  Vector Symbolic Architectures (VSAs) are high-dimensional vector
representations of objects (eg., words, image parts), relations (eg., sentence
structures), and sequences for use with machine learning algorithms. They
consist of a vector addition operator for representing a collection of
unordered objects, a Binding operator for associating groups of objects, and a
methodology for encoding complex structures.
  We first develop Constraints that machine learning imposes upon VSAs: for
example, similar structures must be represented by similar vectors. The
constraints suggest that current VSAs should represent phrases ("The smart
Brazilian girl") by binding sums of terms, in addition to simply binding the
terms directly.
  We show that matrix multiplication can be used as the binding operator for a
VSA, and that matrix elements can be chosen at random. A consequence for living
systems is that binding is mathematically possible without the need to specify,
in advance, precise neuron-to-neuron connection properties for large numbers of
synapses.
  A VSA that incorporates these ideas, MBAT (Matrix Binding of Additive Terms),
is described that satisfies all Constraints.
  With respect to machine learning, for some types of problems appropriate VSA
representations permit us to prove learnability, rather than relying on
simulations. We also propose dividing machine (and neural) learning and
representation into three Stages, with differing roles for learning in each
stage.
  For neural modeling, we give "representational reasons" for nervous systems
to have many recurrent connections, as well as for the importance of phrases in
language processing.
  Sizing simulations and analyses suggest that VSAs in general, and MBAT in
particular, are ready for real-world applications.
