The hierarchy of equivalence relations on the natural numbers under computable reducibility

The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with the Borel reducibility hierarchy from descriptive set theory. Meanwhile, the notion of computable reducibility appears well suited for an analysis of equivalence relations on the c.e.\ sets, and more specifically, on various classes of c.e.\ structures. This is a rich context with many natural examples, such as the isomorphism relation on c.e.\ graphs or on computably presented groups. Here, our exposition extends earlier work in the literature concerning the classification of computable structures. An abundance of open questions remains.Comment: To appear in Computabilit

Canonical Quantization of (2+1)-Dimensional Gravity

We consider the quantum dynamics of both open and closed two- dimensional universes with ``wormholes'' and particles. The wave function is given as a sum of freely propagating amplitudes, emitted from a network of mapping class images of the initial state. Interference between these amplitudes gives non-trivial scattering effects, formally analogous to the optical diffraction by a multidimensional grating; the ``bright lines'' correspond to the most probable geometries.Comment: 22 pages, Mexico preprint ICN-UNAM-93-1

Foliations for solving equations in groups: free, virtually free, and hyperbolic groups

We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips classification of measured band complexes. Using canonical representatives, we deduce an algorithm for solving equations and inequations in hyperbolic groups (maybe with torsion). Additionnally, we can deal with quasi-isometrically embeddable rational constraints.Comment: 70 pages, 7 figures, revised version. To appear in Journal of Topolog

Fab + Craft: Synthesis of Maker, Machine, Material

Within contemporary architecture a fundamental disjunction exists between design and building facilitated by the use of advanced computational methods, and the relationship between form, material, and maker. The making of buildings demands an expertise that is familiar with the physical and involves a level of skill that many designers cannot claim to fully possess or practice. This doctorate project presents a study of a design-through-making methodology that incorporates craft with the material exploration of sandwich panels, digital technology and fabrication in the process of ‘making’ architecture. A focus is placed on the development of a specific design intent through the manipulation of materials, using skills and techniques guided by the practiced hand. This interaction between technology, material, and the designer-maker referred to as “fab+craft” creates a narrative that allows for the physical translation of ideas into the built environment

Inductive and Functional Types in Ludics

Ludics is a logical framework in which types/formulas are modelled by sets of terms with the same computational behaviour. This paper investigates the representation of inductive data types and functional types in ludics. We study their structure following a game semantics approach. Inductive types are interpreted as least fixed points, and we prove an internal completeness result giving an explicit construction for such fixed points. The interactive properties of the ludics interpretation of inductive and functional types are then studied. In particular, we identify which higher-order functions types fail to satisfy type safety, and we give a computational explanation