Complexity assessments for decidable fragments of set theory. II: A taxonomy for ‘small’ languages involving membership

Cantone, Domenico; Maugeri, Pietro; Omodeo, Eugenio G.

doi:10.1016/j.tcs.2020.08.023

Cited by 7 publications

(7 citation statements)

References 12 publications

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“…We postpone to other reports (see, for the time being, [18]) the treatment of ∈, the membership relation. Adding ∈ to BST does not disrupt its decidability (see Sect.…”

Section: Boolean Set Theorymentioning

confidence: 99%

“…Concerning conjuncts of type ∩Li = ∩Ri in ϕ, we get from (15) that either L i R i or R i L i holds. For definiteness assume that L i R i (the treatment of the other case being symmetrical), so that v (18), v R i ∈ ∩MRi, which proves ∩MLi = ∩MRi. Otherwise, D j ⊆ R i holds for some j such that 1 j p , and hence ∩MRi = ∅ (for, R j ⊆ S implies D j ⊆ S) and, by condition (16), D j L i holds for j = 1, .…”

Section: A Quadratic-time Satisfiability Test For Bst(∩ =∅ =)mentioning

confidence: 99%

“…As discussed in [23,18], the number of distinct fragments of MST is 24: of those, 10 are NP-complete and 14 are polynomial. Much as for BST, the complexity of any fragment of MST can be readily determined once the minimal NP-complete fragments and the maximal polynomial fragments have been singled out.…”

Section: Strength-reduction Examplesmentioning

confidence: 99%

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

Cantone¹,

Domenico²,

Maugeri³

et al. 2021

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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.

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“…We postpone to other reports (see, for the time being, [18]) the treatment of ∈, the membership relation. Adding ∈ to BST does not disrupt its decidability (see Sect.…”

Section: Boolean Set Theorymentioning

confidence: 99%

Section: A Quadratic-time Satisfiability Test For Bst(∩ =∅ =)mentioning

confidence: 99%

See 1 more Smart Citation

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

Cantone¹,

Domenico²,

Maugeri³

et al. 2021

Self Cite

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“…Convexity of MLS is plainly inherited by all of its fragments. In [8] and [10], we recently investigated them with the goal of spotting the polynomial ones, namely the fragments of MLS endowed with polynomial-time satisfiability tests. Specifically, we examined all the sublanguages of the theories BST := BST(∪, ∩, \, =∅, =∅, Disj, ¬Disj, ⊆, ⊆, =, =) and MST := MST(∪, ∩, \, ∈, / ∈),…”

Section: Polynomial Fragments Of Mlsmentioning

confidence: 99%

“…In this paper, we start an investigation for combining decidable fragments of pure Zermelo-Fraenkel set theory (in which sets are recursively built up from other sets) with other theories within the Nelson-Oppen framework. More specifically, our main result is that the theory Multi-Level Syllogistic (the basic language of computable set theory-MLS for short) is convex and therefore its decision procedure (and those of its several polynomial fragments [8,10]) can be efficiently combined with the decision procedures of other basic decidable theories, such as for instance the theory of lists and the theory of linear rational arithmetic, since set theory is plainly stably infinite.…”

Section: Introductionmentioning

confidence: 99%

On the Convexity of a Fragment of Pure Set Theory with Applications within a Nelson-Oppen Framework

Cantone

Domenico

Maugeri

2021

Electron. Proc. Theor. Comput. Sci.

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The Satisfiability Modulo Theories (SMT) issue concerns the satisfiability of formulae from multiple background theories, usually expressed in the language of first-order predicate logic with equality. SMT solvers are often based on variants of the Nelson-Oppen combination method, a solver for the quantifier-free fragment of the combination of theories with disjoint signatures, via cooperation among their decision procedures. When each of the theories to be combined by the Nelson-Oppen method is convex (that is, any conjunction of its literals can imply a disjunction of equalities only when it implies at least one of the equalities) and decidable in polynomial time, the running time of the combination procedure is guaranteed to be polynomial in the size of the input formula. In this paper, we prove the convexity of a fragment of Zermelo-Fraenkel set theory, called Multi-Level Syllogistic, most of whose polynomially decidable fragments we have recently characterized.

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Onset and Today’s Perspectives of Multilevel Syllogistic

Cantone,

Omodeo

2024

Lecture Notes in Computer Science

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Complexity assessments for decidable fragments of set theory. II: A taxonomy for ‘small’ languages involving membership

Cited by 7 publications

References 12 publications

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

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

On the Convexity of a Fragment of Pure Set Theory with Applications within a Nelson-Oppen Framework

Onset and Today’s Perspectives of Multilevel Syllogistic

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