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Complexity Assessments for Decidable Fragments of Set Theory. I : A Taxonomy for the Boolean Case

Cantone Domenico, De Domenico Andrea, Maugeri Pietro, Omodeo Eugenio G.

Fundamenta Informaticae

2021

Vol. 181, nr 1

37--69

We report on an investigation aimed at identifying small fragments of set theory (typically, sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator ∈ for the time being, in this paper we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators ∪, ∩, \, the Boolean relators ⊆, ⊈,=, ≠, and the predicates ‘• = Ø’ and ‘Disj(•, •)’, meaning ‘the argument set is empty’ and ‘the arguments are disjoint sets’, along with their opposites ‘• ≠ Ø and ‘¬Disj(•, •)’. We also examine in detail how to test for satisfiability the formulae of six sample fragments: three sample problems are shown to be NP-complete, two to admit quadratic-time decision algorithms, and one to be solvable in linear time.

Composition Theorem for Generalized Sum

Rabinovich A.

Fundamenta Informaticae

2007

Vol. 79, nr 1-2

137-167

Composition theorems are tools which reduce sentences about some compound structure to sentences about its parts. A seminal example of such a theorem is the Feferman-Vaught Theorem [3] which reduces the first-order theory of generalized products to the first order theory of its factors and the monadic second-order theory of index structure. Shelah [23] used the composition theorem for linear orders as one of the main tools for obtaining very strong decidability results for the monadic second-order theory of linear orders. The main technical contribution of our paper is (1) a definition of a generalized sum of structures and (2) a composition theorem for first-order logic over the generalized sum. One of our objectives is to emphasize the work-out of the composition method.