Epsilon ($\varepsilon$) Transitions in DFA and NFA

Core concept: what are $\varepsilon$-transitions?

[epsilon transition]{def="A transition on the empty string $\varepsilon$, consuming no input symbol"} appear in nondeterministic automata where the transition function may move to another state without reading an input symbol. In contrast, a DFA consumes exactly one input symbol per transition, and its transition function is defined for actual input symbols (not for $\varepsilon$-moves). This distinction is why NFA commonly supports $\varepsilon$-moves via an [epsilon-closure]{def="Set of states reachable from a state using only $\varepsilon$ transitions"}.

A typical formal model is:

• [NFA with $\varepsilon$]{def="NFA whose transition relation allows moves labeled by $\varepsilon$"}: transition relation includes $\varepsilon$ as a label.
• DFA: transition function is defined for input symbols; $\varepsilon$-moves are not part of the standard DFA definition.

Therefore, for the multiple-choice statement:

“Epsilon ($\varepsilon$) transitions are allowed in …”

the correct choice is:

(iii) NFA only.

Visualizing the difference

Why DFA doesn’t allow $\varepsilon$ transitions (standard model)

In the standard formal definition of a DFA, the transition function is defined as $\delta: Q \times \Sigma \to Q$ (or equivalently, exactly one next state per state and input symbol). That model has no place for an empty-string label. So $\varepsilon$ moves would violate the premise that each step corresponds to reading one symbol from $\Sigma$.

Meanwhile, an [NFA with $\varepsilon$-moves]{def="NFA whose transition relation allows moves labeled by $\varepsilon$"} uses a transition relation allowing edges labeled by $\varepsilon$, enabling the automaton to “move for free” before (or between) consuming input symbols. This behavior is handled by [epsilon-closure]{def="Set of states reachable from a state using only $\varepsilon$ transitions"} during simulation and subset construction.