Abstract

We continue our investigation aimed at spotting small fragments of Set Theory (in this paper, sublanguages of Boolean Set Theory) that might be of use in automated proof-checkers based on the set-theoretic formalism. Here we propose a method that leads to a cubic-time satisfiability decision test for the language involving, besides variables intended to range over the von Neumann set-universe, the Boolean operator ∪ and the logical relators = and ≠. It can be seen that the dual language involving the Boolean operator ∩ and, again, the relators = and ≠, also admits a cubic-time satisfiability decision test; noticeably, the same algorithm can be used for both languages. Suitable pre-processing can reduce richer Boolean languages to the said two fragments, so that the same cubic satisfiability test can be used to treat the relators ⊆ and ⊈, and the predicates ‘▪’ and ‘▪’, meaning ‘the argument is empty’ and ‘the arguments are disjoint sets’, along with their opposites ‘▪’ and ‘▪’. Those richer languages are ‘polynomial maximal’, in the sense that each language strictly containing either of them and whose formulae are conjunctions of literals has an NP-hard satisfiability problem.A generalized version of the two said satisfiability tests can treat the relator ⊄, though at the price of a worsening of the algorithmic complexity (from cubic to quintic time).

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