Almost every simply typed λ-Term has a long β-Reduction sequence

It is well known that the length of a β-reduction sequence of a simply typed λ-term of order k can be huge; it is as large as k-fold exponential in the size of the λ-term in the worst case. We consider the following relevant question about quantitative properties, instead of the worst case: how many simply typed λ-terms have very long reduction sequences? We provide a partial answer to this question, by showing that asymptotically almost every simply typed λ-term of order k has a reduction sequence as long as (k−1)-fold exponential in the term size, under the assumption that the arity of functions and the number of variables that may occur in every subterm are bounded above by a constant. To prove it, we have extended the inβnite monkey theorem for words to a parameterized one for regular tree languages, which may be of independent interest. The work has been motivated by quantitative analysis of the complexity of higher-order model checking.