It is well known that the length of a beta-reduction sequence of a simply typed lambda-term of order k can be huge; it is as large as k-fold exponential in the size of the lambda-term in the worst case. We consider the following relevant question about quantitative properties, instead of the worst case: how many simply typed lambda-terms have very long reduction sequences? We provide a partial answer to this question, by showing that asymptotically almost every simply typed lambda-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 infinite monkey theorem for strings to a parametrized 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.

@article{paperbot825,
  title = {Almost Every Simply Typed Lambda-Term Has a Long Beta-Reduction Sequence},
  author = {Kazuyuki Asada and Naoki Kobayashi and Ryoma Sin'ya and Takeshi Tsukada},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 15, Issue 1},
  year = {2019},
  doi = {10.23638/lmcs-15(1:16)2019}
}