An algorithm for road coloring

A coloring of edges of a finite directed graph turns the graph into a finite-state automaton. A synchronizing word of a deterministic automaton is a word in the alphabet of colors of its edges (regarded as letters) which maps the automaton to a single state. A coloring of edges of a directed graph of uniform outdegree (constant outdegree of any vertex) is synchronizing if the corresponding deterministic finite automaton possesses a synchronizing word.The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph of uniform outdegree if the greatest common divisor of lengths of all its cycles is one. Posed in 1970, it has evoked noticeable interest among the specialists in the theory of graphs, automata, codes, symbolic dynamics, and well beyond these areas.We present an algorithm for the road coloring of cubic worst-case time complexity and quadratic time complexity in the majority of studied cases. It is based on the recent positive solution of the road coloring problem. The algorithm was implemented in the freeware package TESTAS.     

A recognition algorithm for the intersection graphs of directed paths in directed trees

Consider a finite family of non-empty sets. The intersection graph of this family is obtained by representing each set by a vertex, two vertices being connected by an edge if and only if the corresponding sets intersect. The intersection graph of a family of directed paths in a directed tree is called a directed path graph. In this paper we present an efficient algorithm which constructs to a given graph a representation by a family of directed paths on a directed tree, if one exists. Also, we prove that a graph is a proper directed path graph if and only if it is a directed path graph.     

Fuzzy正则表达式与Fuzzy有限态自动机的关系

首先给出了Fuzzy正则表达式的定义,接着通过研究Fuzzy正则表达式与Fuzzy有限态自动机的关系,得到了两个重要性质,即：每一个Fuzzy正则表达式,都有一个非确定性的Fuzzy有限态自动机接受其代表的语言;每一个被确定性的Fuzzy有限态自动机接受的语言,都能被一个Fuzzy正则表达式表示．     

The road coloring problem

A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. The positive solution of the road coloring problem is presented.     

On minimal realization of fuzzy behaviour and associated categories

This paper is towards the study of minimal deterministic automata with fuzzy output which realizes the given fuzzy behaviour. The construction of such automaton is based on Myhill-Nerode’s theory, and it is shown that this automaton is a canonical realization of given fuzzy behaviour. Meanwhile, we introduce the categories of deterministic automata with fuzzy output and fuzzy behaviour. The relationship between both the categories through functors is also studied.     

Decision-making in a fuzzy environment with fuzzy termination time

The multistage control of a deterministic and stochastic system in a fuzzy environment is considered. The fuzzy environment is meant as fuzzy constraints imposed on subsequent controls and a fuzzy goal to be attained. The fuzzy decision is assumed to be the intersection of fuzzy constraints and a fuzzy goal. The problem is to find a maximizing decision. The termination time is given as a specified fuzzy set in the space of control stages. For solving the problem, the dynamic programming is applied.     

The complexity of social groups and social systems described by graph structures

This paper presents methodology which permits the complete ranking of nondirected graphs (NDG's) on an attribute labelled ‘complexity.’ The technique applies to both small and large systems as might arise in studies of group or organization behavior. The methodology extends to cover the complexity of directed graphs (DG's) and permits the detailed specification of individual and group behavior.For the NDG an abstract automaton representing the participants' interaction or communications function is sited at each node. Each automaton is constructed so its internal complexity is sufficient to realize the minimal social action (e.g. transmission of a rumor and the path followed by the rumor) within the framework of the NDG. It is shown that the complexity of each node automaton depends upon the order of the graph, the degree of the node and the longest path parameter of the graph. The combined complexity of node automata constitutes the complexity of the NDG. The complexity of a DG is specified as a composition of complexities computed for the associated NDG and logical devices which produce the observed behavior. Illustrative examples pertaining to the committee-subcommittee problem and to organizational structures are presented.     

A dynamic programming algorithm for the tree mapping problem

In the graph mapping problem two graphs and a cost function are given as input and the goal is to find a minimum cost mapping of the vertex set of one graph into the vertex set of the other graph. In this paper we introduce a dynamic programming algorithm for the tree mapping problem (i.e., the variant of the problem in which the input graphs are trees), which is still NP-hard, and evaluate its performance with computational experiments.     

Stable graphs for a family of endomorphisms

We prove several fixed subgraph properties. In particular it is shown that if is a commuting family of contractions of a connected graph G without infinite path and infinite interval, then there exists a nonempty finite subgraph F which is invariant under any element of . In particular this subgraph F is a simplex if G is moreover a strongly dismantlable graph or a ball-Helly graph without infinite block, or if it is chordal. This implies that for any commuting family of contractions of a tree without infinite path, there is a common fixed vertex or a common fixed edge.     

Fuzzy local ω-systems

The natural language computing today demands for the study of ω-languages. Therefore in this respect it is convenient to consider fuzzy ω-languages. In this paper, the concept of fuzzy local ω-language, Büchi fuzzy local ω-language, and some closure properties of fuzzy local ω-languages are presented. We introduce deterministic fuzzy finite automaton with different acceptance mode on fuzzy ω-languages and establish the relationship between these various classes of fuzzy ω-languages. We have defined deterministic fuzzy local automaton and also establish relationships between deterministic fuzzy local automaton, fuzzy local ω-language and Büchi fuzzy local ω-language. Further we show that every fuzzy regular ω-language is a projection of a Büchi fuzzy local ω-language.     

