The Pumping Lemma. for all n 0, u 1 v n u 2 L (i.e. u 1 u 2 L, u 1 vu 2 L [but we knew that anyway], u 1 vvu 2 L, u 1 vvvu 2 L, etc.

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1 The Pumping Lemma For every regular language L, there is a number l 1 satisfying the pumping lemma property: All w L with w l can be expressed as a concatenation of three strings, w = u 1 vu 2, where u 1, v and u 2 satisfy: v 1 (i.e. v = ε) u 1 v l for all n 0, u 1 v n u 2 L (i.e. u 1 u 2 L, u 1 vu 2 L [but we knew that anyway], u 1 vvu 2 L, u 1 vvvu 2 L, etc.) Note similarity to construction in Kleene (b)

2 Suppose L = L(M) for a DFA M = (Q, Σ, δ, s, F). Taking l to be the number of elements in Q, if n l, then in a s = 1 a q 0 2 q1 q2 al q }{{} l l+1 states an q n F q 0,..., q l can t all be distinct states. So q i = q j for some 0 i < j l. So the above transition sequence looks like v where s = q 0 u 1 q i = q j u 2 q n F u 1 a 1... a i v a i+1... a j u 2 a j+1... a n

3 How to use the Pumping Lemma to prove that a language L is not regular For each l 1, find some w L of length l so that no matter how w is split into three, w = u 1 vu 2, with u 1 v l and v 1, there is some n 0 for which u 1 v n ( ) u 2 is not in L

4 Examples None of the following three languages are regular: (i) L 1 {a n b n n 0}

5 L 1 = {a n b n n 0} For each l 1, take w = a l b l L 1 If w = u 1 vu 2 with u 1 v l & v 1, then for some r and s: u 1 = a r

6 L 1 = {a n b n n 0} For each l 1, take w = a l b l L 1 If w = u 1 vu 2 with u 1 v l & v 1, then for some r and s: u 1 = a r v = a s, with r+s l and s 1

7 L 1 = {a n b n n 0} For each l 1, take w = a l b l L 1 If w = u 1 vu 2 with u 1 v l & v 1, then for some r and s: u 1 = a r v = a s, with r+s l and s 1 u 2 = a l r s b l

8 L 1 = {a n b n n 0} For each l 1, take w = a l b l L 1 If w = u 1 vu 2 with u 1 v l & v 1, then for some r and s: u 1 = a r v = a s, with r+s l and s 1 u 2 = a l r s b l so u 1 v 0 u 2 =

9 L 1 = {a n b n n 0} For each l 1, take w = a l b l L 1 If w = u 1 vu 2 with u 1 v l & v 1, then for some r and s: u 1 = a r v = a s, with r+s l and s 1 u 2 = a l r s b l so u 1 v 0 u 2 = a r ǫ a l r s b l =

10 L 1 = {a n b n n 0} For each l 1, take w = a l b l L 1 If w = u 1 vu 2 with u 1 v l & v 1, then for some r and s: u 1 = a r v = a s, with r+s l and s 1 u 2 = a l r s b l so u 1 v 0 u 2 = a r ǫ a l r s b l = a l s b l

11 L 1 = {a n b n n 0} For each l 1, take w = a l b l L 1 If w = u 1 vu 2 with u 1 v l & v 1, then for some r and s: u 1 = a r v = a s, with r+s l and s 1 u 2 = a l r s b l so u 1 v 0 u 2 = a r ǫ a l r s b l = a l s b l But a l s b l L 1

12 L 1 = {a n b n n 0} For each l 1, take w = a l b l L 1 If w = u 1 vu 2 with u 1 v l & v 1, then for some r and s: u 1 = a r v = a s, with r+s l and s 1 u 2 = a l r s b l so u 1 v 0 u 2 = a r ǫ a l r s b l = a l s b l But a l s b l L 1, so, by the Pumping Lemma, L 1 is not a regular language

13 Examples None of the following three languages are regular: (i) L 1 {a n b n n 0} [For each l 1, a l b l L 1 is of length l and has property ( ).]

14 Examples None of the following three languages are regular: (i) L 1 {a n b n n 0} [For each l 1, a l b l L 1 is of length l and has property ( ).] (ii) L 2 {w {a, b} w a palindrome}

15 Examples None of the following three languages are regular: (i) L 1 {a n b n n 0} [For each l 1, a l b l L 1 is of length l and has property ( ).] (ii) L 2 {w {a, b} w a palindrome} [For each l 1, a l ba l L 1 is of length l and has property ( ).]

16 Examples None of the following three languages are regular: (i) L 1 {a n b n n 0} [For each l 1, a l b l L 1 is of length l and has property ( ).] (ii) L 2 {w {a, b} w a palindrome} [For each l 1, a l ba l L 1 is of length l and has property ( ).] (iii) L 3 {a p p prime}

17 L 3 = {a p p prime} For each l 1 let w = a p L 3, p prime & p > 2l If w = u 1 vu 2 with the usual...

18 L 3 = {a p p prime} For each l 1 let w = a p L 3, p prime & p > 2l If w = u 1 vu 2 with the usual... then u 1 = a r v = a s u 2 = a p r s with s 1 & r+s l

19 L 3 = {a p p prime} For each l 1 let w = a p L 3, p prime & p > 2l If w = u 1 vu 2 with the usual... then u 1 = a r v = a s u 2 = a p r s with s 1 & r+s l so u 1 v p s u 2 =

20 L 3 = {a p p prime} For each l 1 let w = a p L 3, p prime & p > 2l If w = u 1 vu 2 with the usual... then u 1 = a r v = a s u 2 = a p r s with s 1 & r+s l so u 1 v p s u 2 = a r a s(p s) a p r s =

21 L 3 = {a p p prime} For each l 1 let w = a p L 3, p prime & p > 2l If w = u 1 vu 2 with the usual... then u 1 = a r v = a s u 2 = a p r s with s 1 & r+s l so u 1 v p s u 2 = a r a s(p s) a p r s = a (p s)(s+1)

22 L 3 = {a p p prime} For each l 1 let w = a p L 3, p prime & p > 2l If w = u 1 vu 2 with the usual... then u 1 = a r v = a s u 2 = a p r s with s 1 & r+s l so u 1 v p s u 2 = a r a s(p s) a p r s = a (p s)(s+1) but s 1 s+1 2 and (p s) > (2l l) 1 (p s) 2

23 L 3 = {a p p prime} For each l 1 let w = a p L 3, p prime & p > 2l If w = u 1 vu 2 with the usual... then u 1 = a r v = a s u 2 = a p r s with s 1 & r+s l so u 1 v p s u 2 = a r a s(p s) a p r s = a (p s)(s+1) but s 1 s+1 2 and (p s) > (2l l) 1 (p s) 2 so a (p s)(s+1) L 3

24 Examples None of the following three languages are regular: (i) L 1 {a n b n n 0} [For each l 1, a l b l L 1 is of length l and has property ( ).] (ii) L 2 {w {a, b} w a palindrome} [For each l 1, a l ba l L 1 is of length l and has property ( ).] (iii) L 3 {a p p prime} [For each l 1, we can find a prime p with p > 2l and then a p L 3 has length l and has property ( ).]

25 Pumping Lemma property is necessary for a language to be regular It is not sufficient

26 Example of a non-regular language with the pumping lemma property L {c m a n b n m 1 & n 0} {a m b n m, n 0} satisfies the pumping lemma property with l = 1. [For any w L of length 1, can take u 1 = ε, v = first letter of w, u 2 = rest of w.] But L is not regular see Exercise 5.1.

27 L is not regular: (sketch).

28 L is not regular: (sketch) If L is regular there is a DFA M with L = L(M). Let s build a new machine, M from it..

29 L is not regular: (sketch) If L is regular there is a DFA M with L = L(M). Let s build a new machine, M from it. Take a c transition from the start state of M. Make the state you reach the start state of M.

30 L is not regular: (sketch) If L is regular there is a DFA M with L = L(M). Let s build a new machine, M from it. Take a c transition from the start state of M. Make the state you reach the start state of M. Delete all transitions involving c (and remove c from the alphabet). But don t remove any states and keep the same accept states.

31 L is not regular: (sketch) If L is regular there is a DFA M with L = L(M). Let s build a new machine, M from it. Take a c transition from the start state of M. Make the state you reach the start state of M. Delete all transitions involving c (and remove c from the alphabet). But don t remove any states and keep the same accept states. What language does M recognise?

32 The way ahead, in THEORY What does is mean for a function to be computable? [ Ib Computation Theory ]

33 The way ahead, in THEORY What does is mean for a function to be computable? [ Ib Computation Theory ] Are some computational tasks intrinsiclaly unfeasible? [ Ib ComplexityTheory ]

34 The way ahead, in THEORY What does is mean for a function to be computable? [ Ib Computation Theory ] Are some computational tasks intrinsiclaly unfeasible? [ Ib ComplexityTheory ] How do we specify and reason about program behaviour? [ Ib Logic and Proof, 1b Semantics of PLs ]

35 The way ahead, in FORMAL LANGUAGES Are there other useful language classes?

36 The way ahead, in FORMAL LANGUAGES Are there other useful language classes? Are there other useful automata classes that have a correspondence to them?

37 The way ahead, in FORMAL LANGUAGES Are there other useful language classes? Are there other useful automata classes that have a correspondence to them? What if we ask the same questions about them that we asked about regular languages?

38 Chomsky Hierarchy of Languages Regular Languages Context Free Languages Context Sensitive Languages Recursively Enumerable Languages

39 Grammars Grammars are a shorthand way of expressing the inductive definition of subset inclusion for strings in a Language.

40 Grammars Grammars are a shorthand way of expressing the inductive definition of subset inclusion for strings in a Language. Often by convention we use capitals for non-terminal symbols (which are disjoint from symbols in the alphabet used by the language).

41 Grammars Grammars are a shorthand way of expressing the inductive definition of subset inclusion for strings in a Language. Often by convention we use capitals for non-terminal symbols (which are disjoint from symbols in the alphabet used by the language). We also have productions (or production rules) of the form e.g. A a which says that the non-terminal symbol A can be replaced by the (terminal) symbol a. More complex productions are allowed.

42 Grammars Grammars are a shorthand way of expressing the inductive definition of subset inclusion for strings in a Language. Often by convention we use capitals for non-terminal symbols (which are disjoint from symbols in the alphabet used by the language). We also have productions (or production rules) of the form e.g. A a which says that the non-terminal symbol A can be replaced by the (terminal) symbol a. More complex productions are allowed. There is also a distinguished non-terminal called the goal symbol (we ll use G)

43 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6

44 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6 so, e.g. G 0 E

45 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6 so, e.g. G 0 E 1 E+T

46 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6 so, e.g. G 0 E 1 E+T 4 E+P

47 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6 so, e.g. G 0 E 1 E+T 4 E+P 6 E+x

48 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6 so, e.g. G 0 E 1 E+T 4 E+P 6 E+x 2 T + x

49 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6 so, e.g. G 0 E 1 E+T 4 E+P 6 E+x 2 T + x 4 P+x

50 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6 so, e.g. G 0 E 1 E+T 4 E+P 6 E+x 2 T + x 4 P+x 5 (E)+x

51 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6 so, e.g. G 0 E 1 E+T 4 E+P 6 E+x 2 T + x 4 P+x 5 (E)+x... (x+x)+x

52 Everybody s favourite grammar G E 0 E E+T 1 E T 2 T T P 3 T P 4 P (E) 5 P x 6 so, e.g. G 0 E 1 E+T 4 E+P 6 E+x 2 T + x 4 P+x 5 (E)+x... (x+x)+x is a derivation of (x+x)+x

53 Language classes by forms of production α, β, γ any strings of terminals and non-terminals

54 Language classes by forms of production α, β, γ any strings of terminals and non-terminals Context Free Languages: [ Type 2 ] productions of form N β

55 Language classes by forms of production α, β, γ any strings of terminals and non-terminals Context Free Languages: [ Type 2 ] productions of form N β Context Sensitive Languages: [ Type 1 ] productions of the form αnβ αγβ

56 Language classes by forms of production α, β, γ any strings of terminals and non-terminals Context Free Languages: [ Type 2 ] productions of form N β Context Sensitive Languages: [ Type 1 ] productions of the form αnβ αγβ Recursively Enumerable Languages: [ Type 0 ] productions of the form α β

57 Language classes by forms of production

58 Language classes by forms of production How about Regular Languages? [ Type 3 ]

59 Language classes by forms of production How about Regular Languages? [ Type 3 ] A, B any non-terminals, a any terminal symbol, S any non-terminal that doesn t appear on right side

60 Language classes by forms of production How about Regular Languages? [ Type 3 ] A, B any non-terminals, a any terminal symbol, S any non-terminal that doesn t appear on right side production of the form A a or S ε or A ab (right regular)

61 Language classes by forms of production How about Regular Languages? [ Type 3 ] A, B any non-terminals, a any terminal symbol, S any non-terminal that doesn t appear on right side production of the form A a or S ε or A ab (right regular) or of the form A a or S ε or A Ba (left regular)

62 Language classes by forms of production How about Regular Languages? [ Type 3 ] A, B any non-terminals, a any terminal symbol, S any non-terminal that doesn t appear on right side production of the form A a or S ε or A ab (right regular) or of the form A a or S ε or A Ba (left regular) but not both left and right regular in the same grammar

63 Machines????? Regular Languages: Deterministic Finite Automata

64 Machines????? Regular Languages: Deterministic Finite Automata Context Free Languages: Nondeterministic Push-Down Automata

65 Machines????? Regular Languages: Deterministic Finite Automata Context Free Languages: Nondeterministic Push-Down Automata Context Sensitive Languages: Linear Bounded Nondeterministic Turing Machine

66 Machines????? Regular Languages: Deterministic Finite Automata Context Free Languages: Nondeterministic Push-Down Automata Context Sensitive Languages: Linear Bounded Nondeterministic Turing Machine Recursively Enumerable Languages: Turing Machine

67 Machines????? Regular Languages: Deterministic Finite Automata Context Free Languages: Nondeterministic Push-Down Automata Context Sensitive Languages: Linear Bounded Nondeterministic Turing Machine Recursively Enumerable Languages: Turing Machine Context Free Languages (and particularly the subset that can be recognised by deterministic push-down automata) are important since most programming languages are deterministic context free languages.

68 Deterministic Push-Down Automata (Sketch) Idea: need some way to remember arbitrary number of things that we have seen, eg a n b n

69 Deterministic Push-Down Automata (Sketch) Idea: need some way to remember arbitrary number of things that we have seen, eg a n b n Slightly modified DFA along with a stack which stores pairs of states and symbols.

70 Deterministic Push-Down Automata (Sketch) Idea: need some way to remember arbitrary number of things that we have seen, eg a n b n Slightly modified DFA along with a stack which stores pairs of states and symbols. DPDA looks at top of stack as well as input to decide what to do

71 Deterministic Push-Down Automata (Sketch) Idea: need some way to remember arbitrary number of things that we have seen, eg a n b n Slightly modified DFA along with a stack which stores pairs of states and symbols. DPDA looks at top of stack as well as input to decide what to do on state transitions, DPDA can pop and/or push things on the stack as well as (perhaps) reading symbol

72 What about our "questions"?

73 What about our "questions"? Given two DPDA, M 1 and M 2, can we determine if L(M 1 ) = L(M 2 )?

74 What about our "questions"? Given two DPDA, M 1 and M 2, can we determine if L(M 1 ) = L(M 2 )? Yes.

75 What about our "questions"? Given two DPDA, M 1 and M 2, can we determine if L(M 1 ) = L(M 2 )? Yes. Proved in 1997.

76 What about our "questions"? Given two DPDA, M 1 and M 2, can we determine if L(M 1 ) = L(M 2 )? Yes. Proved in Earned 2002 Gödel Prize.

77 What about our "questions"? Given two DPDA, M 1 and M 2, can we determine if L(M 1 ) = L(M 2 )? Yes. Proved in Earned 2002 Gödel Prize. But for NPDA, the question of equivalence is

78 What about our "questions"? Given two DPDA, M 1 and M 2, can we determine if L(M 1 ) = L(M 2 )? Yes. Proved in Earned 2002 Gödel Prize. But for NPDA, the question of equivalence is undecidable