1

Compiler ConstructionParsing Part I

제 4 주 Parsing

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What We Did Last Time

The cycle in lexical analysis RE → NFA NFA → DFA DFA → Minimal DFA DFA → RE

Engineering issues in building scanners

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Today’s Goals

Parsing Part I Context-free grammars Sentence derivations Grammar ambiguity Left recursion problem with top-down parsing

and how to fix it Predictive top-down parsing

LL(1) condition Recursive descent parsing

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Compilers

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The Front End

ParserChecks the stream of words and their parts of speech(produced by the scanner) for grammatical correctnessDetermines if the input is syntactically well formedGuides checking at deeper levels than syntaxBuilds an IR representation of the code

Think of this as the mathematics of diagramming sentences

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The Study of Parsing (syntax analysis)

The process of discovering a derivation for some sentence Need a mathematical model of syntax — a grammar G Need an algorithm for testing membership in L(G) Need to keep in mind that our goal is building parsers, not studying the math

ematics of arbitrary languages

Roadmap1. Context-free grammars and derivations2. Top-down parsing

Hand-coded recursive descent parsers LL(1) parsers

LL(1) parsed top-down, left to right scan, leftmost derivation, 1 symbol lookahead3. Bottom-up parsing

Operator precedence parsing LR(1) parsers

LR(1) parsed bottom-up, left to right scan, reverse rightmost derivation, 1 symbol lookahead

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Syntax analysis

Every PL has rules for syntactic structure.

The rules are normally specified by a CFG (Context-Free Grammar) or BNF (Backus-Naur Form)

Usually, we can automatically construct an efficient parser from a CFG or BNF.

Grammars also allow SYNTAX-DIRECTED TRANSLATION.

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Specifying Syntax with a Grammar

Context-free syntax is specified with a context-free grammarSheepNoise → SheepNoise baa

| baa

This CFG defines the set of noises sheep normally make

It is written in a variant of Backus–Naur form

Formally, a grammar is a four tuple, G = (S,N,T,P) S is the start symbol (set of strings in L(G)) N is a set of non-terminal symbols (syntactic variables) T is a set of terminal symbols (words) P is a set of productions or rewrite rules (P :N →(N ∪T)+)

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The Big Picture

Chomsky Hierarchy of Language Grammars (1956)

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Deriving Syntax

We can use the SheepNoise grammar to create sentences use the productions as rewriting rules

While it is cute, this example quickly runs out of intellectual steam ...

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A More Useful Grammar

Such a sequence of rewrites is called a derivation Process of discovering a derivation is called parsing

To explore the uses of CFGs, we need a more complex grammar

We denote this derivation: Expr ⇒* id – num * id

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Derivations

At each step, we choose a non-terminal to replace Different choices can lead to different derivations

Two derivations are of interest Leftmost derivation — replace leftmost NT at each step Rightmost derivation — replace rightmost NT at each step

These are the two systematic derivations(We don’t care about randomly-ordered derivations!)

The example on the preceding slide was a leftmost derivation Of course, there is also a rightmost derivation Interestingly, it turns out to be different

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The Two Derivations for x – 2 * y

In both cases, Expr ⇒* id – num * id The two derivations produce different parse trees

Actually, each of two different derivations produces both parse trees as the grammar itself is ambiguous

The parse trees imply different evaluation orders!

Leftmost derivation Rightmost derivation

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Derivations and Parse Trees

Leftmost derivation

This evaluates as x – ( 2 * y )

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Derivations and Parse Trees

Rightmost derivation

This evaluates as ( x – 2 ) * y

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Ambiguity Ambiguity

Definitions

If a grammar has more than one leftmost derivation for a single sentential form, the grammar is ambiguous

If a grammar has more than one rightmost derivation for a single sentential form, the grammar is ambiguous

The leftmost and rightmost derivations for a sentential form may differ, even in an unambiguous grammar

Examples Examples Associativity and precedenceAssociativity and precedence Dangling elseDangling else

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Ambiguous Grammars

This grammar allows multiple leftmost derivations for x - 2 * y Hard to automate derivation if > 1 choice The grammar is ambiguous

different choicethan the first time

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Two Leftmost Derivations for x – 2 * y

The Difference: Different productions chosen on the second step

Both derivations succeed in producing x - 2 * y

Original choice New choice

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Derivations and Precedence/Association

These two derivations point out a problem with the grammar:It has no notion of precedence, or implied order of evaluation

To add precedence Create a non-terminal for each level of precedence Isolate the corresponding part of the grammar Force the parser to recognize high precedence subexpressions first

For algebraic expressions Multiplication and division, first (level one) Subtraction and addition, next (level two)

To add association On same precedenceOn same precedence Left-associative : The next-level (higher) nonterminal places at the last of a productionLeft-associative : The next-level (higher) nonterminal places at the last of a production

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Derivations and PrecedenceAdding the standard algebraic precedence produces:

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Derivations and Precedence

This produces x – ( 2 * y ), along with an appropriate parse tree.Both the leftmost and rightmost derivations give the same expression,because the grammar directly encodes the desired precedence.

