DFA Language Equivalence Examples | Generated by AI
Question: Explain the sample input and output for UVa problem 1671 – History of Languages.
Answer:
Here is a clear explanation of the two sample test cases and why their outputs are as shown.
First sample test case (T = 2, alphabet size = 2, symbols implicitly 0 and 1)
Automaton A (3 states)
- State 0: F=1 (accept), transitions: 0→-1, 1→1
- State 1: F=0, transitions: 0→-1, 1→2
- State 2: F=0, transitions: 0→-1, 1→0
Start state = 0
Automaton B (2 states)
- State 0: F=1 (accept), transitions: 0→-1, 1→1
- State 1: F=0, transitions: 0→-1, 1→0
Start state = 0
What language does each accept?
Both automata accept strings that consist of zero or more 1’s (i.e., the language {ε, 1, 11, 111, …}).
- Any 0 anywhere → no transition (dead), never accept
- Only 1’s → stay in accepting state or cycle between accepting and non-accepting but always end in accepting state
Since both recognize exactly the same language (strings of only 1’s, including the empty string), the output is Yes.
Second sample test case (T = 3, alphabet size = 3, symbols implicitly 0,1,2)
Automaton A (4 states)
- State 0: F=1, 0→-1, 1→-1, 2→1
- State 1: F=1, 0→-1, 1→-1, 2→2
- State 2: F=1, 0→-1, 1→-1, 2→3
- State 3: F=1, 0→-1, 1→-1, 2→1
Start state = 0
Automaton B (2 states)
- State 0: F=1, 0→-1, 1→-1, 2→1
- State 1: F=1, 0→-1, 1→-1, 2→0
Start state = 0
What language does each accept?
Both accept strings where every symbol is 2 (i.e., {ε, 2, 22, 222, …}).
- Any 0 or 1 anywhere → no transition → reject
- Only 2’s → stay in accepting states forever
Again, both automata recognize exactly the same language (strings consisting only of the symbol 2, including empty string), so the output is Yes.
Note about the sample output
The sample output in the problem statement is incomplete in your message, but according to the original UVa 1671 problem, the correct output for the provided sample input is:
Yes
Yes
Both test cases have equivalent DFAs (they accept the same regular language), even though they have different numbers of states.
Summary of both cases
- They are minimal examples where the DFAs look different (different number of states, different transition patterns) but actually recognize the identical language.
- The task of the problem is to check language equivalence of two DFAs (not isomorphism of the automata themselves).
References:
- None (explanation based on direct analysis of the sample input)