Separating Data Examples by Description Logic Concepts with Restricted Signatures

We study the separation of positive and negative data examples in terms of description logic concepts in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols that can be used for separation, and we admit individual names in that signature. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated and we distinguish between separation with and without helper symbols. Within this framework, we compare the separating power of different languages and investigate the complexity of deciding separability. While weak separability is shown to be closely related to conservative extensions, strongly separating concepts coincide with Craig interpolants, for suitably defined encodings of the data and ontology. This enables us to transfer known results from those fields to separability. Conversely, we obtain original results on separability that can be transferred backward. For example, rather surprisingly, conservative extensions and weak separability in ALCO are both 3ExpTime-complete.

ON LOCAL WEAK τ-DENSITY OF TOPOLOGICAL SPACES

A topological space X is locally weakly separable [3] at a point x∈X if x has a weakly separable neighbourhood. A topological space X is locally weakly separable if X is locally weakly separable at every point x∈X. The notion of local weak separability can be generalized for any cardinal τ ≥ℵ0 . A topological space X is locally weakly τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a weak τ-dense neighborhood in X [4]. The local weak density at a point x is denoted as lwd(x). The local weak density of a topological space X is defined in following way: lwd ( X ) = sup{ lwd ( x) : x∈ X } . A topological space X is locally τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a τ-dense neighborhood in X [4]. The local density at a point x is denoted as ld(x). The local density of a topological space X is defined in following way: ld ( X ) = sup{ ld ( x) : x∈ X } . It is known that for any topological space we have ld(X ) ≤ d(X ) . In this paper, we study questions of the local weak τ-density of topological spaces and establish sufficient conditions for the preservation of the property of a local weak τ-density of subsets of topological spaces. It is proved that a subset of a locally τ-dense space is also locally weakly τ-dense if it satisfies at least one of the following conditions: (a) the subset is open in the space; (b) the subset is everywhere dense in space; (c) the subset is canonically closed in space. A proof is given that the sum, intersection, and product of locally weakly τ-dense spaces are also locally weakly τ-dense spaces. And also questions of local τ-density and local weak τ-density are considered in locally compact spaces. It is proved that these two concepts coincide in locally compact spaces.

CONSUMPTION, LEISURE, AND MONEY

This paper takes a parametric approach to demand analysis and tests the weak separability assumptions that are often implicitly made in representative agent models of modern macroeconomics. The approach allows estimation and testing in a systems-of-equations context, using the minflex Laurent flexible functional form for the underlying utility function and relaxing the assumption of fixed consumer preferences by assuming Markov regime switching. We generate inference consistent with both theoretical and econometric regularity. We strongly reject weak separability of consumption and leisure from real money balances as well as weak separability of consumption from leisure and real money balances, meaning that the inclusion of a money in economic models would be of quantitative importance. We also investigate the substitutability/complementarity relationship among different categories of personal consumption expenditure (nondurables, durables, and services), leisure, and money. We find that the goods are net Morishima substitutes, but because of positive income effects they are gross complements. The implications for monetary policy are also briefly discussed.

Pareto Optimality on Compact Spaces in a Preference-Based Setting under Incompleteness

We characterize the existence of Pareto optimal elements for a family of not necessarily total preorders on a compact topological space. We identify a rather general semi-continuity assumption, called weak upper semicontinuity, under which there exist Pareto optimal elements. We also show that weak upper semicontinuity of each individual preorder is a necessary and sufficient condition for determining the Pareto optimal elements by solving the classical multi-objective optimization problem in case that each function is upper semicontinuous and order-preserving for the respective preorder, and each preorder satisfies a condition of weak separability.

INDEX NUMBERS AND REVEALED PREFERENCE RANKINGS

For previously identified weakly separable blockings of goods and assets, we construct aggregates using four superlative index numbers, the Fisher, Sato-Vartia, Törnqvist, and Walsh, two non-superlative indexes, the Laspeyres and Paasche, and the atheoretical simple summation. We conduct several tests to examine how well each of these aggregates “fit” the data. These tests are how close the aggregates come to solving the revealed preference conditions for weak separability, how often each aggregate gets the direction of change correct, and how well the aggregates mimic the preference ranking from revealed preference tests. We find that, as the number of goods and assets being aggregated increases, the problems with simple summation manifest.