Language of DFA and NFA. NFA with ε-transitions

The language recognized by a DFA (Deterministic Finite Automaton) and an NFA (Nondeterministic Finite Automaton) can be defined based on the strings that cause the automaton to reach an accept state. Let’s define the language of each:

Language of a DFA:
The language recognized by a DFA is the set of all strings that, when input to the DFA, cause it to reach an accept state. Formally, the language of a DFA

$M$ is denoted as

$L (M)$ , and it’s defined as follows:

L(M)={w∈Σ∗∣M accepts w}

Where:
- $Σ$ is the input alphabet.
- $Σ^{*}$ is the set of all possible strings over the alphabet
  
  $Σ$ .
- $M$ is the DFA.
- $M$ accepts a string
  
  $w$ if, when started in its initial state and given the input string
  
  $w$ , it ends up in an accept state.
Language of an NFA:
The language recognized by an NFA is defined similarly to that of a DFA. It’s the set of all strings that, when input to the NFA, cause it to reach an accept state. Formally, the language of an NFA

$N$ is denoted as

$L (N)$ , and it’s defined as follows:

L(N)={w∈Σ∗∣N accepts w}

Where:
- $Σ$ is the input alphabet.
- $Σ^{*}$ is the set of all possible strings over the alphabet $Σ$
- $N$ is the NFA.
- $N$ accepts a string
- $w is input to$
NFA with ε-Transitions:
When dealing with an NFA that has ε-transitions (transitions that can be made without consuming any input), we extend the definition of the language to account for the possibility of taking these transitions. The language recognized by an NFA with ε-transitions

$N_{ε}$ is defined as follows:

$there exists a path in from the initial state to an accept state by consuming symbols and ε-transitions}$

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

In this case, the NFA can transition between states without consuming any input symbols, following ε-transitions. This adds flexibility to the transitions, potentially allowing for the recognition of a broader set of languages compared to a regular NFA.