Nondeterministic Finite-state Automata (NFA)

Nondeterministic Finite-state Automata

Description:

• May have several possible next states for each pair of input value and state
• A Nondeterministic finite-state automata $M=(S,I,f,s_0,F)$:
  • consists of
    • $S$: a finite set of states
    • $I$: a finite input alphabet
    • $f$: a transition function
      • assigns a next state set of states to each pair of state and input
      • such that $f:S\times I\to \mathcal{P}(S)$
      • $\mathcal{P}(S)$ is the power set of S
    • an starting state, $s_0$
    • a set of possible accepting final states, $F$
      • also a subset of $S$

NFA’s recognized language:

• Let a string $x=x_1{x}_2\dots x_k$ input to an NFA.
• The first input symbol $x_1$ takes the starting state $s_0$ to a set $S_1$ of states
• The next input symbol $x_2$ takes each of the states in $S_1$ to a set of new states.
  • Let $S_2$ be the union of these sets.
• Repeat, at each stage, all states obtained using a state obtained at the previous stage and the current input symbol
• The string $x$ is accepted if there is at least 1 final state in the set of all states using $x$.
• The language recognized by a NFA is the set of all strings recognized by this NFA.
• theorem
  • If the language $L$ is recognized by a NFA $M_0$, then $L$ is also recognized by a DFA $M_1$
    • where $M_1$ (deterministic) is converted from $M_0$ (Non-deterministic)
    • $M_1$‘s states are created from combinations of possible states reached by $M_0$‘s states
      • with OR operation
      • combination states where each state can have multiple possible states, the new state of $M_1$ is all of those possible states.
• 1. Q&A - find language recognized by NFA; question & answer
• 2. Q&A - find a DFA recognized same language as the NFA; question, answer

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