27,528 research outputs found
Decidability of Identity-free Relational Kleene Lattices
National audienceFamilies of binary relations are important interpretations of regular expressions, and the equivalence of two regular expressions with respect to their relational interpretations is decidable: the problem reduces to the equality of the denoted regular languages.Putting together a few results from the literature, we first make explicit a generalisation of this reduction, for regular expressions extended with converse and intersection: instead of considering sets of words (i.e., formal languages), one has to consider sets of directed and labelled graphs.We then focus on identity-free regular expressions with intersection—a setting where the above graphs are acyclic—and we show that the corresponding equational theory is decidable. We achieve this by defining an automaton model, based on Petri Nets, to recognise these sets of acyclic graphs, and by providing an algorithm to compare such automata
Regular Expressions and Transducers over Alphabet-invariant and User-defined Labels
We are interested in regular expressions and transducers that represent word
relations in an alphabet-invariant way---for example, the set of all word pairs
u,v where v is a prefix of u independently of what the alphabet is. Current
software systems of formal language objects do not have a mechanism to define
such objects. We define transducers in which transition labels involve what we
call set specifications, some of which are alphabet invariant. In fact, we give
a more broad definition of automata-type objects, called labelled graphs, where
each transition label can be any string, as long as that string represents a
subset of a certain monoid. Then, the behaviour of the labelled graph is a
subset of that monoid. We do the same for regular expressions. We obtain
extensions of a few classic algorithmic constructions on ordinary regular
expressions and transducers at the broad level of labelled graphs and in such a
way that the computational efficiency of the extended constructions is not
sacrificed. For regular expressions with set specs we obtain the corresponding
partial derivative automata. For transducers with set specs we obtain further
algorithms that can be applied to questions about independent regular
languages, in particular the witness version of the independent property
satisfaction question
The Semantics of Graph Programs
GP (for Graph Programs) is a rule-based, nondeterministic programming
language for solving graph problems at a high level of abstraction, freeing
programmers from handling low-level data structures. The core of GP consists of
four constructs: single-step application of a set of conditional
graph-transformation rules, sequential composition, branching and iteration. We
present a formal semantics for GP in the style of structural operational
semantics. A special feature of our semantics is the use of finitely failing
programs to define GP's powerful branching and iteration commands
Vector Reachability Problem in
The decision problems on matrices were intensively studied for many decades
as matrix products play an essential role in the representation of various
computational processes. However, many computational problems for matrix
semigroups are inherently difficult to solve even for problems in low
dimensions and most matrix semigroup problems become undecidable in general
starting from dimension three or four.
This paper solves two open problems about the decidability of the vector
reachability problem over a finitely generated semigroup of matrices from
and the point to point reachability (over rational
numbers) for fractional linear transformations, where associated matrices are
from . The approach to solving reachability problems
is based on the characterization of reachability paths between points which is
followed by the translation of numerical problems on matrices into
computational and combinatorial problems on words and formal languages. We also
give a geometric interpretation of reachability paths and extend the
decidability results to matrix products represented by arbitrary labelled
directed graphs. Finally, we will use this technique to prove that a special
case of the scalar reachability problem is decidable
Equational reasoning with context-free families of string diagrams
String diagrams provide an intuitive language for expressing networks of
interacting processes graphically. A discrete representation of string
diagrams, called string graphs, allows for mechanised equational reasoning by
double-pushout rewriting. However, one often wishes to express not just single
equations, but entire families of equations between diagrams of arbitrary size.
To do this we define a class of context-free grammars, called B-ESG grammars,
that are suitable for defining entire families of string graphs, and crucially,
of string graph rewrite rules. We show that the language-membership and
match-enumeration problems are decidable for these grammars, and hence that
there is an algorithm for rewriting string graphs according to B-ESG rewrite
patterns. We also show that it is possible to reason at the level of grammars
by providing a simple method for transforming a grammar by string graph
rewriting, and showing admissibility of the induced B-ESG rewrite pattern.Comment: International Conference on Graph Transformation, ICGT 2015. The
final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-21145-9_
!-Graphs with Trivial Overlap are Context-Free
String diagrams are a powerful tool for reasoning about composite structures
in symmetric monoidal categories. By representing string diagrams as graphs,
equational reasoning can be done automatically by double-pushout rewriting.
!-graphs give us the means of expressing and proving properties about whole
families of these graphs simultaneously. While !-graphs provide elegant proofs
of surprisingly powerful theorems, little is known about the formal properties
of the graph languages they define. This paper takes the first step in
characterising these languages by showing that an important subclass of
!-graphs--those whose repeated structures only overlap trivially--can be
encoded using a (context-free) vertex replacement grammar.Comment: In Proceedings GaM 2015, arXiv:1504.0244
The Joys of Graph Transformation
We believe that the technique of graph transformation offers a very natural way to specify semantics for languages that have dynamic allocation and linking structure; for instance, object-oriented programming languages, but also languages for mobility. In this note we expose, on a rather informal level, the reasons for this belief. Our hope in doing this is to raise interest in this technique and so generate more interest in the fascinating possibilities and open questions of this area.\u
Full Semantics Preservation in Model Transformation – A Comparison of Proof Techniques
Model transformation is a prime technique in modern, model-driven software design. One of the most challenging issues is to show that the semantics of the models is not affected by the transformation. So far, there is hardly any research into this issue, in particular in those cases where the source and target languages are different.\ud
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In this paper, we are using two different state-of-the-art proof techniques (explicit bisimulation construction versus borrowed contexts) to show bisimilarity preservation of a given model transformation between two simple (self-defined) languages, both of which are equipped with a graph transformation-based operational semantics. The contrast between these proof techniques is interesting because they are based on different model transformation strategies: triple graph grammars versus in situ transformation. We proceed to compare the proofs and discuss scalability to a more realistic setting.\u
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
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