Verifying the Fisher-Yates Shuffle Algorithm in Dafny

TL;DR

This paper presents a formal verification of the Fisher-Yates shuffle algorithm in Dafny, ensuring its correctness and unbiased distribution through a rigorous proof framework.

Contribution

It introduces a novel verification approach combining functional and imperative models to validate the Fisher-Yates shuffle's correctness and distribution.

Findings

Verified the Fisher-Yates shuffle's correctness in Dafny

Established the algorithm's unbiased permutation distribution

Provided a blueprint for verifying complex algorithms

Abstract

The Fisher-Yates shuffle is a well-known algorithm for shuffling a finite sequence, such that every permutation is equally likely. Despite its simplicity, it is prone to implementation errors that can introduce bias into the generated permutations. We verify its correctness in Dafny as follows. First, we define a functional model that operates on sequences and streams of random bits. Second, we establish that the functional model has the desired distribution. Third, we define an executable imperative implementation that operates on arrays and prove it equivalent to the functional model. The approach may serve as a blueprint for the verification of more complex algorithms.