Consider a finite state machine (FSM) with one input $X$ and one output $f$, represented by the given state transition table. The minimum number of states required to realize this FSM is _______. (Answer in integer)

To find the minimum number of states required to realize the FSM, we apply the state minimization (partitioning) algorithm.

### Step 1: Initial Partition ($P_0$) Partition the states based on their output $(f_{X=0}, f_{X=1})$:

• Output $(0, 0)$: $\{A, B, C, E\}$

• Output $(1, 0)$: $\{D, H\}$

• Output $(1, 1)$: $\{F\}$

• Output $(0, 1)$: $\{G\}$

### Step 2: Refining partition to $P_1$

1. Check block $\{A, B, C, E\}$:

   • For input $X = 0$:

     • Next states for $A$ and $C$ are $F$ (in group $\{F\}$).

     • Next states for $B$ and $E$ are $D$ (in group $\{D, H\}$).

   • For input $X = 1$:

     • Next state for $A$ is $B$, and for $C$ is $E$ (both in $\{B, E\}$).

     • Next state for $B$ is $C$, and for $E$ is $C$ (both in $\{A, C\}$).

   • Therefore, $\{A, B, C, E\}$ splits into two equivalence classes: $\{A, C\}$ and $\{B, E\}$.

2. Check block $\{D, H\}$:

   • For $X = 0$: $D \to G$ and $H \to G$ (both in group $\{G\}$).

   • For $X = 1$: $D \to A$ and $H \to A$ (both in group $\{A, C\}$).

   • Thus, $D$ and $H$ remain in the same equivalence class.

### Step 3: Check stability ($P_2$) Checking each block of $P_1$:

• For $\{A, C\}$:

  • $X=0 \implies F \in \{F\}$

  • $X=1 \implies B, E \in \{B, E\}$

  (Equivalent)

• For $\{B, E\}$:

  • $X=0 \implies D \in \{D, H\}$

  • $X=1 \implies C \in \{A, C\}$

  (Equivalent)

• For $\{D, H\}$:

  • $X=0 \implies G \in \{G\}$

  • $X=1 \implies A \in \{A, C\}$

  (Equivalent)

Since no further splits occur, $P_2 = P_1$.

The equivalent states are:

• $[A] = \{A, C\}$

• $[B] = \{B, E\}$

• $[D] = \{D, H\}$

• $[F] = \{F\}$

• $[G] = \{G\}$

The minimum number of states is 5.