  Holographic reduced representations (HRR) are based on superpositions of
convolution-bound $n$-tuples, but the $n$-tuples cannot be regarded as vectors
since the formalism is basis dependent. This is why HRR cannot be associated
with geometric structures. Replacing convolutions by geometric products one
arrives at reduced representations analogous to HRR but interpretable in terms
of geometry. Variable bindings occurring in both HRR and its geometric analogue
mathematically correspond to two different representations of
$Z_2\times...\times Z_2$ (the additive group of binary $n$-tuples with addition
modulo 2). As opposed to standard HRR, variable binding performed by means of
geometric product allows for computing exact inverses of all nonzero vectors, a
procedure even simpler than approximate inverses employed in HRR. The formal
structure of the new reduced representation is analogous to cartoon
computation, a geometric analogue of quantum computation.