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Ambiguous Grammars by dangling else

Classic example — the if-then-else problem Stmt → if Expr then Stmt | if Expr then Stmt else Stmt | … other stmts …This ambiguity is entirely grammatical in nature

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Ambiguity

This sentential form has two derivations if Expr1 then if Expr2 then Stmt1 else Stmt2

production 2, thenproduction 1

production 1, thenproduction 2

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Ambiguity

Removing the ambiguity Must rewrite the grammar to avoid generating the problem Match each else to innermost unmatched if (common sense rule)

Intuition: a NoElse always has no else on its last cascaded else if statement

With this grammar, the example has only one derivation

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Ambiguityif Expr1 then if Expr2 then Stmt1 else Stmt2

This binds the else controlling S2 to the inner if

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Deeper Ambiguity

Ambiguity usually refers to confusion in the CFG

Overloading can create deeper ambiguitya = f(17)

In many Algol-like languages, f could be either a function or a subscripted variable

Disambiguating this one requires context Need values of declarations Really an issue of type, not context-free syntax Requires an extra-grammatical solution (not in CFG) Must handle these with a different mechanism

Step outside grammar rather than use a more complex grammar

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Ambiguity - The Final Word

Ambiguity arises from two distinct sources Confusion in the context-free syntax (if-then-else) Confusion that requires context to resolve (overloading)Resolving ambiguity To remove context-free ambiguity, rewrite the grammar To handle context-sensitive ambiguity takes cooperation

Knowledge of declarations, types, … Accept a superset of L(G) & check it by other means†

This is a language design problemSometimes, the compiler writer accepts an ambiguous

grammar Parsing techniques that “do the right thing” i.e., always select the same derivation

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Parsing Techniques

Top-down parsers (LL(1), recursive descent) Start at the root of the parse tree and grow toward leaves Pick a production & try to match the input Bad “pick” ⇒ may need to backtrack Some grammars are backtrack-free (predictive parsing)

Bottom-up parsers (LR(1), operator precedence) Start at the leaves and grow toward root As input is consumed, encode possibilities in an internal

state Start in a state valid for legal first tokens Bottom-up parsers handle a large class of grammars

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Top-down Parsing

A top-down parser starts with the root of the parse tree The root node is labeled with the goal symbol of thegrammar

Top-down parsing algorithm:Construct the root node of the parse treeRepeat until the fringe of the parse tree matches the input string1. At a node labeled A, select a production with A on its lhs and, for each

symbol on its rhs, construct the appropriate child2. When a terminal symbol is added to the fringe and it doesn’t match the

fringe, backtrack3. Find the next node to be expanded (label ∈ NT)

The key is picking the right production in step 1 That choice should be guided by the input string

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The Expression GrammarVersion with precedence derived last lecture

And the input x – 2 * y

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Example

Let’s try x – 2 * y :

Leftmost derivation, choose productions in an order that exposes problems

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Example

Let’s try x – 2 * y :

This worked well, except that “–” doesn’t match “+”The parser must backtrack to here

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ExampleContinuing with x – 2 * y :

⇒ Now, we need to expand Term - the last NT on the fringe

This time, “–”and “–” matched

We can advance past“–” to look at “2”

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Example

Where are we? “2” matches “2” We have more input, but no NTs left to expand The expansion terminated too soon⇒ Need to backtrack

Trying to match the “2” in x – 2 * y :

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ExampleTrying again with “2” in x – 2 * y :

This time, we matched & consumed all the input⇒ Success!

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Another Possible Parse

This doesn’t terminate (obviously) Wrong choice of expansion leads to non-termination Non-termination is a bad property for a parser to have Parser must make the right choice

Other choices for expansion are possible

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Left Recursion

Top-down parsers cannot handle left-recursive grammars

Formally,A grammar is left recursive if ∃ A ∈ NT such that∃ a derivation A ⇒+ Aα, for some string α ∈ (NT ∪ T )+

Our expression grammar is left recursive This can lead to non-termination in a top-down parser For a top-down parser, any recursion must be right recursion We would like to convert the left recursion to right recursion

Non-termination is a bad property in any part of a compiler

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Eliminating Left Recursion

To remove left recursion, we can transform the grammar

Consider a grammar fragment of the form Fee → Fee α | βwhere neither α nor β start with Fee

We can rewrite this as Fee → β Fie Fie → α Fie | εwhere Fie is a new non-terminal

This accepts the same language, but uses only right recursion

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Eliminating Left Recursion

The expression grammar contains two cases of left recursion

Applying the transformation yields

These fragments use only right recursionThey retain the original left associativity

Expr → Expr + Term | Expr – Term | Term

Term → Term * Factor | Term / Factor | Factor

Expr → Term Expr′Expr′ | + Term Expr′ | – Term Expr′ | ε

Term → Factor Term′Term′ | * Factor Term′ | / Factor Term′ | ε

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Eliminating Left RecursionSubstituting them back into the grammar yields

• This grammar is correct, if somewhat non-intuitive.

