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The word problem for categories with free products and coproducts (sums),SP-categories, is directly related to the problem of determining theequivalence of certain processes. Indeed, the maps in these categories may bedirectly interpreted as processes which communicate by two-way channels. Themaps of an SP-category may also be viewed as a proof theory for a simple logicwith a game theoretic intepretation. The cut-elimination procedure for thislogic determines equality only up to certain permuting conversions. As theequality classes under these permuting conversions are finite, it is easy tosee that equality between cut-free terms (even in the presence of the additiveunits) is decidable. Unfortunately, this does not yield a tractable decisionalgorithm 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 forequality. We show that, restricted to cut-free terms s,t : X --> A, thedecision procedure runs in time polynomial on |X||A|, the product of the sizesof the domain and codomain type.

On the word problem for SP-categories, and the properties of two-way communication