Parsing Jaruloj Chongstitvatana Department of Mathematics and Computer

Parsing Jaruloj Chongstitvatana Department of Mathematics and Computer Science Chulalongkorn University

Outline Top-down parsing l l Recursive-descent parsing LL(1) parsing l LL(1) parsing algorithm l First and follow sets l Constructing LL(1) parsing table l Error recovery Bottom-up parsing l l l Shift-reduce parsers LR(0) parsing l LR(0) items l Finite automata of items l LR(0) parsing algorithm l LR(0) grammar SLR(1) parsing l SLR(1) parsing algorithm l SLR(1) grammar l Parsing conflict 2

Introduction Parsing is a process that constructs a syntactic structure (i. e. parse tree) from the stream of tokens. We already learn how to describe the syntactic structure of a language using (context-free) grammar. So, a parser only need to do this? Stream of tokens Context-free grammar Parse tree 3

Top–Down Parsing Bottom–Up Parsing A parse tree is created from root to leaves from leaves to root The traversal of parse trees is a preorder trees is a reversal of traversal postorder traversal Tracing leftmost Tracing rightmost derivation Try different Two types: Morestructures powerfuland than top -down parsing if it does not matched l Backtracking parser backtrack l Predictive parser the input Guess the structure of the parse tree 2301373 from the next input 4

Parse Trees and Derivations E E + id E * E id id Top-down parsing E E id E + E * E id id Bottom-up parsing E+E id + E * E id + id * id E E+E E + E * id E + id * id 5

Top-down Parsing What does a parser need to decide? l Which production rule is to be used at each point of time ? How to guess? What is the guess based on? l What is the next token? l l Reserved word if, open parentheses, etc. What is the structure to be built? l If statement, expression, etc. 6

Top-down Parsing Why is it difficult? l Cannot decide until later l l l Next token: if Structure to be built: St St Matched. St | Unmatched. St if (E) St| if (E) Matched. St else Unmatched. St Matched. St if (E) Matched. St else Matched. St. . . | Production with empty string Next token: id l par. List | l l Structure to be built: par. List exp , par. List | exp 7

Recursive-Descent Write one procedure for each set of productions with the same nonterminal in the LHS Each procedure recognizes a structure described by a nonterminal. A procedure calls other procedures if it need to recognize other structures. A procedure calls match procedure if it need to recognize a terminal. 8

Recursive-Descent: Example For this grammar: E : : = F {O F} l We cannot decide which O : : = + | rule to use for E, and F : : = ( E ) | id l If we choose E E O F, procedure E procedure F it leads to infinitely E; O; F; } recursive loops. { switch token { { case (: match(‘(‘); Rewrite the grammar E; into EBNF match(‘)’); case id: match(id); procedure E default: error; { F; } while (token=+ or token=-) } { O; F; } } E EOF|F O +|F ( E ) | id 9

Match procedure match(exp. Tok) { if (token==exp. Tok) then get. Token else error } The token is not consumed until get. Token is executed. 10

Problems in Recursive-Descent Difficult to convert grammars into EBNF Cannot decide which production to use at each point Cannot decide when to use -production A 11

LL(1) Parsing LL(1( Read input from (L) left to right l Simulate (L) leftmost derivation l 1 lookahead symbol l Use stack to simulate leftmost derivation Part of sentential form produced in the leftmost derivation is stored in the stack. l Top of stack is the leftmost nonterminal symbol in the fragment of sentential form. l 12

Concept of LL(1) Parsing Simulate leftmost derivation of the input. Keep part of sentential form in the stack. If the symbol on the top of stack is a terminal, try to match it with the next input token and pop it out of stack. If the symbol on the top of stack is a nonterminal X, replace it with Y if we have a production rule X Y. l Which production will be chosen, if there are both X Y and X Z? 13

Example of LL(1) Parsing E TX FNX (E)NX (TX)NX (FNX)NX (n. ATX)NX (n+FNX)NX (n+(E)NX)NX (n+(TX)NX)NX (n+(FNX)NX)NX (n+(n)X)NX (n+(n))MFNX (n+(n))*n. X (n+(n))*n n F T N ( ( n + ( n ) ) * n $ X E A n F + ) E T X ( T N X A T X| A - | + E X Finished M * T F N F ) n N M F N| M * T N F (E | ( n E X $ 14