• It is left associative, as was the original

• A top-down parser will terminate using it.

• A top-down parser may need to backtrack with it.

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Eliminating Left Recursion

The transformation eliminates immediate left recursionWhat about more general, indirect left recursion ?

The general algorithm: arrange the NTs into some order A1, A2, …, An

for i ← 1 to n for s ← 1 to i – 1 replace each production Ai → Asγ with Ai→ δ1γ |δ2γ|…|δkγ, where As→ δ1|δ2|…|δk are all the current productions for As

eliminate any immediate left recursion on Ai using the direct transformation

This assumes that the initial grammar has no cycles (Ai ⇒+ Ai ), and no epsilon productions

And back

Must start with 1 to ensure that A1 →A1 β is transformed

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Eliminating Left Recursion

How does this algorithm work?1. Impose arbitrary order on the non-terminals2. Outer loop cycles through NT in order3. Inner loop ensures that a production expanding Ai has no non-termin

al As in its rhs, for s < i4. Last step in outer loop converts any direct recursion on Ai to right rec

ursion using the transformation showed earlier5. New non-terminals are added at the end of the order & have no left r

ecursion

At the start of the ith outer loop iterationFor all k < i, no production that expands Ak contains a non-terminalAs in its rhs, for s < k

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Example

1. Ai = G 3. Ai = T, As = E

G →EE → T E'E' → + T E'E' → εT → T E' ~ TT → id

2. Ai = E

G →EE → T E'E' → + T E'E' → εT → E ~ TT → id

4. Ai = T

G →EE → T E'E' → + T E'E' → εT → id T'T' →E' ~ T T'T' → ε

Order of symbols: G, E, T

G →EE → E + TE → TT → E ~ TT → id

Go toAlgorith

m

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Roadmap (Where are We?)

We set out to study parsing Specifying syntax

Context-free grammars Ambiguity

Top-down parsers Algorithm & its problem with left recursion Left-recursion removal

Predictive top-down parsing The LL(1) condition Simple recursive descent parsers

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Picking the “Right” Production

If it picks the wrong production, a top-down parser may backtrackAlternative is to look ahead in input & use context to pick correctly

How much lookahead is needed? In general, an arbitrarily large amount Use the Cocke-Younger, Kasami algorithm or Earley’s algorithm

Fortunately, Large subclasses of CFGs can be parsed with limited lookahead Most programming language constructs fall in those subclasses

Among the interesting subclasses are LL(1) and LR(1) grammars

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Predictive Parsing

Basic ideaGiven A → α | β, the parser should be able to choose between α & β

FIRST setsFor some rhs α∈G, define FIRST(α) as the set of tokens that appear as the fir

st symbol in some string that derives from αThat is, x ∈ FIRST(α) iff α ⇒* x γ, for some γ

We will defer the problem of how to compute FIRST sets until we look at the LR(1) table construction algorithm

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Predictive Parsing

Basic ideaGiven A → α | β, the parser should be able to choose between α & β

FIRST setsFor some rhs α∈G, define FIRST(α) as the set of tokens that appearas the first symbol in some string that derives from αThat is, x ∈ FIRST(α) iff α ⇒* x γ, for some γ

The LL(1) PropertyIf A → α and A → β both appear in the grammar, we would like

FIRST(α) ∩ FIRST(β) = ∅This would allow the parser to make a correct choice with a lookahead o

f exactly one symbol !

This is almost correctSee the next slide

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Predictive Parsing

What about ε-productions?⇒ They complicate the definition of LL(1)

If A → α and A → β and ε ∈ FIRST(α), then we need to ensure that FIRST(β) is disjoint from FOLLOW(α), too

Define FIRST+(α) as FIRST(α) ∪ FOLLOW(α), if ε ∈ FIRST(α) FIRST(α), otherwise

Then, a grammar is LL(1) iff A → α and A → β implies

FIRST+(α) ∩ FIRST+(β) = ∅ FOLLOW(α) is the set ofall words in the grammarthat can legally appearimmediately after an α

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Predictive Parsing

Given a grammar that has the LL(1) property Can write a simple routine to recognize each lhs Code is both simple & fastConsider A → β1 | β2 | β3, with

FIRST+(β1) ∩ FIRST+ (β2) ∩ FIRST+ (β3) = ∅

/* find an A */if (current_word ∈ FIRST(β1)) find a β1 and return trueelse if (current_word ∈ FIRST(β2)) find a β2 and return trueelse if (current_word ∈ FIRST(β3)) find a β3 and return trueelse report an error and return false

Grammars with the LL(1)property are called predictivegrammars because the parsercan “predict” the correctexpansion at each point in theparse.

