Introduction to the Theory of Computation. Automata 1VO + 1PS. Lecturer: Dr. Ana Sokolova.

Transcription

1 Introduction to the Theory of Computation Automata 1VO + 1PS Lecturer: Dr. Ana Sokolova

2 Setup and Dates Lectures and Instructions Books Introduction to the Theory of Computation by M. Sipser Introduction to Automata Theory, Languages, and Computation by J. E. Hopcroft, R. Motwani, and J.D. Ullman

3 Setup and Dates Lectures and Instructions Books automata Introduction to the Theory of Computation by M. Sipser Introduction to Automata Theory, Languages, and Computation by J. E. Hopcroft, R. Motwani, and J.D. Ullman

4 Setup and Dates Lectures and Instructions Books automata grammars Introduction to the Theory of Computation by M. Sipser Introduction to Automata Theory, Languages, and Computation by J. E. Hopcroft, R. Motwani, and J.D. Ullman

5 Setup and Dates Lectures and Instructions Books automata grammars pushdown automata Introduction to the Theory of Computation by M. Sipser Introduction to Automata Theory, Languages, and Computation by J. E. Hopcroft, R. Motwani, and J.D. Ullman

6 Setup and Dates Lectures and Instructions Books automata grammars pushdown automata Turing machines Introduction to the Theory of Computation by M. Sipser Introduction to Automata Theory, Languages, and Computation by J. E. Hopcroft, R. Motwani, and J.D. Ullman

7 Setup and Dates Lectures and Instructions Books automata grammars pushdown automata Turing machines Introduction to the Theory of Computation by M. Sipser Introduction to Automata Theory, Languages, and Computation by J. E. Hopcroft, R. Motwani, and J.D. Ullman computability

8 The Rules... Instructions Starting in 2 weeks Instruction exercises on the web Automata2015.html after class To be solved by you, the students, (in groups of at most 3 people) and handed in as homework at the next meeting. In class I will present a sample solution and you, the students, will be asked to present solutions/discuss the exercises

9 The Rules... Instructions One randomly chosen exercise will be graded each week The graded exercise will be returned at the next meeting. Grade based on (1) exam (2) the grades of the corrected exercise and (3) activity in class (ability to present solutions) All information about the course / rules / exams / grading is / will be on the course webpage

10 The Rules... Grading Written exam on December 18, 10 am - 12:30 pm Grade based on the number of points on the written exam (80%), homework grades, and activity in class (20%) For better grade oral exam after the written one upon appointment 55% of the maximal points are needed to pass.

11 Finite Automata

12 Alphabets and Languages Alphabet and words if y is not free in P and Q

13 Alphabets and Languages - alphabet (finite set) n = {a1a2..an ai } is the set of words of length n * = {w n N. a1, a2,.., an. w = a1a2..an} is the set of all words over Alphabet and words if y is not free in P and Q

14 Alphabets and Languages 0 = {ℇ} contains only the empty word - alphabet (finite set) n = {a1a2..an ai } is the set of words of length n * = {w n N. a1, a2,.., an. w = a1a2..an} is the set of all words over Alphabet and words if y is not free in P and Q

15 Alphabets and Languages 0 = {ℇ} contains only the empty word - alphabet (finite set) n = {a1a2..an ai } is the set of words of length n * = {w n N. a1, a2,.., an. w = a1a2..an} is the set of all words over Alphabet and words A language L over is a subset L * if y is not free in P and Q

16 Deterministic Automata (DFA) Informal example = {0,1} M1: q0 q1 1 if y is not free in P and Q

17 Deterministic Automata (DFA) alphabet Informal example = {0,1} M1: q0 q1 1 if y is not free in P and Q

18 Deterministic Automata (DFA) alphabet Informal example = {0,1} M1: q0, q1 are states q0 q1 1 if y is not free in P and Q

