Simplification of CFG and Normal Forms Wen-Guey Tzeng Computer Science Department National Chiao Tung University
Normal Forms • We want a cfg with either Chomsky or Greibach normal form – Chomsky normal form • A a, A BC – Greibach normal form • A ax, x V* 2017 Spring 2
• CFG with normal forms are easier for parsing – The membership problem – Given a grammar G and a string w, find the parsing tree for w if a parsing tree exists. w = x+y*z 2017 Spring 3
• -free languages – A language that does not contain • We consider CFG G such that L(G) is -free • For any cfg G, there is G’ such that L(G’)=L(G)-{ } 2017 Spring 4
Transformation to normal forms: steps Remove (1) -productions CFG G=(V, T, P, S) Convert G’ to ( -free context- (2) unit-productions normal forms (3) useless productions free language) from P to get G’ 2017 Spring 5
A substitution rule • For A B, A x 1 Bx 2, B y 1 |y 2 |…|y n is equivalent to A x 1 y 1 x 2 |x 1 y 2 x 2 |…|x 1 y n x 2 , B y 1 |y 2 |…|y n • Example – A a|aaA|abBc, B abbA|b is equivalent to A a|aaA|ababbAc|abbc, B abbA|b 2017 Spring 6
Remove -productions • -production: A • Nullable variable A: A * • Steps 1. Find the nullable variable set V N 2. For each A x 1 x 2 …x m , x i V T, • For each combination x i , x j , …, x k of variables in V N add A x 1 …x i-1 x i+1 … x j-1 x j+1 ... x k-1 x k+1 …x m Note: don’t add A , if all x i are in V N • 2017 Spring 7
Example • S ABaC, A BC, B b| , C D| , D d • Nullable set V N ={A, B, C} • Add productions 2017 Spring 8
Remove unit-productions • unit-production: A B • Steps – Remove A A immediately – Draw dependency graph for variables A and B with: A *B – For A *B and B y 1 |y 2 |…|y n • Add A y 1 |y 2 |…|y n – Remove all A B, where A and B are in dependency graph 2017 Spring 9
Example • S Aa|B, B A|bb, A a|bc|B 1. Draw dependency graph 2. Add S bb|a|bc A bb B a|bc 3. Remove S B, B A, A B 4. Finally S a|bc|bb|Aa A a|bc|bb B a|bc|bb 2017 Spring 10
Remove useless productions • A variable A V is useful if S can generate some terminal string through it. – That is, S * xAy * w, w T* • Example – S aSb|AB|Ba, A aA, B b|Bb, C cB|c – S Ba ba. Thus, B is useful. – S is useful. – But, A and C are not useful (useless) 2017 Spring 11
• Two cases for useless variables – Case 1: variables that cannot generate strings in T* • S aSb|AB|Ba, A aA, B b|Bb, C cB|c • Algorithm (finding variables that generate strings) 1. V 1 ={} 2. For rule A x, x (T V 1 )*, add A to V 1 3. Repeat 2 until no rules can be added to V 1 • V 1 ={S, B, C} • S aSb|Ba, B b|Bb, C cB|c 2017 Spring 12
– Case 2: variables that cannot be reached from S • S aSb|Ba, B b|Bb, C cB|c • Algorithm: dependency graph S B C • C is un-reachable from S. • S aSb|Ba, B b|Bb 2017 Spring 13
• Algorithm (removing useless productions) Input: G=(V, T, P, S) 1. Find the useless variables in Case 1 and remove related useless productions. 2. Find the useless (un-reachable) variables in Case 2 and remove the related useless productions 2017 Spring 14
Chomosy normal form • A cfg is in Chomsky normal form (CNF) if all productions are of form A BC, or A a • Example – S AS|a, A SA|b • Every cfg G, with L(G), has an equivalent CNF grammar. 2017 Spring 15
Converting into CNF 1. Apply the rules of removing -, unit-, and useless-productions 2. Convert the productions into the form A C 1 C 2 …C n , or A a 3. Convert A C 1 C 2 …C n into A C 1 D 1 , D 1 C 2 D 2 , …, D n-2 C n-1 C n 2017 Spring 16
Example • S ABa, A aab, B Ac • Step 2: • Step 3: 2017 Spring 17
Greibach normal form • A cfg is in Greibach normal form (GNF) if all productions are of form A aB 1 B 2 …B n, n 0 • Example – S aBC, B aBA, A a|bBSC • Every cfg G, with L(G), has an equivalent GNF grammar. 2017 Spring 18
Example • Example – S AB, A aA|bB|b, B b – Result • S aAB|bBB|bB, A aA|bB|b, B b • Example – S abSb|aa – Result • S aBSB|aA, B b, A a 2017 Spring 19
Parsing (membership) • Question: Given a CFG G in Chomsky normal form and a string w, determine whether w L(G) • Idea: the dynamic programming technique – A large problem is decomposed into smaller problems – Combine solutions to smaller problems into a solution for the large problem 2017 Spring 20
• Assume w=a 1 a 2 …a n • Main problem: V 1n ={V: V * w} • w L(G) if and only S V 1n • Use the dynamic programming technique – V ij ={ V : V * a i a i+1 …a j }: variables that can generate substring a i a i+1 …a j • Solve smaller problems V ik , V k+1,j , for k=i, i+1,…, j-1 • Combine them to compute V ij – V ij = {A ： A BC, B V ik , C V k+1,j , i k<j} 2017 Spring 21
w = a 1 a 2 a 3 … a i a i+1 … a j-1 a j … a n V ij contains the variables that generate a i a i+1 …a j-1 a j a i a i+1 … a k a k+1 a k+2 … a j-1 a j V i i V i+1 i . . . V k+1 j V i k V i k+1 V k+2 j . . . 2017 Spring 22
CYK Algorithm • Input: G=(V, T, S, P) is in CNF and w=a 1 a 2 …a n – Compute V ij ={ A V : A * a i a i+1 …a j } • V 11 , V 22 , …, V nn • V 12, V 23 , …, V n-1n • … • V 1n 1. Smallest problem: add A to V ii • if A a i is a production in P 2. Bigger problem: add to A to V ij if • For some k, i k j-1, A BC in P, B in V ik , C in V k+1 j 3. w L(G) if and only if S V 1n 2017 Spring 24
Example • S AB, A BB|a, B AB|b • w=aabbb • Steps – V 11 ={A}, V 22 ={A}, V 33 ={B}, V 44 ={B}, V 55 ={B} – V 12 = , V 23 ={S, B}, V 34 ={A}, V 45 ={A} – V 13 ={S, B}, V 24 ={A}, V 35 ={S, B} – V 14 ={A}, V 25 ={S, B} – V 15 ={S, B} 2017 Spring 25
• Triangular table (n=5) V 1,5 V 1,4 V 2,5 V 1,3 V 2,4 V 3,5 V 1,2 V 2,3 V 3,4 V 4,5 V 1,1 V 2,2 V 3,3 V 4,4 V 5,5 2017 Spring 23
Sum up • Context-free grammars are used in designing programming languages, such as , C, PSACAL, etc. • Membership problem in CFG is equivalent to the parsing problem in programming languages • Normal forms are needed for “automatically” generating a “parser” for the programming language 2017 Spring 26
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