CSCI 3130: Automata theory and formal languages

Fall 2011 The Chinese University of Hong Kong CSCI 3130: Automata theory and formal languages AmbiguityParsing algorithms Andrej Bogdanov http://www.cse.cuhk.edu.hk/~andrejb/csc3130

Ambiguity E  E + E | E * E | (E)| N N  1N | 2N | 1 | 2  E 1+2*2 E E E * E + E + E E N = 5 = 6 N E * E N N 2 N N 1 A CFG is ambiguous if some string has more than one parse tree 1 2 2 2

Example Is S  SS | x ambiguous? Yes, because S S S S S S S S S S xxx x x x x x x

Disambiguation S  SS | x S  Sx | x S S S x x x Sometimes we can rewrite the grammar to remove ambiguity

Disambiguation E  E + E | E * E | (E)| N N  1N | 2N | 1 | 2 same precedence! F F T T Divide expression into terms and factors F F 2 * (1 + 2 * 2)

Disambiguation E  E + E | E * E | (E)| N N  1N | 2N | 1 | 2 An expression is a sum of one or more terms E  T | E + T Each term is a product of one or more factors T  F | T * F Each factor is a parenthesized expression or a number F  (E) |1 | 2

Parsing example E E  T | E + T T  F | T * F F  (E) |1 | 2 E T + T F F T * E ( ) F E + T E T T F + * T F F F 2 * (1 + 1 + 2 * 2) + 1

Disambiguation • Disambiguation is not always possible because • There exist inherently ambiguous languages • There is no general procedure for disambiguation • In programming languages, ambiguity comes from precedence rules, and we can do like in example • In English, ambiguity is sometimes a problem: He ate the cookies on the floor

Ambiguity in English He ate the cookies on the floor

Parsing S → 0S1 | 1S0S | T T → S | e input: 0011 How would we program the computer to build a parse tree for us?

Parsing S → 0S1 | 1S0S | T T → S | e input: 0011 ... S 0S1 00S11 000S111 ⇒ ⇒ ⇒ ⇒ 01S0S1 ⇒ ... ⇒ 0T1 00T11 ⇒ 00S11 ✔ ⇒ 0011 1S0S 10S10S ⇒ ⇒ ... ... T S ⇒ ⇒ ⇒  First idea: Try all derivations

Problems Trying all derivations may take a very long time  If input is not in the language, parsing will never stop 

When to stop Idea 2: Stop when S → 0S1 | 1S0S | T T → S | e |derived string| > |input| Problems: S  T  S  T  … S  0S1 0T1  01 Derived strings may shrink because of “e-productions” 1 3 3 Derivation may loop because of “unit productions” 2 Task: remove e and unit productions

Removal of -productions • A variable N is nullable if it derives the empty string * N • Identify all nullable variables N  • Remove nullable variables carefully  • If start variable S is nullable: Add a new start variable S’Add special productionsS’ → S |  

Example grammar nullable variables B C D S  ACD A a B   C  ED |  D  BC | b E  b • Repeat the following: • If X  , mark X as nullable • If X  YZ…W, all marked nullable, • mark X as nullable also. • Identify all nullable variables 

Eliminating e-productions D  C S  AD D  B D  e S  AC S  A C  E S  ACD A a B   C  ED |  D  BC | b E  b • For every nullable N: • If you see X → N, add X →  nullable:B, C, D • If you see N → , remove it. • Remove nullable variables carefully 

Eliminating unit productions • A unit production is a production of the form A → B grammar: unit productions graph: S → 0S1 | 1S0S | T T → S | R |  R → 0SR S T R

Removal of unit productions • If there is a cycle of unit productionsdelete it and replace everything with A  A → B→ ... → C→ A S T  S → 0S1 | 1S0S | T T → S | R |  R → 0SR S → 0S1 | 1S0S S → R |  R → 0SR  R replace T by S

Removal of unit productions • Replace every chainby A → , B→ ,... , C →   A → B→ ... → C→  S S → 0S1 | 1S0S | R |  R → 0SR S → 0S1 | 1S0S | 0SR |  R → 0SR R S → R → 0SR is replaced by S → 0SR, R → 0SR

Recap If input is not in the language, parsing will never stop Problem: Solution: Eliminate  productions Eliminate unit productions important to do in this order Try all possible derivations but stop parsing when |derived string| > |input|

Example S → 0S1 | 0S0S | T T → S | 0 S → 0S1 | 1S0S | 0 input:0011 conclusion: 0011∉ L S 0 ⇒ ✘ 001 0S1 ✘ ⇒ ⇒ too long 00S11 too long 00S0S1 0S0S 000S 0000 ⇒ ✘ ⇒ ⇒ too long too long 1000S1 00S10S too long too long 1000S0S 00S0S0S

Problems Trying all derivations may take a very long time  If input is not in the language, parsing will never stop 

Preparations • To use it we must prepare the CFG: A faster way to parse:the Cocke-Younger-Kasami algorithm Eliminate  productions Eliminate unit productions Convert CFG to Chomsky Normal Form

Chomsky Normal Form • A CFG is in Chomsky Normal Formif every production* has the form A → a A → BC or Noam Chomsky Convert to Chomsky Normal Form: A → BcDE A → BX X → CY Y → DE A → BCDE C → c break up sequences with new variables replace terminals with new variables C → c * Exception: We allow S→ e for start variable only

Cocke-Younger-Kasami algorithm SAC S  AB | BC A  BA | a B  CC | b C  AB | a – SAC – B B SA B SC SA B AC AC B AC x = baaba b a a b a Idea: We generate each substring of x bottom up

SAC – SAC – B B SA B SC SA B AC AC B AC b a a b a Parse tree reconstruction S  AB | BC A  BA | a B  CC | b C  AB | a x = baaba Tracing back the derivations, we obtain the parse tree

Cocke-Younger-Kasami algorithm Grammar without e and unit productions in Chomsky Normal Form table cells Input stringx = x1…xk 1k … … • For all cells in last rowIf there is a production A  xiPutA in table cell ii • For cells st in other rows If there is a production A  BC whereB is in cell sj and C is in cell (j+1)tPutA in cell st 23 12 22 kk 11 x1 x2 … xk s j t k 1 Cell ij remembers all possible derivations of substring xi…xj

CSCI 3130: Automata theory and formal languages