445 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
Probabilistic regular graphs
Deterministic graph grammars generate regular graphs, that form a structural
extension of configuration graphs of pushdown systems. In this paper, we study
a probabilistic extension of regular graphs obtained by labelling the terminal
arcs of the graph grammars by probabilities. Stochastic properties of these
graphs are expressed using PCTL, a probabilistic extension of computation tree
logic. We present here an algorithm to perform approximate verification of PCTL
formulae. Moreover, we prove that the exact model-checking problem for PCTL on
probabilistic regular graphs is undecidable, unless restricting to qualitative
properties. Our results generalise those of EKM06, on probabilistic pushdown
automata, using similar methods combined with graph grammars techniques.Comment: In Proceedings INFINITY 2010, arXiv:1010.611
Expressive Power of Hypergraph Lambek Grammars
Hypergraph Lambek grammars (HL-grammars) is a novel logical approach to
generating graph languages based on the hypergraph Lambek calculus. In this
paper, we establish a precise relation between HL-grammars and hypergraph
grammars based on the double pushout (DPO) approach: we prove that HL-grammars
generate the same class of languages as DPO grammars with the linear
restriction on lengths of derivations. This can be viewed as a complete
description of the expressive power of HL-grammars and also as an analogue of
the Pentus theorem, which states that Lambek grammars generate the same class
of languages as context-free grammars. As a corollary, we prove that
HL-grammars subsume contextual hyperedge replacement grammars
From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality
We study how to relate well-known hypergraph grammars based on the double
pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL
(called HL-grammars). It turns out that DPO rules can be naturally encoded by
types of HL using methods similar to those used by Kanazawa for
multiplicative-exponential linear logic. In order to generalize his reasonings
we extend the hypergraph Lambek calculus by adding the exponential modality,
which results in a new calculus HMEL0; then we prove that any DPO grammar can
be converted into an equivalent HMEL0-grammar. We also define the conjunctive
Kleene star, which behaves similarly to this exponential modality, and
establish a similar result. If we add neither the exponential modality nor the
conjunctive Kleene star to HL, then we can still use the same encoding and show
that any DPO grammar with a linear restriction on the length of derivations can
be converted into an equivalent HL-grammar.Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421
Parsing of Hyperedge Replacement Grammars with Graph Parser Combinators
Graph parsing is known to be computationally expensive. For this reason the construction of special-purpose parsers may be beneficial for particular graph languages. In the domain of string languages so-called parser combinators are very popular for writing efficient parsers. Inspired by this approach, we have proposed graph parser combinators in a recent paper, a framework for the rapid development of special-purpose graph parsers. Our basic idea has been to define primitive graph parsers for elementary graph components and a set of combinators for the flexible construction of more advanced graph parsers. Following this approach, a declarative, but also more operational description of a graph language can be given that is a parser at the same time.
In this paper we address the question how the process of writing correct parsers on top of our framework can be simplified by demonstrating the translation of hyperedge replacement grammars into graph parsers. The result are recursive descent parsers as known from string parsing with some additional nondeterminism
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