Antonios Kalampakas

The syntactic complexity of Eulerian graphs

In this paper we prove that the set of directed Eulerian graphs is not recognizable. On the other hand, the set of directed graphs with an Eulerian underlying graph is shown to be recognizable. Furthermore, we compute the syntactic complexity of this language and we compare it with that of connected graphs.

Fuzzy graphs: algebraic structure and syntactic recognition

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 morphisms 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.

The Path Hyperoperation

Abstract A new class of hypergroupoids, deriving from binary relations, is presented, via the introduced path hyperoperation. Its properties are investigated and its connections with Graph Theory are also delineated. Moreover, we present an application of this hyperoperation to assembly line designing and management.

On the syntactic complexity of tree series

We display a complexity notion based on the syntax of a tree series which yields two distinct hierarchies, one within the class of recognizable tree series and another one in the class of non-recognizable tree series.

On the Complexity of the Syntax of Tree Languages

The syntactic complexity of a tree language is defined according to the number of the distinct syntactic classes of all trees with a fixed yield length. This leads to a syntactic classification of tree languages and it turns out that the class of recognizable tree languages is properly contained in that of languages with bounded complexity. A refined syntactic complexity notion is also presented, appropriate exclusively for the class of recognizable tree languages. A tree language is recognizable if and only if it has finitely many refined syntactic classes. The constructive complexity of a tree automaton is also investigated and we prove that for any reachable tree automaton it is equal with the refined syntactic complexity of its behavior.

Path hypergroupoids

The path hyperoperation, a generalization of Corsinis hyperoperation deriving from graph theory, was recently introduced and applied for assembly line designing. In this paper we study the commutativity property for the corresponding class of hypergroupoids and investigate their connection with Graph Theory. Moreover, we prove that such a (partial) hypergroupoid is commutative if and only if it can be obtained as a disjoint union of non-partial hypergroupoids.

Syntactic Recognizability of Graphs with Fuzzy Attributes

Abstract We present a syntactic recognition theory for graphs with fuzzy edges and fuzzy nodes by employing the algebraic structure of magmoids. Several examples are illustrated and it is proved that the α -cut of a recognizable set of fuzzy graphs is not always recognizable as a set of crisp graphs. The described theory is then applied for the recognition of fuzzy cognitive maps and the recognizability of interesting examples of sets of fuzzy cognitive maps is examined.

A Formal Representation of Video Content with the Picture Hyperoperation

A hyperoperation on the pixels of a two-dimensional picture is introduced and studied. In this setup pictures are defined as a specific type of rectangular graphs and the picture hyperoperation is given by virtue of the notion of the path inside such a picture. Using this hyperoperation objects can naturally be defined inside pictures in an algebraic way and this concept can be utilized in order to formally represent and compare video content using algebraic semiotics.

Towards Formal Representation and Comparison of Video Content Using Algebraic Semiotics

The main idea of this paper is to use an extension of algebraic semiotics - a technique that provides systematic ways to evaluate the quality of representations - that can handle dynamic systems, to formally specify systems of moving pictures, and compare their semantic annotated models. Some first steps towards this direction are presented, through a simple but illustrative example.

Automata on Directed Graphs for the Recognition of Assembly Lines

Automata on general directed hypergraphs have been recently introduced based on the algebraic representation of hypergraphs inside graphoids. In this paper, we employ this mechanism for the construction of automata recognizing mixed model assembly lines.