CSC 3130: Automata theory and formal languages

Andrej Bogdanov

http://www.cse.cuhk.edu.hk/~andrejb/csc3130

The Chinese University of Hong Kong

Normal forms and parsing

Fall 2009

Testing membership and parsing

• Given a grammar

• How can we know if a string x is in its language?

• If so, can we obtain a parse tree for x?

• Can we tell if the parse tree is unique?

S → 0S1 | 1S0S1 | TT → S | e

First attempt

• Maybe we can try all possible derivations:

S → 0S1 | 1S0S1 | TT → S | x = 00111

S 0S1

1S0S1

T

00S1101S0S110T1

S

10S10S1...

when do we stop?

Problems

• How do we know when to stop?

S → 0S1 | 1S0S1 | TT → S | x = 00111

S 0S1

1S0S1

00S1101S0S110T1

10S10S1...

when do we stop?

Problems

• Idea: Stop derivation when length exceeds |x|

• Not right because of -productions

• We might want to eliminate -productions too

S → 0S1 | 1S0S1 | TT → S | x = 01011

S 0S1 01S0S11 01S011 010111 3 7 6 5

Problems

• Loops among the variables (S → T → S) might make us go forever

• We want to eliminate such loops

S → 0S1 | 1S0S1 | TT → S | x = 00111

Removal of -productions

• A variable N is nullable if there is a derivation

• How to remove -productions (except from S)

Find all nullable variables N1, ..., Nk

For every production of the form A → Ni,

add another production A → If Ni → is a production, remove it

If S is nullable, add the special production S →

N *

Example

• Find the nullable variables

S ACDA aB C ED | D BC | bE b

B C D

nullable variablesgrammar

Find all nullable variables N1, ..., Nk

Finding nullable variables

• To find nullable variables, we work backwards– First, mark all variables A s.t. A as nullable– Then, as long as there are productions of the form

where all of A1,…, Ak are marked as nullable, mark A as nullable

A → A1… Ak

Eliminating -productions

S ACDA aB C ED | D BC | bE b

nullable variables: B, C, D

For every production of the form A → Ni,

add another production A → If Ni → is a production, remove it

D CS ADD BD S ACS AC E

Dealing with loops

• A unit production is a production of the form

where A1 and A2 are both variables

• Example

A1 → A2

S → 0S1 | 1S0S1 | TT → S | R | R → 0SR

grammar: unit productions:

S T

R

Removal of unit productions

• If there is a cycle of unit productions

delete it and replace everything with A1

• Example

A1 → A2 → ... → Ak → A1

S → 0S1 | 1S0S1 | TT → S | R | R → 0SR

S T

R

S → 0S1 | 1S0S1S → R | R → 0SR

T is replaced by S in the {S, T} cycle

Removal of unit productions

• For other unit productions, replace every chain

by productions A1 → ,... , Ak →

• Example

A1 → A2 → ... → Ak →

S → R → 0SR is replaced by S → 0SR, R → 0SR

S → 0S1 | 1S0S1 | R | R → 0SR

S → 0S1 | 1S0S1 | 0SR | R → 0SR

Recap

• After eliminating -productions and unit productions, we know that every derivation

doesn’t shrink in length and doesn’t go into cycles

• Exception: S → – We will not use this rule at all, except to check if L

• Note -productions must be eliminated before unit

productions

S a1…ak where a1, …, ak are terminals*

Example: testing membership

S → 0S1 | 1S0S1 | TT → S |

x = 00111

S → | 01 | 101 | 0S1 |10S1 | 1S01 | 1S0S1

S 01, 101

10S1

1S01

1S0S1

10011, strings of length ≥ 6

10101, strings of length ≥ 6

unit, -prod

eliminate

only strings of length ≥ 6

0S1 0011, 0101100S11strings of length ≥ 6

only strings of length ≥ 6

Algorithm 1 for testing membership• How to check if a string x ≠ is in L(G)

Eliminate all -productions and unit productionsLet X := SWhile some new rule R can be applied to X

Apply R to XIf X = x, you have found a

derivation for xIf |X| > |x|, backtrack

If no more rules can be applied to X, x is not in L

Practical limitations of Algorithm I

• This method can be very slow if x is long

• There is a faster algorithm, but it requires that we do some more transformations on the grammar

G = CFG of the java programming languagex = code for a 200-line java program

algorithm might take about 10200 steps!

Chomsky Normal Form

• A grammar is in Chomsky Normal Form if every production (except possibly S → ) is of the type

• Conversion to Chomsky Normal Form is easy:

A → BC A → aor

A → BcDEreplace terminalswith new variables

A → BCDEC → c

break upsequenceswith new variables

A → BX1

X1 → CX2

X2 → DEC → c

Exercise

• Convert this CFG into Chomsky Normal Form:

S |ADDAA aC cD bCb

Algorithm 2 for testing membership

S AB | BCA BA | aB CC | bC AB | a

x = baaba

Idea: We generate each substring of x bottom up

ab b aa

ACB B ACAC

BSA SASC

B– B

SAC–

SAC

Parse tree reconstruction

S AB | BCA BA | aB CC | bC AB | a

x = baabaab b aa

ACB B ACAC

BSA SASC

B– B

SAC–

SAC

Tracing back the derivations, we obtain the parse tree

Cocke-Younger-Kasami algorithm

For cells in last row If there is a production A xi

Put A in table cell iiFor cells st in other rows If there is a production A BC where B is in cell sj and C is in cell jt Put A in cell st

x1 x2 … xk

11 22 kk12 23

… …1k

tablecells

s j t k1

Input: Grammar G in CNF, string x = x1…xk

Cell ij remembers all possible derivations of substring xi…xj