Front End
source code
abstract syntax
tree
lexical analyzer
parser
tokens
IRsemantic analyzer
Parsing
The parser translates the source program into abstract syntax trees Token sequence:
from the lexer abstract syntax trees:
compiler internal data structures for programs check (syntactic) validity of programs
Must take account the program syntax
Conceptually
token sequence
abstract
syntax treeparser
language syntax
Syntax: Context-free Grammar
Context-free grammars are (often) given by BNF expressions (Backus-Naur Form) read dragon-book sec 2.2
More powerful than RE in theory Good for defining language syntax
Context-free Grammar (CFG)
A CFG consists of 4 components: a set of terminals (tokens): T a set of nonterminals: N a set of production rules: P
s -> t1 t2 … tn with sN, and t1, …, tn (T ∪N)
a unique start nonterminal: S
Example// SLP as in Tiger book chap. 1 (simplified):
N = {S, E}T = {SEMICOLON, ID, IF, ASSIGN, …}S = S
S -> S SEMICOLON S | ID ASSIGN E | PRINT LPAREN E RPARENE -> ID | NUM | E PLUS E | E TIMES E
Derivation
A derivation: Starts with the unique start nonterminal S repeatedly replacing a right-hand nonter
minal s by the body of a production rule of the nonterminal s
stop when right-hand are all terminals The final string consists of terminals o
nly and is called a sentence (program)
ExampleS -> S ; S | id := E | print (E)E -> …
x := 5;print (x)
derive me
S -> … (a choice)
Example
x := 5;print (x)
derive meS -> S ; S -> x := E ; S -> x := 5 ; S -> x := 5 ; print (E) -> x := 5 ; print (x)
S -> S ; S | id := E | print (E)E -> id | num | E + E | E * E
Another Try to Derive the same Program
x := 5;print (x)
derive me
S -> x := E -> x := 5 -> // stuck! :-(
S -> S ; S | id := E | print (E)E -> …
Derivation For same string, there may exist
many different derivations left-most derivation right-most derivation
Parsing is the problem of taking a string of terminals and figure out whether it could be derived from a CFG error-detection
Parse Trees Derivation can also be represented as trees
useful to understand AST (discussed later) Idea:
each internal node is labeled with a nonterminal
each leaf node is labeled with a terminal each use of a rule in a derivation explains how t
o generate children in the parse tree from the parent
ExampleS -> S ; S
| …
x := 5;print (x)
derive me
S
S ; S
x := E
5
print E
x
( )
Parse Tree has Meanings:post-order traversalS -> S ; S
| …
x := 5;print (x)
derive me
S
S ; S
x := E
5
print E
x
( )
Ambiguous Grammars
A grammar is ambiguous if the same sequence of tokens can give rise to two or more different parse trees
ExampleE -> num | id | E + E | E * E
3+4*5
derive me
E -> E + E -> 3 + E -> 3 + E * E -> 3 + 4 * E -> 3 + 4 * 5E -> E * E -> E + E * E -> 3 + E * E -> 3 + 4 * E -> 3 + 4 * 5
ExampleE -> num | id | E + E | E * E
E -> E + E -> 3 + E -> 3 + E * E -> 3 + 4 * E -> 3 + 4 * 5E -> E * E -> E + E * E -> 3 + E * E -> 3 + 4 * E -> 3 + 4 * 5
E
E + E
3 E * E
54
E
E * E
5E + E
43
Ambiguous Grammars Problem: compilers make use of parse trees
to interpret the meaning of parsed programs different parse trees have different meanings eg: 4 + 5 * 6 is not (4 + 5) * 6 languages with ambiguous grammars are DISAST
ROUS; the meaning of programs isn’t well-defined! You can’t tell what your program might do!
Solution: rewrite grammar to equivalent forms
Eliminating ambiguity In programming language syntax, am
biguity often arises from missing operator precedence or associativity * is of high precedence than + both + and * are left-associative Why or why not?
Rewrite grammar to take account of this
ExampleE -> num | id | E + E | E * E
E -> E + T | TT -> T * F | FF -> num | id
Q: is the right grammar ambiguous? Why or why not?
