1. CS 3240: Languages and Computation
Introducing Regular Languages:
Deterministic Finite Automata
and Regular Expressions

2. Automata Finite Automata model many design and analysis tasks, e.g.
Lexical analyzer in a compiler
Digital cicuit design
Keywork searching in texts or on the web.
Software for verifying finite state systems,
such as communication protocols.
Your ATM, vendig machine, Etc.
McCulloch&Pitts, Rabin&Scott, Moore, Huffman

7. Finite State Machine and Finite Automata

8. Deterministic Finite Automata
A simplest model for computing
Deterministic: Machine is in a state. Upon receipt of a symbol will go to a unique state.
Finite: Have a finite number of states
Automata: (pl. of automaton) Self-operating machine
DFA: finite-state machine without ambiguity

9. DFA and Strings DFA can recognize strings
String is input
If DFA ends at accept state, string is recognized
A language is called a regular language if some finite automaton recognizes it
Let us look at a few example before giving formal mathematical definition

10. DFA Examples 1 1 start state 1 2 accept state transition1 1
start state 1 2
accept state
transition
Note: The alphabet for this
example is {0, 1}. Each state
has a transition for every symbol
in the alphabet
Accept all strings
that end in 1

11. DFA Examples 5 3 4 2 1 a b
Accept strings of 'a's and 'b's that begin and end with same symbol

12. DFA Examples 1 Start 2 1 2 1 2 2 1 Keep running count of1
Start 2 1 2 1 2 2
Keep running count of
total of symbols read
in mod 3. Accept on 0. 1

13. DFA Examples
Even
Odd 1
Strings with an odd
number of ones.

14. DFA Examples
'001' 1
'0'
'00'
0,1
Strings containing
the substring 001

15. Formal Definition of DFA
A DFA consists of:
Alphabet 
A set of states Q
A transition function δ : Q  Q
One start state q0
One or more accepting states F  Q
Language accepted by a DFA is the set of strings such that DFA ends at an accepting state
Each string is c1c2…cn with ci  
States are qi = δ(qi-1,ci ) for i=1…n
qn is an accepting state

16. Can DFA's be designed to accept any string?
No!

17. Examples
Design a DFA to recognize strings that start out with k zeros followed by k ones.
Impossible
Design a DFA to recognize strings with an equal number of ones and zeros.
Design a DFA to recognize strings with an equal number of strings "01" and "10".
Possible!

18. Actually the third one is regular!1 1 1 1 1 1
DFA to recognize strings with an equal number
of strings "01" and "10" 1

19. DFA More examples

20. DFA More examples A,B are the input into which the
marble is dropped. The x-levers cause
fall either to left or right, but lever reverses
upon a marble passing
Accept if marble exits through D

21. Non-Deterministic Finite Automata

22. NFA and -NFA Nondeterministic Finite Automata 
Same input may produce multiple paths
Allows transition with an empty string or transition from one state to different states given a character q1 q2 1 q3
nondeterministic transition q1 q2 
empty string transition