  This report contains two related sets of results with different assumptions
on synchrony. The first part is about iterative algorithms in synchronous
systems. Following our previous work on synchronous iterative approximate
Byzantine consensus (IABC) algorithms, we provide a more intuitive tight
necessary and sufficient condition for the existence of such algorithms in
synchronous networks1. We believe this condition and the previous results also
hold in partially asynchronous algorithmic model.
  In the second part of the report, we explore the problem in asynchronous
networks. While the traditional Byzantine consensus is not solvable in
asynchronous systems, approximate Byzantine consensus can be solved using
iterative algorithms.
