Order

, Volume 33, Issue 3, pp 503–516 | Cite as

Weakening Additivity in Adjoining Closures

Article

Abstract

In this paper, we weaken the conditions for the existence of adjoint closure operators, going beyond the standard requirement of additivity/co-additivity. We consider the notion of join-uniform (lower) closure operators, introduced in computer science, in order to model perfect lossless compression in transformations acting on complete lattices. Starting from Janowitz’s characterization of residuated closure operators, we show that join-uniformity perfectly weakens additivity in the construction of adjoint closures, and this is indeed the weakest property for this to hold. We conclude by characterizing the set of all join-uniform lower closure operators as fix-points of a function defined on the set of all lower closures of a complete lattice.

Keywords

Residuated closures Uniformity Sdjoint functions 

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References

  1. 1.
    Blyth, T., Janowitz, M.: Residuation theory. Pergamon Press (1972)Google Scholar
  2. 2.
    Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. J. ACM 50(5), 752–794 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Conference Record of the 4th ACM Symposium on Principles of Programming Languages (POPL ’77), pp 238–252. ACM Press (1977)Google Scholar
  4. 4.
    Cousot, P., Cousot, R.: A constructive characterization of the lattices of all retractions, preclosure, quasi-closure and closure operators on a complete lattice. Portug. Math. 38(2), 185–198 (1979)MathSciNetMATHGoogle Scholar
  5. 5.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Conference Record of the 6th ACM Symposium on Principles of Programming Languages (POPL ’79), pp 269–282. ACM Press (1979)Google Scholar
  6. 6.
    Cousot, P., Cousot, R., Feret, J., Mauborgne, L., Miné, A., Monniaux, D., Rival, X.: The astreé analyzer. In: Sagiv, S. (ed.) Programming Languages and Systems, 14th European Symposium on Programming, ESOP 2005, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2005, Edinburgh, UK, April 4-8, 2005, vol. 3444, pp 21–30. Springer (2005). doi:10.1007/978-3-540-31987-0_3
  7. 7.
    Dwinger, P.: On the closure operators of a complete lattice. Indag. Math. 16, 560–563 (1954)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Giacobazzi, R., Ranzato, F.: Refining and compressing abstract domains. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) Proceedings of the 24th International Colloquium on Automata, Languages and Programming (ICALP ’97), Lecture Notes in Computer Science, vol. 1256, pp 771–781. Springer-Verlag (1997)Google Scholar
  9. 9.
    Giacobazzi, R., Ranzato, F.: Optimal domains for disjunctive abstract interpretation. Sci. Comput. Program 32(1-3), 177–210 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Giacobazzi, R., Ranzato, F.: Uniform closures: order-theoretically reconstructing logic program semantics and abstract domain refinements. Inform. Comput 145(2), 153–190 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Janowitz, M.F.: Residuated closure operators. Portug. Math. 26(2), 221–252 (1967)MathSciNetMATHGoogle Scholar
  12. 12.
    Morgado, J.: Some results on the closure operators of partially ordered sets. Portug. Math. 19(2), 101–139 (1960)MathSciNetMATHGoogle Scholar
  13. 13.
    Giacobazzi, R., Ranzato, F., Scozzari, F.: Making abstract interpretation complete. J. ACM 47(2), 361–416 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rice, H.: Classes of recursively enumerable sets and their decision problems. Trans. Amer. Math. Soc. 74, 358–366 (1953)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ward, M.: The closure operators of a lattice. Ann. Math. 43(2), 191–196 (1942)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità di VeronaVeronaItaly
  2. 2.IMDEA Software InstituteMadridSpain

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