Abstract
This paper presents a Coq library that lifts an abstract yet precise notion of running-time into the type of a function. Our library is based on a monad that counts abstract steps, controlled by one of the monadic operations. The monad’s computational content, however, is simply that of the identity monad so programs written in our monad (that recur on the natural structure of their arguments) extract into idiomatic OCaml code. We evaluated the expressiveness of the library by proving that red-black tree insertion and search, merge sort, insertion sort, Fibonacci, iterated list insertion, BigNum addition, and Okasaki’s Braun Tree algorithms all have their expected running times.
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Notes
- 1.
The definition of ret, and all other monadic operations, are in the supplementary material and our public Github repo. The types are the most interesting part, however, so we focus on them.
- 2.
This is the case if BigNums are represented as lists of bits.
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Acknowledgments
Thanks to reviewers of previous versions of this paper. Thanks to Neil Toronto for help with the properties of integer logarithms (including efficient implementations of them). This work grew out of a PL seminar at Northwestern; thanks to Benjamin English, Michael Hueschen, Daniel Lieberman, Yuchen Liu, Kevin Schwarz, Zach Smith, and Lei Wang for their feedback on early versions of the work.
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McCarthy, J., Fetscher, B., New, M., Feltey, D., Findler, R.B. (2016). A Coq Library for Internal Verification of Running-Times. In: Kiselyov, O., King, A. (eds) Functional and Logic Programming. FLOPS 2016. Lecture Notes in Computer Science(), vol 9613. Springer, Cham. https://doi.org/10.1007/978-3-319-29604-3_10
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