On Ordinal VC-Dimension and Some Notions of Complexity

We generalize the classical notion of Vapnik–Chernovenkis (VC) dimension to ordinal VC-dimension, in the context of logical learning paradigms. Logical learning paradigms encompass the numerical learning paradigms commonly studied in Inductive Inference. A logical learning paradigm is defined as a set W of structures over some vocabulary, and a set D of first-order formulas that represent data. The sets of models of ϕ in W, where ϕ varies over D, generate a natural topology W over W. We show that if D is closed under boolean operators, then the notion of ordinal VC-dimension offers a perfect characterization for the problem of predicting the truth of the members of D in a member of W, with an ordinal bound on the number of mistakes. This shows that the notion of VC-dimension has a natural interpretation in Inductive Inference, when cast into a logical setting. We also study the relationships between predictive complexity, selective complexity—a variation on predictive complexity—and mind change complexity. The assumptions that D is closed under boolean operators and that W is compact often play a crucial role to establish connections between these concepts. We then consider a computable setting with effective versions of the complexity measures, and show that the equivalence between ordinal VC-dimension and predictive complexity fails. More precisely, we prove that the effective ordinal VC-dimension of a paradigm can be defined when all other effective notions of complexity are undefined. On a better note, when W is compact, all effective notions of complexity are defined, though they are not related as in the noncomputable version of the framework.

Destination Positioning Slogans - Analysis of Themes used by New Zealand Regional Tourism Organisations

A significant gap in the tourism and travel literature exists in the area of tourism destination branding. Although brands have been used as sources of differentiation in consumer goods markets for over a century, academic research attention towards destination branding has only been reported since the late 1990s. Three important components of the brand construct are brand identity, brand position and brand image. While interest in applications of brand theory to practise in tourism is increasing, there is a paucity of published research in the literature to guide destination marketing organisations (DMOs). In particular there have been few reported analyses of destination brand positioning slogans. The focus of this paper is on destination brand position slogans, which represent the interface between brand identity and brand image. Part of a wider investigation of DMO slogans worldwide, and in keeping with the conference location, the paper focuses on analysis of slogans used by New Zealand RTOs. The slogans are examined in terms of the extent to which they have been limited to ephemeral indifference. In other words, have they stood the test of time and do they effectively differentiate through a meaningful proposition? Analysis of the slogans indicates very few could be characterised as memorably distinctive. This reflects the complexity involved in capturing the essence of a multi-attributed destination in a succinct and focused positioning slogan, in a way that is both meaningful to the target audience and effectively differentiates the destination from competitors offering the same benefits.