lecture 5 - university of richmond4 beyond regular languages • difficulty with regular languages...

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Introduction to Parsing

Lecture 5

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Outline

•  Regular languages revisited

•  Parser overview

•  Context-free grammars (CFG’s)

•  Derivations

•  Ambiguity

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Languages and Automata

•  Formal languages are very important in CS –  Especially in programming languages

•  Regular languages –  The weakest formal languages widely used –  Many applications (as we’ve seen)

•  We will also study context-free languages, tree languages

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Beyond Regular Languages

•  Difficulty with regular languages is that many languages are not regular –  Some are very important –  They can’t be expressed using REs and FAs

•  Ex. Strings of balanced parentheses are not regular: –  Note this is fairly representative of lots of

programming constructs

{ }( ) | 0i i i ≥

Note: given as set not RE

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Beyond Regular Languages

•  Ex. Nested arithmetic expressions –  ((1+2) * 3)

•  Ex. Nested if then else statements if then if then if then … fi fi fi

“if” here acts like “(“ in previous example Note that even if language doesn’t have the “fi” like Cool, it is usually implied

An Example To Help Understand the Limitations

•  Consider the following DFA

•  What does it recognize? •  Note: doesn’t have any way of knowing length

of input string 6

0

0

1

1

Ex: 1111111

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Beyond Regular Languages

•  In general: Nesting constructs cannot be handled by regular expressions

•  Raises the questions: –  What can be expressed? –  Why are REs insufficient for recognizing arbitrary

nesting constructs?

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What Can Regular Languages Express?

•  Languages requiring counting modulo a fixed integer –  E.g., parity

•  Intuition: A finite automaton that runs long enough must repeat states

•  Finite automaton can’t remember # of times it has visited a particular state

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The Functionality of the Parser

•  Input: sequence of tokens from lexer

•  Output: parse tree of the program (But some parsers never produce a parse tree . . .)

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Example

•  Cool if x = y then 1 else 2 fi

•  Parser input (from lexical analyzer) IF ID = ID THEN INT ELSE INT FI

•  Parser output IF-THEN-ELSE

=

ID ID

INT INT

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Example

•  Note: nesting structure has been made explicit by tree

•  Also the three components of the if then else –  Predicate –  Then branch –  Else branch

IF-THEN-ELSE

=

ID ID

INT INT

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Comparison with Lexical Analysis

Phase Input Output

Lexer String of characters

String of tokens

Parser String of tokens

Parse tree

Couple of things

•  As mentioned, sometimes parse tree is only implicit –  More on this later –  Many compilers do build full parse tree, many do not

•  There are compilers that combine lexer and parser phases into one phase –  Everything done by the parser –  Parsing technology powerful enough to express

lexical analysis in addition to parsing –  But most compilers use two phases, because REs are

such a good match for lexical analysis 13

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The Role of the Parser

•  Not all strings of tokens are programs . . . •  . . . parser must distinguish between valid and

invalid strings of tokens –  And give error messages for the invalid ones

•  We need –  A language for describing valid strings of tokens –  An algorithm for distinguishing valid from invalid

strings of tokens

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Context-Free Grammars

•  Programming language constructs have recursive structure

•  An EXPR in Cool can be… if EXPR then EXPR else EXPR fi while EXPR loop EXPR pool …

•  Context-free grammars are a natural notation

for this recursive structure

Note: Recursively composed of other expressions

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What is a Context-Free Grammar (CFG)?

•  A CFG consists of –  A set of terminals T –  A set of non-terminals N –  A start symbol S (a non-terminal) –  A set of productions

X →Y1Y2!Yn

where X ∈ N and Yi ∈ T ∪N∪ ε{ }

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Notational Conventions

•  In these lecture notes –  Non-terminals are written upper-case –  Terminals are written lower-case –  The start symbol is the left-hand side of the first

production •  This is standard for CFGs

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Examples of CFGs

S ( S ) S ε

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Examples of CFGs

S ( S ) S ε

What are the parts of the grammar: N = ? T = ? Start = ?

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Examples of CFGs

S ( S ) S ε

What are the parts of the grammar: N = { S } T = { ( , ) } Start = S (the only nonterminal)

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Examples of CFGs

S ( S ) S ε

What are the parts of the grammar: N = { S } T = { ( , ) } Start = S (the only nonterminal) Productions?

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Examples of CFGs

A fragment of Cool:

EXPR if EXPR then EXPR else EXPR fi| while EXPR loop EXPR pool| id

→

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Examples of CFGs (cont.)

