Finite Automata, or Finite State Machine

• See section 6.2
• A finite automata can be used to determine if words are in a language.
• A Finite Automata consists of
  • a finite set of states. Q
  • a start state, s, s∈ Q
  • a set of accepting states A, A⊆ Q
  • an input alphabet Σ
  • a function δ from Q×Σ to Q
• The automata takes a letter and transisitions from one state to another based on the function δ
• It starts in the start state.
• When it is in an accepting state, that is one of the words in the language.
• We can picture an automata
  • Each state is a circle.
  • The start state generally has an arrow pointing to it.
  • The accepting states are generally marked with a double circle.
  • The transisiton function are arrows from one state to another
• An example from the book
  • Q = { 1,2,3,4,}
  • s = 1
  • A = {4}
  • Σ = {a,b}
  • δ =

    Input\State	1	2	3	4
    a	2	1	4	3
    b	3	4	1	2

  • 
  • If we trace the action of this machine via the use of an ordered pair.
    • (state, remaining string)
  • Consider the trace of ababba
    • (1, ababba)
    • (2, babba)
    • (4, abba)
    • (3, bba)
    • (1, ba)
    • (3, a)
    • (4, ∅)
  • So ababba is in the language accepted by this FA.
  • But so is ab, and ba too!
  • Consdier the string abba
    • (1, abba)
    • (2, bba)
    • (4, ba)
    • (2, a)
    • (1, ∅)
  • In this case, the string was rejected.
  • Thus abba is not in the language accepted by this FA.
• The transition function can be implemented as a table lookup.
  • newState = TransitionTable[input][currentState]