An Introduction to Finite Automata

1
An Introduction to Finite Automata

• By Andrew Milne
• Mentor Professor Cindy Fry
• February 1, 2008

2
A Finite Automaton

• Has some number of states
• Has a start state and at least one end state
• Accepts input that advances it through its states
• Can be Deterministic (DFA) or Non-Deterministic
  (NFA)

3
Deterministic Finite Automata

• A DFA can be written as a quintuple M (Q, E, d,
  q0, F)
• Q A finite set of states
• E A finite set known as the alphabet
• q0 The start state (an element of Q)
• F The final / accepting states (a subset of Q)

4
d

• d is a total function from Q x E to Q called the
  transition function.
• Total function There exists a function f such
  that for each element of E, e, as input to f,
  there is an element of Q, q, such that q, e is
  an element of f.
• From any state there is a path for each possible
  input.

5
Representing Finite Automata

• Finite Automata can be represented visually by
  state diagrams
• M
• Q q0, q1
• E 0, 1
• F q2

d(q0, 0) q1 d(q0, 1) q0 d(q1, 0) q0 d(q1,
1) q1
6
DFA State Diagram

• Consider as input 1001
• q0, 1001 - q0, 001
• q0, 001 - q1, 01
• q1, 01 - q1, 1
• q1, 1 - q0,
• The input is not accepted

7
Nondeterministic Finite Automata

• An NFA can have several possible paths from a
  single node.
• An NFA processes all possible paths to attempt to
  find an accepting state.
• A string is accepted by an NFA if there is at
  least one possible path that
• Processes the entire string, and
• Ends in an accepting state

8
The d of the NFA

• d(q0, ew) q0, q1, qn

9
An NFA State Diagram
10
Nondeterministic Finite Automata

• Two NFA can be joined by using the lambda-arc.
• Processing follows a lambda-arc no matter what
  its input is (even null input)

11
An NFA With a Lambda-Arc
12

• Reference
• Sudkamp, Thomas A
• Languages and Machines, 3rd Edition