided with unambiguous morphic images
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