Simple arithmetic expressions:

( )

E E E| E + E| E| id

→ ∗

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The Language of a CFG

Read productions as rules:

Means can be replaced by That is, in general, the right hand side can replace

the left hand side.

X →Y1!Yn

X Y1!Yn

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Key Idea

1.  Begin with a string consisting of the start symbol “S”

2.  Replace any non-terminal X in the string by a the right-hand side of some production

3.  Repeat (2) until there are no non-terminals in the string

X →Y1!Yn

So note, the string is changing over time

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The Language of a CFG (Cont.)

More formally, write if there is a production and say that the left hand side “derives” the

right, or “can derive” the right hand side, etc.

X1!Xi!Xn → X1!Xi−1Y1!YmXi+1!Xn

Xi →Y1!Ym

This is one step of a context-free derivation.

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The Language of a CFG (Cont.)

Write if in 0 or more steps We say “the left hand side rewrites in zero or

more steps to the right hand side”

X1!Xn→∗ Y1!Ym

X1!Xn →!→!→Y1!Ym

So, in general…

When we write it is shorthand for saying that there is some sequence of individual productions (rules) that get us from X0 to Xn in zero or more steps

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X 0→∗ Xn

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The Language of a CFG

Let G be a context-free grammar with start symbol S. Then the language, L(G), of G is:

a1…an | S→∗ a1…an and every ai is a terminal

#$%

&'(

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Terminals

•  Terminals are so-called because there are no rules for replacing them

•  Once generated, terminals are permanent feature of the string

•  Terminals ought to be tokens of the language

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Recall earlier Example

L(G) is the language of CFG G Strings of balanced parentheses Two grammars: ( )S SS ε

→

→

( )|

S Sε

→

{ }( ) | 0i i i ≥

OR

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Cool Example

A fragment of COOL: Recall:

–  Non-terminals are written upper-case –  Terminals are written lower-case

Also, could have written as three productions

EXPR if EXPR then EXPR else EXPR fi| while EXPR loop EXPR pool| id

→

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Cool Example (Cont.)

Some elements of the language (why?) idif id then id else id fiwhile id loop id poolif while id loop id pool then id else idif if id then id else id fi then id else id fi

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Arithmetic Example

Simple arithmetic expressions: Some elements of the language:

E E+E | E E | (E) | id→ ∗

id id + id(id) id id(id) id id (id)

∗

∗ ∗

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Notes

The idea of a CFG is a big step. But:

•  Membership in a language is “yes” or “no”; also need parse tree of the input

•  Must handle errors gracefully

•  Need an implementation of CFG’s (e.g., bison)

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More Notes

•  Form of the grammar is important –  Many grammars generate the same language –  Tools are sensitive to the grammar

–  Note: Tools for regular languages (e.g., flex) are sensitive to the form of the regular expression, but this is rarely a problem in practice

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Derivations and Parse Trees

A derivation is a sequence of productions A derivation can be drawn as a tree

–  Start symbol is the tree’s root –  For a production add children

to node

S→!→!→!

X →Y1!Yn X

Y1!Yn

X

Y1 Yn …

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Derivation Example

•  Grammar

•  String

E E+E | E E | (E) | id→ ∗

id id + id∗We wish to “parse” the string

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Derivation Example (Cont.)

EE+EE E+Eid E + Eid id + Eid id + id

→

→ ∗

→ ∗

→ ∗

→ ∗

E

E

E E

E +

id *

id id

parse tree (of the input string)

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Derivation in Detail (1)

E

E

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Derivation in Detail (2)

EE+E→

E

E E +

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Derivation in Detail (3)

E E

EE+EE +→ ∗

→

E

E

E E

E +

*

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Derivation in Detail (4)

EE+EE E+Eid E + E→ ∗

→

→ ∗

E

E

E E

E +

*

id

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Derivation in Detail (5)

EE+EE E+Eid E + id id +

EE→ ∗

→

→ ∗

→ ∗

E

E

E E

E +

*

id id

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Derivation in Detail (6)

EE+EE E+Eid E + Eid id + Eid id + id

→

→ ∗

→ ∗

→

→ ∗

∗

E

E

E E

E +

id *

id id

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Some Interesting Things About Parse Trees

•  A parse tree has –  Terminals at the leaves –  Non-terminals at the interior nodes

•  An in-order traversal of the leaves is the original input –  Let’s go back and take a look

•  The parse tree shows the association of operations, the input string does not –  Note * binds more tightly than + because * is a

subtree of the parse tree

An Interesting Question…

How did I know to pick this particular parse tree for the derivation? It turns out that there is more than one…

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Left-most and Right-most Derivations

