  In this paper we address the decision problem for a fragment of set theory
with restricted quantification which extends the language studied in [4] with
pair related quantifiers and constructs, in view of possible applications in
the field of knowledge representation. We will also show that the decision
problem for our language has a non-deterministic exponential time complexity.
However, for the restricted case of formulae whose quantifier prefixes have
length bounded by a constant, the decision problem becomes NP-complete. We also
observe that in spite of such restriction, several useful set-theoretic
constructs, mostly related to maps, are expressible. Finally, we present some
undecidable extensions of our language, involving any of the operators domain,
range, image, and map composition.
  [4] Michael Breban, Alfredo Ferro, Eugenio G. Omodeo and Jacob T. Schwartz
(1981): Decision procedures for elementary sublanguages of set theory. II.
Formulas involving restricted quantifiers, together with ordinal, integer, map,
and domain notions. Communications on Pure and Applied Mathematics 34, pp.
177-195