Parser A program to check whether a program is d
erivable from a given grammar expensive in general must be fast
to compile a 2000k lines of kernel even for small application code, speed may be a conce
rn Theorists have developed specialized kind
of grammar which may be parsed efficiently LL(k) and LR(k)
Recursive Decedent Parsing
Predictive parsing A.K.A: Recursive descent parsing, top-down
parsing simple to code by hand efficient can parse a large set of grammar your Tiger compiler will use this
Key idea: one (recursive) function for each nonterminal one clause for each right-hand production rule
Connecting with the lexer(* step #1: represent tokens *)token = ID | IF | NUM | ASSIGN | SEMICOLON | LPAREN | RPAREN | …(* step #2: connect with lexer *)token current_token; /* external var */
void eat (token t) = if (current_token = t) current_token = Lex_nextToken (); else error (“want “, t, “but got”, current_token)
stm -> stm ; stm | id := exp | print (exp)exp -> ID | NUM | exp + exp | exp * exp
(* step #1: cook a lexer, including tokens *)struct token current_token = lex ();(* step #2: build the parser *)void parse_stm () = switch (current_token) case ID => eat (ID); eat (ASSIGN); parse_exp (); case PRINT => eat (PRINT); eat (LPAREN); parse_exp (); eat (RPAREN); default =>error(“want ID, PRINT”);
void parse_exp () = switch (current_token) case ID: ??? case NUM: ??? // backtracking!!
parse_stm()parse_exp()
How to handle precedence?void parse_stm_all ()
parse_stm();
while (current_token == “;”)
eat (;);
parse_stm ();
void parse_exp_plus ()
parse_exp_times();
while (current_token == “+”)
eat (+);
parse_exp_times();
stm ; stm ; stm ; stm
2+3*4+5*7
Generally: if there are n level of precedence, one may write n parsing functions.
Moral The key point in predicative parsing is
to determine the production rule to use (recursive function to call) must know the “start” symbols of each rule “start” symbol must not overlap e.g.:
exp -> NUM | ID
This motivates the idea of first and follow sets
First & Follow
Moral
S -> w1
-> w2
-> …
-> wn
For nonterminal S, and current input token t if wk starts with t, then choose wk,
or if wk derives empty string, and the
string follow S starts with t First symbol sets of wi (1<=i<=n)
don’t overlap to avoid backtracking
Nullable, First and Follow sets To use predicative parsing, we must compu
te: Nullable: nonterminals that derive empty string First(ω) : set of terminals that can begin any stri
ng derivable from ω Follow(X): set of terminals that can immediately
follow any string derivable from nonterminal X Read tiger sec 3.2
Fixpoint algorithms
Nullable, First and Follow sets Which symbol X, Y and Z
can derive empty string? What terminals may the
string derived from X, Y and Z begin with?
What terminals may follow X, Y and Z?
Z -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Nullable If X can derive an empty string, iff:
base case: X ->
inductive case: X -> Y1 … Yn
Y1, …, Yn are n nonterminals and may all derive empty strings
Computing Nullable/* Nullable: a set of nonterminals */Nullable <- {};while (Nullable still changes) for (each production X -> α) switch (α) case : Nullable = {X};∪ break; case Y1 … Yn: if (Y1Nullable && … && YnNullable) Nullable = {X};∪ break;
Example: NullablesZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round 0 1 2
nullable {}
Example: NullablesZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round 0 1 2
nullable {} {Y, X}
Example: NullablesZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round
0 1 2
Φ {} {Y, X} {Y, X}
First(X) Set of terminals that X begins with:
X => a … Rules
base case: X -> a
First (X) ∪= {a} inductive case:
X -> Y1 Y2 … Yn First (X) ∪= First(Y1) if Y1Nullable, First (X) ∪= First(Y2) if Y1,Y2 Nullable, First (X) ∪= First(Y3) …
Computing First// Suppose Nullable set has been computedforeach (nonterminal X) First(X) <- {}; while (some First set still changes) for (each production X -> α) switch (α) case a: First(X) = {a};∪ break; case Y1 … Yn: First(X) = First(Y1);∪ if (Y1 \not\in Nullable) break; First(X) = First(Y2);∪ …; // Similar as above
Example: FirstZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round
0 1 2 3
First(Z)
{}
First(Y)
{}
First(X)
{}
Nullable = {X, Y}
Example: FirstZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round
0 1 2 3
First(Z)
{} {d}
First(Y)
{} {c}
First(X)
{} {c, a}
Nullable = {X, Y}
Example: FirstZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round
0 1 2 3
First(Z)
{} {d} {d, c, a}
First(Y)
{} {c} {c}
First(X)
{} {c, a} {c, a}
Nullable = {X, Y}
Example: FirstZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round
0 1 2 3
First(Z)
{} {d} {d, c, a}
{d, c, a}
First(Y)
{} {c} {c} {c}
First(X)
{} {c, a} {c, a} {c, a}
Nullable = {X, Y}
Parsing with FirstZ -> d {d}
-> X Y Z {a, c, d}
Y -> c {c}
-> {}
X -> Y {c}
-> a {a}
First(Z)
{d, c, a}
First(Y)
{c}
First(X)
{c, a}Nullable = {X, Y}
Now consider this string: d
Suppose we choose the production: Z -> X Y Z
But we get stuck at:X -> Y -> aBut neither can accept d!What’s the problem?
