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.Comment: In Proceedings GandALF 2017, arXiv:1709.01761. "extended" version at
arXiv:1708.0612
