CS375: Logic and Theory of Computing

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1 CS375: Logic and Theory of Computing Fuhua (Frank) Cheng Department of Computer Science University of Kentucky 1

2 Tale of Contents: Week 1: Preliminaries (set algera, relations, functions) (read Chapters 1-4) Weeks 3-6: Regular Languages, Finite Automata (Chapter 11) Weeks 7-9: Context-Free Languages, Pushdown Automata (Chapters 12) Weeks 10-12: Turing Machines (Chapter 13) 2

3 Tale of Contents (conti): Weeks 13-14: Propositional Logic (Chapter 6), Predicate Logic (Chapter 7), Computational Logic (Chapter 9), Algeraic Structures (Chapter 10) 3

4 Question: can we get a most efficient regular expression for a given NFA? Why? as compact as possile YES, in two steps. First, transform the given NFA to a DFA Then, transform the DFA to a minimum-state DFA 4

5 Transforming an NFA into a DFA The Ʌ-closure of a state s, denoted Ʌ(s), is the set consisting of s together with all states that can e reached from s y traversing Ʌ-edges. The Ʌ-closure of a set S of states, denoted Ʌ(S), is the union of the Ʌ-closure of the states in S. Ʌ(0)=? Ʌ(0)= {0, 1, 2} Ʌ({0, 1})= Ʌ(0) U Ʌ(1) = {0, 1, 2} U {1, 2} = {0, 1, 2} 5

6 The same NFA again: ( 0) {0,1, 2} (1) {1, 2} (2) {2} ( ) ({ 1, 2}) } ({0,1, 2}) 6 {1, 2 {0,1, 2}

7 Algorithm: Transform an NFA into a DFA Construct a DFA tale TD from an NFA tale TN as follows: 1. The start state of the DFA is Ʌ(s), where s is the start state of the NFA. 3. A DFA state is final if one of its elements is an NFA final state. 7

8 Example. Given the following NFA. Construct the DFA transition tale TD and write it in simplified form after renumering states. First: Λ(0) {0,3} (1) {1} (2) {2} (3) {3} Start state 8

9 Then, how to get the States of the DFA? 1. Build the tree on the right 2. Identify all distinct nodes 9

10 Then, how to get the States of the DFA? initially 10

11 Example. Given the following NFA. Λ(0) {0,3} (1) {1} (2) {2} (3) {3} T D ({0,3}, a) ( T (0, a) T ( {3}) ({3}) {3} (3, a)) 11 N N

12 Example. Given the following NFA. Λ(0) {0,3} (1) {1} (2) {2} (3) {3} T D ({0,3}, ) ( T N (0, ) T ({0,1} ) ({0,1}) (0) (1) (3, )) {0,3} {1} {0,1,3} 2/9/ N

13 So, we have 13

14 Then, uild the next level of the tree: 14

15 Example. Given the following NFA. Λ(0) {0,3} (1) {1} (2) {2} (3) {3} T D ({3}, a) ( T (3, a)) ({3}) (3) {3} 15 N

16 Example. Given the following NFA. Λ(0) {0,3} (1) {1} (3) {3} T D ({ 3}, ) ( T (3, )) ( ) (2) {2} 16 N

17 So we have the child nodes of {3}. Then find the child nodes of {0, 1, 3} 17

18 Example. Given the following NFA. Λ(0) {0,3} (1) {1} (2) {2} (3) {3} T D ({0,1, 3}, a) ( T N (0, a) T (1, a) T (3, a)) ( {2} {3}) ({2, 3}) (2) (3) {2} {3} {2,3} 18 N N

19 Example. Given the following NFA. Λ(0) {0,3} (1) {1} (2) {2} (3) {3} T D ({0,1, 3}, ) ( T N (0, ) T (1, ) T (3, )) ({0,1} ) ({0,1}) (0) (1) {0,3} {1} {0,1,3} 19 N N

20 So, we have the child nodes of {0, 1, 3}. Then find child nodes of Φ and {2,3} 20

21 Example. Given the following NFA. Λ(0) {0,3} (1) {1} (2) {2} (3) {3} T D (, a) ( T (, a)) ( ) 21 N

22 Example. Given the following NFA. Λ(0) {0,3} (1) {1} (2) {2} (3) {3} T D (, ) ( T (, )) ( ) 22 N

23 So we have child nodes of Φ now. 23

24 Eventually we would get the following tree: Identify all distinct nodes: 24

25 Then uild the transition tale 25

26 Write in simplified form after renumering the states: 26

27 a 0 2 a a 3 1 a a 5 a, 4 27

28 a a 1 0 a a 3 2 a 5 a, 4 28

29 0 2 a a 3 1 a a 4 a 5 29 a, Language of NFA: *Ʌa*+*a* = *a* + *a* What is the language of the DFA? Ʌ, aa*, *, *a, *aa, *aaa*, *a, *a* = a*, *, *a*, *a* = *a*, *a*

