Costas Busch - RPI1 Context-Free Languages

Costas Busch - RPI2 Regular Languages

Costas Busch - RPI3 Regular Languages Context-Free Languages

Costas Busch - RPI4 Context-Free Languages Pushdown Automata Context-Free Grammars stack automaton

Costas Busch - RPI5 Context-Free Grammars

Costas Busch - RPI6 Example A context-free grammar : A derivation:

Costas Busch - RPI7 A context-free grammar : Another derivation:

Costas Busch - RPI8 (((( )))) Describes parentheses:

Costas Busch - RPI9 A context-free grammar : A derivation: Example

Costas Busch - RPI10 A context-free grammar : Another derivation:

Costas Busch - RPI11

Costas Busch - RPI12 A context-free grammar : A derivation: Example

Costas Busch - RPI13 A context-free grammar : A derivation:

Costas Busch - RPI14 () ((( ))) (( )) Describes matched parentheses:

Costas Busch - RPI15 Definition: Context-Free Grammars Grammar Productions of the form: String of variables and terminals VariablesTerminal symbols Start variable Variable

Costas Busch - RPI16

Costas Busch - RPI17 Definition: Context-Free Languages A language is context-free if and only if there is a context-free grammar with

Costas Busch - RPI18 Derivation Order Leftmost derivation: Rightmost derivation:

Costas Busch - RPI19 Leftmost derivation: Rightmost derivation:

Costas Busch - RPI20 Derivation Trees

Costas Busch - RPI21

Costas Busch - RPI22

Costas Busch - RPI23

Costas Busch - RPI24

Costas Busch - RPI25 Derivation Tree

Costas Busch - RPI26 yield Derivation Tree

Costas Busch - RPI27 Partial Derivation Trees Partial derivation tree

Costas Busch - RPI28 Partial derivation tree

Costas Busch - RPI29 Partial derivation tree sentential form yield

Costas Busch - RPI30 Same derivation tree Sometimes, derivation order doesn’t matter Leftmost: Rightmost:

Costas Busch - RPI31 Ambiguity

Costas Busch - RPI32 leftmost derivation

Costas Busch - RPI33 leftmost derivation

Costas Busch - RPI34 Two derivation trees

Costas Busch - RPI35 The grammar is ambiguous: stringhas two derivation trees

Costas Busch - RPI36 stringhas two leftmost derivations The grammar is ambiguous:

Costas Busch - RPI37 Definition: A context-free grammar is ambiguous if some string has: two or more derivation trees

Costas Busch - RPI38 In other words: A context-free grammar is ambiguous if some string has: two or more leftmost derivations (or rightmost)

Costas Busch - RPI39 Why do we care about ambiguity? take

Costas Busch - RPI40

Costas Busch - RPI41

Costas Busch - RPI42 Correct result:

Costas Busch - RPI43 We want to remove ambiguity Ambiguity is bad for programming languages

Costas Busch - RPI44 We fix the ambiguous grammar: New non-ambiguous grammar:

Costas Busch - RPI45

Costas Busch - RPI46 Unique derivation tree

Costas Busch - RPI47 The grammar : is non-ambiguous: Every string has a unique derivation tree

Costas Busch - RPI48 Another Ambiguous Grammar IF_STMTif EXPR then STMT if EXPR then STMT else STMT

Costas Busch - RPI49 If expr1 then if expr2 then stmt1 else stmt2 IF_STMT expr1then elseifexpr2then STMT stmt1 if IF_STMT expr1thenelse ifexpr2then STMTstmt2 if stmt1 stmt2

Costas Busch - RPI50 Inherent Ambiguity Some context free languages have only ambiguous grammars Example:

Costas Busch - RPI51 The string has two derivation trees

Costas Busch - RPI52 Compilers

Costas Busch - RPI53 Compiler Program v = 5; if (v>5) x = 12 + v; while (x !=3) { x = x - 3; v = 10; } Add v,v,0 cmp v,5 jmplt ELSE THEN: add x, 12,v ELSE: WHILE: cmp x,3... Machine Code

Costas Busch - RPI54 Lexical analyzer parser Compiler program machine code input output

Costas Busch - RPI55 A parser knows the grammar of the programming language

Costas Busch - RPI56 Parser PROGRAM STMT_LIST STMT_LIST STMT; STMT_LIST | STMT; STMT EXPR | IF_STMT | WHILE_STMT | { STMT_LIST } EXPR 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 - RPI57 The parser finds the derivation of a particular input * 5 Parser E -> E + E | E * E | INT E => E + E => E + E * E => 10 + E*E => * E => * 5 input derivation

Costas Busch - RPI58 10 E 25 E => E + E => E + E * E => 10 + E*E => * E => * 5 derivation derivation tree EE EE + *

Costas Busch - RPI59 10 E 25 derivation tree EE EE + * mult a, 2, 5 add b, 10, a machine code

Costas Busch - RPI60 Parsing

Costas Busch - RPI61 grammar Parser input string derivation

Costas Busch - RPI62 Example: Parser derivation input ?

Costas Busch - RPI63 Exhaustive Search Phase 1: All possible derivations of length 1 Find derivation of

Costas Busch - RPI64

Costas Busch - RPI65 Phase 2 Phase 1

Costas Busch - RPI66 Phase 2 Phase 3

Costas Busch - RPI67 Final result of exhaustive search Parser derivation input (top-down parsing)

Costas Busch - RPI68 Time complexity of exhaustive search Suppose there are no productions of the form Number of phases for string :

Costas Busch - RPI69 Time for phase 1: possible derivations For grammar with rules

Costas Busch - RPI70 Time for phase 2: possible derivations

Costas Busch - RPI71 Time for phase : possible derivations

Costas Busch - RPI72 Total time needed for string : Extremely bad!!! phase 1 phase 2 phase 2|w|

Costas Busch - RPI73 There exist faster algorithms for specialized grammars S-grammar: symbolstring of variables appears once Pair

Costas Busch - RPI74 S-grammar example: Each string has a unique derivation

Costas Busch - RPI75 In the exhaustive search parsing there is only one choice in each phase For S-grammars: Total time for parsing string : Time for a phase:

Costas Busch - RPI76 For general context-free grammars: There exists a parsing algorithm that parses a string in time (we will show it in the next class)