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Decision procedures for set/hyperset contexts

  • Eugenio G. Omodeo
  • Alberto Policriti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 722)

Abstract

Pure, hereditarily finite, sets and hypersets are characterized both as an algorithmic data structure and by means of a first-order axiomatization which, although rather weak, suffices to make the following two problems decidable:
  1. (1)

    Establishing whether a conjunction r of formulae of the form ∀y1... ∀ym((y1∃w1 & ...& ymw m )→q), with q unqualified and involving only the relators =, ∃ and propositional connectives, is satisfiable.

     
  2. (2)

    Establishing whether a formula of the form ∀ y q, q as above, is satisfiable.

     

Concerning (1), an explicit decision algorithm is provided; moreover, significantly broad sub-problems of (1) are singled out in which a classification —named the ‘syllogistic decomposition’ of r — of all possible ways of satisfying the input conjunction r can be obtained automatically. For one of these sub-problems, carrying out the decomposition results in producing a finite family of syntactic substitutions that generate the space of all solutions to r. In this sense, one has a unification algorithm. Concerning (2), a technique is provided for reducing it to a sub-problem of (1) for which a decomposition method is available.

Keywords

Free Variable Conjunctive Normal Form Unification Algorithm Finite Family Start Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Eugenio G. Omodeo
    • 1
  • Alberto Policriti
    • 2
  1. 1.Dip. di Informatica e SistemisticaUniversità di Roma “La Sapienza”Roma
  2. 2.Dip. di Matematica e InformaticaUniversità di UdineUdineItaly

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