Language of NFA with ε-transitions, Equivalence of NFA and DFA

The language recognized by an NFA (Nondeterministic Finite Automaton) with ε-transitions is defined as the set of all strings that, when input to the NFA, cause it to reach an accept state, considering the possibility of taking ε-transitions. Let’s denote this language as

$L (N_{ε})$ , where

$N_{ε}$ represents the NFA with ε-transitions. The formal definition of

$L (N_{ε})$ is as follows:

$L (N_{ε}) = {w \in Σ^{*} ∣ there exists a path in N_{ε} from the initial state to an accept state by consuming symbols and ε-transitions}$

In other words,

$L (N_{ε})$ consists of all strings over the input alphabet

$Σ$ for which there exists at least one path in the NFA

$N_{ε}$ from the initial state to an accept state, where the path may involve consuming input symbols and/or taking ε-transitions.

Now, regarding the equivalence of NFAs and DFAs:

Equivalence of NFA and DFA:

An NFA and a DFA are considered equivalent if they recognize the same language, meaning they accept exactly the same set of strings.

NFA to DFA Conversion:
It is known that any language recognized by an NFA can also be recognized by a DFA. This is achieved through the process of converting the NFA to an equivalent DFA. The conversion algorithm typically involves constructing a DFA where each state corresponds to a set of states of the original NFA, and transitions are determined based on the transitions of the NFA. This process ensures that the resulting DFA recognizes the same language as the original NFA.
DFA to NFA Conversion:
Similarly, any language recognized by a DFA can be recognized by an NFA. This can be achieved by constructing an NFA where each state of the DFA corresponds to a single state of the NFA, and transitions are determined based on the transitions of the DFA. The resulting NFA will recognize the same language as the original DFA.

NFAs and DFAs are equivalent in terms of their expressive power, as they can recognize the same set of languages. However, their operational mechanisms differ, with NFAs offering greater flexibility due to nondeterminism and the possibility of ε-transitions.