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									Chap. 4, Formal Grammars
       and Parsing
        J. H. Wang
       Mar. 18, 2011
              Outline
• Introduction
• Context-Free Grammars
• Properties of CFGs
• Transforming Extended Grammars
• Parsers and Recognizers
• Grammar Analysis Algorithms
                    Introduction
• A natural language’s grammar: to capture a
  small but important aspect of a sentence’s
  validity with respect to a natural language
• Regular sets: guiding the actions of
  automatically constructed scanner
  – Chap. 3
• Grammar: guiding the actions of the parsers
  – Chap. 5, 6
• Semantic analysis: enforcing programming
  language rules that are not easily expressed by
  grammars
  – Chap. 7, 8, 9
           The Role of the Parser



source               token               Parse tree
program    Lexical                                    Rest of    Intermediate
                                Parser
          Analyzer                                    Font End   representation
                     Get next
                     token




                             Symbol Table
      Context-Free Grammars
• Components: G=(N,,P,S)
  – A finite terminal alphabet : the set of tokens
    produced by the scanner
  – A finite nonterminal alphabet N: variables of the
    grammar
  – A start symbol S: SN that initiates all derivations
     • Goal symbol
  – A finite set of productions P: AX1…Xm, where AN,
    XiN, 1≤i≤m and m≥0.
     • Rewriting rules
• Vocabulary V=N
  – N=
• CFG: recipe for creating strings
• Derivation: a rewriting step using the
  production A replaces the nonterminal
  A with the vocabulary symbols in 
  – Left-hand side (LHS): A
  – Right-hand side (RHS): 
• Context-free language of grammar G L(G):
  the set of terminal strings derivable from S
                 Notations
Names Beginning       Represent Examples
with                  Symbols In
Uppercase             N          A, B, C, Prefix

Lowercase and                    a, b, c, if, then,
punctuation                       (, ;
X, Y                  N         Xi, Y3

