Department of Computer science – CS 386 Note for Lecture on 04/12/06
Instructor:Prof. Dr J.Miller
Last date for submit the Homework2 is 04/17/06
Language:
Language a set of words, formed by symbols in a given alphabet. May or may not be infinite
Grammar:
Grammar is the study of rules governing the use of language, the set of rules governing a
particular language is also called the grammar of the language.
For example grammar must be context free for computer language in order for it to be
efficient to write in
Formal grammar:
Formal grammar is an abstract structure that describes a formal language precisely, i.e., a set
of rules that mathematically delineates a (usually infinite) set of finite-length strings over a
(usually finite) alphabet. Formal grammars are so named by analogy to grammar in human
languages.
Find the grammar generate this language?
L = {w n a (w) = n b (w)}
Where n a (w) is the # of “a” S in W
G = ({S}, {a.b},S,P)
P: S
aSb
S
bSa
S
SS
S
SS
S
SS
aSbS
aSbbSa
When we can say Equivalent? We have same language
EXAMPEL
What is the language of this grammar?
P: S
aAb 
S
aAb 
the meaning of this
S
aAb
S

Amir Chartab Shargh
1
= OR
abba
Department of Computer science – CS 386 Note for Lecture on 04/12/06
Instructor:Prof. Dr J.Miller
G = ({S,A}, {a,b},S,P)
L = { a n b n : n>= 0}
Automate theory:
Symbol
Symbol an arbitrary datum which has some meaning to or effect on the machine.
Word
A finite string formed by the concatenation of a number of symbols.
Alphabet
A finite set of symbols.
Language
Language a set of words, formed by symbols in a given alphabet. May or may not be infinite.
Automatons are characterized by the following features:
1. Input: read-only
2. Output
3. Temporary storage: CPU
4. Control unit
Automate divided to two parts:
The terminates Automate
None terminates Automate
1
Non-deterministic Automata
a
2
a
3
Now this two kind of automate separately divided to two different parts:
1. Acceptor: An automaton whose output is yes or no is an acceptor.(1,0)
2. Transducer: An automaton whose output is a string of symbols is a transducer.
Amir Chartab Shargh
2
Department of Computer science – CS 386 Note for Lecture on 04/12/06
Instructor:Prof. Dr J.Miller
Nondeterministic finite automaton (NFA)
NFA is a finite state machine where for each pair of state and input symbol there may be
several possible next states.
Deterministic Finite Automata (DFA)
Each state of an automaton of this kind has a transition for every symbol in the alphabet.
M = (Q, ∑ , , qo , F)
Q = finite set of internal states
∑ = alphabet
: Q * ∑ = transition function
Qo  Q = initial stage
F  Q = final stages
0
1
Qo
0
Q1
1
1
Q2
0
Two circles means final stage
M = ({qo , q1 , q2 }, {0,1}, qo q1 )
 qoqo
 q1q2
 qoq1
 q2q2
 q1qo
 q2q1
We have 6 transition here
Now show this 001101
 qoqo
 q1q2
 qoqo
 q2q2
 q0qo1
 q2q1
Amir Chartab Shargh
3
Department of Computer science – CS 386 Note for Lecture on 04/12/06
Instructor:Prof. Dr J.Miller
1 1* (0 1) *
1* = {1, 11, 111 …}
1 = {}
 q0
, aq1
 q1, bq2
 *  q0, abq2
L (M) = {W   * :  * (q0, W)  F}
L (M) = {W   * :  * (q0, W)  F}
b
Qo
a,b
Q1
Q2
a,b
a
Trap state
DFA: L = { a n b : n  0}
a
b
q0
q0
q1
q0
q2
q2
q0
q2
q2
Table show DFA
Amir Chartab Shargh
4
Department of Computer science – CS 386 Note for Lecture on 04/12/06
Instructor:Prof. Dr J.Miller
Example:
a
Qo
∑ = {a, b}
Starting with “ab”
b
Q1
Q2
a
b
a,b
Q3
a,b // a trap state
Example:
∑ = {a, b}
DFA accept all string except those containing the substring 001
1
Q0
0
Q1
0
Q2
1
0
Final Stage
Comments:
1. As long we read a “1” we go back to start state
2. If we get a 0 we have possibility to get a 1.
Amir Chartab Shargh
5
1
Q3