Follow(X) Set of terminals that may follow X:
S => … X a … Rules:
Base case: Follow (X) = {}
inductive case: Y -> ω1 X ω2
Follow(X) ∪= Fisrt(ω2) if ω2 is Nullable, Follow(X) ∪= Follow(Y)
Computing Follow(X)foreach (nonterminal X) Follow(X) <- {};while (some Follow still changes) { for (each production Y -> ω1 X ω2 ) Follow(X) = First (∪ ω2); if (ω2 is Nullable) Follow(X) = Follow (Y);∪
Example: FollowZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round 0 1 2 3
First(Z)Follow(Z)
{d, c, a}{}
First(Y)Follow(Y)
{c}{}
First(X)Follow(X)
{c, a}{}
Nullable = {X, Y}
Example: FollowZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round 0 1 2 3
First(Z)Follow(Z)
{d, c, a}{}
{$}
First(Y)Follow(Y)
{c}{} {d, c,
a}
First(X)Follow(X)
{c, a}{} {d, c,
a}
Nullable = {X, Y}
Example: FollowZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round 0 1 2 3
First(Z)Follow(Z)
{d, c, a}{}
{$} {$}
First(Y)Follow(Y)
{c}{} {d, c,
a}{d, c, a}
First(X)Follow(X)
{c, a}{} {d, c,
a}{d, c, a}
Nullable = {X, Y}
Predicative Parsing Table
With Nullables, First(), and Follow(), we can make a parsing table P(N,T) each entry contains a set of productions
t1 t2 t3 t4 … $(EOF)
N1 ri
N2 rk
N3 rj
…
Predicative Parsing Table
For each rule X -> ω for each aFirst(ω), add X -> ω to P(X, a) if X is nullable, add X -> ω to P(X, b) for ea
ch b Follow (X) all other entries are “error”
t1 t2 t3 t4 … $(EOF)
N1 r1
N2 rk
N3 ri
…
Example: Predicative Parsing Table
First(X)Follow(X)
{c, a}{c, d, a}
First(Y)Follow(Y)
{c}{c, d, a}
First(Z)Follow(Z)
{d, c, a}{$}
Z -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Nullable = {X, Y}
a c d
Z Z->X Y Z Z->X Y Z Z->dZ->X Y Z
Y Y-> Y->cY->
Y->
X X->YX->a
X->Y X->Y
Example: Predicative Parsing Table
First(X)Follow(X)
{c, a}{c, d, a}
First(Y)Follow(Y)
{c}{c, d, a}
First(Z)Follow(Z)
{d, c, a}{$}
Z -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Nullable = {X, Y}
a c d
Z Z->X Y Z Z->X Y Z Z->dZ->X Y Z
Y Y-> Y->cY->
Y->
X X->YX->a
X->Y X->Y
LL(1) A context-free grammar is called LL(1) if it can be parsed this way: Left-to-right parsing Leftmost derivation 1 token lookahead
This means that in the predicative parsing table, there is at most one production in every entry
Speeding up set Construction
All these sets (Nullable, First, Follow) can be computed simultaneously see Tiger book algorithm 3.13
Order the computation: What’s the optimal order to compute th
ese set?
Example: Speeding up set Construction
Z -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Round
0 1 2 3
First(Z)
{}
First(Y)
{}
First(X)
{}
Nullable = {X, Y}
Q1: What’s reasonable order here?
Q2: How to set this order?
Directed Graph ModelZ -> d
-> X Y Z
Y -> c
->
X -> Y
-> a
Nullable = {X, Y}
Q1: What’s reasonable order here?
Q2: How to set this order?
Z
X
Y{c}
{c, a}
{d, c, a}
Order: Y X Z
Reverse Quasi-Topological Sort Quasi-topological sort the directed graph
Quasi: topo-sort general directed graph is impossible
also known as reverse depth-first ordering Reverse: information (here: First) flows fro
m successors backward to predecessors Refer to your favorite algorithm book
Problem
LL(1) can only be used with grammars in which every production rules for a nonterminal start with different terminals
Unfortunately, many grammars don’t have this perfect property
Exampleexp -> NUM -> ID -> exp + exp -> exp * exp
exp -> exp + term -> termterm -> term * factor -> factorfactor -> NUM -> ID
Q: is the right grammar LL(1)? Why or why not?
Solutions
Left-recursion elimination Left-factoring Read:
tiger section 3.2
Example for SLPstm -> id := exp A -> print(exp) AA -> ; stm A ->
Q1: is the right grammar LL(1)?
Q2: are these two grammars equivalent?
stm -> stm ; stm -> id := exp -> print (exp)
LL(k) LL(1) can be further generalized to LL
(k): Left-to-right parsing Leftmost derivation k token lookahead
Q: table size? other problems with this approach?
Summary Context-free grammar is a math tool for spe
cifying language syntax among others…
Writing parsers for general grammar is hard and costly LL(k) and LR(k)
LL(1) grammars can be implemented efficiently table-driven algorithms (again!)