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Efficient Extensional Binary Tries

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Abstract

Lookup tables (finite maps) are a ubiquitous data structure. In pure functional languages they are best represented using trees instead of hash tables. In pure functional languages within constructive logic, without a primitive integer type, they are well represented using binary tries instead of search trees. In this work, we introduce canonical binary tries, an improved binary-trie data structure that enjoys a natural extensionality property, quite useful in proofs, and supports sparseness more efficiently. We provide full proofs of correctness in Coq. We provide microbenchmark measurements of canonical binary tries versus several other data structures for finite maps, in a variety of application contexts; as well as measurement of canonical versus original tries in two big, real systems. The application context of data structures contained in theorem statements imposes unusual requirements for which canonical tries are particularly well suited.

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Notes

  1. All proofs mentioned in this paper are machine-checked unless explicitly noted otherwise.

  2. Persistent arrays give a functional semantics layered on an imperative (destructive-update) implementation, but these are not suitable because they would require extending the logic’s kernel of axioms.

  3. In fact, no programming language has a primitive integer type with constant-time comparisons; they typically have a range-limited integer type \(-2^k\le i < 2^k\) or a modular integer type \((i \bmod 2^k)\). Most versions of Coq 2005-present have not had a built-in integer type with fast comparisons; and relying on them would increase the size of the trusted base, that is, Coq kernel code, Coq axioms, Coq extraction to OCaml, and OCaml operations upon which we rely.

  4. In OCaml, an arity-k constructor applied to arguments takes \(k+1\) words, including one for the constructor-tag. In Coq, an arity-k constructor is represented as one arity-2 OCaml construction plus one length-k array, which take (respectively) 3 words and \(k+1\) words as represented in OCaml within the Coq kernel. The arity-2 OCaml construction also points to a description of the constructor, but that description is shared among all its uses so we don’t count it.

  5. Andreas Abel, personal communications, Aug-Sept 2022. Examples of problematic unifications can be found at https://github.com/agda/agda/issues/483.

  6. In table 1, performance of PTrees in VST could be predicted in row “Coq execution, vm_compute, Sparse test,” column Sigma versus columns Original and Canonical; whereas PTree performance in CompCert could be predicted in row “Extraction to OCaml, Sparse test”, same columns.

  7. This Coq verification is at https://github.com/PrincetonUniversity/VST/releases/tag/v2.9.1 in subdirectory progs/VSUpile.

  8. , , , , , ,

  9. Measurements were performed on a Lenovo t440p laptop with Intel Core i7-4810MQ @ 2.8GHz with 32GB memory, using Coq 8.14/8.15.1 in 32-bit mode with virtual-memory limit of 1GB. The Clight abstract-syntax-tree files were produced by in its default mode.

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Authors and Affiliations

  1. Department of Computer Science, Princeton University, 35 Olden Street, Princeton, NJ, 08540, USA

    Andrew W. Appel

  2. Collège de France, PSL University, 3 rue d’Ulm, Paris, 75005, France

    Xavier Leroy

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  1. Andrew W. Appel
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  2. Xavier Leroy
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Correspondence to Xavier Leroy.

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Code is available at https://github.com/xavierleroy/canonical-binary-tries/tree/v2.

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Appel, A.W., Leroy, X. Efficient Extensional Binary Tries. J Autom Reasoning 67, 8 (2023). https://doi.org/10.1007/s10817-022-09655-x

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