ContextFree Languages Costas Busch RPI 1 Regular Languages

Context-Free Languages Costas Busch - RPI 1

Regular Languages Costas Busch - RPI 2

Context-Free Languages Regular Languages Costas Busch - RPI 3

Context-Free Languages Context-Free Grammars Pushdown Automata stack automaton Costas Busch - RPI 4

Context-Free Grammars Costas Busch - RPI 5

Example A context-free grammar : A derivation: Costas Busch - RPI 6

A context-free grammar : Another derivation: Costas Busch - RPI 7

Describes parentheses: Costas Busch - RPI (((( )))) 8

Example A context-free grammar : A derivation: Costas Busch - RPI 9

A context-free grammar : Another derivation: Costas Busch - RPI 10

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Example A context-free grammar : A derivation: Costas Busch - RPI 12

A context-free grammar : A derivation: Costas Busch - RPI 13

Describes matched parentheses: () ((( ))) (( )) Costas Busch - RPI 14

Definition: Context-Free Grammars Grammar Variables Terminal symbols Start variable Productions of the form: Variable String of variables and terminals Costas Busch - RPI 15

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Definition: Context-Free Languages A language is context-free if and only if there is a context-free grammar with Costas Busch - RPI 17

Derivation Order Leftmost derivation: Rightmost derivation: Costas Busch - RPI 18

Leftmost derivation: Rightmost derivation: Costas Busch - RPI 19

Derivation Trees Costas Busch - RPI 20

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Derivation Tree Costas Busch - RPI 25

Derivation Tree yield Costas Busch - RPI 26

Partial Derivation Trees Partial derivation tree Costas Busch - RPI 27

Partial derivation tree Costas Busch - RPI 28

sentential form Partial derivation tree yield Costas Busch - RPI 29

Sometimes, derivation order doesn’t matter Leftmost: Rightmost: Same derivation tree Costas Busch - RPI 30

Ambiguity Costas Busch - RPI 31

leftmost derivation Costas Busch - RPI 32

leftmost derivation Costas Busch - RPI 33

Two derivation trees Costas Busch - RPI 34

The grammar is ambiguous: string has two derivation trees Costas Busch - RPI 35

The grammar is ambiguous: string has two leftmost derivations Costas Busch - RPI 36

Definition: A context-free grammar if some string is ambiguous has: two or more derivation trees Costas Busch - RPI 37

In other words: A context-free grammar if some string is ambiguous has: two or more leftmost derivations (or rightmost) Costas Busch - RPI 38

Why do we care about ambiguity? take Costas Busch - RPI 39

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Correct result: Costas Busch - RPI 42

• Ambiguity is bad for programming languages • We want to remove ambiguity Costas Busch - RPI 43

We fix the ambiguous grammar: New non-ambiguous grammar: Costas Busch - RPI 44

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Unique derivation tree Costas Busch - RPI 46

The grammar : is non-ambiguous: Every string has a unique derivation tree Costas Busch - RPI 47

Another Ambiguous Grammar IF_STMT if EXPR then STMT else STMT Costas Busch - RPI 48

If expr 1 then if expr 2 then stmt 1 else stmt 2 IF_STMT if expr 1 if then expr 2 STMT then stmt 1 else STMT else stmt 2 IF_STMT if expr 1 if then expr 2 then stmt 1 Costas Busch - RPI 49

Inherent Ambiguity Some context free languages have only ambiguous grammars Example: Costas Busch - RPI 50

The string has two derivation trees Costas Busch - RPI 51

Compilers Costas Busch - RPI 52

Machine Code Program v = 5; if (v>5) x = 12 + v; while (x !=3) { x = x - 3; v = 10; }. . . Compiler Costas Busch - RPI Add v, v, 0 cmp v, 5 jmplt ELSE THEN: add x, 12, v ELSE: WHILE: cmp x, 3. . . 53

Compiler Lexical analyzer input parser output machine code program Costas Busch - RPI 54

A parser knows the grammar of the programming language Costas Busch - RPI 55

Parser PROGRAM STMT_LIST STMT; STMT_LIST | STMT; STMT EXPR | IF_STMT | WHILE_STMT | { STMT_LIST } EXPR + EXPR | EXPR - EXPR | ID IF_STMT if (EXPR) then STMT | if (EXPR) then STMT else STMT WHILE_STMT while (EXPR) do STMT Costas Busch - RPI 56

The parser finds the derivation of a particular input derivation input 10 + 2 * 5 Parser E -> E + E |E*E | INT Costas Busch - RPI E => E + E * E => 10 + E*E => 10 + 2 * 5 57

derivation tree derivation E => E + E * E => 10 + E*E => 10 + 2 * 5 E E 10 + E 2 Costas Busch - RPI E * E 5 58

derivation tree E E 10 machine code + E 2 E * mult a, 2, 5 add b, 10, a E 5 Costas Busch - RPI 59

Parsing Costas Busch - RPI 60

input string Parser grammar Costas Busch - RPI derivation 61

Example: Parser derivation input ? Costas Busch - RPI 62

Exhaustive Search Phase 1: Find derivation of All possible derivations of length 1 Costas Busch - RPI 63

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Phase 2 Phase 1 Costas Busch - RPI 65

Phase 2 Phase 3 Costas Busch - RPI 66

Final result of exhaustive search (top-down parsing) Parser input derivation Costas Busch - RPI 67

Time complexity of exhaustive search Suppose there are no productions of the form Number of phases for string Costas Busch - RPI : 68

For grammar with rules Time for phase 1: possible derivations Costas Busch - RPI 69

Time for phase 2: possible derivations Costas Busch - RPI 70

Time for phase : possible derivations Costas Busch - RPI 71

Total time needed for string phase 1 phase 2 : phase 2|w| Extremely bad!!! Costas Busch - RPI 72

There exist faster algorithms for specialized grammars S-grammar: symbol string of variables appears once Pair Costas Busch - RPI 73

S-grammar example: Each string has a unique derivation Costas Busch - RPI 74

For S-grammars: In the exhaustive search parsing there is only one choice in each phase Time for a phase: Total time for parsing string Costas Busch - RPI : 75

For general context-free grammars: There exists a parsing algorithm that parses a string in time (we will show it in the next class) Costas Busch - RPI 76