For an NFA with k states, there is a DFA with at most __ states.1 pointk^2k^3k2^k

To determine the upper bound on the number of states in a Deterministic Finite Automaton (DFA) that can be constructed from a Non-deterministic Finite Automaton (NFA) with $k$ states, we can use the following reasoning based on subset construction.

1. ### Break Down the Problem

To convert an NFA into a DFA, we need to consider the subsets of states in the NFA. Each state in the DFA represents a possible combination of states that the NFA could be in at any point.

2. ### Relevant Concepts

The maximum number of subsets that can be formed with $k$ NFA states is given by $2^k$, since each state can either be included or not in a subset.

3. ### Analysis and Detail

1. Each state in the NFA has a potential transition to various states based on the input symbols.
2. In the worst case, every possible subset of the NFA states could correspond to a unique state in the DFA.
3. Therefore, the number of states in the resulting DFA can be as many as $2^k$.

4. ### Verify and Summarize

Having deduced that the states of the DFA can correspond to every possible subset of states from the NFA, we conclude that the maximum number of states is $2^k$.

Final Answer

The DFA constructed from an NFA with $k$ states can have at most $2^k$ states.