133,200 research outputs found
Infinite words and universal free actions
This is the second paper in a series of three, where we take on the unified
theory of non-Archimedean group actions, length functions and infinite words.
Here, for an arbitrary group of infinite words over an ordered abelian
group we construct a -tree equipped with a free
action of . Moreover, we show that is a universal tree for in
the sense that it isometrically embeds in every -tree equipped with a
free -action compatible with the original length function on .Comment: 20 pages, 4 figure
Synthesis of Data Word Transducers
In reactive synthesis, the goal is to automatically generate an
implementation from a specification of the reactive and non-terminating
input/output behaviours of a system. Specifications are usually modelled as
logical formulae or automata over infinite sequences of signals
(-words), while implementations are represented as transducers. In the
classical setting, the set of signals is assumed to be finite. In this paper,
we consider data -words instead, i.e., words over an infinite alphabet.
In this context, we study specifications and implementations respectively given
as automata and transducers extended with a finite set of registers. We
consider different instances, depending on whether the specification is
nondeterministic, universal or deterministic, and depending on whether the
number of registers of the implementation is given or not.
In the unbounded setting, we show undecidability for both universal and
nondeterministic specifications, while decidability is recovered in the
deterministic case. In the bounded setting, undecidability still holds for
nondeterministic specifications, but can be recovered by disallowing tests over
input data. The generic technique we use to show the latter result allows us to
reprove some known result, namely decidability of bounded synthesis for
universal specifications
Universal properties of group actions on locally compact spaces
We study universal properties of locally compact G-spaces for countable
infinite groups G. In particular we consider open invariant subsets of the
\beta-compactification of G (which is a G-space in a natural way), and their
minimal closed invariant subspaces. These are locally compact free G-spaces,
and the latter are also minimal. We examine the properies of these G-spaces
with emphasis on their universal properties.
As an example of our resuts, we use combinatorial methods to show that each
countable infinite group admits a free minimal action on the locally compact
non-compact Cantor set.Comment: 42 page
-trees and laminations for free groups II: The dual lamination of an -tree
This is the second part of a series of three articles which introduce
laminations for free groups (see math.GR/0609416 for the first part). Several
definition of the dual lamination of a very small action of a free group on an
-tree are given and proved to be equivalent.Comment: corrections of typos and minor updat
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