Other Greek letters   (N)*      , , 
• Or          • A=>: one step of derivation
  notation:     using the production A
  – A          – =>+: derives in one or more steps
     |          – =>*: derives in zero or more steps
     …
     |       • S=>*:  is a sentential form of
  – A         the CFG
    A       • SF(G): the set of sentential forms
     …          of G
    A
              • L(G)={w*|S=>+w}
                 – L(G)=SF(G)*
• Two conventions that nonterminals are
  rewritten in some systematic order
  – Leftmost derivation: from left to right
  – Rightmost derivation: from right to left
         Leftmost Derivation
• A derivation that always chooses the
  leftmost possible nonterminal at each step
  – =>lm, =>+lm, =>*lm
  – A left sentential form
     • A sentential form produced via a leftmost
       derivation
     • E.g. production sequence in top-down parsers
     • (Fig. 4.1)
• E.g: a leftmost derivation of f ( v + v )
  – E =>lm Prefix ( E )
      =>lm f ( E )
      =>lm f ( v Tail )
      =>lm f ( v + E )
      =>lm f ( v + v Tail )
      =>lm f ( v + v )
      Rightmost Derivations
• The rightmost possible nonterminal is
  always expanded
  – Canonical derivation
  – =>rm, =>+rm, =>*rm
  – A right sentential form
    • A sentential form produced via a rightmost
      derivation
    • E.g. produced by bottom-up parsers (Ch. 6)
    • (Fig. 4.1)
• E.g: a rightmost derivation of f ( v + v )
  – E =>rm Prefix ( E )
      =>rm Prefix ( v Tail )
      =>rm Prefix ( v + E )
      =>rm Prefix ( v + v Tail )
      =>rm Prefix ( v + v )
      =>rm f ( v + v )
                    Parse Trees
• Parse tree: graphical representation of a
  derivation
  – Root: start symbol S
  – Each node: either grammar symbol or λ
  – Interior nodes: nonterminals
     • An interior node and its children: production
  – E.g. Fig. 4.2
• Phrase of the sentential form: a sequence of
  symbols descended from a single
  nonterminal in the parse tree
• A simple or prime phrase: a phrase that
  contains no smaller phrase
• Handle of a sentential form: the leftmost
  simple phrase
• E.g. f ( v Tail ) in Fig. 4.2
    Other Types of Grammars
• Regular grammars: less powerful
• Context-sensitive and unrestricted
  grammars: more powerful
          Regular Grammars
• A CFG that is limited to productions of
  the form AaB or Cd
  – RHS: either a symbol from {λ} followed by
    a nonterminal symbol, or a symbol from {λ}
  – Regular set
     • E.g. {[i]i|i>=1} not regular
        – ST
          T[T]
           |λ
     • Regular sets are a proper subset of the context-free
       languages
Beyond Context-Free Grammars
• Context-sensitive grammar: nonterminals
  are rewritten only when they appear in a
  particular context (A), provided
  the rule never causes the sentential form
  to contract in length
• Unrestricted grammar (type-0 grammar):
  the most general
• More powerful, but less useful
  – Efficient parsers for such grammars do not
    exist
  – It’s difficult to prove properties about such
    grammars
• CFGs: a nice balance between generality
  and practicability
         Properties of CFGs
• Some grammars might have problems:
  – Include useless symbols
  – Allow multiple, distinct derivations for some
    input string
  – Include strings not in the language, or exclude
    strings in the language
          Reduced Grammars
• Each of its nonterminals and productions
  participates in the derivation of some string
  – Useless nonterminals: can be safely removed
  – E.g.
     • SA
        |B
       Aa
       BB b
       Cc
  – Algorithms to detect useless nonterminals
     • Ex.16 and Ex.19
                        Ambiguity
• Allow a derived string to have two or
  more different parse trees
  – E.g.
     • Expr  Expr – Expr
            | id
     • Two different parse trees for id – id – id
           – Fig. 4.3
  – No algorithm for checking an arbitrary CFG
    for ambiguity
     • Undecidable
   Faulty Language Definition
• Terminal strings derivable by the
  grammar do not correspond exactly to the
  strings in the language
• Determining in general whether two CFGs
  generate the same language is an
  undecidable problem
Transforming Extended Grammars
• BNF (Backus-Naur form)
  – Optional symbols: enclosed in square brackets
     • A [X1…Xn] 
  – Repeated symbols: enclosed in braces
     • B {X1…Xm} 
  – E.g. Java-like declaration
     • Declaration  [final][static][const] Type identifier {,
       identifier }
  – Transforming extended BNF grammars into standard
    form
     • Fig. 4.4
EW   ON   ERM




EW   ON   ERM
     Parsers and Recognizers
• Recognizer: to determine if input string x
  L(G)
• Parser: to determine the string’s validity
  and structure (parse tree)
  – Top-down: starting at the root, expanding the
    tree in a depth-first manner
     • Preorder traversal, predictive
  – Bottom-up: starting at the leaves
     • Postorder traversal
• E.g. grammar
  – Program  begin Stmts end $
    Stmts  Stmt; Stmts
           |λ
    Stmt  simplestmt
          | begin Stmts end
  – String: begin simplestmt; simplestmt; end $
    • Top-down parse: Fig. 4.5
    • Bottom-up parse: Fig. 4.6
• Parsing techniques
  – E.g. LL(1), LR(1) are the best-known top-
    down and bottom-up parsing strategies
• L: token sequence is processed from left to
  right
• L,R: Leftmost or Rightmost parse
• 1: the number of lookahead symbols
 Grammar Analysis Algorithms
• Grammar representation
  – Programming language constructs:
     • A set: an unordered collection of distinct entities
     • A list: an ordered collection of entities
     • An iterator: a construct that enumerates the contents of a set
       or list
  – Observations
     • Symbols are rarely deleted from a grammar
     • Transformations can add symbols and productions to a
       grammar
     • Typically visit all rules for a nonterminal, or visit all
       occurrences of a symbol in productions
     • A production’s RHS processed on symbol at a time
                  Grammar Utilities
• Creating or adding:
    –   Grammar(S)
    –   Production(A, rhs)
    –   Nonterminal(A)
    –   Terminal(x)
• Iterators:
    –   Productions()
    –   Noterminals()
    –   Terminals()
    –   RHS(p)
    –   LHS(p)
    –   ProductionsFor(A)
    –   Occurrences(X)
    –   Tail(y)
• Others
    – IsTerminal(X)
    – Production(y)
    Deriving the Empty String
• It’s common to determine which
  nonterminals can derive λ
  – Not trivial because the derivation can take
    more than one step
     • A=>BCD=>BC=>B=> λ
  – Fig. 4.7
ERIVES      MPTY   TRING
       ON    ERMINALS


