CS 3240: Languages and Computation

1
CS 3240 Languages and Computation

• Chomsky Normal Form andPushdown Automata

2
Review

• A context-free grammar (V, S, R, S) is a grammar
  where all rules are of the form A ? x, with
  A?V and x?(V?S)
• A string is accepted if there is a derivation
  from S to the string
• Representation of derivation by ? or parse
  trees
• Left-most and right-most derivations

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Review Ambiguity

• A string w?L(G) is derived ambiguously if it has
  more than one derivation tree (or equivalently
  if it has more than one leftmost derivation (or
  rightmost)).
• A grammar is ambiguous if some strings are
  derived ambiguously.
• Some languages are inherently ambiguous

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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
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Pushdown automata

• Similar to finite automata, but for CFGs
• Finite automata are not adequate for CFGs
  because we cannot keep track of what weve done
• At any point, we only know the current state, not
  previous states
• Need memory
• PDAs are finite automata with a stack

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Finite automata and PDA schematics
State control
FA
State control
PDA
Stack Infinite LIFO (last in first out) device
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Example
read e pop off stack
read e push e on stack
read e push on stack
read 0 push 0 on stack
read 1 pop 0 off stack
Language accepted 0n1n n ? 0
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Differences between PDAs and NFAs

• Transitions read a symbol of the string and push
  a symbol onto or pop a symbol off of the stack
• Stack alphabet is not necessarily the same as the
  alphabet for the language
• e.g., marks bottom of stack in previous (0n1n)
  example

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Equivalence of PDAs and CFGs

• Theorem A language is context free if and only
  if some pushdown automaton recognizes it
• Proved in two lemmas one for the if direction
  and one for the only if direction
• We will only do the only if step i.e., show
  that every context-free language has an
  associated PDA

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CFGs are recognized by PDAs

• Lemma If a language is context free, then some
  pushdown automaton recognizes it
• Proof idea Construct a PDA following CFG rules

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Constructing the PDA

• You can read any symbol in ? when that symbol is
  at the top of the stack
• Transitions of the form a,a?e
• The rules will be pushed onto the stack when a
  variable A is on top of the stack and there is a
  rule A?w, you pop A and push w
• You can go to the accept state only if the stack
  is empty

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Idea of PDA construction for A?xBz
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Actual construction for A?xBz
In an abuse of notation, we say ?(q,e,A)(q,xBz)
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Constructing the PDA

• Q qstart, qloop, qaccept?E, where E is the
  set of states used for replacement rules onto the
  stack
• ? (the PDA alphabet) is the set of terminals in
  the CFG
• ? (the stack alphabet) is the union of the
  terminals and the variables and (or some
  suitable placeholder)

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Constructing the PDA

• ? is comprised of several rules
• ?(qstart,e,e)(qloop,S)
• Start with placeholder on the stack and with the
  start variable
• ?(qloop,a,a)(qloop,e) for every a??
• Terminals may be read off the top of the stack
• ?(qloop,e,A)(qloop,w) for every rule A?w
• Implement replacement rules
• ?(qloop,e,)(qaccept,e)
• Accept when the stack is empty

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CFGs are recognized by PDAs

• Format of the new PDA

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Example

• S ? SS (S) ()

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Example
e,S?SS e,S?(S) e,S?()

• Read (()())

S
e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
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Example
( S )
e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
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Example
S )
e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
(
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Example
SS )
e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
(
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Example
( ) S )
e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
(
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Example
) S )
e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
((
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Example
S )
e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
(()
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Example
( ) )
e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
(()
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Example
) )
e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
(()(
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Example
)
e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
(()()
37
Example

e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qaccept
qloop
(,(?e ),)?e
(()())
38
Example

e,S?SS e,S?(S) e,S?()

• Read (()())

e, e ?S
e, ?e
qstart
qloop
qaccept
(,(?e ),)?e
(()())
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Test

• Test 1 will be 3-355pm on Tuesday, June 6th
  (second half of the class will be lecture)
• Test will cover regular languages and
  context-free languages, including everything we
  discussed in class except for the proof of
  DFA-gtRE
• Test will be close-book and close-note. But you
  can bring one page of cheat sheet.
• Prepare the cheat sheet yourself! Reorganizing
  the material by yourself is a very effective way
  to learning!