LL(1) Parsing Algorithm Push the start symbol into the stack WHILE stack is not empty ($ is not on top of stack) and the stream of tokens is not empty (the next input token is not($ SWITCH (Top of stack, next token( CASE (terminal a, a: ( Pop stack; Get next token CASE (nonterminal A, terminal a: ( IF the parsing table entry M[A, a] is not empty THEN Get A X 1 X 2. . . Xn from the parsing table entry M[A, a] Pop stack ; Push Xn. . . X 2 X 1 into stack in that order ELSE Error CASE ($, $): Accept OTHER: Error 15

LL(1) Parsing Table If the nonterminal N is on the top of stack and the next token is t, which production rule to use? Choose a rule N X such that X * t. Y or l X * and S * WNt. Y l t N Y X Q N X t Y t … … … 16

First Set Let X be or be in V or T. First(X ) is the set of the first terminal in any sentential form derived from X. If X is a terminal or , then First(X ) ={X. { l If X is a nonterminal and X X 1 X 2. . . Xn is a rule, then l First(X 1) -{ } is a subset of First(X( l First(Xi )-{ } is a subset of First(X) if for all j<i First(Xj) contains { { l is in First(X) if for all j ≥n First(Xj)contains l 17

Examples of First Set st ifst | other exp addop term | ifst if ( exp ) st elsepart term elsepart else st | addop - | + 0|1 term mulop factor | exp factor First(exp) = {0, 1{ mulop * First(elsepart) = {else, { factor (exp) | num First(ifst) = {if{ First(addop {- , +} = ( First(st) = {if, other{ First(mulop{*} = ( First(factor) = {(, num { First(term) = {(, num { First(exp) = {(, num{ exp 18

Algorithm for finding First(A( For all terminals a, First(a) = {a{ For all nonterminals A, First(A) : = { { While there are changes to any First(A ( For each rule A X 1 X 2. . . Xn For each Xi in {X 1, X 2, …, Xn{ If for all j<i First(Xj) contains , Then add First(Xi)-{ } to First(A( If is in First(X 1), First(X 2), . . . , and First(Xn ( If A is a terminal or , then First(A) = {A. { If A is a nonterminal, then for each rule A X 1 X 2. . . Xn, First(A) contains First(X 1) - {. { If also for some i<n, First(X 1), First(X 2), . . . , and First(Xi) contain , then First(A) contains First(Xi+1)-{. { If First(X 1), First(X 2), . . . , and First(Xn) contain , then First(A) also contains . 19

Finding First Set: An Example exp term exp’ addop term exp’ | addop - | + term factor term’ mulop factor term’ | mulop * factor ( exp ) | num First exp’ addop term’ mulop factor - + ( num * ( num 20

Follow Set Let $ denote the end of input tokens If A is the start symbol, then $ is in Follow(A. ( If there is a rule B X A Y, then First(Y) { } is in Follow(A. ( If there is production B X A Y and is in First(Y), then Follow(A) contains Follow(B. ( 21

Algorithm for Finding Follow(A( Follow(S{$} = ( If A is the start symbol, then $ is FOR each A in V-{S { in Follow(A. ( Follow(A{}=( If there is a rule A WHILE change is made to some Y X Z, then Follow sets First(Z) - { } is in FOR each production A X 1 X 2. . . Follow(X. ( Xn , If there is production FOR each nonterminal Xi B X A Y and Add First(Xi+1 Xi+2. . . Xn)is in First(Y), then { } into Follow(Xi. ( Follow(A) contains ) NOTE: If i=n, Xi+1 Xi+2. . . Xn= Follow(B. ( ( IF is in First(Xi+1 Xi+2. . . Xn) 22

Finding Follow Set: An Example exp term exp’ addop term exp’ | addop - | + term factor term’ mulop factor term’ | mulop * factor ( exp ) | num First exp’ addop term’ mulop factor Follow ( num $) - + $) ( num - + $ ) * * ( num 23

Constructing LL(1) Parsing Tables FOR each nonterminal A and a production A X FOR each token a in First(X( A X is in M(A, a( IF is in First(X) THEN FOR each element a in Follow(A( Add A X to M(A, a( 24

Example: Constructing LL(1) Parsing Table First Follow exp {(, num{(, $} { ( ) exp’ {+, -, {(, $} { exp addop {+, -} 1 {(, num{ term {(, num{$, (, -, +} { exp’ 3 term’ {*, -, +} { {$, (, addop mulop {*} {(, num{ factor {(, num{$, (, -, +, *} {term 6 1 exp term exp’ 2 exp’ addop term exp ’ 3 exp’ 4 addop + 5 addop 6 term factor term’ 7 term’ mulop factor term’ 8 term’ 9 mulop * 10 factor ( exp ( 11 factor num term’ 8 mulop factor + - * n $ 1 2 2 4 5 3 6 8 8 7 8 9 10 11 25

