Pushdown Automata and ContextFree Languages

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Pushdown Automata and Context-Free Languages

• Jianhua Yang
• Department of Math and Computer Science
• Bennett College

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Goals

• Pushdown automata
• Context-free grammars
• The limits of PDA
• LL(k) parsers
• LR(k) parsers

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2.1 Pushdown Automata (PDA)

• How to recognize L?
• Lxnyn n?N

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1. Definition of PDA
Input tape
Head moves in this direction
tape head
1
8
2
7
3
4
6
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Control mechanism
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Read a symbol by a PDA

• Following the sequence
• Read a symbol from the input
• Pop a symbol from the stack
• Push a symbol into the stack
• Generate a new state.

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Examples

• (p, x, s q, y)

• (p, ?, ? q, ?)

• (p, ?, s q, ?)

• (p, x, ? q, z)

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PDA transition diagram
y, x ?
y, x ?
2
3
?, ?
?, ?
x, ? x
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Formal definition

• P(S, ?,G,T, i, F )
• S finite collection of states.
• ? is the machines alphabet.
• G is the finite collection of stack symbols.
• T is a finite collection of transitions.
• i (an element of S) is the initial state.
• F is the collection of accept states.

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2. Pushdown automata as Language Accepters

• How to accept a language?
• L(M) denotes all the strings accepted by PDA M.
• The languages accepted by PDA include the regular
  languages.

Why? Think about the case (p, x, ? q, ?).
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Example

• Design a PDA to accept language Lxnyn n?N

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One exception

• A string might be accepted by a PDA but without
  emptying its stack.

x,?x
y,?x
y,?x
1
2

• Lxmyn m,n?N, mn

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Theorem 2.1

• For each PDA that accepts strings without
  emptying its stack, there is a PDA that accepts
  the same language but empties its stack before
  reaching an accept state.

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• For this part, please read the book.

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2.2 Context-free Grammars

• The question is what languages can be recognized
  by a PDA.

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1. Definition of CFG

• Definition
• Context-free
• Reflects the fact that since the left side of
  each rewrite rule can contain only a single
  nonterminal, the rule can be applied regardless
  of the context in which that nonterminal is
  found.

Left side of each rule must be a single
nonterminal, there are no restrictions on the
right side of each rule.
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Examples

• xNy?xzy Context-free?

• S?zMNz
• M?aMa
• M?z
• N?bNb
• N?z
• Context-free?

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Derivations

• Leftmost derivation
• Rightmost derivation

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Context-free Languages

• The languages generated by context-free grammars
  are called context-free languages (CFL)

Lxnynn?N
S?xSy S??
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2. CFG and PDA

• For convenience,
• We take the notation (p, a, s q, xyz) to
  represent push more than one symbol on the stack.
• It equals to (p, a, s q1, z), (q1, ?, ? q2, y),
  and (q2, ?, ? q, x)

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Equivalence between CFG and PDA

• We need to show
• For any CFG G, there is a PDA M, such that
  L(M)L(G) (Theorem 2.2)
• And for any PDA M, there is a CFG G such that
  L(G)L(M). (Theorem 2.3)

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Theorem 2.2

• For each CFG G, there is a pushdown automaton M
  such that L(G)L(M).

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Proof

• Given G, we construct M as follows
• 1. ?terminals of G
• Gterminals, nonterminals,
• 2. Assume M has states i, p, q, f.
• i initial state, f accept state
• 3. Introduce the transition (i, ?,? p, )
• 4. Introduce a transition (p, ?,? q, S) where S
  is the start symbol in G.

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Proof (Cont.)

• 5. Introduce a transition of the form (q,?, N
  q, w) for each rule N?w
• 6. Introduce a transition of the form (q, x, x
  q,?) for each terminal x in G.
• 7. Introduce the transition
• (q,?, f,?)

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Example

• We have a CFG G

S?xMNz M?aMa M?z N?bNb N?z
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The designed PDA M
?, ? S
q
p
?, ?
?, ?
?, S zMNz
?, M aMa
i
z, z ?
?, M z
b, b ?
?, N bNb
a, a ?
?, N z
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M accepts zazabzbz
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Theorem 2.3

• For each PDA M, there is a CFG G such that
  L(M)L(G).

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Proof

• The idea of prove this is that for each PDA, we
  can design a CFG, and vise versa.

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2.3 The limits of PDA

• Are there any languages that are not context-free?

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Pumping lemma

• Theorem 2.5 If L is a CFG containing infinitely
  many strings, then there must be a string in L of
  the form svuwt, where s, v, u, w, t are
  substrings, at least one of v and w is nonempty,
  and svnuwnt is in L for each n? N.

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Proof

• skip

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Example

• To prove that the language xnynzn n ?N is not
  CFL by using pumping lemma.

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Deterministic PDA

• It is a PDA in which one and only one transition
  is applicable at any time.

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Definition

• We define a deterministic PDA to be a pushdown
  automata (S,?,G T, i, F ) such that for each
  triple (p, x, y) in S x ? x G, there is one and
  only one transition in T of the form (p, u, v q,
  z), where (u, v) is in (x, y), (x, ?), (?,y),
  (?, ?),, q is in S, and z is in G.

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Theorem 2.6

• There is a context-free language that is not the
  language accepted by any deterministic pushdown
  automata.

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Proof

• Self study.

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2.4 Summary

• Pushdown Automata
• Context-free Languages
• Context-free grammar

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The hierarchy of languages introduced so far