Lifschitz introduced the notion of defining extensions of predicate default theories not as absolute, but relative to a specified domain. We look specifically at default theories over a countable domain and show the set of default theories which possess an ω -extension is Σ 2 1 -complete. That the set is in Σ 2 1 is shown by writing a nearly circumscriptive formula whose ω -models correspond to the ω -extensions of a given default theory; similarly, Σ 2 1 -hardness (...) is established by a method for translating formulas into default theories in such a way that ω -models of the circumscriptive formula correspond to ω -extensions of the default theory. (shrink)

We motivate and formalize the idea of sameness by default: two objects are considered the same if they cannot be proved to be different. This idea turns out to be useful for a number of widely different applications, including natural language processing, reasoning with incomplete information, and even philosophical paradoxes. We consider two formalizations of this notion, both of which are based on Reiter’s Default Logic. The first formalization is a new relation of indistinguishability that is introduced by default. We (...) prove that the corresponding default theory has a unique extension, in which every two objects are indistinguishable if and only if their non-equality cannot be proved from the known facts. We show that the indistinguishability relation has some desirable properties: it is reflexive, symmetric, and, while not transitive, it has a transitive “flavor.” The second formalization is an extension (modification) of the ordinary language equality by a similar default: two objects are equal if and only if their non-equality cannot be proved from the known facts. It appears to be less elegant from a formal point of view. In particular, it gives rise to multiple extensions. However, this extended equality is better suited for most of the applications discussed in this paper. (shrink)