as well as complex sentences. Furthermore, geometric product of two multivectors is geometrically meaningful, unlike the convolution or a binary exclusive OR operation performed on two vectors.

In this paper we consider the C`n algebra generated by orthonormal vectors bi =
{0,...,0,1,0*...,*0} for i ∈ {1*,...,*n}. The inner product used throughout the paper is an extension of the inner product *h·|·i* from the Euclidean space R
n. For blades X<k> = x1 *∧ ··· ∧*xk and Y<l> = y1 *∧ ··· ∧*ylthe inner product · : C`n ×C`n → R is defined as

$$\langle X_{<k>}|Y_{<l>}\rangle=\begin{vmatrix}\langle x_{1}|y_{l}\rangle&\langle x_{1}|y_{l-1}\rangle&\cdots&\langle x_{1}|y_{1}\rangle\\ \langle x_{2}|y_{l}\rangle&\langle x_{2}|y_{l-1}\rangle&\cdots&\langle x_{2}|y_{1}\rangle\\ \vdots&&\ddots&\vdots\\ \langle x_{k}|y_{l}\rangle&\langle x_{k}|y_{l-1}\rangle&\cdots&\langle x_{k}|y_{1}\rangle\end{vmatrix}\ \ \text{for}\ k=l,$$  $$\langle X_{<k>}|Y_{<l>}\rangle=0\ \ \text{for}\ k\neq l$$
$${\mathrm{(1)}}$$
$$(2)$$

and is extended by linearity to the entire algebra.

Originally, the GA model was developed as a geometric analogue of BSC and HRR and was described by Czachor, Aerts and De Moor in [1] and [4]. Before switching from geometric product to BSC and HRR one has to realize, that geometric product is a projective representation of boolean exclusive OR. Let x1 ... xn and y1 ... yn be binary representations of two n-bit numbers x and y and let cx = cx1...xn =
b x1 1
...b xn n and cy = cy1...yn = b y1 1
...b yn n be their corresponding blades, b 0 i being equal to 1. The following examples show that geometric product of two blades cx and cy equals, up to a sign, cx⊕y

$b_{1}b_{1}=c_{10\_0}c_{10\_0}=1=c_{0\_0\_0}=c_{(10\_0)\oplus(10\_0)}$,  $b_{1}b_{12}=c_{10\_0}c_{110\_0}=b_{1}b_{1}b_{2}=b_{2}=c_{010\_0}=c_{(10\_0)\oplus(110\_0)}$,  $b_{12}b_{1}=c_{110\_0}c_{10\_0}=b_{1}b_{2}b_{1}=-b_{2}b_{1}b_{1}=-b_{2}$  $=-c_{010\_0}=-c_{(110\_0)\oplus(10\_0)}=(-1)^{D}c_{(110\_0)\oplus(10\_0)}$,
the number D being calculated as follows

$D=y_{1}(x_{2}+\cdots+x_{n})+y_{2}(x_{3}+\cdots+x_{n})+\cdots+y_{n-1}x_{n}=\sum_{k<l}y_{k}x_{l}$.  
ykxl. (6)
The original BSC is illustrated by an example taken from [11, 4] - atomic objects are represented by randomly chosen strings of bits, "⊕" is a componentwise addition mod 2 and "" represents a thresholded sum producing a binary vector —
the threshold is set at one half of sentence chunks and a random string of bits is added in case of an even number of sentence chunks to break the tie. The encoded record is

$$(6)$$

$$\left(7\right)$$
$\small66$), . 
PSmith = (name⊕Pat)(sex⊕male)(age⊕66), (7)
and the decoding of *name* uses the involutive nature of XOR