Compiler Construction Parsing Part I

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Compiler ConstructionParsing Part I

• ?4? 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

3
Todays 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

• Parser
• Checks the stream of words and their parts of
  speech(produced by the scanner) for grammatical
  correctness
• Determines if the input is syntactically well
  formed
• Guides checking at deeper levels than syntax
• Builds 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 mathematics of
  arbitrary languages
• Roadmap
• Context-free grammars and derivations
• Top-down parsing
• Hand-coded recursive descent parsers
• LL(1) parsersLL(1) parsed top-down, left to
  right scan, leftmost derivation, 1 symbol
  lookahead
• Bottom-up parsing
• Operator precedence parsing
• LR(1) parsersLR(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 grammar
• SheepNoise ? SheepNoise baa
• baa
• This CFG defines the set of noises sheep normally
  make
• It is written in a variant of BackusNaur 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
To explore the uses of CFGs, we need a more
complex grammar

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

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 dont 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
Leftmost derivation
Rightmost derivation

• 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!

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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

• 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
• Associativity and precedence
• Dangling else

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Ambiguous Grammars

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

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Two Leftmost Derivations for x 2 y

• The Difference
• Different productions chosen on the second step

New choice
Original choice

• Both derivations succeed in producing x - 2 y

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Derivations and Precedence/Association

• These two derivations point out a problem with
  the grammarIt 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 precedence
• Left-associative The next-level (higher)
  nonterminal places at the last of a production

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Derivations and Precedence
Adding 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 1, then production 2
production 2, then production 1
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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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Ambiguity
if 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 ambiguity
• a 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 problem
• Sometimes, 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 tree
• Repeat until the fringe of the parse tree matches
  the input string
• At a node labeled A, select a production with A
  on its lhs and, for each symbol on its rhs,
  construct the appropriate child
• When a terminal symbol is added to the fringe and
  it doesnt match the fringe, backtrack
• 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 Grammar
Version with precedence derived last lecture
And the input x 2 y
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Example
Lets try x 2 y
Leftmost derivation, choose productions in an
order that exposes problems
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Example
Lets try x 2 y
This worked well, except that doesnt match
The parser must backtrack to here
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Example
Continuing with x 2 y
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Example
Trying to match the 2 in x 2 y

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

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Example
Trying again with 2 in x 2 y
This time, we matched consumed all the input ?
Success!
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Another Possible Parse
Other choices for expansion are possible

• This doesnt 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

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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 ? Aa, for some string a ? (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 a
• ß
• where neither a nor ß start with Fee
• We can rewrite this as
• Fee ? ß Fie
• Fie ? a Fie
• e
• where 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
Term ? Term Factor Term / Factor
Factor
Expr ? Expr Term Expr Term
Term
Applying the transformation yields
Expr ? Term Expr' Expr' Term Expr'
Term Expr' e
Term ? Factor Term' Term' Factor Term'
/ Factor Term' e
These fragments use only right recursion They
retain the original left associativity
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Eliminating Left Recursion
Substituting 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
  recursion
• What 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 ? As? with
  Ai? d1? d2?dk?,
• where As? d1d2dk 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
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Eliminating Left Recursion

• How does this algorithm work?
• Impose arbitrary order on the non-terminals
• Outer loop cycles through NT in order
• Inner loop ensures that a production expanding Ai
  has no non-terminal As in its rhs, for s lt i
• Last step in outer loop converts any direct
  recursion on Ai to right recursion using the
  transformation showed earlier
• New non-terminals are added at the end of the
  order have no left recursion
• At the start of the ith outer loop iteration
• For all k lt i, no production that expands Ak
  contains a non-terminal
• As in its rhs, for s lt k