19 Deterministic Automata (DFA) alphabet Informal example = {0,1} M1: q0, q1 are states q0 q1 q0 is initial 1 if y is not free in P and Q

20 Deterministic Automata (DFA) alphabet Informal example = {0,1} M1: q0, q1 are states q0 q1 q0 is initial 1 q1 is final if y is not free in P and Q

21 Deterministic Automata (DFA) alphabet Informal example = {0,1} M1: q0, q1 are states q0 is initial q0 1 q1 q1 is final if y is not free in P and Q transitions, labelled by alphabet symbols

22 Deterministic Automata (DFA) alphabet Informal example = {0,1} M1: q0, q1 are states q0 is initial q0 1 q1 q1 is final if y is not free in P and Q transitions, labelled by alphabet symbols Accepts the language L(M1) = {w * w ends with a 0} = * 0

23 Deterministic Automata (DFA) alphabet Informal example = {0,1} M1: q0, q1 are states q0 is initial q0 1 q1 q1 is final if y is not free in P and Q transitions, labelled by alphabet symbols Accepts the language L(M1) = {w * w ends with a 0} = * 0 regular language

24 Deterministic Automata (DFA) alphabet Informal example = {0,1} M1: q0, q1 are states q0 is initial q0 1 q1 q1 is final if y is not free in P and Q transitions, labelled by alphabet symbols Accepts the language L(M1) = {w * w ends with a 0} = * 0 regular language regular expression

25 DFA Definition A deterministic finite automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet δ: Q x Q is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q

26 DFA Definition A deterministic finite automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet δ: Q x Q is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M

27 DFA Definition A deterministic finite automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet δ: Q x Q is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M M1 = (Q,, δ, q0, F) for

28 DFA Definition A deterministic finite automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet δ: Q x Q is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M M1 = (Q,, δ, q0, F) for Q = {q0, q1}

29 DFA Definition A deterministic finite automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet δ: Q x Q is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M M1 = (Q,, δ, q0, F) for Q = {q0, q1} = {0, 1}

30 DFA Definition A deterministic finite automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet δ: Q x Q is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M M1 = (Q,, δ, q0, F) for Q = {q0, q1} = {0, 1} F = {q1}

31 DFA Definition A deterministic finite automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet δ: Q x Q is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M M1 = (Q,, δ, q0, F) for Q = {q0, q1} = {0, 1} F = {q1} δ(q0, 0) = q1,δ(q0, 1) = q0 δ(q1, 0) = q1, δ(q1, 1) = q0

32 DFA The extended transition function if y is not free in P and Q

33 DFA The extended transition function Given M = (Q,, δ, q0, F) we can extend δ: Q x Q to δ * : Q x * Q inductively, by: δ * (q, ε) = q and δ * (q,wa) = δ(δ * (q,w), a) if y is not free in P and Q

34 DFA The extended transition function Given M = (Q,, δ, q0, F) we can extend δ: Q x Q to δ * : Q x * Q In M1, δ * (q0,110010) = q1 inductively, by: if y is not free in P and Q δ * (q, ε) = q and δ * (q,wa) = δ(δ * (q,w), a)

35 DFA The extended transition function Given M = (Q,, δ, q0, F) we can extend δ: Q x Q to δ * : Q x * Q In M1, δ * (q0,110010) = q1 inductively, by: if y is not free in P and Q δ * (q, ε) = q and δ * (q,wa) = δ(δ * (q,w), a) Definition The language recognised / accepted by a deterministic finite automaton M = (Q,, δ, q0, F) is L(M) = {w * δ * (q0,w) F}

36 DFA The extended transition function Given M = (Q,, δ, q0, F) we can extend δ: Q x Q to δ * : Q x * Q In M1, δ * (q0,110010) = q1 inductively, by: if y is not free in P and Q δ * (q, ε) = q and δ * (q,wa) = δ(δ * (q,w), a) Definition The language recognised / accepted by a deterministic finite automaton M = (Q,, δ, q0, F) is L(M) = {w * δ * (q0,w) F} L(M1) = {w0 w {0,1} * }

37 Regular languages and operations Definition Let be an alphabet. A language L over (L * ) is regular iff it is recognised by a DFA.

