4 research outputs found
Fuzzy graphs: Algebraic structure and syntactic recognition
© Springer Science+Business Media Dordrecht 2013. Directed fuzzy hypergraphs are introduced as a generalization of both crisp directed hypergraphs and directed fuzzy graphs. It is proved that the set of all directed fuzzy hypergraphs can be structured into a magmoid with operations graph composition and disjoint union. In this framework a notion of syntactic recognition inside magmoids is defined. The corresponding class is proved to be closed under boolean operations and inverse mor-phisms of magmoids. Moreover, the language of all strongly connected fuzzy graphs and the language that consists of all fuzzy graphs that have at least one directed path from the begin node to the end node through edges with membership grade 1 are recognizable. Additionally, a useful characterization of recognizability through left derivatives is also achieved
Dynamic Programming on Nominal Graphs
Many optimization problems can be naturally represented as (hyper) graphs,
where vertices correspond to variables and edges to tasks, whose cost depends
on the values of the adjacent variables. Capitalizing on the structure of the
graph, suitable dynamic programming strategies can select certain orders of
evaluation of the variables which guarantee to reach both an optimal solution
and a minimal size of the tables computed in the optimization process. In this
paper we introduce a simple algebraic specification with parallel composition
and restriction whose terms up to structural axioms are the graphs mentioned
above. In addition, free (unrestricted) vertices are labelled with variables,
and the specification includes operations of name permutation with finite
support. We show a correspondence between the well-known tree decompositions of
graphs and our terms. If an axiom of scope extension is dropped, several
(hierarchical) terms actually correspond to the same graph. A suitable
graphical structure can be found, corresponding to every hierarchical term.
Evaluating such a graphical structure in some target algebra yields a dynamic
programming strategy. If the target algebra satisfies the scope extension
axiom, then the result does not depend on the particular structure, but only on
the original graph. We apply our approach to the parking optimization problem
developed in the ASCENS e-mobility case study, in collaboration with
Volkswagen. Dynamic programming evaluations are particularly interesting for
autonomic systems, where actual behavior often consists of propagating local
knowledge to obtain global knowledge and getting it back for local decisions.Comment: In Proceedings GaM 2015, arXiv:1504.0244
Free inverse monoids up to rewriting
In this paper, generalizing the study of free partially commutative inverse monoids [5], for any rewriting system T over an alphabet A, we define the notion of T-compatible inverse A-generated monoids, we show there is a free T-compatible monoid FIM(A,T) generated by A and we provide an explicit construction of this monoid. Then, as examples, free partially commutative inverse monoid and free partially semi-commutative inverse monoids are studied and shown to have effective representations
Inverse monoids of higher-dimensional strings
International audienceHalfway between graph transformation theory and inverse semigroup theory, we define higher dimensional strings as bi-deterministic graphs with distinguished sets of input roots and output roots. We show that these generalized strings can be equipped with an associative product so that the resulting algebraic structure is an inverse semigroup. Its natural order is shown to capture existence of root preserving graph mor-phism. A simple set of generators is characterized. As a subsemigroup example, we show how all finite grids are finitely generated. Last, simple additional restrictions on products lead to the definition of subclasses with decidable Monadic Second Order (MSO) language theory