Polynomial Time and Dependent Types

We combine dependent types with linear type systems that soundly and completely capture polynomial time computation. We explore two systems for capturing polynomial time: one system that disallows construction of iterable data, and one, based on the LFPL system of Martin Hofmann, that controls const...

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Vydané v:Proceedings of ACM on programming languages Ročník 8; číslo POPL; s. 2288 - 2317
Hlavný autor: Atkey, Robert
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York, NY, USA ACM 02.01.2024
Predmet:
Categorical semantics
Complexity classes
Complexity theory and logic
Linear logic
Theory of computation
Type theory
linear logic
type theory
implicit computational complexity
ISSN:2475-1421, 2475-1421
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Shrnutí:We combine dependent types with linear type systems that soundly and completely capture polynomial time computation. We explore two systems for capturing polynomial time: one system that disallows construction of iterable data, and one, based on the LFPL system of Martin Hofmann, that controls construction via a payment method. Both of these are extended to full dependent types via Quantitative Type Theory, allowing for arbitrary computation in types alongside guaranteed polynomial time computation in terms. We prove the soundness of the systems using a realisability technique due to Dal Lago and Hofmann. Our long-term goal is to combine the extensional reasoning of type theory with intensional reasoning about the resources intrinsically consumed by programs. This paper is a step along this path, which we hope will lead both to practical systems for reasoning about programs’ resource usage, and to theoretical use as a form of synthetic computational complexity theory.
ISSN:2475-1421
2475-1421
DOI:10.1145/3632918