•  The example we did is a left-most derivation –  At each step, replace the

left-most non-terminal

•  There is an equivalent notion of a right-most derivation

EE+EE E+Eid E + Eid id + Eid id + id

→

→ ∗

→ ∗

→ ∗

→ ∗

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Left-most and Right-most Derivations

•  The example we did is a left-most derivation –  At each step, replace the

left-most non-terminal

•  There is an equivalent notion of a right-most derivation

EE+EE+idE E + idE id + idid id + id

→

→

→ ∗

→ ∗

→ ∗

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Right-most Derivation in Detail (1)

E

E

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Right-most Derivation in Detail (2)

EE+E→

E

E E +

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Right-most Derivation in Detail (3)

id

EE+EE+→

→

E

E E +

id

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Right-most Derivation in Detail (4)

EE+EE+idE E + id

→

∗

→

→

E

E

E E

E +

id *

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Right-most Derivation in Detail (5)

EE+EE+idE E E

+ idid + id

→

→

→

∗

∗

→

E

E

E E

E +

id *

id

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Right-most Derivation in Detail (6)

EE+EE+idE E + idE id + idid id + id→ ∗

→

→

→ ∗

→ ∗

E

E

E E

E +

id *

id id

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Derivations and Parse Trees

•  Note that right-most and left-most derivations have the same parse tree –  In this case –  And this is not an accident

•  The difference is the order in which branches are added

•  Finally, there could be other parse trees that arise from neither left-most or right-most derivation –  But we are most interested in left-most and right-

most

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Summary of Derivations

•  We are not just interested in whether s is in L(G) –  We need a parse tree for s

•  A derivation defines a parse tree –  But one parse tree may have many derivations

•  Left-most and right-most derivations are important in parser implementation

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Ambiguity

•  Grammar

•  String

E E+E | E E | (E) | id→ ∗

id id + id∗

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Ambiguity (Cont.)

This string has two parse trees

E

E

E E

E *

id +

id id

E

E

E E

E +

id *

id id

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Ambiguity (Cont.)

•  A grammar is ambiguous if it has more than one parse tree for some string –  Equivalently, there is more than one right-most or

left-most derivation for some string

•  Ambiguity is BAD –  Leaves meaning of some programs ill-defined

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Dealing with Ambiguity

•  There are several ways to handle ambiguity

•  Most direct method is to rewrite grammar unambiguously

•  Enforces precedence of * over +

' '

'

E E E | EE id E | id | (E) E | (E)

→ +

ʹ′ ʹ′→ ∗ ∗

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Ambiguity in Arithmetic Expressions

•  Recall the grammar E → E + E | E * E | ( E ) | int •  The string int * int + int has two parse trees:

E

E

E E

E *

int +

int int

E

E

E E

E +

int *

int int

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Ambiguity: The Dangling Else

•  Consider the grammar E → if E then E | if E then E else E | OTHER

•  This grammar is also ambiguous

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The Dangling Else: Example

•  The expression if E1 then if E2 then E3 else E4 has two parse trees

if

E1 if

E2 E3 E4

if

E1 if

E2 E3

E4

•  Typically we want the second form

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The Dangling Else: A Fix

•  else matches the closest unmatched then •  We can describe this in the grammar

E → MIF /* all then are matched */ | UIF /* some then is unmatched */ MIF → if E then MIF else MIF | OTHER UIF → if E then E | if E then MIF else UIF

•  Describes the same set of strings

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The Dangling Else: Example Revisited

•  The expression if E1 then if E2 then E3 else E4

if

E1 if

E2 E3 E4

if

E1 if

E2 E3

E4

•  Not valid because the then expression is not a MIF

•  A valid parse tree (for a UIF)

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Ambiguity

•  No general techniques for handling ambiguity

•  Impossible to convert automatically an ambiguous grammar to an unambiguous one

•  Used with care, ambiguity can simplify the grammar –  Sometimes allows more natural definitions –  We need disambiguation mechanisms

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Precedence and Associativity Declarations

•  Instead of rewriting the grammar –  Use the more natural (ambiguous) grammar –  Along with disambiguating declarations

•  Most tools allow precedence and associativity declarations to disambiguate grammars

•  Examples …

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Associativity Declarations

•  Consider the grammar E → E + E | int •  Ambiguous: two parse trees of int + int + int

E

E

E E

E +

int +

int int

E

E

E E

E +

int +

int int

•  Left associativity declaration: %left +

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Precedence Declarations

•  Consider the grammar E → E + E | E * E | int –  And the string int + int * int

E

E

E E

E +

int *

int int

E

E

E E

E *

int +

int int •  Precedence declarations: %left + %left *