How many state bits are necessary to encode the current state of this FSM

How many state bits are necessary to encode the current state of this FSM?

The correct answer and explanation is:

To determine how many state bits are necessary to encode the current state of a finite state machine (FSM), we need to know how many distinct states the FSM has.

General Rule:

To encode NN distinct states, you need at least ⌈log⁡2(N)⌉\lceil \log_2(N) \rceil bits.

This formula tells us the minimum number of binary bits required to represent all states uniquely.

Step-by-Step Explanation:

1. Count the Number of States:
   Suppose the FSM has NN states (e.g., 5 states: S0, S1, S2, S3, S4).
2. Use the formula: Number of state bits=⌈log⁡2(N)⌉\text{Number of state bits} = \lceil \log_2(N) \rceil
   • Example: If N=5N = 5 log⁡2(5)≈2.32⇒⌈2.32⌉=3\log_2(5) \approx 2.32 \Rightarrow \lceil 2.32 \rceil = 3 So, 3 bits are needed to encode the current state.
3. Why do we use log base 2?
   Because each state bit has two possible values (0 or 1), and the number of possible binary combinations of nn bits is 2n2^n. We need the smallest number nn such that: 2n≥N2^n \geq N
4. State Encoding Methods (Optional Context):
   • Binary encoding: Uses the minimum number of bits (efficient in space).
   • One-hot encoding: Uses one bit per state (simpler hardware logic but more bits).

Final Answer:

To give you the exact number of state bits, I need to know how many states your FSM has.

But generally:
If an FSM has NN states, the number of state bits required is ⌈log⁡2(N)⌉\boxed{\lceil \log_2(N) \rceil}

Example Table: