Chapter 4. Syntax Analysis (1)

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Chapter 4.Syntax Analysis (1)
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Application of a production ?A????? in a
derivation step ?i ? ?i1
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Formal grammars (1/3)

• Example Let G1 have N A, B, C, T a, b,
  c and the set of productions
• ? ? A CB ? BC
• A ? aABC bB ? bb
• A ? abC bC ? bc
• cC ? cc
• The reader should convince himself that the word
  akbkck is in L(G1) for all k ? 1 and that only
  these words are in L(G1). That is,
• L(G1) akbkck k ? 1.

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Formal grammars (2/3)

• Example Grammar G2 is a modification of G1
• G2 ? ? A CB ? BC
• A ? aABC bB ? bb
• A ? abC bC ? b
• The reader may verify that L(G2) akbk k ?
  1. Note that the last rule, bC ? b, erases all
  the C's from the derivation, and that only this
  production removes the nonterminal C from
  sentential forms.

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Formal grammars (3/3)

• Example A simpler grammar that generates akbk
  k ? 1 is the grammar G3
• G3 ? ? S
• S ? aSb
• S ? ab
• A derivation of a3b3 is
• ? ? S ? aSb ? aaSbb ? aaabbb
• The reader may verify that L(G3) akbk k ?
  1.

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Type Format of Productions Remarks
0 fA?? f? ? Unrestricted Substitution Rules
1 fA?? f? ?, ??? ??? Context Sensitive Context Free Right Linear Left Linear
2 A ??, ??? ??? Context Sensitive Context Free Right Linear Left Linear
3 A?aB A?a ??? A?Ba A ?a ??? Context Sensitive Context Free Right Linear Left Linear
Contracting
Noncon- tracting
Regular
The four types of formal grammars
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Context-Sensitive Grammars(Type1)
Unrestricted Grammars(Type0)

• Definition A context-sensitive grammar G
  (N,T,P,?) is a formal grammar in which all
  productions are of the form
• fA??f??, ?? ?
• The grammar may also contain the production ?
  ??, if G is a context-sensitive (type1) grammar,
  then L(G) is a context-sensitive (type1) language.

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Context-Free Grammars (Type2)

• Definition A context-free grammar G(N,T,P,?)
  is a formal grammar in which all productions are
  of the form
• A??
• The grammar may also contain the production ?
  ??. If G is a context-free (type2) grammar, then
  L(G) is a context-free (type2) language.

A?N?? ??(N?T)-?
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Regular Grammars (Type3) (1/2)

• Definition A production of the form
• A?aB or A?a
• is called a right linear production. A
  production of the form
• A?Ba or A?a
• is a left linear production. A formal grammar is
  right linear if it contains only right linear
  productions, and is left linear if it contains
  only left linear production ? ??. Left and right
  linear grammars are also known as regular
  grammars. If G is a regular (type3) grammar, then
  L(G) is a regular (type3) language.

A?N?? B?N a?T
A?N?? B?N a?T
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Regular Grammars (Type3) (2/2)

• Example A left linear grammar G1 and a right
  linear grammar G2 have productions as follows
• G1 G2
• The reader may verify that
• L(G1) (10)11(01)L(G2)

? ? 1B ? ? 1 A ? 1B B ? 0A A ? 1
? ? B1 ? ? 1 A ? B1 B ? A0 A ? 1
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Ambiguity (1/2)

• Example Consider the context-free grammar
• G ? ? S
• S ? SS
• S ? ab
• We see that the derivations correspond to
  different tree diagrams. The grammar G is
  ambiguous with respect to the sentence ababab if
  the tree diagrams were used as the basis for
  assigning meaning to the derived string, mistaken
  interpretation could result.

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Ambiguity (2/2)

• Definition A context-free grammar is ambiguous
  if and only if it generates some sentence by two
  or more distinct leftmost derivations.

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Fig. 4.1. Position of parser in compiler model.
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Syntax Error Handling (1/2)

• Probable Errors
• lexical, such as misspelling an identifier,
  keyword, or operator
• syntactic, such as an arithmetic expression with
  unbalanced parentheses
• semantic, such as an operator applied to an
  incompatible operand
• logical, such as an infinitely recursive call

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Syntax Error Handling (2/2)

• The error handler in a parser has simple-to-state
  goals
• It should report the presence of errors clearly
  and accurately.
• It should recover from each error quickly enough
  to be able to detect subsequent errors.
• It should not significantly slow down the
  processing of correct programs.

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Error-Recovery Strategies

• panic mode
• phrase level
• error productions
• global correction

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Example 4.2

• The grammar with the following productions
  defines simple arithmetic expressions.

expr expr expr expr op op op op op ? ? ? ? ? ? ? ? ? expr op expr ( expr ) - expr id - / ?
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Notational Conventions (1/2)

• 1. These symbols are terminals
• i) Lower-case letters early in the alphabet such
  as a, b, c.
• ii) Operator symbols such as , -, etc.
• iii) Punctuation symbols such as parentheses,
  comma, etc.
• iv) The digits 0, 1, . . . , 9.
• v) Boldface strings such as id or if.
• 2. These symbols are nonterminals
• i) Upper-case letters early in the alphabet such
  as A, B, C.
• ii) The letter S, which, when it appears, is
  usually the start symbol.
• iii) Lower-case italic names such as expr or
  stmt.
• 3. Upper-case letters late in the alphabet, such
  as X, Y, Z, represent grammar symbols, that is,
  either nonterminals or terminals.

