求接受模6余5的三进制字符串的DFA最小状态数

Alright, let's break down how to find the minimal number of states for this DFA step by step:

Step 1: Understand the Core Problem #

We need a DFA that accepts base-3 strings (made up of characters 0, 1, 2) whose numeric value meets the condition N ≡ 5 mod 6.

Step 2: Model Initial States by Remainder #

First, we can model states based on the remainder of the current string's value when divided by 6. There are 6 possible remainders: 0, 1, 2, 3, 4, 5 (let's name these states q0 to q5, where q5 is our only accepting state).

Next, we calculate state transitions: for a current state qi (representing remainder i), inputting a digit d (0/1/2) gives a new remainder of (i*3 + d) mod 6. Here's the full transition table:

• q0: 0→q0, 1→q1, 2→q2
• q1: 0→q3, 1→q4, 2→q5
• q2: 0→q0, 1→q1, 2→q2
• q3: 0→q3, 1→q4, 2→q5
• q4: 0→q0, 1→q1, 2→q2
• q5: 0→q3, 1→q4, 2→q5

Step 3: Minimize States Using Equivalence Classes #

We use the Myhill-Nerode theorem to merge equivalent states (states that behave identically for all possible future inputs):

1. Initial grouping: Split states into accepting and non-accepting sets: {q0,q1,q2,q3,q4}, {q5}
2. Refine the non-accepting group:
   • q0, q2, q4 have identical transition behavior: input 0 stays in the non-accepting set, input 1 moves to q1/q3, input 2 stays in the non-accepting set.
   • q1, q3 have identical transition behavior: input 0 stays in their subgroup, input 1 moves to q0/q2/q4, input 2 moves to the accepting state q5.
     Now we have three distinct groups: {q0,q2,q4}, {q1,q3}, {q5}
3. Verify if further merging is possible:
   • The accepting group {q5} can't merge with non-accepting groups (since it's the only state that leads to acceptance).
   • The two non-accepting groups have distinct transitions (e.g., input 2 takes q0/q2/q4 to themselves, but takes q1/q3 to the accepting state), so they can't be merged.

Step 4: Final Minimal State Count #

The minimal DFA has 3 states:

• State A (merges q0,q2,q4): Non-accepting, represents remainders 0, 2, 4
• State B (merges q1,q3): Non-accepting, represents remainders 1, 3
• State C (merges q5): Accepting, represents remainder 5

Here's the simplified transition table for the minimal DFA:

• A: 0→A, 1→B, 2→A
• B: 0→B, 1→A, 2→C
• C: 0→B, 1→A, 2→C

内容的提问来源于stack exchange，提问作者Gowtham Unique