Formative processes with applications to the decision problem in set theory: II. Powerset and singleton operators, finiteness predicate

Cantone, Domenico; Ursino, Pietro

doi:10.1016/j.ic.2014.02.005

Cited by 12 publications

(15 citation statements)

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“…Further decidable existential fragments of set-theory have been studied, such as the fragment with power-set and singleton operators [6,9], which opens the way to a settheoretic definition of other decidable extensions of ALC with well-founded sets. A natural question arising is whether these description logics with power-set, well-founded sets and extensionality, can be translated as well into standard DLs and, in particular, whether the extensionality and well-foundedness assumptions can be captured in standard DLs.…”

Section: Discussionmentioning

confidence: 99%

“…The above definition uniquely determines hypersets in HF 1/2 (A). This follows from the fact that all finite systems of (finite) set-theoretic equations have a solution in HF 1/2 (A) 9 .…”

Section: Proposition 3 (Completeness Of the Translation)mentioning

confidence: 99%

“…Strictly speaking the graph G introduced here is not really necessary: it is just mentioned to single out the membership relation ∈ from e I more clearly 9. More generally, when e I is a well-founded relation, M (·) is a set-theoretic "rendering" of e I : the so-called Mostowski collapse of e I (see[21]).…”

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confidence: 99%

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Adding the power-set to description logics

Giordano

Policriti

2020

Theoretical Computer Science

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We explore the relationships between Description Logics and Set Theory. The study is carried on using, on the set-theoretic side, a very rudimentary axiomatic set theory Ω, consisting of only four axioms characterizing binary union, set difference, inclusion, and the power-set. An extension of ALC, ALC Ω , is then defined in which concepts are naturally interpreted as sets living in Ω-models. In ALC Ω not only membership between concepts is allowed-even admitting circularity-but also the power-set construct is exploited to add metamodeling capabilities. We investigate translations of ALC Ω into standard description logics as well as a set-theoretic translation. A polynomial encoding of ALC Ω in ALCOI proves the validity of the finite model property as well as an EXP-TIME upper bound on the complexity of concept satisfiability. We develop a settheoretic translation of ALC Ω in the theory Ω, exploiting a technique proposed for translating normal modal and polymodal logics into Ω. Finally, we show that the fragment LC Ω of ALC Ω , which does not admit roles and individual names, is as expressive as ALC ΩAdding the Power-Set to Description Logicscompleted with the standard deduction rules of generalization and modus-ponens.The above theory must be intended as a minimal Hilbert-style axiomatic system for set theory. When thinking of specific models of Ω, however, we can clearly think of structures satisfying extra axioms. In particular, for example, the familiar well-founded models of set theories, are perfectly legitimate models of Ω, in which the extra axiom of well-foundedness-implying that ∈ cannot form cycles or infinite descending chainsholds. For instance let x = {∅, {∅}}. x is a finite well-founded set, and the sets ∅ and {∅} are elements of x. Instead, the set y = {∅, {∅, {∅, {. . .}}} is finite but not wellfounded.Whatever the axioms satisfied by the Ω-model under consideration are, however, everything in the domain of such a model is supposed to be a set. As a consequence, a set will have (only) sets as its elements. Moreover, as observed, circular definitions of sets are not forbidden. That is, for example, there are models of Ω in which there are sets admitting themselves as elements. For instance, the set y above could simply be defined as y = {∅, y} and has elements ∅ and y itself.Finally, not postulating in Ω any explicit "axiomatic link" between membership ∈ and equality-more precisely: having no extensionality axiom-, there exist Ω-models in which there are different sets with equal collection of elements. One (elementary) consequence of the extensionality axiom is the familiar fact that if a ⊆ b and b ⊆ a, then a = b. In non-extensional models, instead, there can be pairwise distinct sets included in each other. The set x ′ = {a, b, {b, c}} with a, b and c pairwise distinct and such that a, b, c ⊆ ∅, does not satisfy extensionality as a, b and c are different sets with the same (empty) extension.Definition 3. Ω-models are first order interpretations M = (U, · M ) satisfying the axioms of the theory ...

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Section: Discussionmentioning

confidence: 99%

“…The above definition uniquely determines hypersets in HF 1/2 (A). This follows from the fact that all finite systems of (finite) set-theoretic equations have a solution in HF 1/2 (A) 9 .…”

Section: Proposition 3 (Completeness Of the Translation)mentioning

confidence: 99%

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Adding the power-set to description logics

Giordano

Policriti

2020

Theoretical Computer Science

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“…Nevertheless, we prove in [CU14] that even theories which force a model to be infinite can be proven to be decidable by using the small witness-model property, which is a way to finitely represent the infinity.…”

Section: Introductionmentioning

confidence: 98%

Decidability and NP-completeness for some languages which extend Boolean Set Theory

Ursino¹

2021

Preprint

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“…solutions to the decision problems for the extension MLSSP of MLS with the power set and the singleton operators [COU02] and the extension MLSSPF with the finiteness predicate too [CU14].…”

Section: Introductionmentioning

confidence: 99%

Decidability of the decision problem for Boolean set theory with the unordered Cartesian product operator

Cantone¹,

Ursino²

2021

Preprint

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The satisfiability problem for multilevel syllogistic extended with the Cartesian product operator (MLS×) is a long-standing open problem in computable set theory. For long, it was not excluded that such a problem were undecidable, due to its remarkable resemblance with the well-celebrated Hilbert's tenth problem, as it was deemed reasonable that union of disjoint sets and Cartesian product might somehow play the roles of integer addition and multiplication.To dispense with nonessential technical difficulties, we report here about a positive solution to the satisfiability problem for a slight simplified variant of MLS×, yet fully representative of the combinatorial complications due to the presence of the Cartesian product, in which membership is not present and the Cartesian product operator is replaced with its unordered variant.We are very confident that such decidability result can be generalized to full MLS×, though at the cost of considerable technicalities. * We gratefully acknowledge partial support from the projects STORAGE and MEGABIT -

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Formative processes with applications to the decision problem in set theory: II. Powerset and singleton operators, finiteness predicate

Cited by 12 publications

References 4 publications

Adding the power-set to description logics

Adding the power-set to description logics

Decidability and NP-completeness for some languages which extend Boolean Set Theory

Decidability of the decision problem for Boolean set theory with the unordered Cartesian product operator

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