Implicit Computational Complexity, Non-uniformity and the Lambda-Calculus

Implicit computational complexity. Loosely speaking, the aim of implicit computational complexity is to replace clocks (or other explicit resource bounds) with certificates. For example, if we consider polynomial time computation, the idea is to define a structured programming language whose programs are guaranteed to be polytime by construction, i.e., because they satisfy some syntactic condition, not because their execution is artificially stopped after a polynomial number of steps (which is the “explicit” definition used in computational complexity). At the same time, such a programming language must be expressive enough so that every polynomial time function may be somehow implemented. Notable early examples of such methodology are the work of Bellantoni and Cook [BC92], Leivant and Marion [LM93], and Jones [Jon99]. Although there is an annual international workshop on the topic,1 implicit computational complexity is a niche research subject which rarely appears in mainstream programming languages conferences. Nevertheless, it is thematically relevant, both in terms of techniques (type systems, proof theory, semantics. . . ) and potential applications (mostly program analysis [JHLM10, DLG11], but people like Patrick Baillot, Gilles Barthe and Ugo Dal Lago are starting to develop applications to the verification of cryptosystems).