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Dependently-Typed AARA: A Non-Affine Approach for Resource Analysis of Higher-Order Programs

Han Xu, Di Wang

TL;DR

This paper introduces λ_are^{na}, a non-affine, dependently-typed amortized resource analysis (AARA) for higher-order functional programs. By decoupling potentials from data types and employing a global potential context, it expresses cost behavior with dependent-type potential functions [f1]x T -> [f2]y T, enabling precise reasoning about closures and partial applications. The work formalizes syntax, semantics, and soundness, and demonstrates expressiveness through case studies like List Traverse and Map Append, while discussing algorithmic typing and avenues for automation and type-inference synthesis. Overall, it provides a foundational, compositional framework for verifying resource bounds in higher-order languages without affine constraints, with future work toward refinement types and automated synthesis.

Abstract

Static resource analysis determines the resource consumption (e.g., time complexity) of a program without executing it. Among the numerous existing approaches for resource analysis, affine type systems have been one dominant approach. However, these affine type systems fall short of deriving precise resource behavior of higher-order programs, particularly in cases that involve partial applications. This article presents λ_\ms{amor}^\ms{na}}, a non-affine AARA-style dependent type system for resource reasoning about higher-order functional programs. The key observation is that the main issue in previous approaches comes from (i) the close coupling of types and resources, and (ii) the conflict between affine and higher-order typing mechanisms. To derive precise resource behavior of higher-order functions, λ_\ms{amor}^\ms{na}} decouples resources from types and follows a non-affine typing mechanism. The non-affine type system of λ_\ms{amor}^\ms{na}} achieves this by using dependent types, which allows expressing type-level potential functions separate from ordinary types. This article formalizes λ_\ms{amor}^\ms{na}}'s syntax and semantics, and proves its soundness, which guarantees the correctness of resource bounds. Several challenging classic and higher-order examples are presented to demonstrate the expressiveness and compositionality of λ_\ms{amor}^\ms{na}}'s reasoning capability.

Dependently-Typed AARA: A Non-Affine Approach for Resource Analysis of Higher-Order Programs

TL;DR

This paper introduces λ_are^{na}, a non-affine, dependently-typed amortized resource analysis (AARA) for higher-order functional programs. By decoupling potentials from data types and employing a global potential context, it expresses cost behavior with dependent-type potential functions [f1]x T -> [f2]y T, enabling precise reasoning about closures and partial applications. The work formalizes syntax, semantics, and soundness, and demonstrates expressiveness through case studies like List Traverse and Map Append, while discussing algorithmic typing and avenues for automation and type-inference synthesis. Overall, it provides a foundational, compositional framework for verifying resource bounds in higher-order languages without affine constraints, with future work toward refinement types and automated synthesis.

Abstract

Static resource analysis determines the resource consumption (e.g., time complexity) of a program without executing it. Among the numerous existing approaches for resource analysis, affine type systems have been one dominant approach. However, these affine type systems fall short of deriving precise resource behavior of higher-order programs, particularly in cases that involve partial applications. This article presents λ_\ms{amor}^\ms{na}}, a non-affine AARA-style dependent type system for resource reasoning about higher-order functional programs. The key observation is that the main issue in previous approaches comes from (i) the close coupling of types and resources, and (ii) the conflict between affine and higher-order typing mechanisms. To derive precise resource behavior of higher-order functions, λ_\ms{amor}^\ms{na}} decouples resources from types and follows a non-affine typing mechanism. The non-affine type system of λ_\ms{amor}^\ms{na}} achieves this by using dependent types, which allows expressing type-level potential functions separate from ordinary types. This article formalizes λ_\ms{amor}^\ms{na}}'s syntax and semantics, and proves its soundness, which guarantees the correctness of resource bounds. Several challenging classic and higher-order examples are presented to demonstrate the expressiveness and compositionality of λ_\ms{amor}^\ms{na}}'s reasoning capability.
Paper Structure (43 sections, 21 theorems, 42 equations, 10 figures)

This paper contains 43 sections, 21 theorems, 42 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Resource Analysis
  3. Closure and Resource
  4. Challenge: Go beyond Affine Typing
  5. Challenge: Support Lightweight Dependent Typing
  6. Contributions
  7. Overview
  8. Prior Work: Automatic Amortized Resource Analysis
  9. This Work: Non-Affine Dependently-Typed AARA
  10. Technical Details
  11. Syntax
  12. Types
  13. Expressions
  14. Values and Pre-Values
  15. Type Context and Potential Context
  16. ...and 28 more sections

Key Result

theorem 1

For $\Gamma \sststile{q}{p} e:A$, with $Pure(\cdot)$ maps a AARA type context to a potential free $\lambda_\mathsf{amor}^\mathsf{na}$ type context, $\Phi(\cdot)$ maps a AARA type context to the potential it carries, $h(\cdot)$ a non-trivial mapping from a AARA term to a $\lambda_\mathsf{amor}^\maths

Figures (10)

  • Figure 1: Potential Functions
  • Figure 2: Typing Rules (Selected, full in Appendix)
  • Figure 3: Typing Rules (Selected, full in Appendix)
  • Figure 4: Selected Algorithmic Typing Rules
  • Figure 5: Algorithmic Typing Rules
  • ...and 5 more figures

Theorems & Definitions (36)

  • theorem 1: AARA embedding
  • lemma 1: Progress Weakening
  • theorem 2: Progress
  • lemma 2: Value Substitution
  • theorem 3: Preservation
  • theorem 4
  • theorem 5: Soundness
  • theorem 6: Determinisism
  • lemma 3: Progress Weakening
  • proof
  • ...and 26 more