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Notational Conventions (2/2)

• 4. Lower-case letters late in the alphabet,
  chiefly u, v, . . . , z, represent strings of
  terminals.
• 5. Lower-case Greek letters, ?, ?, ?, for
  example, represent strings of grammar symbols.
  Thus, a generic production could be written as
  A ? ?, indicating that there is a single
  nonterminal A on the left of the arrow (the left
  side of the production) and a string of grammar
  symbols ? to the right of the arrow (the right
  side of the production).
• 6. If A ? ?1, A ? ?2, . . . , A ? ?k are all
  productions with A on the left (we call them
  A-productions), we may write A ? ?1 ?2 . . .
  ?k . We call ?1, ?2, . . . , ?k the alternatives
  for A.
• 7. Unless otherwise stated, the left side of the
  first production is the start symbol.

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Derivations

• We say that ?A? ? ??? if A ? ? is a production
  and ? and ? are arbitrary strings of grammar
  symbols. If
• ?1 ? ?2 ? . . . ? ?n, we say ?1 derives ?n. The
  symbol ? means derives in one step. Often we
  wish to say derives in zero or more steps. For
  this purpose we can use the symbol ?. Thus,
• 1. ? ? ? for any string ?, and
• 2. If ? ? ? and ? ? ?, then ? ? ?.

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Fig. 4.3. Building the parse tree from derivation
(4.4)
(Grammar 4.4 ) E ? -E ? -(E) ? -(EE) ? -(idE) ?
-(idid)
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Eliminating Ambiguity
stmt ? if expr then stmt if expr then stmt else stmt other
stmt matched_stmt unmatched_stmt ? ? ? matched_stmt unmatched_stmt if expr then matched_stmt else matched_stmt other if expr then stmt if expr then matched_stmt else unmatched_stmt
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Elimination of Left Recursion

• No matter how many A-productions there are, we
  can eliminate immediate left recursion from them
  by the following technique. First, we group the
  A-productions as
• A ? A?1 A?2 . . . A?m ?1 ?2 . . .
  ?n
• where no begins with an A. Then, we replace the
  A-productions by
• A ? ?1A' ?2A' . . . ?nA'
• A' ? ?1A' ?2A' . . . ?mA' ?

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Left Factoring

• In general, if A ? ??1 ??2 are two
  A-productions, and the input begins with a
  nonempty string derived from ?, we do not know
  whether to expand A to ??1 or to ??2 . However,
  we may defer the decision by expanding A to ?A'.
  Then, after seeing the input derived from ?, we
  expand A' to ?1 or to ?2 . That is,
  left-factored, original productions become
• A ? ?A' A' ? ?1 ?2
• Example 4.12.
• The language L2 anbmcndm n ? 1 and m ? 1

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Fig. 4.9. Steps in top-down parse.
(a)
(b)
(c)
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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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Fig. 4.13. Model of a nonrecursive predictive
parser.
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Nonrecursive Predictive Parsing

• 1. If X a , the parser halts and announces
  successful completion of parsing.
• 2. If X a ? , the parser pops X off the stack
  and advances the input pointer to the next input
  symbol.
• 3. If X is a nonterminal, the program consults
  entry MX, a of the parsing table M. This entry
  will be either an X-production of the grammar or
  an error entry. If, for example, MX, a X ?
  UVW, the parser replaces X on top of the stack
  by WVU (with U on top). As output, we shall
  assume that the parser just prints the production
  used any other code could be executed here. If
  MX, a error, the parser calls an error
  recovery routine.

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Fig. 4.15. Parsing table M for grammar (4.11).
NONTER-MINAL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL
NONTER-MINAL Id ( )
E E' T T' F E ? TE' T ? FT' F ? id E' ? TE' T' ? ? T' ? FT' E ? TE' T ? FT' F ? (E) E' ? ? T' ? ? E' ? ? T' ? ?
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Fig. 4.16. Moves made by predictive parser on
input id id id.
STACK INPUT OUTPUT
E E' T E' T' F E' T' id E' T' E' E' T E' T E' T' F E' T' id E' T' E' T' F E' T' F E' T' id E' T' E' id id id id id id id id id id id id id id id id id id id id id id id id id id id id E ? T E' T ? F T' F ? id T' ? ? E' ? T E' T ? F T' F ? id T' ? F T' F ? id T' ? ? E' ? ?
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Fig. 4.17. Parsing table M for grammar (4.13).
NONTER-MINAL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL
NONTER-MINAL a b e i t
S S ? a S ? iEtSS'
S' S' ? ? S' ? eS S' ? ?
E E ? b
S E ? ? iEtS iEtSeS a b
(Grammar 4.13 )
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Fig. 4.18. Synchronizing tokens added to parsing
table of Fig. 4.15.
NONTER-MINAL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL INPUT SYMBOL
NONTER-MINAL id ( )
E E' T T' F E ? TE' T ? FT' F ? id E' ? TE' synch T' ? ? synch T' ? FT' synch E ? TE' T ? FT' F ? (E) synch E' ? ? synch T' ? ? synch synch E' ? ? synch T' ? ? synch
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Fig. 4.19. Parsing and error recovery moves made
by predictive parser.
STACK INPUT OUTPUT
E E E' T E' T' F E' T' id E' T' E' T' F E' T' F E' T' E' E' T E' T E' T' F E' T' id E' T' E' ) id id id id id id id id id id id id id id id id id id id error, skip ) id is in FIRST(E) error, MF, synch F has been popped