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Example

• Order of symbols G, E, T

• 1. Ai G

4. Ai T G ?E E ? T E' E' ? T E' E' ? e T ?
id T' T' ?E' T T' T' ? e

• 3. Ai T, As E
• G ?E
• E ? T E'
• E' ? T E'
• E' ? e
• T ? T E' T
• T ? id

2. Ai E G ?E E ? T E' E' ? T E' E' ? e T ? E
T T ? id
G ?E E ? E T E ? T T ? E T T ? id
Go to Algorithm
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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 backtrack
• Alternative 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
  Earleys 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 idea
• Given A ? a ß, the parser should be able to
  choose between a ß
• FIRST sets
• For some rhs a?G, define FIRST(a) as the set of
  tokens that appear as the first symbol in some
  string that derives from a
• That is, x ? FIRST(a) iff a ? x ?, for some ?
• 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 idea
• Given A ? a ß, the parser should be able to
  choose between a ß
• FIRST sets
• For some rhs a?G, define FIRST(a) as the set of
  tokens that appear
• as the first symbol in some string that derives
  from a
• That is, x ? FIRST(a) iff a ? x ?, for some ?
• The LL(1) Property
• If A ? a and A ? ß both appear in the grammar, we
  would like
• FIRST(a) n FIRST(ß) Ø
• This would allow the parser to make a correct
  choice with a lookahead of exactly one symbol !

This is almost correct See the next slide
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Predictive Parsing

• What about e-productions?
• ? They complicate the definition of LL(1)
• If A ? a and A ? ß and e ? FIRST(a), then we need
  to ensure that FIRST(ß) is disjoint from
  FOLLOW(a), too
• Define FIRST(a) as
• FIRST(a) ? FOLLOW(a), if e ? FIRST(a)
• FIRST(a), otherwise
• Then, a grammar is LL(1) iff A ? a and A ? ß
  implies
• FIRST(a) n FIRST(ß) Ø

FOLLOW(a) is the set of all words in the
grammar that can legally appear immediately after
an a
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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 fast
• Consider A ? ß1 ß2 ß3, with
• FIRST(ß1) n FIRST (ß2) n FIRST (ß3) Ø

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Recursive Descent Parsing
Recall the expression grammar, after
transformation
This produces a parser with six mutually
recursive routines Goal Expr EPrime
Term TPrime Factor Each recognizes one NT or
T The term descent refers to the direction in
which the parse tree is built.
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Fig. 4.10. Transition diagrams for grammar (4.11).
(Grammar 4.11 )
?
E E' T T' F ? ? ? ? ? TE' TE' ? FT' FT' ? (E) id
?
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Fig. 4.11. Simplified transition diagrams.
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Fig. 4.12. Simplified transition diagrams for
arithmetic expressions.
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Recursive Descent Parsing
A 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 using 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 order

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 a that
occurs in two or more right-hand
sides of A if a ? e then replace all of the A
productions, A ? aß1 aß2 aßn ?
, with A ? aZ ? Z ? ß1 ß2
ßn where Z is a new element of
NT Repeat until no common prefixes remain
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Left Factoring
A graphical explanation for the same idea
A ? aß1 aß2 aß3
becomes
A ? a Z Z ? ß1 ß2 ßn
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Left Factoring An Example
Consider the following fragment of the expression
grammar
Factor ? Identifier Identifier
ExprList Identifier ( ExprList )
FIRST(rhs1) Identifier FIRST(rhs2)
Identifier FIRST(rhs3) Identifier
After left factoring, it becomes
Factor ? Identifier Arguments Arguments ?
ExprList ( ExprList )
e
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
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Recursive Descent (Summary)

• Build FIRST (and FOLLOW) sets
• Massage grammar to have LL(1) condition
• Remove left recursion
• Left factor it
• Define a procedure for each non-terminal
• Implement a case for each right-hand side
• Call procedures as needed for non-terminals
• Add extra code, as needed
• Perform context-sensitive checking
• Build an IR to record the code
• Can we automate this process?

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Summary

• Parsing Part I
• Introduction to parsinggrammar, 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