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Section: New Results

From Sets to Bits in Coq

Sets form the building block of mathematics, while finite sets are a fundamental data structure of computer science. In the world of mathematics, finite sets enjoy appealing mathematical properties, such as a proof-irrelevant equality and extensionality of functions. Computer scientists, on the other hand, have devised efficient algorithms for set operations based on the representation of finite sets as bit vectors and on bit twiddling, exploiting the hardware's ability to efficiently process machine words.

With interactive theorem provers, sets are reinstituted as mathematical objects. While there are several finite set libraries in Coq , these implementations are far removed from those used in efficient code. Recent work on modeling low-level architectures, such as x86 [41] processors, however, have brought the world of bit twiddling within reach of our proof assistants. We are now able to specify and reason about low-level programs.

In this work, we have implemented bitsets and their associated operations in the Coq proof assistant, thus allowing us to transparently navigate between the concrete world of bit vectors and the abstract world of finite sets. This work grew from a puzzled look at the first page of Warren's Hacker's Delight [77] , where lies the cryptic formula $x\&(x-1)$ to turn off the rightmost bit in a word. How do we translate the English specification given in the book into a formal definition? How do we prove that this formula meets its specification? Could Coq generate efficient and trustworthy code from it? And how efficiently could we simulate it within Coq itself?

In our work, we have established a bijection between bitsets and sets over finite types. Following a refinement approach, we have shown that a significant part of SSReflect finset library can be refined to operations manipulating bitsets. We have also developed a trustworthy extraction of bitsets down to OCaml's machine integers. While we were bound to axiomatize machine integers, we adopted a methodology based on exhaustive testing to gain greater confidence in our model. Finally, we have demonstrateed the usefulness of our library through two applications, a certified implementation of Bloom filters and a verified implementation of the $n$-queens algorithm.