38 Definition Regular languages and operations Let be an alphabet. A language L over (L * ) is regular iff it is recognised by a DFA. L(M1) = {w0 w {0,1} * } is regular

39 Definition Regular languages and operations Let be an alphabet. A language L over (L * ) is regular iff it is recognised by a DFA. L(M1) = {w0 w {0,1} * } is regular Regular operations

40 Definition Regular languages and operations Let be an alphabet. A language L over (L * ) is regular iff it is recognised by a DFA. L(M1) = {w0 w {0,1} * } is regular Regular operations Let L, L1, L2 be languages over. Then L1 L2, L1 L2, and L * are languages, where L1 L2 = {w1 w2 w1 L1,w2 L2} L * = {w n N. w1, w2,.., wn L. w = w1w2..wn}

41 Definition Regular languages and operations Let be an alphabet. A language L over (L * ) is regular iff it is recognised by a DFA. L(M1) = {w0 w {0,1} * } is regular Regular operations Let L, L1, L2 be languages over. Then L1 L2, L1 L2, and L * are languages, where L1 L2 = {w1 w2 w1 L1,w2 L2} L * = {w n N. w1, w2,.., wn L. w = w1w2..wn} ℇ L * always

42 Definition Regular expressions

43 finite representation of infinite languages Regular expressions Definition

44 finite representation of infinite languages Regular expressions Definition Let be an alphabet. The following are regular expressions 1. a for a 2. ε (R1 R2) for R1, R2 regular expressions 5. (R1 R2) for R1, R2 regular expressions 6. (R1) * for R1 regular expression

45 Regular expressions inductive finite representation of infinite languages Definition Let be an alphabet. The following are regular expressions 1. a for a 2. ε (R1 R2) for R1, R2 regular expressions 5. (R1 R2) for R1, R2 regular expressions 6. (R1) * for R1 regular expression

46 finite representation of infinite languages Regular expressions Definition inductive example: (ab a) * Let be an alphabet. The following are regular expressions 1. a for a 2. ε (R1 R2) for R1, R2 regular expressions 5. (R1 R2) for R1, R2 regular expressions 6. (R1) * for R1 regular expression

47 finite representation of infinite languages Regular expressions Definition inductive example: (ab a) * Let be an alphabet. The following are regular expressions 1. a for a corresponding languages L(a) = {a} 2. ε L(ε) = {ε} 3. L( ) = 4. (R1 R2) for R1, R2 regular expressions L(R1 R2) = L(R1) L(R2) 5. (R1 R2) for R1, R2 regular expressions L(R1 R2) = L(R1) L(R2) 6. (R1) * for R1 regular expression L(R1 * ) = L(R1) *

48 Equivalence of regular expressions and regular languages if y is not free in P and Q

49 Equivalence of regular expressions and regular languages Theorem (Kleene) A language is regular (i.e., recognised by a finite automaton) iff it is the language of a regular expression. if y is not free in P and Q

50 Equivalence of regular expressions and regular languages Theorem (Kleene) A language is regular (i.e., recognised by a finite automaton) iff it is the language of a regular expression. Proof easy, via the closure if y is not free in P and Q properties discussed next, not so easy, we ll skip it for now

51 Closure under regular operations

52 Closure under regular operations Theorem C1 The class of regular languages is closed under union

53 Closure under regular operations also under intersection Theorem C1 The class of regular languages is closed under union

54 Closure under regular operations also under intersection Theorem C1 The class of regular languages is closed under union Theorem C2 The class of regular languages is closed under complement

55 Closure under regular operations also under intersection Theorem C1 The class of regular languages is closed under union Theorem C2 The class of regular languages is closed under complement Theorem C3 The class of regular languages is closed under concatenation

