MPI-I-1999-3-001

1. We analyze expressiveness of the simply typed lambda calculus (STLC) over a single base type, and show how k-DEXPTIME computations can be simulated in the order k+6 STLC. This gives a double order improvement over the lower bound of [Hillebrand \& Kanellakis LICS'96], reducing k-DEXPTIME to the order 2k+3 STLC. 2. We show that k-DEXPTIME is linearly reducible to the higher-order matching problem (in STLC) of order k+7. Thus, order k+7 matching requires (lower bound) k-level exponential time. This refines over the best previously known lower bound (a stack of twos growing almost linearly, O(n / log(n)) in the length of matched terms) from [Vorobyov LICS'97], which holds in assumption that orders of types are UNBOUNDED, but does not imply any nontrivial lower bounds when the order of variables is FIXED. 3. These results are based on the new simplified and streamlined proof of Statman's famous theorem. Previous proofs in [Statman TCS'79, Mairson TCS'92, Vorobyov LICS'97] were based on a two-step reduction: proving a non-elementary lower bound for Henkin's higher-order theory Omega of propositional types and then encoding it in the STLC. We give a direct generic reduction from k-DEXPTIME to the STLC, which is conceptually much simpler, and gives stronger and more informative lower bounds for the fixed-order STLC, in contrast with the previous proofs.