CS 3240: Languages and Computation

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CS 3240 Languages and Computation

• Ambiguity and
• Chomsky Norm Form

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Parsing

• Generative aspect of CFG it is easy to derive
  strings w?L(G) from a CFG G
• Analytical aspect Given a CFG G and strings w,
  decide if w?L(G) and if so how do you determine
  the derivation tree or the sequence of rules that
  produce w?
• This is the problem of parsing.

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Derivation Examples

• Grammar
• S?SS SS (S) id
• String
• int int int
• Derivation
• S ? S S ? S S S ? int S S ? int int
  S ? int int int
• Can be represented using a tree structure

Numbers indicate orderof substitution.
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Left-Most and Right-Most

• Left-most derivations
• At each step, replace the left-most non-terminal
• Right-most derivations
• At each step, replace the left-most non-terminal
• E.g., S ? S S ? S int ? S S int ? S int
  int ? int int int
• Note Different traversal order of a tree

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Defining a Parse Tree

• For a CFG G (V, S, R, S) a derivationtree has
  the following properties
• The root is labeled S
• Each leaf is from S ??
• Each interior node is in V
• If node has label A?V and its children a1an
  (from L to R),then P must have the ruleA? a1an
  (with aj?V?T??)
• A leaf labeled ? is a singlechild (has no
  siblings).
• Let G be a CFG. We have w?L(G) if and only if
  there exists a derivation tree of G that yields w.

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Another Example

• Consider CFG S ? 0 1 ?(S) (S)?(S) (S)?(S)
  for terminals 0, 1, ?, ?, ?, ( and
  ).
• Derivations of (0)?((0)?(1))
• leftmost S ? (S)?(S) ? (0)?(S) ?
  (0)?((S)?(S)) ? (0)?((0)?(S)) ?
  (0)?((0)?(1))
• rightmost S ? (S)?(S) ? (S)?((S)?(S)) ?
  (S)?((S)?(1)) ? (S)?((0)?(1)) ? (0)?((0)?(1))
• something else S ? (S)?(S) ? (0)?(S) ?
  (0)?((S)?(S)) ? (0)?((S)?(1)) ? (0)?((0)?(0))
• Note We are typically use either leftmost or
  rightmost

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Its Parse Tree
The derivation S ? (0)?((0)?(1)) can be
expressed by the following parse tree
S
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Ambiguity
A string w?L(G) is derived ambiguously if it has
more than one derivation tree (or equivalently
if it hasmore than one leftmost derivation (or
rightmost)). A grammar is ambiguous if some
strings are derived ambiguously.
Example rule S ? 0 1 SS S?S S ? SS ?
S?SS ? 0?SS ? 0?1S ? 0?11 versus S ? S?S ?
0?S ? 0?SS ? 0?1S ? 0?11
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Ambiguity and Parse Trees
The ambiguity of 0?11 is shown by the
twodifferent parse trees
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Inherently Ambiguous

• Languages that can only be generated by ambiguous
  grammars are inherently ambiguous.
• Example L aibjck, ij or jk.
• The way to make a CFG for this L somehow has to
  involve the step S ? S1S2 where S1 produces the
  strings anbncm and S2 the strings anbmcm.
• This will be ambiguous on strings anbncn.
• Proving this rigorously is difficult.

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Abstract Syntax Trees

• (Abstract) syntax trees are simplified
  representations of parse trees
• Example 324

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Abstract syntax tree
Parse tree
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Resolution of Ambiguities

• Some ambiguities are inessential but some others
  must be resolved
• The following grammar is ambiguous
• exp ? exp op exp ( exp ) numberop ? -
  
• Sample ambiguous strings 123 and 1-2-3
• Resolution of ambiguity
• Precedence has higher precedence than and -
• Left-association perform ops from left to right
• Full parenthesization
• Can we revise the grammar rules to incorporate
  these techniques?

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Resolution of Ambiguities

• Precedence Group operators into different groups
  make operations with lower precedence closer to
  the root
• exp ? exp addop exp termaddop ? -term ?
  term mulop term factormulop ? factor ? ( exp
  ) number
• Associativity allow recursion only on left
• exp ? exp addop term termterm ? term mulop
  factor factor

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Dangling Else Statement

• Ambiguous grammar
• statement ? if-stmt other
• if-stmt ? if ( exp) statement if ( exp)
  statement else statement
• exp ? ...
• Resolution
• Bracketing with endif (e.g., shell script)
• Revise the grammar
• statement ? matched-stmt unmatched-stmt
• matched-stmt ? if ( exp) matched-stmt else
  matched-stmt other
• unmatched-stmt ? if ( exp) statement if ( exp)
  matched-stmt else unmatched-stmt
• In practice, compilers typically use
  disambiguating rules instead of changing grammar
  rules

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Chomsky normal form

• Method of simplifying a CFG
• Definition A context-free grammar is in Chomsky
  normal form if every rule is of one of the
  following forms
• A ? BC
• A ? a
• where a is any terminal and A is any variable,
  and B, and C are any variables or terminals other
  than the start variable
• if S is the start variable then
• the rule S ? e is the only permitted ? rule

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CFGs and Chomsky normal form

• Theorem Any context-free language is generated
  by a context-free grammar in Chomsky normal form.
• Proof idea Convert any CFG to one in Chomsky
  normal form by removing or replacing rules in
  wrong form
• Add a new start symbol
• Eliminate e rules of the form A ? e
• Eliminate unit rules of the form A ? B
• Convert remaining rules into proper form
• Example (work out on whiteboard)
• S ?ASA aB
• A ?B S
• B ?b ?

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Convert a CFG to Chomsky normal form

• Add a new start symbol
• Create the following new rule
• S0 ? S
• where S is the start symbol and S0 is not used
  in the CFG

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Convert a CFG to Chomsky normal form

• Eliminate all e rules A ? e, where A is not the
  start variable
• For each rule with an occurrence of A on the
  right-hand side, add a new rule with the A
  deleted
• R ? uAv becomes R ? uAv uv
• R ? uAvAw becomes R ? uAvAw uvAw uAvw uvw
• If we have R ? A, replace it with R ? e unless
  we had already removed R ? e

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Convert a CFG to Chomsky normal form

• Eliminate all unit rules of the form A ? B
• For each rule B ? u, add a new rule A ? u, where
  u is a string of terminals and variables, unless
  this rule had already been removed
• Repeat until all unit rules have been replaced

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Convert a CFG to Chomsky normal form

• Convert remaining rules into proper form
• Whats left?
• Replace each rule A ? u1u2uk, where k ? 3 and ui
  is a variable or a terminal with k-1 rules
• A ? u1A1 A1 ? u2A2 Ak-2 ? uk-1uk

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Chomsky Hierarchy

• Chomsky Hierarchy of grammars
• Type 0 unrestricted grammars
• Type 1 context-senstive grammars
• Type 2 context-free grammars
• Type 3 regular
• Named after Noam Chomsky who made seminal
  contributions to the field of theoretical
  linguisticsin the 60s. (cf. Chomsky hierarchyof
  languages).

1928- Linguist / Political theorist