Abstract
As a case-study in machine-checked reasoning about the complexity of algorithms in type theory, we describe a proof of the average-case complexity of Quicksort in Coq. The proof attempts to follow a textbook development, at the heart of which lies a technical lemma about the behaviour of the algorithm for which the original proof only gives an intuitive justification.
We introduce a general framework for algorithmic complexity in type theory, combining some existing and novel techniques: algorithms are given a shallow embedding as monadically expressed functional programs; we introduce a variety of operation-counting monads to capture worst- and average-case complexity of deterministic and nondeterministic programs, including the generalization to count in an arbitrary monoid; and we give a small theory of expectation for such non-deterministic computations, featuring both general map-fusion like results, and specific counting arguments for computing bounds.
Our formalization of the average-case complexity of Quicksort includes a fully formal treatment of the ‘tricky’ textbook lemma, exploiting the generality of our monadic framework to support a key step in the proof, where the expected comparison count is translated into the expected length of a recorded list of all comparisons.
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van der Weegen, E., McKinna, J. (2009). A Machine-Checked Proof of the Average-Case Complexity of Quicksort in Coq. In: Berardi, S., Damiani, F., de’Liguoro, U. (eds) Types for Proofs and Programs. TYPES 2008. Lecture Notes in Computer Science, vol 5497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02444-3_16
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DOI: https://doi.org/10.1007/978-3-642-02444-3_16
Publisher Name: Springer, Berlin, Heidelberg
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