Quantitative Models and Implicit Complexity

TL;DR

This paper introduces a unified semantic framework using resource-bounded realizability to prove the soundness of various light logics and functional languages within specific complexity classes, simplifying existing proofs.

Contribution

It presents the first entirely semantic proof of polynomial-time soundness for Light Logics and extends LFPL with polymorphism and a modality for inductive datatypes.

Findings

Semantic framework based on resource-bounded realizability

Simplification of LFPL's soundness proof

Extension of LFPL with polymorphism and inductive datatypes

Abstract

We give new proofs of soundness (all representable functions on base types lies in certain complexity classes) for Elementary Affine Logic, LFPL (a language for polytime computation close to realistic functional programming introduced by one of us), Light Affine Logic and Soft Affine Logic. The proofs are based on a common semantical framework which is merely instantiated in four different ways. The framework consists of an innovative modification of realizability which allows us to use resource-bounded computations as realisers as opposed to including all Turing computable functions as is usually the case in realizability constructions. For example, all realisers in the model for LFPL are polynomially bounded computations whence soundness holds by construction of the model. The work then lies in being able to interpret all the required constructs in the model. While being the first entirely semantical proof of polytime soundness for light logi cs, our proof also provides a notable simplification of the original already semantical proof of polytime soundness for LFPL. A new result made possible by the semantic framework is the addition of polymorphism and a modality to LFPL thus allowing for an internal definition of inductive datatypes.