Efficient algorithms for isomorphisms of simple types

The <i>first order isomorphism problem</i> is to decide whether two non-recursive types using product- and function-type constructors, are isomorphic under the axioms of commutative and associative products, and currying and distributivity of functions over products. We show that this problem can be solved in <i>O</i>(<i>n</i> log² <i>n</i>) time and <i>O</i>(<i>n</i>) space, where is <i>n</i> the input size. This result improves upon the <i>O</i>(<i>n</i> log² <i>n</i>) time and <i>O</i>(<i>n</i>²) space bounds of the best previous algorithm. We also describe an <i>O</i>(<i>n</i>) time algorithm for the <i>linear isomorphism problem</i>, which does not include the distributive axiom, whereby improving upon the <i>O</i>(<i>n</i> log <i>n</i>) time of the best previous algorithm for this problem.