This thesis is concerned with impossibility results, i.e., proofs of the fact that certain classes of algorithms cannot exist. The algorithms investigated are from the field of fault-tolerant distributed computing, which is devoted to the formal study of processing entities, modeled as communicating state machines, that may possibly fail and communicate with each other by either exchanging messages or via access to a shared memory. We investigate the problem of k-set agreement, a natural generalization of consensus. While consensus concerns itself with the task in which all processes eventually have to decide on a common value that was originally some process- input value, k-set agreement allows up to k different decision values. Hence, for k = 1, k-set agreement is equivalent to consensus. Although there exist impossibility results for deterministic consensus in systems prone to failures, relying solely on combinatoric arguments that might be considered classical today, the corresponding impossibility results for k-set agreement require complex arguments from algebraic topology. Nevertheless, there has been recent research on finding "easy" or non-topological impossibility proofs for k-set agreement, which may also provide a new handle on solving some long-standing open problems like the weakest failure detector for k-set agreement in message-passing systems. The focus of this thesis lies on such non-topological impossibilities for k-set agreement. We present and discuss existing approaches and results and provide rigorous proofs for new results regarding various models and scenarios, including the important class of dynamic systems that may evolve over time.
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