扩展理性选择行为：具有选择对应的布尔集合论决策问题
Extending rational choice behavior: The decision problem for Boolean set theory with a choice correspondence
摘要：
给定备选项集合$U$的所有非空子集的族$P$，对$U$的选择是函数$c \colon Ω\to P$，使得对于所有菜单$B \in Ω$为$Ω\subseteq P$和$c(B) \subseteq B$。如果选择$Ω= P$，则选择全部，否则选择部分。在经济学中，当一个代理人的选择行为满足适当的一致性公理时，她就被认为是理性的，一致性公理是在菜单上量化的属性。在这里，我们解决以下提升问题：给定一个部分选择满足一个或多个公理的一致性，是否有可能将其扩展到一个总的选择满足相同的公理？在刻画了经济学文献中一些著名的选择性质的提升之后，我们研究了集合论的一个基本片段的未量化公式的连通可满足性问题的可判定性，其中涉及到选择函数符号、布尔集合算子、单例、等式和包含谓词以及命题连接词.在两种情况下，我们证明了可满足性问题是NP-完全的，而在其余的情况下，我们得到NP-完全的额外的假设下，选择条款的数量是常数。
英文摘要：
Given the family $P$ of all nonempty subsets of a set $U$ of alternatives, a choice over $U$ is a function $c \colon Ω\to P$ such that $Ω\subseteq P$ and $c(B) \subseteq B$ for all menus $B \in Ω$. A choice is total if $Ω= P$, and partial otherwise. In economics, an agent is considered rational whenever her choice behavior satisfies suitable axioms of consistency, which are properties quantified over menus. Here we address the following lifting problem: Given a partial choice satisfying one or more axioms of consistency, is it possible to extend it to a total choice satisfying the same axioms? After characterizing the lifting of some choice properties that are well-known in the economics literature, we study the decidability of the connected satisfiability problem for unquantified formulae of an elementary fragment of set theory, which involves a choice function symbol, the Boolean set operators, the singleton, the equality and inclusion predicates, and the propositional connectives. In two cases we prove that the 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.
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