dvanhorn @ λ-calcul.us

We’ve been working on two techniques for systematically deriving abstract interpretations approximating canonical machines for higher-order languages. The first allocates continuations in a bounded store to achieve a finite state-space. We demonstrate the generality of this approach by transforming classical abstract machines into abstract interpreters.

Abstracting Abstract Machines.
David Van Horn and Matthew Might.
The 15th ACM SIGPLAN International Conference on Functional Programming (ICFP’10), Baltimore, Maryland, September, 2010.

We describe a derivational approach to abstract interpretation that yields novel and transparently sound static analyses when applied to well-established abstract machines. To demonstrate the technique and support our claim, we transform the CEK machine of Felleisen and Friedman, a lazy variant of Krivine’s machine, and the stack-inspecting CM machine of Clements and Felleisen into abstract interpretations of themselves. The resulting analyses bound temporal ordering of program events; predict return-flow and stack-inspection behavior; and approximate the flow and evaluation of by-need parameters. For all of these machines, we find that a series of well-known concrete machine refactorings, plus a technique we call store-allocated continuations, leads to machines that abstract into static analyses simply by bounding their stores. We demonstrate that the technique scales up uniformly to allow static analysis of realistic language features, including tail calls, conditionals, side effects, exceptions, first-class continuations, and even garbage collection.

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The second technique keeps continuations on the stack to achieve a push-down model of abstract interpretation. The resulting abstract interpreter always matches calls and returns, achieving a higher level of precision by never conflating call and return pairs. Although this technique produces abstract interpreters with infinite state-spaces, we demonstrate how basic static analysis questions remain decidable by casting them as language inclusion problems answered by push-down automata.

Pushdown Control-Flow Analysis of Higher-Order Programs.
Christopher Earl, Matthew Might, and David Van Horn.
The 2010 Workshop on Scheme and Functional Programming (SFP 2010), Montréal, Québec, Canada, August, 2010.

Context-free approaches to static analysis gain precision over classical approaches by perfectly matching returns to call sites—a property that eliminates spurious interprocedural paths. Vardoulakis and Shivers’s recent formulation of CFA2 showed that it is possible (if expensive) to apply context-free methods to higher-order languages and gain the same boost in precision achieved over first-order programs.

To this young body of work on context-free analysis of higher-order programs, we contribute a pushdown control-flow analysis framework, which we derive as an abstract interpretation of a CESK machine with an unbounded stack. One instantiation of this framework marks the first polyvariant pushdown analysis of higher-order programs; another marks the first polynomial-time analysis. In the end, we arrive at a framework for control-flow analysis that can efficiently compute pushdown generalizations of classical control-flow analyses.

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There are also slides (PDF) covering both techniques from a talk given at the Harvard PL seminar.