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’ The input is not accepted
[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 }
The transitions of an NFA are very similar to those of a DFA, except that each [state, input] pair has a set of result states, not just one.

9. An NFA State Diagram
This NFA determines whether an input string from the language {a, b} contains the string ‘abba.’

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