LL(1) Grammar A grammar is an LL(1) grammar if its LL(1) parsing table has at most one production in each table entry. 26

LL(1) Parsing Table for non-LL(1) Grammar 1 exp addop term 2 exp term 3 term mulop factor 4 term factor 5 factor ( exp ( 6 factor num 7 addop + 8 addop 9 mulop * First(exp) = { (, num{ First(term) = { (, num{ First(factor) = { (, num{ First(addop{ - , + } = ( First(mulop{ * } = ( 27

Causes of Non-LL(1) Grammar What causes grammar being non-LL(1? ( Left-recursion l Left factor l 28

Left Recursion Immediate left recursion l l A A X | Y A=Y X* A A X 1 | A X 2 |…| A Xn | Y 1 | Y 2 |. . . | Ym Can be removed very easily l l A Y A’, A’ X A’| A Y 1 A’ | Y 2 A’ |. . . | Ym A’, A’ X 1 A’| X 2 A’|…| Xn A’| A={Y 1, Y 2, …, Ym} {X 1, X 2, …, Xn}* General left recursion l A => X =>* A Y Can be removed when there is no empty-string production and no cycle in the grammar 29

Removal of Immediate Left Recursion exp + term | exp - term | term * factor | factor ( exp ) | num Remove left recursion exp = term ( term)* exp term exp’ + term exp’ | - term exp’ | term factor term’ term = factor (* factor)* term’ * factor term’ | factor ( exp ) | num 30

General Left Recursion Bad News! l Can only be removed when there is no emptystring production and no cycle in the grammar. Good News!!!! l Never seen in grammars of any programming languages 31

Left Factoring Left factor causes non-LL(1( l Given A X Y | X Z. Both A X Y and A X Z can be chosen when A is on top of stack and a token in First(X) is the next token. A XY|XZ can be left-factored as A X A’ and A’ Y | Z 32

Example of Left Factor if. St if ( exp ) st else st | if ( exp ) st can be left-factored as if. St if ( exp ) st else. Part else st | seq st ; seq | st can be left-factored as seq st seq ’ seq’ ; seq | 33

Outline Top-down parsing l l Recursive-descent parsing LL(1) parsing l LL(1) parsing algorithm l First and follow sets l Constructing LL(1) parsing table l Error recovery Bottom-up parsing l l l Shift-reduce parsers LR(0) parsing l LR(0) items l Finite automata of items l LR(0) parsing algorithm l LR(0) grammar SLR(1) parsing l SLR(1) parsing algorithm l SLR(1) grammar l Parsing conflict 34

Bottom-up Parsing Use explicit stack to perform a parse Simulate rightmost derivation (R) from left (L) to right, thus called LR parsing More powerful than top-down parsing l Left recursion does not cause problem Two actions Shift: take next input token into the stack l Reduce: replace a string B on top of stack by a nonterminal A, given a production A B l 35

Example of Shift-reduce Parsing Grammar S’ S S (S)S | Parsing actions Stack Input $(()) $ $(() )$ $(( ))$ ) ) $S ) S )$ ) $S ) S $ $S $ Reverse of Action shift reduce reduce accept rightmost derivation from left to right S S S 1 2 3 4 5 (S)S 6 7 8 (S)S 9 10 S’ ( ( ) ) ( ( S( ( S ) S( ( S( (S)S S 36

Example of Shift-reduce Parsing Grammar S’ S S (S)S | Parsing actions Stack Input $(()) $ $(() )$ $(( ))$ ) ) $S ) S )$ ) $S ) S $ $S $ Viable prefix Action shift reduce reduce accept S S S 1 2 3 4 5 (S)S 6 7 8 (S)S 9 10 S’ ( ( ) ) ( ( S( ( S ) S( ( S( (S)S S handle 37

Terminologies Right sentential form l sentential form in a rightmost derivation Viable prefix l sequence of symbols on the parsing stack Handle l right sentential form + position where reduction can be performed + production used for reduction LR(0) item l production with distinguished position in its RHS Right sentential form l l )S ) S( Viable prefix l l )S ) S, ( S ), ( S) , ) )S ) S, ( ( S ) , ) ) , Handle l l l )S ) S. with S ) )S ) S. ) with S ( S ) S LR(0) item l l l S S S ( S ) S. (S). S (S. )S (. S)S. (S)S 38

