A Computational complexity via finite types

We address computational complexity writing polymorphic functions between finite types (i.e. types with a finite number of canonical elements), expressing costs in terms of the cardinality of these types. This allows us to rediscover, in a more syntactical setting, the known result that the different levels in the hierarchy of higher-order primitive recursive functions (Gödel system T), when interpreted over finite structures, precisely capture basic complexity classes: functions of rank-1 characterize LOGSPACE, rank-2 PTIME, rank-3 PSPACE, rank-4 EXPTIME = DTIME(2poly), and so on.