The Geometry of Types (Long Version)

TL;DR

This paper introduces an efficient method for analyzing the time complexity of higher-order functional programs by translating the problem into first-order inequalities, enabling automated verification with promising initial results.

Contribution

It presents a novel inference algorithm for linear dependent types that simplifies complexity analysis to first-order inequality verification, ensuring accuracy and facilitating automation.

Findings

Algorithm produces valid type judgments iff proof obligations are valid

Method simplifies complexity analysis to first-order inequalities

Initial experiments show promising results

Abstract

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.