72,045 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
Regular realizability problems and context-free languages
We investigate regular realizability (RR) problems, which are the problems of
verifying whether intersection of a regular language -- the input of the
problem -- and fixed language called filter is non-empty. In this paper we
focus on the case of context-free filters. Algorithmic complexity of the RR
problem is a very coarse measure of context-free languages complexity. This
characteristic is compatible with rational dominance. We present examples of
P-complete RR problems as well as examples of RR problems in the class NL. Also
we discuss RR problems with context-free filters that might have intermediate
complexity. Possible candidates are the languages with polynomially bounded
rational indices.Comment: conference DCFS 201
Hairdressing in groups: a survey of combings and formal languages
A group is combable if it can be represented by a language of words
satisfying a fellow traveller property; an automatic group has a synchronous
combing which is a regular language. This article surveys results for combable
groups, in particular in the case where the combing is a formal language.Comment: 17 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper24.abs.htm
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 B\"uchi-Elgot-Trakhtenbrot theorem for automata with MSO graph storage
We introduce MSO graph storage types, and call a storage type MSO-expressible
if it is isomorphic to some MSO graph storage type. An MSO graph storage type
has MSO-definable sets of graphs as storage configurations and as storage
transformations. We consider sequential automata with MSO graph storage and
associate with each such automaton a string language (in the usual way) and a
graph language; a graph is accepted by the automaton if it represents a correct
sequence of storage configurations for a given input string. For each MSO graph
storage type, we define an MSO logic which is a subset of the usual MSO logic
on graphs. We prove a B\"uchi-Elgot-Trakhtenbrot theorem, both for the string
case and the graph case. Moreover, we prove that (i) each MSO graph
transduction can be used as storage transformation in an MSO graph storage
type, (ii) every automatic storage type is MSO-expressible, and (iii) the
pushdown operator on storage types preserves the property of
MSO-expressibility. Thus, the iterated pushdown storage types are
MSO-expressible
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