A cost-aware logical framework

TL;DR

This paper introduces calf, a cost-aware logical framework that models cost as a computational effect, enabling formal analysis of algorithm complexity within a dependent type theory, and demonstrates its effectiveness through case studies in Agda.

Contribution

It presents calf, a novel dependent type theory framework that integrates cost analysis with program behavior, allowing formal verification of algorithmic complexity.

Findings

Proved tight bounds for Euclid's algorithm

Verified amortized complexity of batched queues

Derived bounds for merge sort's complexity

Abstract

We present $\textbf{calf}$, a $\textbf{c}$ost-$\textbf{a}$ware $\textbf{l}$ogical $\textbf{f}$ramework for studying quantitative aspects of functional programs. Taking inspiration from recent work that reconstructs traditional aspects of programming languages in terms of a modal account of \emph{phase distinctions}, we argue that the cost structure of programs motivates a phase distinction between $\textit{intension}$ and $\textit{extension}$. Armed with this technology, we contribute a synthetic account of cost structure as a computational effect in which cost-aware programs enjoy an internal noninterference property: input/output behavior cannot depend on cost. As a full-spectrum dependent type theory, $\textbf{calf}$ presents a unified language for programming and specification of both cost and behavior that can be integrated smoothly with existing mathematical libraries available in type theoretic proof assistants. We evaluate $\textbf{calf}$ as a general framework for cost analysis by implementing two fundamental techniques for algorithm analysis: the $\textit{method of recurrence relations}$ and $\textit{physicist's method for amortized analysis}$. We deploy these techniques on a variety of case studies: we prove a tight, closed bound for Euclid's algorithm, verify the amortized complexity of batched queues, and derive tight, closed bounds for the sequential and $\textit{parallel}$ complexity of merge sort, all fully mechanized in the Agda proof assistant. Lastly we substantiate the soundness of quantitative reasoning in $\textbf{calf}$ by means of a model construction.