# AlgebrasΒΆ

NengoSPA uses elementwise addition for superposition and circular convolution for binding (HrrAlgebra) by default. However, other choices are viable. In NengoSPA we call such a specific choice of such operators an algebra. It is easy to change the algebra that is used by NengoSPA as it is tied to the vocabulary. To use a different algebra, it suffices to manually create a vocabulary with the desired algebra and use this in your model:

import nengo
import nengo_spa as spa

vocab = spa.Vocabulary(64, algebra=spa.algebras.VtbAlgebra())

with spa.Network() as model:
a = spa.State(vocab)
b = spa.State(vocab)
c = spa.State(vocab)
a * b >> c


In this example the VtbAlgebra (vector-derived transformation binding, VTB) is used to bind a and b.

Note that circular convolution is commutative, i.e. $$a \circledast b = b \circledast a$$, but this is not true for all algebras. In particular, the VTB is not commutative. That means you have to pay attention from which side vectors are bound and unbound. For such non-commutative algebras, special elements like the identity, absorbing element, zero element, and inverse might only fulfil their properties on a specific side of the binding operation. In these cases, the method to obtain the element usually has a sidedness argument to specify for which side of the binding operation the element is requested. For the inverse, however, you have the choice between ~ for the two-sided inverse, linv() and rinv() for the left and right inverse, respectively. Moreover, when given $$\mathcal{B}(\mathcal{B}(a, b), c)$$ in VTB, it is not possible to directly unbind $$a$$, but $$c$$ has to be unbound first because VTB is not associative.

Custom algebras can be implemented by implementing the AbstractAlgebra interface. The process involves implementing math versions of the superposition and binding operator, functions for obtaining specific matrices (such as inverting a vector), functions for obtaining special elements like the identity vector, and functions to provide neural implementations of the superposition and binding. A partial implementation is possible, but will prevent the usage of certain parts of NengoSPA. For example, when not providing neural implementations, only non-neural math can be performed.