The condition-based approach for consensus solvability (that we have introduced in a previous paper, ACM STOC'01) consists in identifying sets of input vectors for which it is possible to design a protocol solving the consensus problem for n processes despite the occurrence of up to f process crashes. For each value of f these conditions actually defines a hierarchy. This paper continues our investigation of this approach. It has two main contributions. It first show that it is possible to define conditions from very simple weight functions. Interestingly any weight function define an acceptable condition, i.e., a condition that allows to solve the consensus problem. The second contribution is an efficient protocol whose wait-free part has a number of shared memory accesses upper bounded by O