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Tutorial 05 -- CSC3130 : Formal Languages and Automata Theory

Tutorial 05 -- CSC3130 : Formal Languages and Automata Theory. Tu Shikui ( sktu@cse.cuhk.edu.hk ) SHB 905, Office hour: Thursday 2:30pm-3:30pm 2008-10-06. Outline. Context-free Languages, Context-free grammars (CFG), Push-down Automata (PDA) Grammar Transformations Remove ε - production;

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Tutorial 05 -- CSC3130 : Formal Languages and Automata Theory

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  1. Tutorial 05-- CSC3130 : Formal Languages and Automata Theory Tu Shikui (sktu@cse.cuhk.edu.hk) SHB 905, Office hour: Thursday 2:30pm-3:30pm 2008-10-06

  2. Outline • Context-free Languages, Context-free grammars (CFG), Push-down Automata (PDA) • Grammar Transformations • Remove ε- production; • Remove unit-production; • Covert to Chomsky-Normal-Form (CNF) • Cocke-Younger-Kasami (CYK) algorithm

  3. Relations L = L(G) Context-free Languages L Context-free Grammars G L(G) = L(M) L = L(M) Push-down Automata M

  4. Example (I) • Given the following CFG S  X | Y X  aXb | aX | a Y  aYb | Yb | b • (1) L(G) = ? • (2) Design an equivalent PDA for it. Σ={a, b}

  5. Example (I) --- solution: L(S) S  X |Y X aXb | aX | a Y aYb | Yb | b Try to write some strings generated by it: SXaXbaaXbbaaaXbbaaaabb SYaYbaYbbaaYbbbaabbbb more a’s than b’s more b’s than a’s • Observations: • Start from S, we can enter two States X & Y, and X, Y are “independent”; • In X state, always more a are generated; • In Y state, always more b are generated. Ls = Lx U Ly L(S) = { aibj; i≠j } Lx = { aibj; i>j } Lx = { aibj; i<j }

  6. a,e/A b,A/e b,#/# a,e/A b,A/e e,A/e e,#/e e,#/e e,e/e e,e/# e,e/# e,e/e e,A/e b,#/# q0 q’0 q1 q’1 q2 q’2 q3 q’3 Example (I) --- solution: PDA L(S) = { aibj; i≠j } S  X | Y X aXb | aX | a Y aYb | Yb | b = { aibj; i>j } U { aibj; i<j } PDA = NFA + a stack (infinite memory) A possible way: “divide and conquer” Lx = { aibj; i>j } LY = { aibj; i<j } e, e /e Combine both …

  7. Example (II) • Given the following language: • (1) design a CFG for it; • (2) design a PDA for it. • L = {0i1j: i ≤ j ≤ 2i, i=0,1,…}, S = {0, 1}

  8. Example (II) -- solution: CFG L = {0i1j: i ≤ j ≤ 2i, i=0,1,…}, S = {0, 1} Consider two extreme cases: (a). if j = i, then L1 = { 0i1j: i=j }; (b). if j = 2i, then L2 = { 0i1j: 2i=j }. S  0S1 S  ε S  0S11 S  ε “blue-rule” “red-rule” If i ≤ j ≤ 2i , then randomly choose “red-rule” or “blue-rule” in the generation. S  0S1 S  0S11 S  ε

  9. Example (II) -- solution: CFG L = {0i1j: i ≤ j ≤ 2i, i=0,1,…}, S = {0, 1} ? Need to verify L = L(G) G = S  0S1 S  0S11S  ε 1). L(G) is a subset of L: The “red-rule” and “blue-rule” guarantee that in each derivation, the number of 1s generated is one or two times larger than that of 0s. So, L(G) is a subset of L. 2). L is a subset of L(G): For any w = 0i1j, i ≤ j ≤ 2i, we use “red-rule” (2i - j) times and then “blue-rule” ( j - i ) times, i.e., S =*=> 02i-jS12i-j =*=> 02i-j0j-iS12(j-i)12i-j==> 0i1j = w

  10. 1,X/e 0,e/X q1 e,e/e e,#/e e,e/# 1,X/X 1,X/e q0 q2 q3 Example (II) -- solution: PDA L = {0i1j: i ≤ j ≤ 2i, i=0,1,…}, S = {0, 1} Similar idea: “randomly choose two extreme cases”

  11. Example (III) • Given the following language: • (1). Design a CFG for it; • (2). Design a PDA for it. L = { aibjckdl: i,j,k,l=0,1,…; i+k=j+l }, where the alphabet Σ= {a, b, c, d}

  12. w an dn ajbj ckdk Example (III) – solution: CFG L = { aibjckdl: i,j,k,l=0,1,…; i+k=j+l }, Note that i + k = j + l ==> | i – j | = | l – k |. Assume n = i – j = l – k > 0, then i = n + j, l = n + k, and w = aibjckdl = anajbj ckdkdn S  aSd | XY X  aXb |ε Y  cYd|ε Three blocks come from the same template: N  x N x S=*=>anSdn=*=> anXYdn =*=>anajbjYdn =*=>anajbj ckbkdn aj bj ck dk = an+jbj ckbn+k (n+j) + k = j + (n+k)

  13. Example (III) – solution: PDA L = { aibjckdl: i,j,k,l=0,1,…; i+k=j+l }, • Main idea: • use X to record an a or c; use Y to record an b or d. • Compare #X and #Y: by cancellation. How to realize the comparison by cancellation? Action1: Push an X, when a or c was read; Action2: Pop an X (if any, otherwise push a Y), when b or d was read. Action3: Pop an Y (if any, otherwise push an X), when a or c was read.

