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COSC 3340: Introduction to Theory of Computation

COSC 3340: Introduction to Theory of Computation. University of Houston Dr. Verma Lecture 10. Chomsky Normal Form (CNF). Rules of CFG G are in one of two forms: (i) A  a (ii) A  BC, B  S and C  S + Only one rule of the form S   is allowed if  in L(G).

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COSC 3340: Introduction to Theory of Computation

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  1. COSC 3340: Introduction to Theory of Computation University of Houston Dr. Verma Lecture 10 UofH - COSC 3340 - Dr. Verma

  2. Chomsky Normal Form (CNF) • Rules of CFG G are in one of two forms: (i) A  a (ii) A  BC, B  S and C  S + Only one rule of the form S  is allowed if  in L(G). • Easier to reason with in proofs • Leads to more efficient algorithms • Credited to Prof. Noam Chomsky at MIT Reading Assignment: Converting a CFG to CNF. UofH - COSC 3340 - Dr. Verma

  3. Exercises • Are the following CFG's in CNF? (i) SaSb |  (ii) SaS | Sb |  (iii) SAS | SB |  Aa Bb (iv) SAS | SB Aa |  Bb UofH - COSC 3340 - Dr. Verma

  4. Closure properties of CFL's • CFL's are closed under: (i) Union (ii) Concatenation (iii) Kleene Star • What about intersection and complement? UofH - COSC 3340 - Dr. Verma

  5. The setting • L1= L(G1) where G1 = (V1, T, P1, S1) • L2 = L(G2) where G2 = (V2, T, P2, S2) • Assume wlog that V1V2 =  UofH - COSC 3340 - Dr. Verma

  6. Closure under Union -- Example • L1 = { anbn | n  0 } • L2 = { bnan | n  0 } • G1 ? • Ans: S1aS1b |  • G2? • Ans: S2bS2a |  • How to make grammar for L1L2? • Ans: Idea: Add new start symbol S and rules SS1| S2 UofH - COSC 3340 - Dr. Verma

  7. Closure under Union General construction • Let G = (V, T, P, S) where • V = V1V2 { S }, • SV1V2 • P = P1P2 { SS1 | S2 } UofH - COSC 3340 - Dr. Verma

  8. Closure under concatenation Example • L1 = { anbn | n  0 } • L2 = { bnan | n  0 } • Is L1L2 = { anb{2n}an | n >= 0 } ? • Ans: No! It is { anb{n+m}am | n, m  0 } • How to make grammar for L1L2 ? • Idea: Add new start symbol and rule SS1S2 UofH - COSC 3340 - Dr. Verma

  9. Closure under concatenation General construction • Let G = (V, T, P, S) where • V = V1V2 { S }, • SV1V2 • P = P1P2 { SS1S2 } UofH - COSC 3340 - Dr. Verma

  10. Closure under kleene star Examples • L1 = {anbn | n 0} • What is (L1)*? • Ans: { a{n1}b{n1} ... a{nk}b{nk} | k 0 and ni 0 for all i } • L2 = { a{n2} | n 1 } • What is (L2)*? • Ans: a*.Why? • How to make grammar for (L1)*? • Idea: Add new start symbol S and rules SSS1 | . UofH - COSC 3340 - Dr. Verma

  11. Closure under kleene star General construction • Let G = (V, T, P, S) where • V = V1 { S }, • SV1 • P = P1 { SSS1| } UofH - COSC 3340 - Dr. Verma

  12. Tips for Designing CFG's • Use closure properties -- divide and conquer • Analyze strings – Is order important? Number? Do we need recursion? • Flat versus hierarchical? • Are any possibilities (strings) missing? • Is the grammar generating too many strings? UofH - COSC 3340 - Dr. Verma

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