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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Published in:Proceedings of ACM on programming languages Vol. 8; no. POPL; pp. 2288 - 2317
Main Author: Atkey, Robert
Format: Journal Article
Language:English
Published: New York, NY, USA ACM 02.01.2024
Subjects:
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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Abstract 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.
AbstractList 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 .
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.
ArticleNumber 76
Author Atkey, Robert
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Snippet We combine dependent types with linear type systems that soundly and completely capture polynomial time computation. We explore two systems for capturing...
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SubjectTerms Categorical semantics
Complexity classes
Complexity theory and logic
Linear logic
Theory of computation
Type theory
SubjectTermsDisplay Theory of computation -- Categorical semantics
Theory of computation -- Complexity classes
Theory of computation -- Complexity theory and logic
Theory of computation -- Linear logic
Theory of computation -- Type theory
Title Polynomial Time and Dependent Types
URI https://dl.acm.org/doi/10.1145/3632918
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