A Combinatorial Bit Bang Leading to Quaternions

This paper describes in detail how (discrete) quaternions - ie. the abstract structure of 3-D space - emerge from, first, the Void, and thence from primitive combinatorial structures, using only the exclusion and co-occurrence of otherwise unspecified events. We show how this computational view supplements and provides an interpretation for the mathematical structures, and derive quark structure. The build-up is emergently hierarchical, compatible with both quantum mechanics and relativity, and can be extended upwards to the macroscopic. The mathematics is that of Clifford algebras emplaced in the homology-cohomology structure pioneered by Kron. Interestingly, the ideas presented here were originally developed by the author to resolve fundamental limitations of existing AI paradigms. As such, the approach can be used for learning, planning, vision, NLP, pattern recognition; and as well, for modelling, simulation, and implementation of complex systems, eg. biological.Comment: 23 pages, 4 figure

Distributed Computation as Hierarchy

This paper presents a new distributed computational model of distributed systems called the phase web that extends V. Pratt's orthocurrence relation from 1986. The model uses mutual-exclusion to express sequence, and a new kind of hierarchy to replace event sequences, posets, and pomsets. The model explicitly connects computation to a discrete Clifford algebra that is in turn extended into homology and co-homology, wherein the recursive nature of objects and boundaries becomes apparent and itself subject to hierarchical recursion. Topsy, a programming environment embodying the phase web, is available from www.cs.auc.dk/topsy.Comment: 16 pages, 3 figure

A different perspective on canonicity

One of the most interesting aspects of Conceptual Structures Theory is the notion of canonicity. It is also one of the most neglected: Sowa seems to have abandoned it in the new version of the theory, and most of what has been written on canonicity focuses on the generalization hierarchy of conceptual graphs induced by the canonical formation rules. Although there is a common intuition that a graph is canonical if it is "meaningful'', the original theory is somewhat unclear about what that actually means, in particular how canonicity is related to logic. This paper argues that canonicity should be kept a first-class notion of Conceptual Structures Theory, provides a detailed analysis of work done so far, and proposes new definitions of the conformity relation and the canonical formation rules that allow a clear separation between canonicity and truth

Homeomorphic Embedding for Online Termination of Symbolic Methods

Well-quasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of techniques for program analysis, specialisation, transformation, and verification. In this paper we survey and discuss this use of homeomorphic embedding and clarify the advantages of such an approach over one using well-founded orders. We also discuss various extensions of the homeomorphic embedding relation. We conclude with a study of homeomorphic embedding in the context of metaprogramming, presenting some new (positive and negative) results and open problems