An intersection problem for finite automata

A finite automaton with state space S and alphabet A can be thought of as a directed graph with vertex set S such that for each vertex t ϵ S the edges leaving t are in one-to-one correspondence with A. Motivated by a problem in logic, we propose a problem about intersections of languages accepted by finite automata and give some partial results.     

Finitely nonstationary nondeterministic automata with random input

Properties of generalized finitely nonstationary nondeterministic automata with an additional random input over a Boolean lattice are considered which are related to the definition of the class of languages represented by such automaton models. New notions of an elementary nondeterministic automatic structure with a random input, of a generalized finitely nonstationary nondeterministic automaton with a random input, of the generalized mapping induced by such an automaton, and of a generalized language represented by such an automaton are introduced. A number of statements substantiating synthesis for any given generalized finitely nonstationary nondeterministic automaton with a random input of an abstract probabilistic finite automaton equivalent to the given one relative to the represented generalized language probabilistic language of the stationary abstract probabilistic finite automaton. The number of states of the synthesized probabilistic automaton is estimated and a synthesis algorithm is developed in detail and illustrated by an example.     

Optimal stopping in a fuzzy environment

The problem under consideration is that of optimally controlling and stopping either a deterministic or a stochastic system in a fuzzy environment. The optimal decision is the sequence of controls that maximizes the membership function of the intersection of the fuzzy constraints and a fuzzy goal. The fuzzy goal is a fuzzy set in the cartesian product of the state space with the set of possible stopping times. Dynamic programming is applied to yield a numerical solution. This approach yields an algorithm that corrects a result of Kacprzyk.     

Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also "randomizing" given graphs. The main specific result that we prove here (Theorem 1) is that if is the smallest vertex degree in G, then almost all n-covers of G are -connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. Received June 21, 1999/Revised November 16, 2000 RID="*" ID="*" Work supported in part by grants from the Israel Academy of Aciences and the Binational Israel-US Science Foundation.     

The Automata that Define Representations of Monomial Algebras

It is well known that the sets of strings that define all representations of string algebras and many representations of other quotients of path algebras form a regular set, and hence are defined by finite state automata. This short article aims to explain this connection between representation theory and automata theory in elementary terms; no technical background in either representation theory or automata theory is assumed. The article describes the structure of the set of strings of a monomial algebra as a locally testable and hence regular set, and describes explicitly the construction of the automaton, illustrating the construction with an elementary example. Hence it explains how the sets of strings and bands of a monomial algebra correspond to the sets of paths and closed (non-powered) circuits in a finite graph, and how the growth rate of the set of bands is immediately visible from that graph. Presented by C. Ringel.     

直觉模糊变换半群

首先定义了直觉模糊变换半群的概念,给出了一种特殊的直觉模糊变换半群.其次,引入了直觉模糊变换半群上的直觉容许关系,讨论了两个直觉模糊变换半群间的关系,为直觉模糊有限自动机进一步的理论研究提供了代数方法.     

Pareto Simulated Annealing for Fuzzy Multi-Objective Combinatorial Optimization

The paper presents a metaheuristic method for solving fuzzy multi-objective combinatorial optimization problems. It extends the Pareto simulated annealing (PSA) method proposed originally for the crisp multi-objective combinatorial (MOCO) problems and is called fuzzy Pareto simulated annealing (FPSA). The method does not transform the original fuzzy MOCO problem to an auxiliary deterministic problem but works in the original fuzzy objective space. Its goal is to find a set of approximately efficient solutions being a good approximation of the whole set of efficient solutions defined in the fuzzy objective space. The extension of PSA to FPSA requires the definition of the dominance in the fuzzy objective space, modification of rules for calculating probability of accepting a new solution and application of a defuzzification operator for updating the average position of a solution in the objective space. The use of the FPSA method is illustrated by its application to an agricultural multi-objective project scheduling problem.     

Vertex transversals that dominate

For any graph, there is a largest integer k such that given any partition of the vertex set with at most k elements in each class of the partition, there is transversal of the partition that is a dominating set in the graph. Some basic results about this parameter, the partition domination number, are obtained. In particular, it is shown that its value is 2 for the two-dimensional infinite grid, and that determining whether a given vertex partition admits a dominating transversal is NP-complete, even for a graph which is a simple path. The existence of various dominating transversals in certain partitions in regular graphs is studied as well. © 1996 John Wiley & Sons, Inc.     

Finding a minimum path cover of a distance-hereditary graph in polynomial time

A path cover of a graph G=(V,E) is a set of pairwise vertex-disjoint paths such that the disjoint union of the vertices of these paths equals the vertex set V of G. The path cover problem is, given a graph, to find a path cover having the minimum number of paths. The path cover problem contains the Hamiltonian path problem as a special case since finding a path cover, consisting of a single path, corresponds directly to the Hamiltonian path problem. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. The complexity of the path cover problem on distance-hereditary graphs has remained unknown. In this paper, we propose the first polynomial-time algorithm, which runs in O(|V|⁹) time, to solve the path cover problem on distance-hereditary graphs.     

笛卡尔乘积图里的最小k-路点覆盖

对于子集$S\subseteq V(G)$,如果图$G$里的每一条$k$路都至少包含$S$中的一个点,那么我们称集合$S$是图$G$的一个$k$-路点覆盖.很明显,这个子集并不唯一.我们称最小的$k$-路点覆盖的基数为$k$-路点覆盖数, 记作$\psi_k(G)$.本文给出了一些笛卡尔乘积图上$\psi_k(G)$值的上界或下界.