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We show that, given an ordinary lambda-term and a normal resource lambda-term which appears in the normal form of its Taylor expansion, the unique resource term of the Taylor expansion of the ordinary lambda-term reducing to this normal resource term can be obtained by running a version of the Krivine abstract machine.

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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.

Abstract. The relational semantics for Linear Logic induces a semantics for the type free Lambda Calculus. This one is built on non-idempotent intersection types. We give a principal typing property for this type system. We then prove that the size of the derivations is closely related to the execution time of lambda-terms in a particular environment machine, Krivine’s machine.

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This paper defines head linear reduction, a reduction strategy of #-terms that performs the minimal number of substitutions for reaching a head normal form. The definition relies on an extended notion of redex, and head linear reduction is therefore not a strategy in the exact usual sense. Krivine 's Abstract Machine is proved to be sound by relating it both to head linear reduction and to usual head reduction. The first proof suggests a variant machine, the Pointer Abstract Machine, which is also proved to be sound with respect to head linear reduction.

We derive two big-step abstract machines, a natural semantics, and the valuation function of a denotational semantics based on the small-step abstract machine for Core Scheme presented by Clinger at PLDI’98. Starting from a functional implementation of this small-step abstract machine, (1) we fuse its transition function with its driver loop, obtaining the functional implementation of a big-step abstract machine; (2) we adjust this big-step abstract machine so that it is in defunctionalized form, obtaining the functional implementation of a second big-step abstract machine; (3) we refunctionalize this adjusted abstract machine, obtaining the functional implementation of a natural semantics in continuation style; and (4) we closure-unconvert this natural semantics, obtaining a compositional continuation-passing evaluation function which we identify as the functional implementation of a denotational semantics in continuation style. We then compare this valuation function with that of Clinger’s original denotational semantics of Scheme.

2 DIAL∀l µ l 4 2.1 The dual typing system............................ 4 2.2 Data structures and notations......................... 5

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