Closure and Causality

We present a model of causality which is defined by the intersection of two distinct closure systems, ${cal I}$ and ${cal T}$. Next we present empirical evidence to demonstrate that this model has practical validity by examining computer trace data to reveal causal dependencies between individual code modules. From over 498,000 events in the transaction manager of an open source system we tease out 66 apparent causal dependencies. Finally, we explore how to mathematically model the transformation of a causal topology resulting from unforlding events

Convexity in directed graphs

AbstractIn this paper the concept of convexity in directed graphs is described. It is shown that the set of convex subgraphs of a directed graph G partially ordered by inclusion forms a complete, semimodular, A-regular lattice, denoted ℒG. The lattice theoretic properties of the convex subgraph lattice lead to inferences about the path structure of the original graph G. In particular, a graph factorization theorem is developed. In Section 4, several graph homomorphism concepts are investigated in relation to the preservation of convexity properties. Finally we characterize an interesting class of locally convex directed graphs

ESTABLISHING LOGICAL RULES FROM EMPIRICAL DATA

We review a method of generating logical rules, or axioms, from empirical data. This method, using closed set properties of formal concept analysis, has been previously described and tested on rather large sets of deterministic data. In spite of the fact that formal concept techniques have been used to prune frequent set data mining results, frequency and/or statistical significance are totally irrelevant to this method. It is strictly logical and deterministic. The contribution of this paper is a completely new extension of this method to create implications involving numeric inequalities. That is, numerical inequalities such as “age> 39” can be treated as logical predicates that have been extracted from the data itself and not postulated apriori

A Category of Discrete Closure Spaces

Discrete systems such as sets, monoids, groups are familiar categories. The internal strucutre of the latter two is defined by an algebraic operator. In this paper we describe the internal structure of the base set by a closure operator. We illustrate the role of such closure in convex geometries and partially ordered sets and thus suggestthe wide applicability of closure systems. Next we develop the ideas of closed and complete functions over closure spaces. These can be used to establish criteria for asserting that "the closure of a functional image under $f$ is equal to the functional image of the closure". Functions with these properties can be treated as categorical morphisms. Finally, the category "CSystem" of closure systems is shown to be cartesian closed

Finding the Mule in the Network

Abstract—There exist a variety of procedures for identifying clusters in large networks. This paper fo-cuses on finding the connections between such clusters. We employ the concept of closed sets to reduce a network down to its fundamental cycles. These cycles begin to capture the global structure of the network by eliminating a great deal of the fine detail. Nevertheless, the reduced version is completely faithful to the original. No connection in the reduced version exists unless it was in the original network; connectivity is preserved. Reductions of as much as 80 % can be observed in real networks. Just reducing the size makes compre-hension of the network much easier. I