Representing geometrical knowledge

Abstract

SU M M A RYThis paper introduces perspex algebra which is being developed as a common representation of geometrical knowledge. A perspex can currently be interpreted in one of four ways. First, the algebraic perspex is a generalization of matrices, it provides the most general representation for all of the interpretations of a perspex. The algebraic perspex can be used to describe arbitrary sets of coordinates. The remaining three interpretations of the perspex are all related to square matrices and operate in a Euclidean model of projective space-time, called perspex space. Perspex space di¡ers from the usual Euclidean model of projective space in that it contains the point at nullity. It is argued that the point at nullity is necessary for a consistent account of perspective in top-down vision. Second, the geometric perspex is a simplex in perspex space. It can be used as a primitive building block for shapes, or as a way of recording landmarks on shapes. Third, the transformational perspex describes linear transformations in perspex space that provide the a¤ne and perspective transformations in space-time. It can be used to match a prototype shape to an image, even in so called`accidental' views where the depth of an object disappears from view, or an object stays in the same place across time. Fourth, the parametric perspex describes the geometric and transformational perspexes in terms of parameters that are related to everyday English descriptions. The parametric perspex can be used to obtain both continuous and categorical perception of objects. The paper ends with a discussion of issues related to using a perspex to describe logic.

Perspex machine: VII. The universal perspex machine

“…When nullity arises as the result of an exact calculation it means that there is no unique solution on the number line -though there may be infinitely many non-unique solutions on the line. 2 It can also be that nullity itself, which lies off the number line, is a unique solution 2 or else one of many. 2 Thus, nullity itself gives no information about whether a unique solution exists or where any non-unique solutions lie on or off the number line.…”

“…2 It can also be that nullity itself, which lies off the number line, is a unique solution 2 or else one of many. 2 Thus, nullity itself gives no information about whether a unique solution exists or where any non-unique solutions lie on or off the number line. We summarise this by saying that nullity means "no information," even though additional calculations might allow us to classify the nullity solutions.…”

“…Firstly, the machine uses a general-linear instruction, not a linear one. This makes it easier to operate on individual elements of a perspex by isolating a single element by pre-and post-multiplication by j-matrices, 2,15 and by composing a perspex from individual elements via matrix addition. Secondly, the universal perspex-machine has a cellular structure that allows fractal machines to be constructed with cardinality .…”

“…5 We now modify that proof so that it proves that the general-linear perspexmachine can do everything that the Turing Machine can do. In developing this proof we take advantage of j-matrices 2,15 to isolate an element of a matrix. We use the addition term in the perspex instruction to compose a matrix from corresponding elements.…”

Transreal Logical Space of All Propositions

“…We use transreal arithmetic, which is a generalisation of real arithmetic. Transreal arithmetic was originally developed [1] [2] from a subset of the algorithms used in the arithmetic of fractions. It has been axiomatised and a machine proof of consistency has been given [7].…”