  The word problem for categories with free products and coproducts (sums),
SP-categories, is directly related to the problem of determining the
equivalence of certain processes. Indeed, the maps in these categories may be
directly interpreted as processes which communicate by two-way channels. The
maps of an SP-category may also be viewed as a proof theory for a simple logic
with a game theoretic intepretation. The cut-elimination procedure for this
logic determines equality only up to certain permuting conversions. As the
equality classes under these permuting conversions are finite, it is easy to
see that equality between cut-free terms (even in the presence of the additive
units) is decidable. Unfortunately, this does not yield a tractable decision
algorithm as these equivalence classes can contain exponentially many terms.
However, the rather special properties of these free categories -- and, thus,
of two-way communication -- allow one to devise a tractable algorithm for
equality. We show that, restricted to cut-free terms s,t : X --> A, the
decision procedure runs in time polynomial on |X||A|, the product of the sizes
of the domain and codomain type.
