Peter Hines

Measuring category intuitiveness in unconstrained categorization tasks

What makes a category seem natural or intuitive? In this paper, an unsupervised categorization task was employed to examine observer agreement concerning the categorization of nine different stimulus sets. The stimulus sets were designed to capture different intuitions about classification structure. The main empirical index of category intuitiveness was the frequency of the preferred classification, for different stimulus sets. With 169 participants, and a within participants design, with some stimulus sets the most frequent classification was produced over 50 times and with others not more than two or three times. The main empirical finding was that cluster tightness was more important in determining category intuitiveness, than cluster separation. The results were considered in relation to the following models of unsupervised categorization: DIVA, the rational model, the simplicity model, SUSTAIN, an Unsupervised version of the Generalized Context Model (UGCM), and a simple geometric model based on similarity. DIVA, the geometric approach, SUSTAIN, and the UGCM provided good, though not perfect, fits. Overall, the present work highlights several theoretical and practical issues regarding unsupervised categorization and reveals weaknesses in some of the corresponding formal models.

A NON-PARAMETRIC APPROACH TO SIMPLICITY CLUSTERING

The simplicity principle—an updating of Ockhams razor to take into account modern information theory—states that the preferred theory for a set of data is the one that allows for the most efficient encoding of the data. We consider this in the context of classification, or clustering, as a data reduction technique that helps describe a set of objects by dividing the objects into groups. The simplicity model we present favors clusters such that the similarity of the items in the clusters is maximal, while the similarity of items between clusters is minimal. Several novel features of our clustering criterion make it especially appropriate for clustering of data derived from, psychological procedures (e.g., similarity ratings): It is non-parametric, and may be applied in situations where the metric axioms are violated without requiring (information-forgetting) transformation procedures. We illustrate the use of the criterion with a selection of data sets. A distinctive aspect of this research is that it motivates a clustering algorithm from psychological principles.

Quantum circuit oracles for Abstract Machine computations

This paper considers a very general model of computation via conditional iteration, the abstract machines of Hines (2008) [23], and studies the conditions under which these describe reversible computations. Using this, we demonstrate how to construct quantum circuits that act as oracles for these Abstract Machines. For a classical computation with worst-case performance T, the resulting quantum circuit requires an ancilla of 1+log(T) qubits, and takes O(T) steps. The ancilla starts and finishes in the constant state |0>, so garbage collection is performed automatically.

A Framework for Heterotic Computing

Computational devices combining two or more different parts, one controlling the operation of the other, for example, derive their power from the interaction, in addition to the capabilities of the parts. Non-classical computation has tended to consider only single computational models: neural, analog, quantum, chemical, biological, neglecting to account for the contribution from the experimental controls. In this position paper, we propose a framework suitable for analysing combined computational models, from abstract theory to practical programming tools. Focusing on the simplest example of one system controlled by another through a sequence of operations in which only one system is active at a time, the output from one system becomes the input to the other for the next step, and vice versa. We outline the categorical machinery required for handling diverse computational systems in such combinations, with their interactions explicitly account ed for. Drawing on prior work in refinement and retrenchment, we suggest an appropriate framework for developing programming tools from the categorical framework. We place this work in the context of two contrasting concepts of “efficiency”: theoretical comparisons to determine the relative computational power do not always reflect the practical comparison of real resources for a finite-size d computational task, especially when the inputs include (approximations of) real numbers. Finally we outline the limitations of our simple model, and identify some of the extensions that will be required to treat more complex interacting computational systems.

Classical Structures Based on Unitaries

Starting from the observation that distinct notions of copying have arisen in different categorical fields (logic and computation, contrasted with quantum mechanics) this paper addresses the question of when, or whether, they may coincide. Provided all definitions are strict in the categorical sense, we show that this can never be the case. However, allowing for the defining axioms to be taken up to canonical isomorphism, a close connection between the classical structures of categorical quantum mechanics, and the categorical property of self-similarity familiar from logical and computational models becomes apparent. The required canonical isomorphisms are non-trivial, and mix both typed (multi-object) and untyped (single-object) tensors and structural isomorphisms; we give coherence results that justify this approach. We then give a class of examples where distinct self-similar structures at an object determine distinct matrix representations of arrows, in the same way as classical structures determine matrix representations in Hilbert space. We also give analogues of familiar notions from linear algebra in this setting such as changes of basis, and diagonalisation.

A categorical analogue of the monoid semiring construction

This paper introduces and studies a categorical analogue of the familiar monoid semiring construction. By introducing an axiomatisation of summation that unifies notions of summation from algebraic program semantics with various notions of summation from the theory of analysis, we demonstrate that the monoid semiring construction generalises to cases where both the monoid and the semiring are categories. This construction has many interesting and natural categorical properties, and natural computational interpretations.

Identities in modular arithmetic from reversible coherence operations

This paper investigates some issues arising in categorical models of reversible logic and computation. Our claim is that the structural (coherence) isomorphisms of these categorical models, although generally overlooked, have decidedly non-trivial computational content. The theory of categorical coherence is based around reversible structural operations (canonical isomorphisms) that allow for transformations between related, but distinct, mathematical structures. A number of coherence theorems are commonly used to treat these transformations as though they are identity maps, from which point onwards they play no part in computational models. We simply wish to point out that doing so overlooks some significant computational content. We give a single example (taken from an uncountably infinite set of similar examples, and based on structures used in models of reversible logic and computation) of a category whose structural isomorphisms manipulate modulo classes of natural numbers. We demonstrate that the coherence properties that usually allow us to ignore these structural isomorphisms in fact correspond to countably infinite families of non-trivial identities in modular arithmetic. Further, proving the correctness of these equalities without recourse to the theory of categorical coherence appears to be a hard task.

Quantum Speedup and Categorical Distributivity

This paper studies one of the best-known quantum algorithms — Shor’s factorisation algorithm — via categorical distributivity. A key aim of the paper is to provide a minimal set of categorical requirements for key parts of the algorithm, in order to establish the most general setting in which the required operations may be performed efficiently.