Certification of Breadth-First Algorithms by Extraction

Abstract

By using pointers, breadth-first algorithms are very easy to implement efficiently in imperative languages. Implementing them with the same bounds on execution time in purely functional style can be challenging, as explained in Okasaki's paper at ICFP 2000 that even restricts the problem to binary trees but considers numbering instead of just traversal. Okasaki's solution is modular and factors out the problem of implementing queues (FIFOs) with worst-case constant time operations. We certify those FIFO-based breadth-first algorithms on binary trees by extracting them from fully specified Coq terms, given an axiomatic description of FIFOs. In addition, we axiomatically characterize the strict and total order on branches that captures the nature of breadthfirst traversal and propose alternative characterizations of breadth-first traversal of forests. We also propose efficient certified implementations of FIFOs by extraction, either with pairs of lists (with amortized constant time operations) or triples of lazy lists (with worst-case constant time operations), thus getting from extraction certified breadth-first algorithms with the optimal bounds on execution time. Keywords: Breadth-first algorithms • Queues in functional programming • Correctness by extraction • Coq The first author got financial support by the TICAMORE joint ANR-FWF project. The second author got financial support by the COST action CA15123 EUTYPES.

The Braga Method: Extracting Certified Algorithms from Complex Recursive Schemes in Coq

“…For instance, fact b algorithm of Section 2.6 or breadth-first search algorithms can be implemented using measure based induction. 12 Notice that although it is a very common strategy, it is not always applicable, e.g. the decreasing measure might simply not be total computable, as in the case of the Tortoise and the Hare algorithm.…”

“…extraction when dealing with somewhat complicated schemes. 12 Also for termination, it is based on well-founded recursion and thus, not always suitable for partial algorithms or else algorithms that are better manipulated as partial, typically nested ones. That said, Equations can perfectly be used when deploying the Braga method and it is our hope that the method will one day find its way for full integration in the Equations framework, thus allowing a seamless treatment of partial recursive functions.…”