如何绘制可识别语言L(n)={a^n b^n : n=0,1,2}的DFA状态图？

Alright, let's walk through constructing this DFA step by step. The target language only includes three types of strings: the empty string, ab, and aabb—so our automaton needs to strictly accept these and reject everything else (like ba, aaa, abb, etc.).

Step 1: Define State Meanings #

First, we need states that track our progress through valid strings. Here's what each state represents:

• S0: Initial state (no characters read yet). This is an accepting state because the empty string is part of the language.
• S1: We've read exactly 1 a and no bs.
• S2: We've read exactly 2 as and no bs.
• S3: We've read 1 a followed by 1 b (this matches the string ab, so it's an accepting state).
• S4: We've read 2 as followed by 1 b.
• S5: We've read 2 as followed by 2 bs (this matches aabb, so it's an accepting state).
• D: Dead state (error state). Any string that enters this state is invalid, and we'll never leave this state no matter what input comes next.

Step 2: Define State Transitions #

Now let's map out what happens for each state when we read an a or b:

• From S0:
  • On input a: Move to S1 (we've started a valid string with one a)
  • On input b: Move to D (a string starting with b can never be in our language)
• From S1:
  • On input a: Move to S2 (now we have two as, which is allowed for n=2)
  • On input b: Move to S3 (we've completed the valid string ab)
• From S2:
  • On input a: Move to D (three as exceed our max n=2)
  • On input b: Move to S4 (we've got two as and one b, halfway to aabb)
• From S3:
  • On input a or b: Move to D (adding any character to ab makes it invalid—we can't have more as after bs, and an extra b would make abb which isn't in the language)
• From S4:
  • On input a: Move to D (we can't have an a after a b in valid strings)
  • On input b: Move to S5 (we've completed the valid string aabb)
• From S5:
  • On input a or b: Move to D (adding any character to aabb makes it invalid)
• From D:
  • On input a or b: Stay in D (once we hit an invalid string, we stay in the error state forever)

Step 3: Draw the State Transition Diagram #

To visualize this:

1. Draw a circle for each state. Use a double circle for accepting states (S0, S3, S5).
2. Draw an arrow pointing to S0 to mark it as the initial state.
3. Add arrows between states labeled with the input character (a or b) based on the transitions above.
4. Include the dead state D to make invalid paths explicit (you can omit it if you want, but it makes the automaton clearer).

Step 4: Test the DFA with Example Strings #

Let's verify a few cases to make sure it works:

• Empty string: Start and stay in S0 → Accepted
• ab: S0 → S1 → S3 → Accepted
• aabb: S0 → S1 → S2 → S4 → S5 → Accepted
• ba: S0 → D → Rejected
• aaa: S0 → S1 → S2 → D → Rejected
• abb: S0 → S1 → S3 → D → Rejected

内容的提问来源于stack exchange，提问作者JIsam