Axiomatics of Restricted Choices by Linear Orders of Sets with Minimum as Fallback

TL;DR

This paper explores how linear orders can be used to define choice functions under restricted conditions, especially when fallback options are involved, with implications for knowledge representation and argumentation.

Contribution

It introduces a method to construct choice functions via linear orders on sets, accommodating fallback options, and provides an axiomatics framework for these restricted choice structures.

Findings

Choice functions can be constructed using linear orders with fallback as minimal element.

Axiomatic characterization of restricted choice functions is provided.

Applications discussed include theory change and abstract argumentation.

Abstract

We study how linear orders can be employed to realise choice functions for which the set of potential choices is restricted, i.e., the possible choice is not possible among the full powerset of all alternatives. In such restricted settings, constructing a choice function via a relation on the alternatives is not always possible. However, we show that one can always construct a choice function via a linear order on sets of alternatives, even when a fallback value is encoded as the minimal element in the linear order. The axiomatics of such choice functions are presented for the general case and the case of union-closed input restrictions. Restricted choice structures have applications in knowledge representation and reasoning, and here we discuss their applications for theory change and abstract argumentation.