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We give the first linear time (randomised) algorithm for the first order isomorphism problem, that is, the
isomorphism of non-recursive types involving product- and function-type
constructors, under the axioms of commutativity and associativity of products,
currying and distributivity of functions over products. This problem can also be
thought of as the problem of formal equality-testing of multi-variate
expressions involving only multiplications and exponentiation. Previous work
gave a deterministic O(n log2n) time and O(n) space algorithm for the problem (n being the input size). Our specific contribution
includes two randomised algorithms for the problem: an O(n) time Monte Carlo
algorithm (that is, with a small probability it may decide
erroneously that the two types are isomorphic), and an O(n log n) expected
time and O(n) space Las
Vegas algorithm (that is, with a small probability it
may execute long).
The algorithms rely on a preprocessing stage, which computes the
sequence of the first n primes in O(n log
n/log log n) time and space.
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