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

UofH - COSC 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.

Lecture 10UofH - COSC 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

Lecture 10UofH - COSC 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?

Lecture 10UofH - COSC Dr. Verma 5 The setting L 1 = L(G 1 ) where G 1 = (V 1, T, P 1, S 1 ) L 2 = L(G 2 ) where G 2 = (V 2, T, P 2, S 2 ) Assume wlog that V 1  V 2 = 

Lecture 10UofH - COSC Dr. Verma 6 Closure under Union -- Example L 1 = { a n b n | n  0 } L 2 = { b n a n | n  0 } G 1 ? – Ans: S 1  aS 1 b |  G 2 ? – Ans: S 2  bS 2 a |  How to make grammar for L 1  L 2 ? – Ans: Idea: Add new start symbol S and rules S  S 1 | S 2

Lecture 10UofH - COSC Dr. Verma 7 Closure under Union General construction Let G = (V, T, P, S) where – V = V 1  V 2  { S }, – S  V 1  V 2 – P = P 1  P 2  { S  S 1 | S 2 }

Lecture 10UofH - COSC Dr. Verma 8 Closure under concatenation Example L 1 = { a n b n | n  0 } L 2 = { b n a n | n  0 } Is L 1 L 2 = { a n b {2n} a n | n >= 0 } ? – Ans: No! It is { a n b {n+m} a m | n, m  0 } How to make grammar for L 1 L 2 ? – Idea: Add new start symbol and rule S  S 1 S 2

Lecture 10UofH - COSC Dr. Verma 9 Closure under concatenation General construction Let G = (V, T, P, S) where – V = V 1  V 2  { S }, – S  V 1  V 2 – P = P 1  P 2  { S  S 1 S 2 }

Lecture 10UofH - COSC Dr. Verma 10 Closure under kleene star Examples L 1 = {a n b n | n  0} What is (L 1 )*? – Ans: { a {n1} b {n1}... a {nk} b {nk} | k  0 and ni  0 for all i } L 2 = { a {n 2 } | n  1 } What is (L 2 )*? – Ans: a*. Why? How to make grammar for (L 1 )*? – Idea: Add new start symbol S and rules S  SS 1 | .

Lecture 10UofH - COSC Dr. Verma 11 Closure under kleene star General construction Let G = (V, T, P, S) where – V = V 1  { S }, – S  V 1 – P = P 1  { S  SS 1 |  }

Lecture 10UofH - COSC 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?