Parsers that capitalize on theLL(1) property are calledpredictive parsers.

One kind of predictive parseris the recursive descentparser.

Of course, there is more detail to“find a βi” (§ 3.3.4 in EAC)

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Recursive Descent ParsingRecall the expression grammar, after transformation

This produces a parser with sixmutually recursive routines: • Goal • Expr • EPrime • Term • TPrime • Factor

Each recognizes one NT or T

The term descent refers to thedirection in which the parse treeis built.

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Fig. 4.10. Transition diagrams for grammar (4.11).Fig. 4.10. Transition diagrams for grammar (4.11).

0 102E :T

1E'

3E' :+

4T

1065E'

7 109T :F

8T'

10T' : * 11F

101312T'

14F :(

15E

101716)

id

EEEE''TTTT''FF

TETE''+TE+TE' ' | | FTFT''*FT*FT' ' | | ((EE)) | | idid

(Grammar 4.11 )

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Fig. 4.11. Simplified transition diagrams.Fig. 4.11. Simplified transition diagrams.

3E' :+

4T

5

106

3E' :+

4

T

106

3E :+

4

T

106

0T

3E :

+

106

0T

(a) (b)

(c) (d)

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Fig. 4.12. Simplified transition diagrams for Fig. 4.12. Simplified transition diagrams for arithmetic expressions.arithmetic expressions.

*

7 1013T :F

8

14F :(

15E

101716)

id

+

0 106E :T

3

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Recursive Descent ParsingA couple of routines from the expression parser

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Recursive Descent Parsing

To build a parse tree: Augment parsing routines to build

nodes Pass nodes between routines usi

ng a stack Node for each symbol on rhs Action is to pop rhs nodes, make

them children of lhs node, and push this subtree

To build an abstract syntax tree Build fewer nodes Put them together in a different or

der

Expr( ) result ←true; if (Term( ) = false) then return false; else if (EPrime( ) = false) then result ←false; else build an Expr node pop EPrime node pop Term node make EPrime & Term children of Expr push Expr node return result;

Success ⇒ build a piece of the parse tree

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Left Factoring

What if my grammar does not have the LL(1) property?⇒ Sometimes, we can transform the grammar

The Algorithm

∀A ∈ NT, find the longest prefix α that occurs in two or more right-hand sides of A if α ≠ ε then replace all of the A productions, A → αβ1 | αβ2 | … | αβn | γ , with A → αZ | γ Z → β1 | β2 | … | βn

where Z is a new element of NT

Repeat until no common prefixes remain

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Left FactoringA graphical explanation for the same idea

becomes …

A → αβ1

| αβ2

| αβ3

A → α ZZ → β1

| β2

| βn

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Left Factoring: An ExampleConsider the following fragment of the expression grammar

After left factoring, it becomes

Factor → Identifier | Identifier [ ExprList ] | Identifier ( ExprList )

Factor → Identifier ArgumentsArguments → [ ExprList ] | ( ExprList ) | ε

FIRST(rhs1) = { Identifier }FIRST(rhs2) = { Identifier }FIRST(rhs3) = { Identifier }

FIRST(rhs1) = { Identifier }FIRST(rhs2) = { [ }FIRST(rhs3) = { ( }FIRST(rhs4) = FOLLOW(Factor)⇒ It has the LL(1) property

This form has the same syntax, with the LL(1) property

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Left Factoring

Factor

ε

[ ExprList ]

[ ExprList ]Word determines

correct choice

Identifier

Becomes …

Factor Identifier

Identifier

Identifier

[ Identifier ]

[ Identifier ]No basis for choice

Graphically

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Recursive Descent (Summary)

1. Build FIRST (and FOLLOW) sets2. Massage grammar to have LL(1) condition

a. Remove left recursionb. Left factor it

3. Define a procedure for each non-terminala. Implement a case for each right-hand sideb. Call procedures as needed for non-terminals

4. Add extra code, as neededa. Perform context-sensitive checkingb. Build an IR to record the code

Can we automate this process?

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Summary

Parsing Part I Introduction to parsing

grammar, derivation, ambiguity, left recursion Predictive top-down parsing

LL(1) condition Recursive descent parsing

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Next Class

Table-driven LL(1) parsing Bottom-up parsing