  14. c,#/X# b,X/e a,e/X d,X/e c,X/XX q1 q3 q4 q2 e,e/e e,e/e e,e/e e,#/e e,e/# b,#/Y# c,Y/e d,#/Y# q5 b,Y/YY d,Y/YY Example (III) – solution: PDA L = { aibjckdl: i,j,k,l=0,1,…; i+k=j+l }, Action1: Push an X, when a or c was read; Action2: Pop an X (if any, otherwise push a Y), when b or d was read. Action3: Pop an Y (if any, otherwise push an X), when a or c was read.

  15. Outline • Context-free Languages, Context-free grammars (CFG), Push-down Automata (PDA) • Grammar Transformations • Remove ε- production; • Remove unit-production; • Covert to Chomsky-Normal-Form (CNF) • Cocke-Younger-Kasami (CYK) algorithm

  16. Remove ε-production • Example: Remove ε-productions of the following grammar G: S ABaC A  BC B  b |ε C  D |ε D  d

  17. For A: S  BaC For B: S  AaC | aC A  C For C: S  ABa | Ba | Aa | a A  B Remove ε-production Nullable variables: A, B, C Add Delete S ABaC | BaC | AaC | aC | ABa | Ba | Aa | a A  BC | C | B B  b |ε C  D |ε D  d B  ε C  ε • For i = 1 to k • For every production of the form A → Ni, • add another production A →  • If Ni →  is a production, remove it

  18. Remove unit-productions • Example: Remove the unit-productions of the following grammar: S  S + T | T T  T * F | F F  (S) | a

  19. For T T*F S  T*F For T F  a S  a For T F  (S) S  (S) For F  a S  a For F  (S) S  (S) Remove unit-productions Delete Add S  T S S+T | T | T*F | a | (S) T  T*F | F | a | (S) F  (S) | a T  F A1 → A2 → ... → Ak→ A1 delete it and replace everything with A1 Rule 1: A1 → A2 → ... → Ak→  Rule 2: A1 → ,... , Ak → 

  20. Convert CFG to Chomsky-Normal-Form (CNF) • Example: Convert the following CFG to CNF: S  0AB A  0D | 1AD B  0 D  1

  21. S  XAB; X  0 A  XD | YAD; Y  1 Convert CFG to Chomsky-Normal-Form (CNF) A → BcDE A → BX1 X1→ CX2 X2→ DE A → BCDE C → c break up sequences with new variables replace terminals with new variables C → c S  0AB A  0D | 1AD B  0 D  1

  22. S  XN1; N1  AB A  YN2; N2  AD Convert CFG to Chomsky-Normal-Form (CNF) A → BcDE A → BX1 X1→ CX2 X2→ DE A → BCDE C → c break up sequences with new variables replace terminals with new variables C → c S  XAB X  0 A  XD |YAD Y  1 B  0 D  1

  23. Convert CFG to Chomsky-Normal-Form (CNF) A → BcDE A → BX1 X1→ CX2 X2→ DE A → BCDE C → c break up sequences with new variables replace terminals with new variables C → c S  XN1 N1  AB A  XD A  YN2 N2  AD X  0 Y  1 B  0 D  1

  24. Outline • Context-free Languages, Context-free grammars (CFG), Push-down Automata (PDA) • Grammar Transformations • Remove ε- production; • Remove unit-production; • Covert to Chomsky-Normal-Form (CNF) • Cocke-Younger-Kasami (CYK) algorithm

  25. Cocke-Younger-Kasami (CYK) algorithm • Constructing the parse table • Input:w = a1a2…an ∈ Σ+ • Output: The parse table T for w such that tij contains A ⇔ A +⇒ aiai+1…ai+j-1 • Steps: • Step(1):cell(i,1) = { A| A→ai ∈ P, 1≤i≤n } • Step(2): for 1 ≤ k < j, cell(i,j) = { A| for some k, A→BC ∈ P, B is in cell(i,k), C is in cell(i+k, j-k)} • Step(3): repeat Step(2) for 1≤ i≤ n, 1≤ j ≤ n-i+1

  26. Cocke-Younger-Kasami (CYK) algorithm Cell (i,k), Cell (i+k, j-k) k = 1, …, j-1 x1 x2 x3 x4 x5

  27. Cocke-Younger-Kasami (CYK) algorithm • S → AA | AS | b • A → SA | AS | a Test: w = abaab a b a a b

  28. Cocke-Younger-Kasami (CYK) algorithm Trace back to find a parse tree • S → AA | AS | b • A → SA | AS | a Test: w = abaab a b a a b

  29. Cocke-Younger-Kasami (CYK) algorithm We can find another parse tree. • S → AA | AS | b • A → SA | AS | a Test: w = abaab a b a a b

  30. End of this tutorial!Thanks for coming!

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