56 Closure under regular operations also under intersection Theorem C1 The class of regular languages is closed under union Theorem C2 The class of regular languages is closed under complement Theorem C3 The class of regular languages is closed under concatenation Theorem C4 The class of regular languages is closed under Kleene star

57 Closure under regular operations also under intersection Theorem C1 The class of regular languages is closed under union We can already prove these Theorem C2 The class of regular languages is closed under complement Theorem C3 The class of regular languages is closed under concatenation Theorem C4 The class of regular languages is closed under Kleene star

58 Closure under regular operations also under intersection Theorem C1 The class of regular languages is closed under union We can already prove these Theorem C2 The class of regular languages is closed under complement Theorem C3 The class of regular languages is closed under concatenation But not yet these two Theorem C4 The class of regular languages is closed under Kleene star

59 Nondeterministic Automata (NFA) Informal example = {0,1} M2: 0,1 q0 1 q1 0,ε q2 1 q3 0,1

60 Nondeterministic Automata (NFA) Informal example = {0,1} M2: 0,1 q0 1 q1 0,ε q2 1 q3 0,1 sources of nondeterminism

61 Nondeterministic Automata (NFA) Informal example = {0,1} M2: 0,1 q0 1 q1 0,ε q2 1 q3 0,1 sources of nondeterminism

62 Nondeterministic Automata (NFA) Informal example = {0,1} M2: 0,1 q0 1 q1 0,ε q2 1 q3 0,1 sources of nondeterminism

63 Nondeterministic Automata (NFA) no 1 transition Informal example = {0,1} M2: 0,1 q0 1 q1 0,ε q2 1 q3 0,1 sources of nondeterminism

64 Nondeterministic Automata (NFA) no 1 transition Informal example no 0 transition = {0,1} M2: 0,1 q0 1 q1 0,ε q2 1 q3 0,1 sources of nondeterminism

65 Nondeterministic Automata (NFA) no 1 transition Informal example no 0 transition = {0,1} M2: 0,1 q0 1 q1 0,ε q2 1 q3 0,1 sources of nondeterminism Accepts a word iff there exists an accepting run

66 NFA Definition A nondeterministic automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet δ: Q x ε P(Q) is the transition function q0 is the initial state, q0 Q F is a set of final states, F Q if y is not free in P and Q

67 NFA Definition A nondeterministic automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet ε = {ε} δ: Q x ε P(Q) is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q

68 NFA Definition A nondeterministic automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet ε = {ε} δ: Q x ε P(Q) is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M

69 NFA Definition A nondeterministic automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet ε = {ε} δ: Q x ε P(Q) is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M M2 = (Q,, δ, q0, F) for

70 NFA Definition A nondeterministic automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet ε = {ε} δ: Q x ε P(Q) is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M M2 = (Q,, δ, q0, F) for Q = {q0, q1, q2, q3}

71 NFA Definition A nondeterministic automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet ε = {ε} δ: Q x ε P(Q) is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M M2 = (Q,, δ, q0, F) for Q = {q0, q1, q2, q3} = {0, 1}

72 NFA Definition A nondeterministic automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet ε = {ε} δ: Q x ε P(Q) is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M M2 = (Q,, δ, q0, F) for Q = {q0, q1, q2, q3} = {0, 1} F = {q3}

73 NFA Definition A nondeterministic automaton M is a tuple M = (Q,, δ, q0, F) where Q is a finite set of states is a finite alphabet ε = {ε} δ: Q x ε P(Q) is the transition function q0 is the initial state, q0 Q if y is not free in P and Q F is a set of final states, F Q In the example M Q = {q0, q1, q2, q3} = {0, 1} F = {q3} M2 = (Q,, δ, q0, F) for δ(q0, 0) = {q0} δ(q0, 1) = {q0,q1} δ(q0, ε) =..