The sizes of Markov equivalence classes of directed acyclic graphs play important roles in measuring the uncertainty and complexity in causal learning. A Markov equivalence class can be represented by an essential graph and its undirected subgraphs determine the size of the class. In this paper, we develop a method to derive the formulas for counting the sizes of Markov equivalence classes. We ...

For an operator T in the class Bn(Ω), introduced in [4], the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle ET determine the unitary equivalence class of the operator T . In the paper [6], the authors ask if there exists some pair of inequivalent oprators T1 and T2 for which the simultaneous unitary equival...

Previous studies have shown that after being trained on A-B and A-C match-to-sample tasks, adults match not only same-class Band C stimuli (equivalence) but also BC compounds with sameclass elements and with different-class elements (BC-BC) . The assumption was that the BC-BC performances are based on matching equivalence and nonequivalence relations (equivalenceequivalence) . The present study...

A class K of structures is controlled if for all cardinals λ, the relation of L∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled (see [4]).

It is well known that there may be many causal explanations that are consistent with a given set of data. Recent work has been done to represent the common aspects of these explanations into one representation. In this paper, we address what is less well known: how do the relationships common to every causal explanation among the observed variables of some DAG process change in the presence of ...

A class K of structures is controlled if, for all cardinals λ, the relation of L∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that the class of doubly transitive linear orders is controlled, while any pseudo-elementary class with the ω-independence property is not controlled.

Group classification of a class of Benjamin–Bona–Mahony (BBM) equations with time dependent coefficients is carried out. Two equivalent lists of equations possessing Lie symmetry extensions are presented: up to point equivalence within the class of BBM equations and without the simplification by equivalence transformations. It is shown that the complete results can be achieved using either the ...

The variability of structure in a finite Markov equivalence class of causally sufficient mod­ els represented by directed acyclic graphs has been fully characterized. Without causal suf­ ficiency, an infinite semi-Markov equivalence class of models has only been characterized by the fact that each model in the equiva­ lence class entails the same marginal statis­ tical dependencies. In this pap...

UML is a complex language with many modeling features. Especially the modeling of static structures with class diagrams is supported by a rich set of description primitives. We show how to transfrom UML class diagrams involving cardinality constraints, qualifiers, association classes, aggregations, compositions, and generalizations into equivalent UML class diagrams employing only n-ary associa...

The conventional understanding of genetic algorithms depends upon analysis by schemata and the notion of intrinsic parallelism. For this reason, only k-ary string representations have had any formal basis and non-standard representations and operators have been regarded largely as heuristics, rather than principled algorithms. This paper extends the analysis to general representations through i...