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

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.

• Yes, the outputs are the same, but that is not the point of my question. My point is that given the fact that the only difference between states A and B is that their next states are each other, how do you prove that they are actually equivalent? I used the term "statement" because this also relates to mathematical logic. As for cardinality, I only show two states because only these two are relevant. Commented May 23, 2023 at 15:36
• That is good advice, thank you and apologies for the ambiguity. Commented May 24, 2023 at 2:59
• Done, thanks for the reminder. Commented May 24, 2023 at 4:43