Abstract
The problem of computing dominators in a control flow graph is central to numerous modern compiler optimizations. Many efficient algorithms have been proposed in the literature, but mechanizing the correctness of the most sophisticated algorithms is still considered as too hard problems, and to this date, verified compilers use less optimized implementations. In contrast, production compilers, like GCC or LLVM, implement the classic, efficient Lengauer-Tarjan algorithm [12], to compute dominator trees. And subsequent optimization phases can then determine whether a CFG node dominates another node in constant time by using their respective depth-first search numbers in the dominator tree. In this work, we aim at integrating such techniques in verified compilers. We present a formally verified validator of untrusted dominator trees, on top of which we implement and prove correct a fast dominance test following these principles. We conduct our formal development in the Coq proof assistant, and integrate it in the middle-end of the CompCertSSA verified compiler. We also provide experimental results showing performance improvement over previous formalizations.
This work was supported by Agence Nationale de la Recherche, grant number ANR-14-CE28-0004 DISCOVER.
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Notes
- 1.
Such a property would be required to prove completeness: if a node d dominates a node n then the dominance test on (d, n) should succeed. To our experience in verified compilation, we never make usage of such a completeness property. The property holds, but we do not need to prove it in Coq.
- 2.
Not to be confused with the control flow graph here.
- 3.
is a dictionary implementation using Patricia trees provided in CompCert. Type \(\mathtt {(Ptree.t~a)}\) denotes an associative, partial map with keys of type \(\mathtt {positive}\) – binary encoding of strictly positive integers – with associated data of type \(\mathtt {a}\). In this paper, types \(\mathtt {node}\) and \(\mathtt {positive}\) are synonyms.
- 4.
We write m!n the lookup of a key
in a map
.
- 5.
Recall that we rely on the standard implementation of positive integers,
.
- 6.
The impact of building and using the dominance test is currently negligible compared to the whole compilation time, as, currently, certain compiler passes (such as the SSA deconstruction) would need performance improvement.
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Blazy, S., Demange, D., Pichardie, D. (2015). Validating Dominator Trees for a Fast, Verified Dominance Test. In: Urban, C., Zhang, X. (eds) Interactive Theorem Proving. ITP 2015. Lecture Notes in Computer Science(), vol 9236. Springer, Cham. https://doi.org/10.1007/978-3-319-22102-1_6
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