The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence

TL;DR

This paper investigates the computational complexity of the satisfiability problem in a logical fragment involving choice functions and set operations, revealing NP-completeness results under various axioms.

Contribution

It characterizes the complexity of satisfiability for a set-theoretic logic with choice functions, including cases with axioms like WARP, and establishes NP-completeness results.

Findings

Satisfiability is NP-complete for certain axioms of choice consistency.

NP-completeness holds even when the number of choice terms is fixed.

The study links logical satisfiability with choice rationalizability conditions.

Abstract

Given a set U of alternatives, a choice (correspondence) on U is a contractive map c defined on a family Omega of nonempty subsets of U. Semantically, a choice c associates to each menu A in Omega a nonempty subset c(A) of 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.