Higher-order Interpretations and Program Complexity (Long Version)

Polynomial interpretations and their generalizations like quasi-interpretations have been used in the setting of first-order functional languages to design criteria ensuring statically some complexity bounds on programs [1]. This fits in the area of implicit computational complexity, which aims at giving machine-free characterizations of complexity classes. Here we extend this approach to the higher-order setting. For that we consider the notion of simply typed term rewriting systems [2], we define higher-order polynomial interpretations (HOPI) for them and give a criterion based on HOPIs to ensure that a program can be executed in polynomial time. In order to obtain a criterion which is flexible enough to validate some interesting programs using higher-order primitives, we introduce a notion of polynomial quasiinterpretations, coupled with a simple termination criterion based on linear types and path-like orders.