Introduction to Computability
Theory
Discussion1: Non-Deterministic
Finite Automatons
Prof. Amos Israeli
1
Reminder
An NFA is a finite automaton in which:
1. 0, 1, many transitions with the same label
emanating out of the same state are allowed.
2.  transitions are allowed.
2
Exercise (omitted)
Construct the following NFA-s:
1. An NFA accepting all words ending with
abbab .
2. An NFA accepting all words whose third or
second word from the end is a .
3. An NFA accepting all words of the form
a i b j c k satisfying
a  0, b  0, c  0 a  b  c mod 2  0
3
Reminder
A computation an NFA a tree like collection of
paths induced by following all possible
transitions and forking when either the
current input symbol allows several transitions
more than a single transition or when an
transition is enabled.

4
Exercise
Construct the
1
a
computation
of this NFA on
some short words.
5
b

a
3
2
a,b
Reminder
Proposition
DFA-s and NFA-s are equivalent.
Meaning: DFA-s and NFA-s recognize the same
class of languages, called the Regular
Languages.
6
Reminder
Every DFA is (a special case of) an NFA, thus
 La | La is recognized   La | La is recognized 



by some DFA  
by some NFA 

In order to prove equality we have to prove:
 La | La is recognized   La | La is recognized 



by some DFA  
by some NFA 

7
Reminder
The proof works as follows:
Given some arbitrary NFA N, we construct a DFA
N, such that
LM   LN 
Yesterday we started to look at the proof.
Today we will demonstrate the construction on
one particular NFA:
8
Exercise
Given NFA N
Construct the
1
a
b
satisfying LM   LN  .
a
3
2
a DFA M
9

a,b
Reminder
For each set S, the Power Set of S, P( S ) , is the
set containing all subsets of S.
The set S satisfy:
| PS  | 2|S|
Why:
Each subset corresponds to a binary vector
of | S | elements.
10
Reminder
For each set S, the Power Set of S, P( S ) , is the
set containing all subsets of S.
The set S satisfy:
| PS  | 2|S|
Why:
Each subset corresponds to a binary vector
of | S | elements.
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Reminder
The state set of
M is the
power set of
the state set
of N .
1
a
b

a
3
2
a,b
Q  1,2,3
PQ  , 
1 , 2
, 3
, 1,2
, 1,3
, 2,3
, 1,2,3
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