On the Existence of Extension-Based Proofs of Impossibility for Set-Agreement

A recent breakthrough by Alistarh, Aspnes, Ellen, Gelashvili, and Zhu [STOC 2019] established that there are no extension-based proofs of impossibility for set-agreement within the class of non-uniform iterated immediate snapshot (NIIS) algorithms. An extension-based proof can be modeled as a game between a prover and an algorithm claiming to solve set-agreement, where this algorithm can be from a class C of algorithms. Note that the non-existence of extension-based proofs of impossibility for a class C of algorithms implies the non-existence of extension-based proofs of impossibility for all classes of algorithms containing C. This result has then been revisited by Attiya, Castañeda, and Rajsbaum [OPODIS 2020] who showed that the same holds for the smaller class of uniform iterated immediate snapshot (IIS) algorithms, and even with a slightly stronger prover. The main takeaway message of our work is that these previous results, which show the non-existence of extension-based proofs of impossibility for set-agreement within smaller and smaller universal classes of algorithms, do not necessarily preclude the existence of an extension-based proof of impossibility for set-agreement for an even smaller class of algorithms. To illustrate this, we exhibit an extension-based proof of impossibility for set-agreement for the class of memoryless IIS algorithms. This latter class may not be universal but is strong enough to solve non-trivial tasks such as approximate agreement and renaming. Moreover, we show that the result by Attiya et al. for IIS algorithms, with the stronger prover, does not extend to colorless IIS algorithms. These two results underline the fact that the existence of an extension-based proof of impossibility for a task strongly depends on the considered class of algorithms, and is still open even for set-agreement.