A General Input-Dependent Colorless Computability Theorem and Applications to Core-Dependent Adversaries

Distributed computing tasks can be presented with a triple $(\I,\Ou,\Delta)$. The solvability of a colorless task on the Iterated Immediate Snapshot model (IIS) has been characterized by the Colorless Computability Theorem \cite[Th.4.3.1]{HKRbook}. A recent paper~\cite{CG-24} generalizes this theorem for any message adversaries $\ma \subseteq IIS$ by geometric methods. In 2001, Most\'efaoui, Rajsbaum, Raynal, and Roy \cite{condbased} introduced \emph{condition-based adversaries}. This setting considers a particular adversary that will be applied only to a subset of input configurations. In this setting, they studied the $k$-set agreement task with condition-based $t$-resilient adversaries and obtained a sufficient condition on the conditions that make $k$-Set Agreement solvable. In this paper we have three contributions: -We generalize the characterization of~\cite{CG-24} to \emph{input-dependent} adversaries, which means that the adversaries can change depending on the input configuration. - We show that core-resilient adversaries of $IIS_n$ have the same computability power as the core-resilient adversaries of $IIS_n$ where crashes only happen at the start. - Using the two previous contributions, we provide a necessary and sufficient characterization of the condition-based, core-dependent adversaries that can solve $k$-Set Agreement. We also distinguish four settings that may appear when presenting a distributed task as $(\I,\Ou,\Delta)$. Finally, in a later section, we present structural properties on the carrier map $\Delta$. Such properties allow simpler proof, without changing the computability power of the task. Most of the proofs in this article leverage the topological framework used in distributed computing by using simple geometric constructions.