A decidable quantified fragment of set theory with ordered pairs and some undecidable extensions

TL;DR

This paper investigates a decidable fragment of set theory with ordered pairs, analyzing its computational complexity and expressing useful constructs, while also identifying undecidable extensions involving set operators.

Contribution

It introduces a new decidable set theory fragment with pair quantifiers, analyzes its complexity, and explores its expressive power and limitations with undecidable extensions.

Findings

Decision problem is NEXPTIME-complete for the general case.

Restricted formulas with bounded quantifier prefixes are NP-complete.

Several set-theoretic constructs related to maps are expressible within the fragment.

Abstract

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