Axiomatics of Restricted Choices by Linear Orders of Sets with Minimum as Fallback
This provides theoretical foundations for restricted choice structures in knowledge representation and reasoning, with applications in theory change and abstract argumentation, though it appears incremental in extending existing axiomatic frameworks.
The paper tackles the problem of constructing choice functions when the set of potential choices is restricted, showing that a linear order on sets of alternatives can always be used to define such functions, even with a fallback value as the minimal element. It presents axiomatics for general cases and union-closed restrictions.
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.