Rough analysis in lattices

An outline of an algebraie generalization of the rough set theory is presented in the paper. It is shown that the majority of the basic concepts of this theory has an immediate algebraic generalization, and that some rough set facts are true in general algebraic structures. The formalism employed is that of lattice theory. New concepts of rough order, approximation space and rough (quantitative) approximation space are introduced and investigated. It is shown that the original Pawlak's theory of rough sets and information systems is a model of this general approach

Rough analysis in lattices.

An outline of an algebraie generalization of the rough set theory is presented in the paper. It is shown that the majority of the basic concepts of this theory has an immediate algebraic generalization, and that some rough set facts are true in general algebraic structures. The formalism employed is that of lattice theory. New concepts of rough order, approximation space and rough (quantitative) approximation space are introduced and investigated. It is shown that the original Pawlak's theory of rough sets and information systems is a model of this general approach.Rough set; Information system; Rough dependenee; Rough lattiee; Approximation spaee;

On external presentations of infinite graphs

The vertices of a finite state system are usually a subset of the natural numbers. Most algorithms relative to these systems only use this fact to select vertices. For infinite state systems, however, the situation is different: in particular, for such systems having a finite description, each state of the system is a configuration of some machine. Then most algorithmic approaches rely on the structure of these configurations. Such characterisations are said internal. In order to apply algorithms detecting a structural property (like identifying connected components) one may have first to transform the system in order to fit the description needed for the algorithm. The problem of internal characterisation is that it hides structural properties, and each solution becomes ad hoc relatively to the form of the configurations. On the contrary, external characterisations avoid explicit naming of the vertices. Such characterisation are mostly defined via graph transformations. In this paper we present two kind of external characterisations: deterministic graph rewriting, which in turn characterise regular graphs, deterministic context-free languages, and rational graphs. Inverse substitution from a generator (like the complete binary tree) provides characterisation for prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We illustrate how these characterisation provide an efficient tool for the representation of infinite state systems

An algebraic analysis of the graph modularity

One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix $M$, defined in terms of the adjacency matrix and a rank one null model matrix. That matrix could be posed inside the set of relevant matrices involved in graph theory, alongside adjacency, incidence and Laplacian matrices. This is the reason we propose a graph analysis based on the algebraic and spectral properties of such matrix. In particular, we propose a nodal domain theorem for the eigenvectors of $M$; we point out several relations occurring between graph's communities and nonnegative eigenvalues of $M$; and we derive a Cheeger-type inequality for the graph optimal modularity

Expansions of MSO by cardinality relations

We study expansions of the Weak Monadic Second Order theory of (N,<) by cardinality relations, which are predicates R(X1,...,Xn) whose truth value depends only on the cardinality of the sets X1, ...,Xn. We first provide a (definable) criterion for definability of a cardinality relation in (N,<), and use it to prove that for every cardinality relation R which is not definable in (N,<), there exists a unary cardinality relation which is definable in (N,<,R) and not in (N,<). These results resemble Muchnik and Michaux-Villemaire theorems for Presburger Arithmetic. We prove then that + and x are definable in (N,<,R) for every cardinality relation R which is not definable in (N,<). This implies undecidability of the WMSO theory of (N,<,R). We also consider the related satisfiability problem for the class of finite orderings, namely the question whether an MSO sentence in the language {<,R} admits a finite model M where < is interpreted as a linear ordering, and R as the restriction of some (fixed) cardinality relation to the domain of M. We prove that this problem is undecidable for every cardinality relation R which is not definable in (N,<).Comment: to appear in LMC