7,856 research outputs found
On external presentations of infinite graphs
The vertices of a finite state system are usually a subset of the natural
numbers. Most algorithms relative to these systems only use this fact to select
vertices.
For infinite state systems, however, the situation is different: in
particular, for such systems having a finite description, each state of the
system is a configuration of some machine. Then most algorithmic approaches
rely on the structure of these configurations. Such characterisations are said
internal. In order to apply algorithms detecting a structural property (like
identifying connected components) one may have first to transform the system in
order to fit the description needed for the algorithm. The problem of internal
characterisation is that it hides structural properties, and each solution
becomes ad hoc relatively to the form of the configurations.
On the contrary, external characterisations avoid explicit naming of the
vertices. Such characterisation are mostly defined via graph transformations.
In this paper we present two kind of external characterisations:
deterministic graph rewriting, which in turn characterise regular graphs,
deterministic context-free languages, and rational graphs. Inverse substitution
from a generator (like the complete binary tree) provides characterisation for
prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We
illustrate how these characterisation provide an efficient tool for the
representation of infinite state systems
An introduction to (Co)algebras and (Co)induction and their application to the semantics of programming languages
This report summarizes operational approaches to the formal
semantics of programming languages and shows that they can be
interpreted inductively by least fixed points as well as
coinductively by greatest fixed points. While the inductive
interpretation gives semantics to all terminating programs,
the coinductive one defines moreover also a semantics for all
non-terminating programs. This is especially important in
areas where programs do not terminate in general, e.g. data
bases, operating systems, or control software in embedded
systems. The semantic foundations described in this report can
be used to verify that transformations (e.g. in compilers) of
such software systems are correct.
In the course of this report, coalgebras and coinduction are
introduced, starting with a gentle intuitive motivation and
ending with a detailed mathematical description within the
notions of category theory
A geometric approach to (semi)-groups defined by automata via dual transducers
We give a geometric approach to groups defined by automata via the notion of
enriched dual of an inverse transducer. Using this geometric correspondence we
first provide some finiteness results, then we consider groups generated by the
dual of Cayley type of machines. Lastly, we address the problem of the study of
the action of these groups in the boundary. We show that examples of groups
having essentially free actions without critical points lie in the class of
groups defined by the transducers whose enriched dual generate a torsion-free
semigroup. Finally, we provide necessary and sufficient conditions to have
finite Schreier graphs on the boundary yielding to the decidability of the
algorithmic problem of checking the existence of Schreier graphs on the
boundary whose cardinalities are upper bounded by some fixed integer
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