Secondary generalisation in categorisation: an exemplar-based account

The parallel rule activation and rule synthesis (PRAS) model is a computational model for generalisation in category learning, proposed by Vandierendonck (1995). An important concept underlying the PRAS model is the distinction between primary and secondary generalisation. In Vandierendonck (1995), an empirical study is reported that provides support for the concept of secondary generalisation. In this paper, we re-analyse the data reported by Vandierendonck (1995) by fitting three different variants of the Generalised Context Model (GCM) which do not rely on secondary generalisation. Although some of the GCM variants outperformed the PRAS model in terms of global fit, they all have difficulty in providing a qualitatively good fit of a specific critical pattern

Generalisation : graphs and colourings

The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe

Landauer's erasure principle in non-equilibrium systems

In two recent papers, Maroney and Turgut separately and independently show generalisations of Landauer's erasure principle to indeterministic logical operations, as well as to logical states with variable energies and entropies. Here we show that, although Turgut's generalisation seems more powerful, in that it implies but is not implied by Maroney's and that it does not rely upon initial probability distributions over logical states, it does not hold for non-equilibrium states, while Maroney's generalisation holds even in non-equilibrium. While a generalisation of Turgut's inequality to non-equilibrium seems possible, it lacks the properties that makes the equilibrium inequality appealing. The non-equilibrium generalisation also no longer implies Maroney's inequality, which may still be derived independently. Furthermore, we show that Turgut's inequality can only give a necessary, but not sufficient, criteria for thermodynamic reversibility. Maroney's inequality gives the necessary and sufficient conditions.Comment: 9 pages, no figure

Generalisation of Scott permanent identity

Scott considered the determinant of 1/(y-z)^2, with y,z running over two sets X,Y of size n, and determined its specialisation when Y and Z are the roots of y^n-a and z^n-b. We give the same specialisation for the determinant 1/\prod_x(xy-z), where {x} is an arbitrary set of indeterminates. The case of the Gaudin-Izergin-Korepin is for {x}={q,1/q}.Comment: 5 page

The Geometry of Stimulus Control

Many studies, both in ethology and comparative psychology, have shown that animals react to modifications of familiar stimuli. This phenomenon is often referred to as generalisation. Most modifications lead to a decrease in responding, but to certain new stimuli an increase in responding is observed. This holds for both innate and learned behaviour. Here we propose a heuristic approach to stimulus control, or stimulus selection, with the aim of explaining these phenomena. The model has two key elements. First, we choose the receptor level as the fundamental stimulus space. Each stimulus is represented as the pattern of activation it induces in sense organs. Second, in this space we introduce a simple measure of `similarity' between stimuli by calculating how activation patterns overlap. The main advantage we recognise in this approach is that the generalisation of acquired responses emerges from a few simple principles which are grounded in the recognition of how animals actually perceive stimuli. Many traditional problems that face theories of stimulus control (e.g. the Spence-Hull theory of gradient interaction or ethological theories of stimulus summation) do not arise in the present framework. These problems include the amount of generalisation along different dimensions, peak-shift phenomena (with respect to both positive and negative shifts), intensity generalisation, and generalisation after conditioning on two positive stimuli