30 Example. Transform the following NFA into a DFA. 30

31 Example. Transform the following NFA into a DFA. First (0) {0, 3} (1) {1} (2) {2} (3) {3} 31

32 Quiz. Transform the following NFA into a DFA. Then, uild the tree 32

33 Example. Transform the following NFA into a DFA. Hence, solution: 33

34 Algorithm: Transform a DFA to a minimum-state DFA 1. Construct the following sequence of sets of possile equivalent pairs of distinct states: where and E E E E k E 0 1 k 1 E0 = {{s, t} s and t are either oth final or oth non-final} i 1 {{ s, t} E i { T ( s, a), T ( t, a)} E or T ( s, a) T ( t, a) for every a A} Ek represents the distinct pairs of equivalent states from which an equivalence relation ~ can e generated. 34 i

35 E0 =? E0 = {{s, t} s & t are either oth final or oth non-final} E0 = { {0,4}, {1,2}, {1,3}, {2,3} } E0 = { {0,0}, {4,4}, {0,4}, {4,0}, {1,1}, {2,2}, {3, 3}, {1,2}, {2,1}, {1,3}, {3,1}, {2,3}, {3,2} } Theoretically 35

36 E1 =? E0 = E i 1 {{ s, t} for every a E i A} { T ( s, a), T ( t, a)} E i or T ( s, a) T ( t, a) To e a pair in Ei+1, s and t must e mapped to the same state or states in the same group in Ei y every a ϵ A. 36

37 E0 = 1 a 2 2 a a 1 4 a 4 E1 = 37

38 E1 = Theoretically, E1 = { {0,0}, {4,4}, {1,1}, {2,2}, {3, 3}, {1,2}, {2,1}, {1,3}, {3,1}, {2,3}, {3,2} } or simply, E1 = { {1,2}, {1,3}, {2,3} } 38

39 E2 =? One-element groups can not e further reduced and the threeelement group will remain the same Hence, E2 = E1 S is partitioned y {0}, {1, 2, 3}, {4}. 39

40 Algorithm: Transform a DFA to a minimum-state DFA 2. The equivalence classes form the states of the minimum state DFA with transition tale Tmin defined y Tmin([s], a) = [T(s, a)]. 3. The start state is the class containing the start state of the given DFA. 4. A final state is any class containing a final state of the given DFA. 40

41 Example. Transform the given DFA into a minimum-state DFA. E2 = E1 E2 = So S is partitioned y {0}, {1, 2, 3}, {4}. 41 The minimum-state DFA has three states: [0], [1], [4].

42 Example. Transform the given DFA into a minimum-state DFA. Tmin([s], a) = [T(s, a)] Tmin([0], a) = [T(0, a)] = [1] Tmin([0], ) = [T(0, )] = [4] Tmin([1], a) = [T(1, a)] = [2] = [1] 2/9/

43 Example. Transform the given DFA into a minimum-state DFA. 43

44 Example. Transform the given DFA into a minimum-state DFA. Question: What regular expression equality arises from the two DFAs? Answer: a + aa + (aaa + aa + a)(a + )* = a(a + )*. 44

45 Prove : a + aa + (aaa + aa + a)(a + )* = a(a + )* LHS = a + aa +aa(a + )(a + )* + a(a + )* = a + aa + aa(a + ) + + a(a + )* = a + aa( Ʌ + (a + ) + ) + a(a + )* = a + aa(a + )* + a(a + )* = a + a(a + )(a + )* = a + a(a + ) + = a(ʌ + (a + ) + ) = a (a + )* = RHS 45

46 Question: Is the following DFA a minimum-state DFA? E0 = Answer. No. Use the minimum-state algorithm. E0 = {{0, 1}}, E1 = {{0, 1}} = E0. The partition is the whole set of states {0, 1} = [0]. Therefore, we have 46

47 Question: Is the following DFA a minimum-state DFA? c c a S 0 a,c E0 = c 1 a a,c 5 2 a 4 3 a 47 c

48 Question: Is the following DFA a minimum-state DFA? a S 0 a,c E0 = c c E1 c =? 1 0 c 3 a 2 a,c 1 c 1 5 c 5 5 a 4 3 a 48 c

49 Question: Is the following DFA a minimum-state DFA? a S 0 a,c E0 = c c E1 c =? a 2 a,c a 4 3 a 49 c

50 Question: Is the following DFA a minimum-state DFA? S E1 = 0 a c a,c a c 1 5 a,c 3 a 50 c 2 c a 4 E2 =? E2 = E1 The minimumstate DFA has 4 states : [0], [1], [2], [3]. Answer: NO

51 Example. S = {0, 1, 2, 3, 4, 5} : states of a DFA 0 : start state ; 2, 5 : final states. The equivalence relation on S for a minimum-state DFA is generated y the following set of equivalent pairs of states: {{0, 1}, {0, 4}, {1, 4}, {2, 5}} Write down the states of the minimum-state DFA 51

52 Example. S = {0, 1, 2, 3, 4, 5} : states of a DFA 0 : start state ; 2, 5 : final states. Equivalent pairs of states: {{0,1}, {0,4}, {1,4}, {2,5}} E0 = So, states of the minimum-state DFA: {0, 1, 4}, {3}, {2, 5} 52

53 End of Regular Language and Finite Automata III 53

CS375: Logic and Theory of Computing

CS375: Logic and Theory of Computing Fuhua (Frank) Cheng Department of Computer Science University of Kentucky 1 Tale of Contents: Week 1: Preliminaries (set alger relations, functions) (read Chapters