Given states as Q and the alphabet of symbols as ∑, a transition function δ for a DFA is a mapping function given by which of the following?

Question

Given states as Q  and the alphabet of symbols as ∑, a  transition function δ for a DFA is a mapping function given by which of the following?

BlackTom AI · Accepted Answer

A DFA's transition function δ maps a current state and an input symbol to a single next state.

1. Relevant knowledge points and concepts
- Deterministic Finite Automaton (DFA) is a 5-tuple (Q, ∑, δ, q0, F) where Q is the finite set of states, ∑ is the input alphabet, q0 is the start state, and F is the set of accepting states.
- The transition function δ must take both a current state and an input symbol and produce exactly one next state (determinism).
- Therefore δ's domain must pair a state with a symbol, and its codomain must be a state from Q.

2. Complete reasoning process step by step
- To determine the correct signature for δ, ask: what information is needed to decide the next state? Answer: the current state (from Q) and the current input symbol (from ∑).
- Combine these two pieces of information as an ordered pair (state, symbol) which is represented by the Cartesian product Q × ∑.
- The output of δ is a single state, so the codomain is Q.
- Thus δ is a function from Q × ∑ to Q, written δ : Q × ∑ → Q.

3. Intermediate steps / examples
- Example: Q = {q0, q1}, ∑ = {0,1}. A possible δ could be δ(q0,0) = q0, δ(q0,1) = q1, δ(q1,0) = q1, δ(q1,1) = q0. Each mapping takes a pair (state, symbol) and returns a state in Q.

4. Why the correct answer is correct
- Mapping from Q × ∑ to Q captures exactly the needed inputs (state and symbol) and yields exactly one next state, satisfying determinism and the DFA definition. This is precisely δ = Q x ∑ ⟶ Q.

5. Why the other options are incorrect
- ∑ ⟶ Q: omits the current state, so behavior would depend only on the symbol, not the state—insufficient for a DFA.
- Q × Q ⟶ Q: inputs two states, which is not the required information; a transition depends on a state and a symbol, not two states.
- Q ⟶ Q: omits the input symbol, so transitions could not depend on the symbol read.

6. Conclusion
Therefore the correct transition function signature is δ = Q x ∑ ⟶ Q, which matches the provided answer.

Given states as Q and the alphabet of symbols as ∑, a transition function δ for a DFA is a mapping function given by which of the following? 单项选择题

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