In Sequential Circuit, Are Two States Equivalent If Their Next States Are Each Other?

Question

When eliminating redundant states in the state table of a sequential circuit, we need to find equivalent states in the same circuit. In Fundamentals of Logic Design, two states are equivalent if and only if the states have the same next states and the same present outputs. The problem is, what if the next states of two states are each other when nothing else is different?

As an example, consider the following state table (X is the circuit input):

The only difference between states A and B is that their next states are each other. Because you need to show two states have the same next states to prove they are equivalent, it follows that you need to show A=B in order to prove A=B, i.e., A=B iff A=B. Given this requirement, I am not sure how to go about proving A=B. The textbook dismisses this as self-implied. I can kind of see that they are equivalent, but I don't know how to get past the conundrum that you need to prove a statement S first before you can show S is true (sounds like circular reasoning).

Edit: I am not asking about whether these two states are equivalent, I am asking about how you would rigorously prove these two states in this specific example are equivalent.

Answer 1

In Sequential Circuit, Are Two States Equivalent If Their Next States Are Each Other?

In general, no. If two states are equivalent, then starting from either state, if the circuit is fed with identical sequences of inputs, then they will have identical output sequences. However, this is not guaranteed merely by the two states having each other as next states.

In your simple example, it happens to be the case that the two states are equivalent. But simply changing the outputs associated with one of the states should be a convincing demonstration that in that case, the states aren't equivalent.