Shift-reduce parsers There are two possible actions : l shift and reduce Parsing is completed when the input stream is empty and l the stack contains only the start symbol l The grammar must be augmented a new start symbol S’ is added l a production S’ S is added l l To make sure that parsing is finished when S’ is on top of stack because S’ never appears on the RHS of any production. 39

LR(0) parsing Keep track of what is left to be done in the parsing process by using finite automata of items l An item A w. B y means: A w B y might be used for the reduction in the future, l at the time, we know we already construct w in the parsing process, l if B is constructed next, we get the new item A w. B. Y l 40

LR(0) items LR(0) item l production with a distinguished position in the RHS Initial Item with the distinguished position on the leftmost of the production Complete Item l Item with the distinguished position on the rightmost of the production Closure Item of x l Item x together with items which can be reached from x via -transition Kernel Item l Original item, not including closure items 41

Finite automata of items Grammar: S’ . S S’ S S (S)S S S’ S. S . (S)S Items: S’ . S S’ S. S . (S)S S (S. )S S (S)S. S . S ) S (. S)S S . S S (S. )S ( S (S). S S S (S)S. 42

DFA of LR(0) Items S’ . S S ) S . S S (. S)S S S (S)S. ) S ( S (. S)S S . (S)S S . S (S. )S ) S (S). S S’ S. S (S. )S S . (S)S S S’ . S S . (S)S S . S’ S. ) ) S (S). S S . (S)S S (S)S. 43

LR(0) parsing algorithm 44

LR(0) Parsing Table A’ . A A . (A( A . a 0 A a a ) A (. A( A . (A( A . a 3 A ) A’ A. 1 A a. 2 A (A(. 4 ( A (A. ( 5 45

Example of LR(0) Parsing Stack ) )0$ ) 3)0$ 3)3)0$a 2 3)3)0$A 4)5 0$A 1 Input a))$ ))$ )$ )$ shift $ $ Action shift reduce accept 46

Non-LR(0)Grammar Conflict l Shift-reduce conflict l l S’ . S S . (S)S S . A state contains a complete item A x. and a shift item A x. By 0 ) S S S Reduce-reduce conflict l A state contains more than one complete items. A grammar is a LR(0) grammar if there is no conflict in the grammar. S ) (. S)S . (S)S . 2 S’ S. 1 S (S. )S 3 S ) ( S S S (S). S . (S)S . 4 S S (S)S. 5 47

SLR(1) parsing Simple LR with 1 lookahead symbol Examine the next token before deciding to shift or reduce If the next token is the token expected in an item, then it can be shifted into the stack. l If a complete item A x. is constructed and the next token is in Follow(A), then reduction can be done using A x. l Otherwise, error occurs. l Can avoid conflict 48

SLR(1) parsing algorithm 49

SLR(1) grammar Conflict l Shift-reduce conflict l l A state contains a shift item A x. Wy such that W is a terminal and a complete item B z. such that W is in Follow(B). Reduce-reduce conflict l A state contains more than one complete item with some common Follow set. A grammar is an SLR)1 (grammar if there is no conflict in the grammar. 50

SLR(1) Parsing Table A (A) | a A’ . A A . (A( A . a 0 A a A’ A. 1 A a. 2 ) a A (. A( A . (A( A . a 3 A A (A(. 4 ( ) A (A. ( 5 51

SLR(1) Grammar not LR(0) S’ . S S . (S)S S . 0 ) S S S (. S)S S . (S)S S . 2 ) S’ S. 1 S (S)S | S (S. )S 3 ( ) S (S). S S . (S)S S . 4 S S (S)S. 5 52

Disambiguating Rules for Parsing Conflict Shift-reduce conflict l Prefer shift over reduce l In case of nested if statements, preferring shift over reduce implies most closely nested rule for dangling else Reduce-reduce conflict l Error in design 53

Dangling Else S’ . S 0 S . I S . other I . if S else S other S I I . other if S. 5 if S. else S S S’ I S S. I. if other 3 other S if 1 2 I I else if I if. S 4 I if. S else S S . I S . other I . if S else S I if S else. S 6 S . I S . other I . if S else S state if 0 S 4 else S I . if S else S 7 other $ S 3 1 R 1 3 R 2 S 4 5 6 7 I 1 2 5 2 7 2 ACC 2 4 S S 3 S 6 S 4 R 3 S 3 R 4 54

End 55