We show that time complexity analysis of higher-order functional programs can be effectively reduced to an arguably simpler (although computationally equivalent) verification problem, namely checking first-order inequalities for validity. This is done by giving an efficient inference algorithm for linear dependent types which, given a PCF term, produces in output both a linear dependent type and a cost expression for the term, together with a set of proof obligations. Actually, the output type judgement is derivable iff all proof obligations are valid. This, coupled with the already known relative completeness of linear dependent types, ensures that no information is lost, i.e., that there are no false positives or negatives. Moreover, the procedure reflects the difficulty of the original problem: simple PCF terms give rise to sets of proof obligations which are easy to solve. The latter can then be put in a format suitable for automatic or semi-automatic verification by external solvers. Ongoing experimental evaluation has produced encouraging results, which are briefly presented in the paper.

Linear dependent types [11] allow to precisely capture both the extensional behavior and the time complexity of λ-terms, when the latter are evaluated by Krivine's abstract machine. In this work, we show that the same paradigm can be applied to call-by-value computation. A system of linear dependent types for Plotkin's PCF is introduced, called dlPCFv whose types reflect the complexity of evaluating terms in the so-called CEK machine. dlPCFv is proved to be sound, but also relatively complete: every true statement about the extensional and intentional behavior of terms can be derived, provided all true index term inequalities can be used as assumptions.