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

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

💡 Research Summary

The paper investigates the satisfiability problem for a new two‑sorted fragment of set theory, denoted ∀π⁰,₂, which extends the previously studied restricted‑quantifier fragment ∀₀ by adding ordered‑pair constructors and map‑related quantifiers. The authors begin by reviewing the background of Computable Set Theory, highlighting the importance of decidable set‑theoretic fragments such as MLS and its extensions, and noting that the fragment ∀₀ already allows a surprisingly rich set of constructions while keeping quantifier nesting limited.

In ∀π⁰,₂ the language consists of an infinite supply of set variables (SVars) and map variables (MVars). Maps are interpreted as sets of ordered pairs, and a generic pairing function π (e.g., Kuratowski’s pair) is assumed to be injective and to produce sets that belong to the von Neumann hierarchy. Atomic formulas are limited to four forms: x ∈ y, x = y,