672021 Dr Sandip Mal 1 ContextFree Languages 672021

6/7/2021 Dr. Sandip Mal 1

Context-Free Languages 6/7/2021 Dr. Sandip Mal 2

Context-Free Languages Regular Languages 6/7/2021 Dr. Sandip Mal 3

Context-Free Languages Context-Free Grammars Pushdown Automata stack automaton 6/7/2021 Dr. Sandip Mal 4

Context-Free Grammars 6/7/2021 Dr. Sandip Mal 5

Example Sequence of terminals and variables Grammar: Variable 6/7/2021 The right side may be Dr. Sandip Mal 6

Grammar: Derivation of string 6/7/2021 : Dr. Sandip Mal 7

Grammar: Derivation of string 6/7/2021 : Dr. Sandip Mal 8

Grammar: Other derivations: 6/7/2021 Dr. Sandip Mal 9

Grammar: Language of the grammar: 6/7/2021 Dr. Sandip Mal 10

A Convenient Notation We write: for zero or more derivation steps Instead of: 6/7/2021 Dr. Sandip Mal 11

In general we write: If: in zero or more derivation steps Trivially: 6/7/2021 Dr. Sandip Mal 12

Example Grammar 6/7/2021 Possible Derivations Dr. Sandip Mal 13

Another convenient notation: 6/7/2021 Dr. Sandip Mal 14

Formal Definitions Grammar: Set of variables 6/7/2021 Set of terminal symbols Start variable Dr. Sandip Mal Set of productions 15

Context-Free Grammar: All productions in are of the form Variable 6/7/2021 String of variables and terminals Dr. Sandip Mal 16

Example of Context-Free Grammar productions variables 6/7/2021 terminals Dr. Sandip Mal start variable 17

Language of a Grammar: For a grammar with start variable String of terminals or 6/7/2021 Dr. Sandip Mal 18

Example: context-free grammar : Since, there is derivation for any 6/7/2021 Dr. Sandip Mal 19

Context-Free Language: A language is context-free if there is a context-free grammar with 6/7/2021 Dr. Sandip Mal 20

Example: is a context-free language since context-free grammar : generates 6/7/2021 Dr. Sandip Mal 21

Another Example Context-free grammar : Example derivations: Palindromes of even length 6/7/2021 Dr. Sandip Mal 22

Another Example Context-free grammar : Example derivations: Describes matched parentheses: 6/7/2021 () ((( ))) (( )) Dr. Sandip Mal 23

Derivation Order and Derivation Trees 6/7/2021 Dr. Sandip Mal 24

Derivation Order Consider the following example grammar with 5 productions: 6/7/2021 Dr. Sandip Mal 25

Leftmost derivation order of string : At each step, we substitute the leftmost variable 6/7/2021 Dr. Sandip Mal 26

Rightmost derivation order of string : At each step, we substitute the rightmost variable 6/7/2021 Dr. Sandip Mal 27

Leftmost derivation of Rightmost derivation of 6/7/2021 Dr. Sandip Mal : : 28

Derivation Trees Consider the same example grammar: And a derivation of 6/7/2021 : Dr. Sandip Mal 29

yield 6/7/2021 Dr. Sandip Mal 30

yield 6/7/2021 Dr. Sandip Mal 31

yield 6/7/2021 Dr. Sandip Mal 32

yield 6/7/2021 Dr. Sandip Mal 33

Derivation Tree (parse tree) yield 6/7/2021 Dr. Sandip Mal 34

Sometimes, derivation order doesn’t matter Leftmost derivation: Rightmost derivation: Give same derivation tree 6/7/2021 Dr. Sandip Mal 35

Ambiguity 6/7/2021 Dr. Sandip Mal 36

Grammar for mathematical expressions Example strings: Denotes any number 6/7/2021 Dr. Sandip Mal 37

A leftmost derivation for 6/7/2021 Dr. Sandip Mal 38

Another leftmost derivation for 6/7/2021 Dr. Sandip Mal 39

Two derivation trees for 6/7/2021 Dr. Sandip Mal 40

take 6/7/2021 Dr. Sandip Mal 41

Good Tree Bad Tree Compute expression result using the tree 6/7/2021 Dr. Sandip Mal 42

Two different derivation trees may cause problems in applications which use the derivation trees: • Evaluating expressions • In general, in compilers for programming languages 6/7/2021 Dr. Sandip Mal 43

Ambiguous Grammar: A context-free grammar if there is a string is ambiguous which has: two different derivation trees or two leftmost derivations (Two different derivation trees give two different leftmost derivations and vice-versa) 6/7/2021 Dr. Sandip Mal 44

Example: this grammar is ambiguous since string 6/7/2021 has two derivation trees Dr. Sandip Mal 45

this grammar is ambiguous also because string 6/7/2021 has two leftmost derivations Dr. Sandip Mal 46

Another ambiguous grammar: IF_STMT if EXPR then STMT else STMT Variables Terminals Very common piece of grammar in programming languages 6/7/2021 Dr. Sandip Mal 47

If expr 1 then if expr 2 then stmt 1 else stmt 2 IF_STMT if expr 1 if then expr 2 STMT then expr 1 if 6/7/2021 then expr 2 else stmt 2 Two derivation trees IF_STMT if stmt 1 STMT then else stmt 2 stmt 1 Dr. Sandip Mal 48

In general, ambiguity is bad and we want to remove it Sometimes it is possible to find a non-ambiguous grammar for a language But, in general we cannot do so 6/7/2021 Dr. Sandip Mal 49

A successful example: Equivalent Ambiguous Grammar Non-Ambiguous Grammar generates the same language 6/7/2021 Dr. Sandip Mal 50

Unique derivation tree for 6/7/2021 Dr. Sandip Mal 51

An un-successful example: is inherently ambiguous: every grammar that generates this language is ambiguous 6/7/2021 Dr. Sandip Mal 52

Example (ambiguous) grammar for 6/7/2021 Dr. Sandip Mal : 53

The string has always two different derivation trees (for any grammar) For example 6/7/2021 Dr. Sandip Mal 54