       RODUCTIONS




HECK    OR    MPTY




             CCURRENCES
   RODUCTION


   HECK      OR    MPTY


HECK    OR    MPTY
• The algorithm establishes two structures
  – RuleDerivesEmpty(p)
  – SymbolDerivesEmpty(A)
  – Useful in grammar analysis and parsing
    algorithms in Chap.4, 5, & 6
                      First Sets
• The set of all terminal symbols that can
  begin a sentential form derivable from the
  string 
  – First()={ a| =>*a }
  – We never include λ in First() even if =>λ
  – E.g. (in Fig.4.1)
     • First(Tail) = {+}
     • First(Prefix) = {f}
     • First(E) = {v, f, (}
  – Fig.4.8, Fig. 4.9, Fig. 4.10
IRST
          ON    ERMINALS
NTERNAL        IRST




NTERNAL    IRST




                 NTERNAL   IRST


                NTERNAL    IRST
               Follow Sets
• The set of terminals that can follow a
  nonterminal A in some sentential form
  – For AN,
     • Follow(A) = {b|S=>+Ab}
  – The right context associated with A
  – Fig. 4.11
OLLOW
          ON   ERMINALS


NTERNAL OLLOW



NTERNAL OLLOW




             CCURRENCES
                  IRST    AIL
     LL   ERIVE    MPTY


                      RODUCTION
                      NTERNAL OLLOW



LL   ERIVE     MPTY
• First and Follow sets can be generalized to
  include strings of length k
  – Firstk(), Followk(A)
  – Useful in parsing techniques that use k-
    symbol lookaheads (e.g. LL(k), LR(k))
  More on FIRST and FOLLOW
• Two functions FIRST and FOLLOW allow us to
  choose which production to apply, based on the
  next input symbol
• FIRST(): the set of terminals that begin strings
  derived from 
  – Ex: (Fig. 4.15) A=>* c, c is in FIRST(A)
• FOLLOW(a): the set of terminals a that can
  appear immediately to the right of A in some
  sentential form
  – Ex: S =>* Aa
• To compute FIRST(X) for all grammar
  symbols X
  – If X is a terminal, FIRST(X)={X}
  – If X is a nonterminal and XY1Y2…Yk, then
    place a in FIRST(X) if for some i, a is in
    FIRST(Yi) and Y1…Yi-1=>* 
  – If Xe is a production, add  to FIRST(X)
• To compute FOLLOW(A) for all
  nonterminals A
  – Place $ in FOLLOW(S)
  – If there’s a production AB, then
    everything in FIRST() except  is in
    FOLLOW(B)
  – If there’s a production AB, or AB,
    where FIRST() contains , then everything in
    FOLLOW(A) is in FOLLOW(B)
                   Example
• Ex: (4.28)
   – E  T E’
     E’  + T E’ | 
     T  F T’
     T’  * F T’ | 
     F  (E) | id
   – FIRST(F)=FIRST(T)=FISRT(E)={(,id}
   – FIRST(E’)={+,e}
   – FIRST(T’)={*,e}
   – FOLLOW(E)=FOLLOW(E’)={),$}
   – FOLLOW(T)=FOLLOW(T’)={+,),$}
   – FOLLOW(F)={+,*,),$}
Thanks for Your Attention!

								
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