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well for one, the problem that you answered would be considered...complete.

the other one which is not done, would be incomplete.

DUH!

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14y ago

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Closure analysis determines statically which function de¯nitions reach which programpoints. This information is used for many di®erent purposes; e.g., type inference forobject-oriented programming languages [PS91], globalization analysis of functional pro-grams [Ses89], partial evaluation [Bon90], type recovery in Scheme [Shi90], and others.The reason why closure analysis is such a fundamental analysis in di®erent applications isthat it is (sort of) the universal data °ow analysis problem for monomorphic (data °oworiented) analyses for higher-order (functional) languages. As such it may be viewed asthe analogue of path analysis, which is universal for (continuous) ¯rst-order (classical)data °ow analysis problems [Tar81].General closure analysis is, unlike its ¯rst-order counterpart, expensive in the worstcase: the best known algorithms take time £(n3) in the worst case [Hen91a,PS92]. 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The computed abstract value °owinformation1 is coarser than normal closure analysis exactly in the following sense: exactclosure analysis keeps track of uni-directional °ow of (abstract) values from one programpoint to another whereas our simple closure analysis works with the assumption that °owsare reversible; that is, that any abstract value that °ows from point p to point p0 can also°ow (backwards!) from p0 to p.In the practice of partial evaluation this loss of information appears to be insubstantial.Since the simple closure analysis algorithm runs in almost-linear time [Hen91b] and is very¤DIKU Semantics Report D-1931We prefer to call the abstractions of values corresponding to expressions in a program abstract valuesrather than (abstract) closures, since these values represent values other than (run-time) closures, includingpairs, integers, etc.12 BASIC IDEA OF SIMPLE CLOSURE ANALYSIS2e±cient in practice [Hen91c] this appears to be an attractive alternative to computingcomplete closure information.2 Basic idea of simple closure analysisIn the binding-time analysis of [Gom89], on which [Hen91b] is based, we begin by asso-ciating a unique abstract value (also called a token, label or a type variable depending onthe intention of their use) with every (sub)expression in a program, and constraints areextracted that capture the °ow of actual values represented by these abstract values in theprogram. 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(Of course,\exact" closure analysis is itself a conservative approximation of the actual dynamic °owof run-time values, including concrete closures.)3 Simple closure analysis exempli¯edIn this section we exemplify the steps above by considering a simple example.Consider the following code (fragment), representing Turner's tautology checker andtwo calls to it:taut = fn f => fn n. if n = 0 then f else taut (f true) (n-1) and taut (f false) (n-1)g = fn x => fn y => (x and not y) or (not x or y)h = fn z => ztaut g 2 taut h 13.1 Constraint extractionIn the ¯rst step we associate a distinct abstract value with every occurrence of a subex-pression occurring in this code; for simplicity's sake all occurrences of a variable havethe same abstract value. We shall refer to the abstract value of an (occurrence of an)expression by the special variable ® indexed by the expression; e.g., the abstract valuesof the three function de¯nitions are ®taut; ®g and ®h. 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Inparticular, there is no need for an occurs check rule since we don't have to interpret theresult as ¯nite type expressions; and there is no need for a special type Dynamic (or ¤)representing either possible type errors or run-time computable expressions or both, sincewe are interested neither in the former nor in the latter. In particular, the rewriting rulesmanipulating Dynamic can be omitted. We end up with the rewriting rules in Figure 2.As proved in [Hen91b] a rewriting system normalizing constraints with additionalrewriting rules can be implemented in time O(n®(n; n) where ®(n; n)