that tensor product algebra can be used in some architectures as long as the size of a tensor is restricted. In 1994 Plate [13] worked up his Holographic Reduced Representation (HRR) that uses circular convolution and vector addition to combine vectors representing elements of a domain in hierarchical structures. Elements are represented by randomly chosen high-dimensional vectors. A vector representing a structure is of the same size as the vectors representing the elements it contains. In 1997 Pentti Kanerva [10, 11] introduced Binary Spatter Code (BSC) that is very similar to HRR and is often referred to as a form of HRR. In BSC objects are represented by binary vectors and the boolean exclusive OR is used instead of convolution. The clean-up memory is an important part of any distributed representation model as an auto-associative collection of all atomic objects and complex statements produced by that system. Given a noisy extracted vector such structure must be able to recall the most similar item stored or indicate, that no matching object had been found.

The geometric algebra (GA) model, which is the focus of this paper, is an alternative to models developed so far. It has been inspired by a well-known fact, that most people think in pictures, i.e. two- and three-dimensional shapes, not by using sequences of ones and zeroes. As far as brain functions are concerned, geometric computing has been applied thus far only in the context of primate visual system
([5], Chapters 1 and 2).

In the GA model convolutions are replaced by geometric products and superposition is performed by ordinary addition. Sentences are represented by multivectors
- superpositions of blades. The concept of GA first appeared in the 19th century works of Grassmann and Clifford, but was abandoned for almost a century until Hestenes brought up the subject in [8] and [9]. The Hestenes system has recently found applications in quantum computation (Czachor *et al.* [1]–[4], [6]), which appears to be a promising leap from cognitive systems based on traditional computing.

Section 2 of this paper recalls basic operations that can be performed on blades and multivectors, using the example Kanerva [11] gave to illustrate BSC. For further details on multivectors as well as interesting exercises the reader may refer to
[7, 12]. Section 3 gives rise to discussion about various ways of asking questions and investigates the percentage of correctly recognized items under two possible constructions. Section 4 introduces measures of similarity based not on only the inner product of a multivector, but also on its matrix representation. Finally, Section 5 studies the influence of accidental blade equality on recognition and Section 6 compares the performance of the GA model with HRR and BSC.

## 2 Geometric Algebra Model

Distributed representation models developed so far were based on long binary or real vectors. However, most people tend to think by pictures, not by sequences of numbers. Therefore geometric algebra with its ability to describe shapes is the most natural language to mimic human thought process and to represent atomic objects