Igor Litovsky

Different local controls for graph relabeling systems

We are interested in models to encode and to prove decentralized and distributed computations on graphs. In this paper we define and compare six models of graph relabeling systems. These systems do not change the underlying structure of the graph on which they work, but only the labeling of its components (edges or vertices). Each relabeling step is fully determined by the knowledge of a fixed-size subgraph, the local context of the relabeled occurrence. The families studied are based on the relabeling of partial or induced subgraphs and we use two kinds of mechanisms to control the applicability of rules locally: a priority relation on the set of rules or a set of forbidden contexts associated with each rule. We show that these two basic (i.e., without local control) families of graph relabeling systems are distinct, but whenever we consider the local controls of the relabeling, the four families so obtained are equivalent.

Computing with graph rewriting systems with priorities

Abstract In this paper, the computational power of the noetherian graph rewriting systems with priorities (PGRSs) is studied. We define the notion of safe PGRS with respect to a given property. The PGRSs are considered as recognizers for sets of graphs (1-graphs). The classes of sets of graphs (1-graphs) so defined are compared with the classes definable by logic formulas. We end with the particular cases of trees and words.

The Power and the Limitations of Local Computations on Graphs

This paper is a contribution to understanding the power and the limitations of asynchronous local computations on graphs and networks. We use local computations to define a notion of graph recognition, in particular our model enables a simulation of finite automata on words and on trees. We introduce the notion of k-covering to examine limitations of such systems. For example we prove that we cannot recognize the families of series parallel graphs and planar graphs by means of local computations.

Finite acceptance of infinite words

Abstract In this paper we consider the following two types of finite acceptance of infinite words by finite automata: An infinite word ξ is accepted if and only if there is a run on input ξ for which 1. (1) an accepting state is visited at least once, or 2. (2) an accepting state is visited at least once but only finitely often. The resulting classes of regular ω-languages are characterized by language-theoretic means, and they are positioned into the known hierarchies of regular ω-languages.

Characterizations of rational Ω-languages by means of right congruences

Abstract By means of right congruences, we characterize in a unified way different classical classes of rational ω-languages. A new congruence associated with an ω-automaton is introduced named cycle congruence. The family of rational ω-languages which have, by morphism, a unique minimal recognizing ω-automaton is characterized. It appears that for such a minimal ω-automaton, the cycle congruence coincides with the syntactic congruence of the recognized ω-language. We prove that the other rational ω-languages have an infinite number of minimal automaton.

Stability for the zigzag submonoids

Abstract A notion of stability for the zigzag submonoids is introduced and called Z-stability. We prove that for the zigzag submonoids the Z-stability and the Z-freeness are equivalent, like the stability and the freeness for the ordinary submonoids. The class of Z-free zigzag submonoids is, however, not closed under intersection. We prove also that the property of being Z-stable is decidable for the regular zigzag submonoids.

A more efficient notion of zigzag stability

In [6] a definition of zigzag stability for zigzag submonoids is given. We give here a new definition of zigzag stability, which is simpler and very close to the usual definition of stability for ordinary submonoids. We give a short proof that zigzag stability is decidable in the rational case. Moreover this notion of zigzag stability enables one to obtain an algorithm to decide whether a rational language is a zigzag code. Despite being in exponential time, the complexity of this algorithm is better than that of every other known algorithm [3], [8], [6].

Definitions and Comparisons of Local Computations on Graphs

We are interested in models to encode and to prove decentralized and distributed computations on graphs or networks. In this paper, we define and compare six models of graph rewriting systems. These systems do not change the underlying structure of the graph on which they work, but only the labelling of its components (edges or vertices). Each rewriting step is fully determinated by the knowledge of a fixed size subgraph, the local context of the rewritten occurrence. The studied families are based on the rewriting of partial or induced subgraphs and we use two kinds of mechanisms to locally control the applicability of rules: a priority relation on the set of rules or a set of forbidden contexts associated with each rule. We show that these two basic (i.e. without local control) families of graph rewriting systems are distinct, but whenever we consider the local controls of the rewriting, the four so-obtained families are equivalent.

Computing with Graph Relabelling Systems with Priorities

In this paper, the computational power of the noetherian Graph Relabelling systems with Priorities (PGRS for short) is studied. The PGRSs are considered as recognizers for sets of graphs and for sets of 1-sourced graphs. We show that the PGRSs are strictly more powerful for the 1-sourced graphs than for the graphs. Furthermore every set of 1-sourced graphs definable in First Order Logic is recognizable by some PGRS.

On the minimization problem for ω-automata

The family of rational ω-languages which are accepted by a unique minimal ω-automaton — using deterministic automaton morphism reductions — is characterized. All the other rational ω-languages have an infinite number of minimal ω-automata.