How to interpret the concept of adjacent initial states in the FLP impossibility paper?

Question

Link: https://dl.acm.org/doi/pdf/10.1145/3149.214121

In the proof for Lemma 2, the authors define two initial states to be adjacent if the only difference between them is the value in the input register $x_p$ for a single process $p$. Then, they claim that any two initial configurations can be joined by a chain of initial configurations, each adjacent to the next.

However, if two initial states differ in the internal storage of a single process $p$, then there is no way to make them adjacent. The definition of initial state seems to include the internal state as suggested from the following excerpt.

The values in the input and output registers, together with the program counter and internal storage, comprise the internal state. Initial states prescribe fixed starting values for all but the input register; in particular, the output register starts with value b.

I know this doesn't change the point of the Lemma, as the decision value cannot depend solely on the internal state of some process in an initial configuration, but I was wondering what the authors intended with the precise definition of adjacent initial configurations.

Answer 1

I think the authors assume that the processes start with identical (or empty) internal storage. This seems a reasonable assumption: if the internal storage is arbitrary, a process that reads it could be correct by coincidence, or if it were random, then the process can succeed with a certain probability. And if the storage is a function of the input value, then the process might as well compute that function later.

Answer 2

My reading of

Initial states prescribe fixed starting values for all but the input register; in particular, the output register starts with value $b$.

on page 376 is that, for any consensus protocol $P$ considered in the paper, the program counter and the internal storage of every process $p$ are uniquely determined. Otherwise the proof of Lemma 2 does indeed not work.