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

The word problem for categories with free products and coproducts (sums), ΣΠ-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 ΣΠ-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.