An Introduction to the Technique of Formative Processes in Set Theory

Cantone, Domenico; Ursino, Pietro

doi:10.1007/978-3-319-74778-1

Cited by 11 publications

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“…The semantics of BSTC follows exactly the definition of many other languages already treated (see for example [CU18,CU21]). Other operators and relators of BSTC are interpreted according to their usual semantics as well as satisfiability by partitions and normalization of BSTC-formulae [CU21].…”

Section: Introductionmentioning

confidence: 93%

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

Ursino¹

2021

Preprint

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Section: Introductionmentioning

confidence: 93%

“…In the above cited reduction to HTP, membership operator plays no role [CU18], then, by extending BST⊗ or BST× 2 with the two-place predicate | • | | • | for cardinality comparison, you get again a problem reducible to HTP.…”

Section: Introductionmentioning

confidence: 99%

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

Ursino¹

2021

Preprint

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“…Its NP-completeness (and that of its extension MLSS with the singleton operator) has been later proved in [11]. Several extensions of MLS have been proved decidable over the years, giving rise to the field of Computable Set Theory (see [9,12,18,13] for an in-depth account).…”

Section: Syntax and Semantics Of Mlsmentioning

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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“…In Section 4.3 we rediscover such result as a by-product of the solution to the satisfiability problem of BSTC under the WARP-semantics: the latter is based on a novel term-oriented non-clausal approach. The reader can find extensive information on Computable Set Theory in the monographs [3,8,18,9].For our purposes, it will be relevant to solve the following lifting problem: Given a partial choice satisfying some axioms of consistency, can we suitably characterize whether it is extendable to a total choice satisfying the same axioms? The lifting problem for the various combinations of axioms (α) and (β ) is addressed in depth in Section 3.…”

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The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence

Cantone

Giarlotta

Watson

2017

Electron. Proc. Theor. Comput. Sci.

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Given a set U of alternatives, a choice (correspondence) on U is a contractive map c defined on a family Ω of nonempty subsets of U. Semantically, a choice c associates to each menu A ∈ Ω a nonempty subset c(A) ⊆ A comprising all elements of A that are deemed selectable by an agent. A choice on U is total if its domain is the powerset of U minus the empty set, and partial otherwise. According to the theory of revealed preferences, a choice is rationalizable if it can be retrieved from a binary relation on U by taking all maximal elements of each menu. It is well-known that rationalizable choices are characterized by the satisfaction of suitable axioms of consistency, which codify logical rules of selection within menus. For instance, WARP (Weak Axiom of Revealed Preference) characterizes choices rationalizable by a transitive relation. Here we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory involving a choice function symbol c, the Boolean set operators and the singleton, the equality and inclusion predicates, and the propositional connectives. In particular, we consider the cases in which the interpretation of c satisfies any combination of two specific axioms of consistency, whose conjunction is equivalent to WARP. In two cases we prove that the related satisfiability problem is NP-complete, whereas in the remaining cases we obtain NP-completeness under the additional assumption that the number of choice terms is constant.In this paper we examine the decidability of the satisfiability problem connected to rational choice theory, which is a framework to model social and economic behavior. A choice on a set U of alternatives is a correspondence B → c(B) associating to "feasible menus" B ⊆ U nonempty "choice sets" c(B) ⊆ B. This choice can be either total (or full) -i.e, defined for all nonempty subsets of the ground set U of alternatives -or partial -i.e., defined only for suitable subsets of U .According to the Theory of Revealed Preferences pioneered by the economist Paul Samuelson [17], preferences of consumers can be derived from their purchasing habits: in a nutshell, an agent's choice behavior is observed, and her underlying preference structure is inferred. The preference revealed by a primitive choice is typically modeled by a binary relation on U . The asymmetric part of this relation is informative of a "strict revealed preference" of an item over another one, whereas its symmetric part codifies a "revealed similarity" of items. Then a choice is said to be rationalizable when the observed behavior can be univocally retrieved by maximizing the relation of revealed preference.Since the seminal paper of Samuelson, a lot of attention has been devoted to notions of rationality within the framework of choice theory: see, among the many contributions to the topic, the classical papers [15,2,16,14,19]. (See also the book [1] for the analysis of the links among the theories of * choice, preference, and utility. For a very recent contribution witnessing the ferve...

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An Introduction to the Technique of Formative Processes in Set Theory

Cited by 11 publications

References 0 publications

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

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

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

The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence

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