Characterizing polynomial time complexity of stream programs using interpretations

Abstract

This paper provides a criterion based on interpretation methods on term rewrite systems in order to characterize the polynomial time complexity of second order functionals. For that purpose it introduces a first order functional stream language that allows the programmer to implement second order functionals. This characterization is extended through the use of exp-poly interpretations as an attempt to capture the class of Basic Feasible Functionals (bff). Moreover, these results are adapted to provide a new characterization of polynomial time complexity in computable analysis. These characterizations give a new insight on the relations between the complexity of functional stream programs and the classes of functions computed by Oracle Turing Machine, where oracles are treated as inputs.

Polynomial Time over the Reals with Parsimony

“…These only focus on soundness though. The paper [17] provides a characterization of polynomial time over the reals on first order programs using type-2 polynomial interpretations, which are however known to be untractable both at the level of inference and checking. In [18], it is shown that algebras and coalgebras may be encoded in the light affine lambda calculus while preserving polynomial time normalization properties.…”

“…We have provided a first linear logic based characterization of polynomial time over the reals avoiding the use of untractable tools such as type-2 polynomials as in [17].…”

A tier-based typed programming language characterizing Feasible Functionals

“…Nevertheless, these characterizations are faced by at least two problems: (1) Characterizations using a general model of computation (whether machine-or programbased) require externally imposed and explicit resource bounding, either by a type-2 polynomial [KC91, KC96,FHHP15] or a bounding function within the class of [Con73,Meh76]. This is analogous to a shortcoming in Cobham's characterization of the class the light logic approach can deal with programs at higher types, its applications are restricted to type-1 complexity classes such as FP [BT04,BM10] or polynomial space [GMR08].…”

A tier-based typed programming language characterizing Feasible Functionals

“…1. Characterizations using a general model of computation (whether machine-or program-based) require externally imposed and explicit resource bounding, either by a type-2 polynomial [Férée et al 2015;Cook 1991, 1996] or a bounding function within the class [Constable 1973;Mehlhorn 1976]. This is analogous to a shortcoming in Cobham's characterization of the class of (type 1) polynomial time computable functions FP [Cobham 1965].…”

“…Whereas the light logic approach can deal with programs at higher types, its applications are restricted to type-1 complexity classes such as FP [Baillot and Mazza 2010;Baillot and Terui 2004] or polynomial space [Gaboardi et al 2008]. Interpretations were extended to higher-order polynomials in [Baillot and Lago 2016] to study FP and adapted in [Férée et al 2015;Hainry and Péchoux 2017] to BFF 2 . However, by essence, all these characterizations use (at least) type-2 polynomials and cannot be considered as tractable.…”