1

Introduction to Computability Theory

Discussion1: Conversion ofA DFA to a Regular Expression

Prof. Amos Israeli

Given a GNFA G, any state of G, not including and , can be ripped off G, while preserving .This is demonstrated in the next slide by considering a general state, denoted by , and an arbitrary pair of states, and :

Ripping a state from a GNFA(rem.)

2

GLstartq acceptq

ripq

iq jq

Removing a state from a GNFA

3

1R

iqjq

ripq

4R

3R

2R

43*

21 RRRR iq jq

Before Ripping After Ripping

Note: This should be done for every pair of incoming and outgoing transitions.

Ellaboration

Assume the following situation:In order to rip , all pairsof incoming and outgoingtransitions should be considered in the way showed on the previous slide namely consider one after the other. After that can be ripped while preserving .

4

1t 2t 3t

5t4t 53

4352425141

,

,,,,,,,,,,

tt

tttttttttt

ripq

ripq

ripq

GL

The Conversion Algorithm - Outline

The conversion algorithm has 3 stages:1. Converting a DFA D with k states to an

equivalent GNFA G with states .2. Repeatedly ripping an arbitrarily chosen state

of G while preserving its functionality until remaining with a 2 states equivalent GNFA with two states.

3. Return the RE labeling remaining transition.

5

2k

Exercise

Apply the algorithm to obtain the regular expression equivalent to D:

What is the equivalent Regular expression?

6

** 10101

0aq bq

1cq

1 0 1,0

Stage 1: Convert D to a GNFA

1.0 Start with D

7

0

aq

bq

1

cq

1

0

1,0

Stage 1: Convert D to a GNFA

1.1 Add 2 new states

8

0

aq

bq

1

cq

1

0

1,0

acceptqstartq

Stage 1: Convert D to a GNFA

1.2 Make the initial state and the finalstate.

9

0

aq

bq

1

cq

1

0

1,0

acceptqstartq

startq acceptq

Stage 1: Convert D to a GNFA

1.3 Replace multi label transitions by their union.

10

0

aq

bq

1

cq

1

0

10

acceptqstartq

Stage 1: Convert D to a GNFA

1.4 Add all missing transitions and label them .

11

0

aq

bq

1

cq

1

0

startq

acceptq

10

Stage 2: Rip a state

2.0 Start with G.

12

0

aq

bq

1

cq

1

0

startq

acceptq

10

Stage 2: Rip a state

2.1 Choose an arbitrary state to be ripped.

13

0

aq

bq

1

cq

1

0

startq

acceptq

10

Stage 2: Rip a state

2.1 Remove all -labeled incoming and outgoing transitions. (Note: This stage does not appear in the book).

2.3 Replace each pair of incoming and outgoing transitions using the procedure we showed before.

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Stage 2: Rip a state

Reminder: if the incoming transition from to is labeled , the self-loop of , , the transition from to with , and the transition from to is labeled with

then the new label from to is labeled .Also note: for any regular expression R,

15

43*

21 RRRR

1R2Rripq

ripqiq

ripq jq3R

iq jq4R

RR

iq jq

Stage 2: Rip a state

2.1 Remove all - labeled incoming transitions.

16

0

aq

bq

1

cq

1

0

10

startq

acceptq

Stage 2: Rip a state

2.1 Remove all - labeled outgoing transitions.

17

0

aq

bq

1

cq

1

0

10

startq

acceptq

Stage 2: Rip a state

2.1 Remove all - labeled outgoing transitions.

18

0

aq

bq

1

cq

1

0

10

startq

acceptq

Stage 2: Rip a state

2.1 Remove transitions new label is .

19

0

aq

bq

1

cq

1

0

startq

acceptq

10100*

10

10

Stage 2: Rip a state

2.1 Now all incoming transitions are removed.

20

aq

bq

1

cq

1

0

startq

acceptq

10

10

Stage 2: Rip a state

2.2 Remove outgoing transitions.

21

aq

bq

cq

1

startq

acceptq

10

10

Stage 2: Rip a state

2.3 choose a new .

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aq cq

1

startq

acceptq

10

ripq

10

Stage 2: Rip a state

2.1 Remove all - labeled incoming transitions.

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aq cq

1

startq

acceptq

10

10

Stage 2: Rip a state

2.1 Remove all - labeled outgoing transitions.

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aq cq

1

startq

acceptq

10

10

Stage 2: Rip a state

2.1 Remove all - labeled outgoing transitions.

25

aq cq

1

startq

acceptq

10

10

Stage 2: Rip a state

2.2 Remove transitions new label is . .

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aq cq

1

startq

acceptq

** 10101010

10

10

*1010

Stage 2: Rip a state

2.3 Remove and all its transitions.

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aq

1

startq

acceptq

** 10101010

ripq

Stage 2: Rip last state

2.3 Choose the last remaining state to be ripped.

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aq

1

startq

acceptq

** 10101010

Stage 2: Rip a state

2.3 choose a new and repeat procedure.

29

startq acceptq

ripq

**** 1010110101

Stage 3: Return the remaining RE

.

30

startq acceptq

** 10011