Converting NFAs to DFAs

Note: to view the symbols on this page correctly, you should be using
a PC with the Windows TrueType Symbol font loaded.

Converting a nfa to a dfa
Defn:  The e-closure of a state is the set of all states,
including S itself, that you can get to via e-transitions.
The e-closure of state S is denoted:  S

Example:  
The e-closure of state 1:  1 = { 1, 2, 4 }
The e-closure of state 3:  3 = { 3, 2, 4 }
Defn:  The e-closure of a set of states S1, ... Sn is 
       S1  S2  ...  Sn.
Example:  The e-closure for above states 1 and 3 is 
       { 1, 2, 4 }  { 3, 2, 4 } = { 1, 2, 3, 4 }

To construct a dfa from a nfa:
Step 1:  Let the start state of the dfa be formed from the 
         e-closure of the start state of the nfa.
Subsequent steps:  If S is any state that you have previously
         constructed for the dfa and it is formed from say
         states t1, ... , tr of the nfa, then for any symbol x
         for which at least one of the states t1, ... , tr has a
         x-successor, the x-successor of S is the e-closure of
         the x-successors of t1, ... , tr.
         Any state of the dfa which is formed from an accepting 
         state, among others, of the nfa becomes an accepting state.  
Example 1:  To convert the following nfa:
we get:
This constructs a dfa that has no epsilon-transitions and a
single accepting state. 
Example 2:  To convert the nfa for an identifier to a dfa
we get:
	

Minimizing the Number of States in a DFA
Step 1:  Start with two sets of states
         (a) all the accepting states, and 
         (b) all the nonaccepting states
Subsequent steps:
         Given the sets of states S1, ... Sr consider
         each state S and each symbol x in turn.  If any
         member of S has a x-successor and this x-successor 
         is in say S', then unless all the members of S 
         have x-successors that are in S', split up S into 
         those members whose x-successors are in S' and the 
         others (which don't have x-successors in S').
Example 1.  Consider the dfa we constructed for an identifier 
          (with renumbered states):
The sets of states for this dfa are:
   S1             S2
Nonaccepting	Accepting 
states:		states:

   1		   2          All states in S2 have the successors
		   3          letter-successor and digit-successor, and
		   4          the successor states are all in the set of
                              states S2.
Combine all the states of S2 to get:
Example 2.  Consider the dfa:
All of the states (1, 2, and 3) are accepting states and all their
successors are also accepting states, but state 1 has an a-successor
whereas states 2 and 3 do not.  

So, we split the set of accepting states into two sets S1 and S2 where:
S1 consists of state 1, and  
S2 consists of states 2, 3 
to get: