Indexed Languages and why you care. Presented by Pieter Hooimeijer

Transcription

1 Indexed Languages and why you care Presented by Pieter Hooimeijer

2 Think Way Back

3 ...Far Enough:... CS NCF DCF REG REG 3

4 Formal Language Classes... CS NCF DCF REG REG PDA 4

5 Formal Language Classes... CS NCF PDA DCF REG REG DFA 5

6 Formal Language Classes... CS NCF Machines PDA Formalism S asb z DCF REG REG DFA a(ab)*b 6

7 Formal Language Classes... CS NCF Pump DCF REG REG Pump 7

8 The Context-Free Pumping Lemma Suppose L = { anbncn n = 1...} is contextfree. By the pumping lemma, any string s with s p can be 'pumped.' p 8

9 The Context-Free Pumping Lemma What does 'can be pumped' mean? s = uvxyz 1) vy >0 2) vxy p 3) uvixyiz is in L for all i 0 9

10 The Context-Free Pumping Lemma Suppose L = { anbncn n = 1...} is context-free. Consider s = apbpcp; for any split s = uvxyz, we have: - if v contains a's, then y cannot contain c's - if y contains c's, then v cannot contain a's - a problem: uv0xy0z = uxz will never be in L 10

11 Moving Right Along 11

12 Motivation Some Examples Can solve problems by phrasing them as 'language' problems: Finding valid control flow graph paths Solving set constraint problems Static string analysis Lexing, parsing For fun... 12

13 Example String Variables 13

14 The Nugget Some Code: x = 'z'; while(n < 5) { x = '('. x. ')'; n ++; } We want a context free grammar to model x Suppose we don't know anything about n 14

15 The Nugget Some Code: >x = 'z'; Grammar: A -> z while(n < 5) { x = '('. x. ')'; n ++; } 15

16 The Nugget Some Code: x = 'z'; while(n < 5) { > x = '('. x. ')'; n ++; } Grammar: A -> z B -> (A) [True] [n < 5] 16

17 The Nugget Some Code: x = 'z'; while(n < 5) { x = '('. x. ')'; n ++; >>} Grammar: A -> z B -> (A) C -> A B [True] [n < 5] [n < 5] 17

18 The Nugget Some Code: x = 'z'; while(n < 5) { x = '('. x. ')'; n ++; >>} Grammar: A -> z B -> (C) C -> A B [True] [n < 5] [True] 18

19 The Nugget Some Code: x = 'z'; while(n < 5) { x = '('. x. ')'; n ++; >>} Grammar: X -> C A -> z B -> (C) C -> A B [True] [n < 5] [True] 19

20 Indexed Languages... CS *I* NCF DCF REG REG 20

21 Definition: Indexed Grammar G = (N, T, F, P, S) P : { N ((NF*) T)* } F : { { N (N T)*}f } 21

22 Derivation Rule Example 1 S Sf S ABgC S Sf Sff Aff Bgff Cff 22

23 Derivation Rule Example 1 S Sf S ABgC 'normal' production: copy index stack S Sf Sff Aff Bgff Cff 23

24 Derivation Rule Example 2 S Af F = { { A B}f } S Af B 24

25 Derivation Rule Example 2 S Af F = { { A B}f } S Af B 25

26 Derivation Rule Example 2 S Af F = { { A B}f } 'index' production: pop leftmost index (for current nonterminal only) S Af B 26

27 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} f = {A a B b C c} 27

28 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} f = {A a B b C c} What is the language of this grammar? 28

29 An Example Grammar S Df D Dg ABC S g = {A Aa B Bb C Cc} f = {A a B b C c} 29

30 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} S Df f = {A a B b C c} 30

31 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} S Df Dgf f = {A a B b C c} 31

32 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} S Df Dgf Agf Bgf Cgf f = {A a B b C c} 32

33 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} S Df Dgf Agf Bgf Cgf f = {A a B b C c} 33

34 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} f = {A a B b C c} S Df Dgf Agf Bgf Afa Cgf 34

35 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} f = {A a B b C c} S Df Dgf Agf Bgf Afa a Cgf aa 35

36 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} f = {A a B b C c} S Df Dgf Agf Bgf Afa Bfb a b aa bb Cgf 36

37 An Example Grammar S Df D Dg ABC g = {A Aa B Bb C Cc} f = {A a B b C c} S Df Dgf Agf Bgf Cgf Afa Bfb Cfc c a b aa bb cc 37

38 Where am I? { anbncn n = 1...}... CS *I* NCF DCF REG REG 38

39 ... CS *I* DCF NCF Machines PDA Formalisms S asb z Pumps REG REG DFA a(ab)*b 39

40 ... CS *I* DCF NCF NSA PDA S asfb z S asb z? REG REG DFA a(ab)*b 40

41 Associated Automaton 41

42 Associated Automaton 42

43 (Actual NSA) [Gilman, Shapiro 1998] 43

44 But will it pump? It's complicated on derivation trees rather than strings [Hayashi 1973] Several corollaries exist [Gilman 1996] (less general; easier to use) 44

45 The Lemma L : indexed language m : positive integer There is a constant k > 0 so that each w in L can be split (w = w1w2...wr) subject to:

46 The Lemma L : indexed language m : positive integer There is a constant k > 0 so that each w in L can be split (w = w1w2...wr) subject to: m<r k each wi > 0

47 The Lemma L : indexed language m : positive integer There is a constant k > 0 so that each w in L can be split (w = w1w2...wr) subject to: m<r k each wi > 0 any m-sized set of wi's is a subset of some w' in L; w' is a subproduct of w

48 Lemma Example

49 Lemma Example

50 (Montage Time) 50

51 { anbncn n = 1...} { (abn)n n = 1...}... CS *I* DCF NCF NSA PDA S asfb z S asb z? REG REG DFA a(ab)*b 51

52 52

53 References Indexed Grammars An Extension to Context-Free Grammars (Aho) Nested Stack Automata (Aho) Sequentially Indexed Grammars (van Eijck) A Shrinking Lemma for Indexed Languages (Gilman) On Groups Whose Word Problem is Solved by a Nested Stack Automaton (Gilman and Shapiro) On Derivation Trees of Indexed Grammars (Hayashi) 53