1. Topics Automata Theory Grammars and Languages Complexities
Formal Languages and Automata Theory

2. Why Automata Theory?
To study abstract computing devices which are closely related to today’s computers. A simple example of finite state machine: 1 on
start
off 1
There are many different kinds of machines.
Formal Languages and Automata Theory

3. Another Example 1 off on start off 1 1 When will this be on?
off on
start
off 1 1
When will this be on?
Try 100, 1001, 1000, 111, 00, …
Formal Languages and Automata Theory

4. Grammar and Languages
Grammars and languages are closely related to automata theory and are the basis of many important software components like:
Compilers and interpreters
Text editors and processors
Search engines
System verification components
Formal Languages and Automata Theory

5. Complexities
Study the limits of computations. What kinds of problems can be solved with a computer? What kinds of problems can be solved efficiently?
Formal Languages and Automata Theory

6. Preliminaries Alphabets Strings Languages Problems
Formal Languages and Automata Theory

7. Alphabets An alphabet is a finite set of symbols.
Usually, use  to represent an alphabet.
Examples:
 = {0,1}, the set of binary digits.
 = {a, b, … , z}, the set of all lower-case letters.
 = {(, )}, the set of open and close parentheses.
Formal Languages and Automata Theory

8. Strings A string is a finite sequence of symbols from an alphabet.
Examples:
0011 and 11 are strings from  = {0,1}
abc and bbb are strings from  = {a, b, … , z}
(()(())) and )(() are strings from  = {(, )}
Formal Languages and Automata Theory

9. Strings Empty string:  Length of string: |0010| = 4, |aa| = 2, ||=0
Prefix of string: aaabc, aaabc, aaabc
Proper prefix of string: aaabc, aaabc
Suffix of string: aaabc, aaabc, aaabc
Proper suffix of string: aaabc, aaabc
Substring of string: aaabc, aaabc, aaabc
Formal Languages and Automata Theory

10. Strings What is 1? Is 1 different from ? How?
Concatenation: =abd, =ce, =abdce
Exponentiation: =abd, 3=abdabdabd, 0=
Reversal: =abd, R = dba
k = set of all k-length strings formed by symbols in 
e.g., ={a,b}, 2={ab, ba, aa, bb}, 0={}
What is 1? Is 1 different from ? How?
Formal Languages and Automata Theory

11. Strings Kleene Closure * = 012… = k0 k
e.g., ={a, b}, * = {, a, b, ab, aa, ba, bb, aaa, aab, abb, … } is the set of all strings formed by a’s and b’s.
+ = 123… = k>0 k
i.e., * without the empty string.
Formal Languages and Automata Theory

12. Languages A language is a set of strings over an alphabet. Examples:
={(, )}, L1={(), )(, (())} is a language over .
={a, b, c, … , z}, the set L of all legal English words is a language over .
The set {} is a language over any alphabet.
What is the difference between  and {}?
Formal Languages and Automata Theory

13. Languages Other Examples:
={0, 1}, L={0n1n | n1} is a language over  consisting of the strings {01, 0011, , … }
={0, 1}, L = {0i1j | ji0} is a language over  consisting of the strings with some 0’s (possibly none) followed by at least as many 1’s.
Formal Languages and Automata Theory

14. Problems
In automata theory, a problem is to decide whether a given string is a member of some particular language.
This formulation is general enough to capture the difficulty levels of all problems.
Formal Languages and Automata Theory

15. Finite Automata ( or Finite State Machines)
This is the simplest kind of machine.
We will study 3 types of Finite Automata:
Deterministic Finite Automata (DFA)
Non-deterministic Finite Automata (NFA)
Finite Automata with -transitions (-NFA)
Formal Languages and Automata Theory

16. Deterministic Finite Automata (DFA)
We have seen a simple example before: 1 on
start
off 1
There are some states and transitions (edges)
between the states. The edge labels tell when
we can move from one state to another.
Formal Languages and Automata Theory

17. Definition of DFA A DFA is a 5-tuple (Q, , , q0, F) where
Q is a finite set of states
 is a finite input alphabet
 is the transition function mapping Q   to Q
q0 in Q is the initial state (only one)
F  Q is a set of final states (zero or more)
Formal Languages and Automata Theory

18. Definition of DFA Q is the set of states: {on, off}
For example: 1 on
start
off 1
Q is the set of states: {on, off}
 is the set of input symbols: {1}
 is the transitions: off  1  on; on  1  off
q0 is the initial state: off
F is the set of final states (double circle): {on}
Formal Languages and Automata Theory

19. Definition of DFA What are Q, , , q0 and F in this DFA?
Another Example: 1 q2 q1 q0
start 1 1
What are Q, , , q0 and F in this DFA?
Formal Languages and Automata Theory

20. Transition Table
For the previous example, the DFA is (Q,,,q0,F) where Q = {q0,q1,q2},  = {0,1}, F = {q2} and  is such that
States
Inputs 1 q0 q1
*q2 q2
Note that there is one transition only for each input
symbol from each state.
Formal Languages and Automata Theory

21. Language of a DFA
Given a DFA M, the language accepted (or recognized) by M is the set of all strings that, starting from the initial state, will reach one of the final states after the whole string is read.
For example, the language accepted by the previous example is the string that ends with 00
Formal Languages and Automata Theory

22. DFA Example
Consider the DFA M=(Q,,,q0,F) where Q = {q0,q1,q2,q3},  = {0,1}, F = {q0} and  is:
Inputs 1
Start q0 q1
States 1 1 q0 q2 q1 q1 q3 q0 OR 1 q2 q0 q3 q2 q3 1 q3 q1 q2
We can use a transition table or a transition diagram to specify
the transitions. What input can take you to the final state in M?
Formal Languages and Automata Theory