Analyzing Automaton State Complexity for Regular Languages

This problem delves into the relationship between the number of states in a Non-deterministic Finite Automaton (NFA) and the states required by Deterministic Finite Automata (DFAs) that accept the same regular language. We are given a regular language $L$ accepted by an NFA with $n$ states and need to identify which statement(s) about its acceptance are FALSE.

Key Concepts: Regular Languages, NFAs, and DFAs

• Regular Languages: These are classes of languages that can be described by regular expressions and accepted by finite automata (both NFAs and DFAs).
• NFA (Non-deterministic Finite Automaton): An automaton that can have multiple transitions for the same input symbol from a state, or transitions on the empty string ($\epsilon$). It accepts a string if there exists at least one path from the start state to an accepting state.
• DFA (Deterministic Finite Automaton): An automaton where each state has exactly one transition for each input symbol, and no transitions on $\epsilon$. It accepts a string if there is a unique path from the start state to an accepting state upon reading the string.
• Equivalence: A fundamental result states that NFAs and DFAs accept the exact same class of languages (the regular languages). This means for any NFA, there exists an equivalent DFA, and vice versa.

NFA to DFA Conversion and State Count

The standard method to convert an NFA with $n$ states into an equivalent DFA is the subset construction algorithm. In this method, each state in the resulting DFA corresponds to a subset of the states in the original NFA. Since an NFA with $n$ states has $2^n$ possible subsets of states, the equivalent DFA constructed using this method can have at most $2^n$ states.

It's also important to note that both NFAs and DFAs can often be minimized. A minimal DFA for a given language is the DFA with the smallest possible number of states. Minimization might result in a DFA with significantly fewer states than $2^n$, and potentially even fewer states than the original NFA ($n$). Similarly, an NFA might not be minimal, and a simpler NFA (possibly with fewer states) could exist.

Evaluating Each Statement

Statement 1: $L$ may have an accepting NFA with $< n$ states.

This statement is TRUE. An NFA provided for a language might not be minimal. It's possible to construct an equivalent NFA with fewer than $n$ states by removing redundant states or simplifying transitions, especially if the initial NFA was constructed directly from a regular expression without minimization.

Statement 2: $L$ may have an accepting DFA with $< n$ states.

This statement is TRUE. While the subset construction guarantees a DFA with at most $2^n$ states, the minimal DFA for the language $L$ might possess fewer states than the original NFA's state count ($n$). This can occur if the language $L$ is simple enough, or if the original NFA was not minimal.

Statement 3: There exists a DFA with $\leq 2^n$ states that accepts $L$.

This statement is TRUE. This is a direct consequence of the subset construction algorithm. For any NFA with $n$ states, we can always construct an equivalent DFA, and the number of states in this DFA is bounded by $2^n$. Therefore, at least one such DFA exists.

Statement 4: Every DFA that accepts $L$ has $> 2^n$ states.

This statement is FALSE. Statement 3 guarantees that there exists a DFA accepting $L$ with at most $2^n$ states. If such a DFA exists, then the claim that *every* DFA accepting $L$ must have *more than* $2^n$ states cannot be true. It is possible for the minimal DFA to have fewer states than $2^n$, or exactly $2^n$ states, but not necessarily more.

Conclusion

Based on the analysis, the only statement that is definitively false is that every DFA accepting language $L$ must have more than $2^n$ states. The existence of a DFA with $\leq 2^n$ states contradicts this universal claim.