DFA Minimization - Advanced Theory of Computation - Lecture Slides, Slides of Theory of Computation

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DFA Minimization

DFA 

Deterministic Finite Automata (DFSA) 

Q

δ

q 0

F

 Q – (finite) set of states  Σ

• alphabet – (finite) set of input symbols  δ
  • transition function  q 0
    • start state  F – set of final / accepting states

DFA Minimization 

Some states can be redundant: 

The following DFA accepts (a|b)+  State s1 is not necessary

DFA Minimization 

So these two DFAs are

equivalent

DFA Minimization 

The task of

DFA minimization

, then, is to

automatically transform a given DFA into astate-minimized DFA 

Several algorithms and variants are known  Note that this also in effect can minimize an NFA(since we know algorithm to convert NFA to DFA)

DFA Minimization Algorithm 

Recall that a DFA

M

Q

q

F



Two states

p

and

q

are distinct if

 p in F and q not in F or vice versa, or  for some α in

δ

p

α ) and δ

q

α ) are distinct 

Using this inductive definition, we cancalculate which states are distinct

Very Simple Example

s0 s1 s s0 s1 s

Very Simple Example

s0 s ε s ε s0 s1 s Label pairs with ε where one is a final state and the other is not

Very Simple Example

s0 s ε s ε s0 s1 s DISTINCT(s1, s2) is empty, so s1 and s2 are equivalent states

Very Simple Example

Merge s1 and s

More Complex Example 

Check for pairs with one state final and one not:

More Complex Example 

First iteration of main loop:

More Complex Example 

Third iteration makes no changes 

Blank cells are equivalent pairs of states

More Complex Example 

Combine equivalent states for minimized DFA: