On the Desirable Link Between Theory

operations are de ned as upgraded versions of previous ones. This ispossible only because of the systematic use of speci cations, since a correctreuse of previous operations can be based on explicit reasoning. Moreover, themathematical framework used by this version is still weaker than the previousones, since there is no abstract xpoint at all, even in theory. Finally, this analysisconsiders properties of higher-order objects (i.e., substitution sequences) thatare not downwards closed, contrary to many frameworks based on denotationalsemantics.A limitation of GAIA. To date [24], more than seven (substantially) di erentversions of GAIA have been completed in order to conduct various experimentsin logic and constraint logic program analysis. They amount to more than 50,000lines of C code. Thus, in its current form, GAIA is not a completely genericsystem, but rather a series of systems that were rather straightforwardly derivedfrom the rst original version. Although the objective of a \real" generic systemseems a priori desirable, we believe that it is in fact di cult to foresee all futureapplications of GAIA, since experience has shown that every new experimentmay entail unexpected variations of the framework; thus we think that it isprobably wiser to continue moving from one version to another one, based onexperiments. However, this is wise only if we keep a clear understanding of thesystem, based on good speci cations of all its parts.5 ConclusionIs abstract interpretation a uni ed framework? In this paper, we have mostlyanswered this question negatively. Much of our argumentation is personal opin-ion, but it is based on actual and largely successful experiments with a practicalsystem. Nevertheless, notions such as Galois insertions, collecting semantics, andwidening operators provide deep insights into how to properly design a programanalysis. Once these notions are well understood, one can relax or modify themin order to nd the framework that is best adapted to one's goals [9]. Findingthe ultimate framework is an endless quest!AcknowledgementsBaudouin Le Charlier wishes to thank Pascal Van Hentenryck for his invaluablecontributions to GAIA.References1. C. Braem, B. Le Charlier, S. Modard, and P. Van Hentenryck. Cardinality anal-ysis of Prolog. In M. Bruynooghe, editor, Proceedings of the International LogicProgramming Symposium (ILPS'94), Ithaca NY, USA, November 1994. MIT Press.2. M. Bruynooghe. A practical framework for the abstract interpretation of logicprograms. Journal of Logic Programming, 10(2):91{124, February 1991. 3. M. Bruynooghe, G. Janssens, A. Callebaut, and B. Demoen. Abstract interpreta-tion: Towards the global optimization of Prolog programs. In Proceedings of the1987 Symposium on Logic Programming, pages 192{204, San Francisco, California,August 1987. Computer Society Press of the IEEE.4. P. 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