Pushdown Automata

1
Pushdown Automata

• There are context-free languages that are not
  regular.
• Finite automata cannot recognize all context-free
  languages.

2
Pushdown Automata

• a, b is regular.
• akbk k is a constant is regular.
• anbn n ? 0 is not regular.

3
Finite Automata
Input file
Control unit q0
yes/no
4
Pushdown Automata
Input file
Control unit q0
Stack
yes/no
5
Non-deterministic Pushdown Automata (NPDA)

• M (Q, ?, ?, ?, q0, z, F)
• Q finite set of internal states
• ? finite set of symbols - input alphabet
• ? finite set of symbols - stack alphabet
• ? Q ? (???) ? ? ? finite subsets of Q ? ?
  transition function
• q0 ? Q initial state
• z ? ? stack start symbol
• F ? Q set of final states

6
Non-deterministic Pushdown Automata (NPDA)

• ? Q ? (???) ? ? ? finite subsets of Q ? ?

stack top
stack top replacement
7
Example

• ?(q1, a, b) (q2, cd), (q3, ?)

c
d
q2
b
q1
q3
8
Example

• M (Q, ?, ?, ?, q0, z, F)
• Q q0, q1, q2, q3 ?(q0, a, 0) (q1, 10),
  (q3, ?)
• ? a, b ?(q0, ?, 0) (q3, ?)
• ? 0, 1 ?(q1, a, 1) (q1, 11)
• z 0 ?(q1, b, 1) (q2, ?)
• F q3 ?(q2, b, 1) (q2, ?)
• ?(q2, ?, 0) (q3, ?)

9
Instantaneous Description

• (q, w, u)
• current state unread part of stack contents
• input string

10
Instantaneous Description

• move (q1, aw, bx) ? (q2, w, yx)
• iff (q2, y) ? ?(q1, a, b)

11
Instantaneous Description

• (q1, x, y) ?M (q2, u, v)
• (q1, x, y) ?M (q2, u, v)

12
Language accepted by NPDA

• Let M (Q, ?, ?, ?, q0, z, F) be an NPDA.
• L(M) w ? ? (q0, w, z) ?M (qf, ?, u), qf ?
  F, u ? ?

13
Example

• L w ? a, b na(w) nb(w)
• M (Q, ?, ?, ?, q0, z, F) ?

14
Example

• L w ? a, b na(w) nb(w)
• M (Q, ?, ?, ?, q0, z, F)
• Q q0, qf ?(q0, ?, z) (qf, z)
• ? a, b ?(q0, a, z) (q0, 0z)
• ? 0, 1, z ?(q0, b, z) (q0, 1z)
• F qf ?(q0, a, 0) (q0, 00)
• ?(q0, b, 0) (q0, ?)
• ?(q0, a, 1) (q0, ?)
• ?(q0, b, 1) (q0, 11)

15
Example

• L wwR w ? a, b
• M (Q, ?, ?, ?, q0, z, F) ?

16
Example

• L wwR w ? a, b
• M (Q, ?, ?, ?, q0, z, F)
• Q q0, q1, q2 ? a, b ? a, b, z F
  q2
• ?(q0, a, a) (q0, aa) ?(q0, ?, a) (q1,
  a)
• ?(q0, b, a) (q0, ba) ?(q0, ?, b) (q1,
  b)
• ?(q0, a, b) (q0, ab)
• ?(q0, b, b) (q0, bb) ?(q1, a, a) (q1,
  ?)
• ?(q0, a, z) (q0, az) ?(q1, b, b) (q1,
  ?)
• ?(q0, b, z) (q0, bz) ?(q1, ?, z) (q2,
  z)

17
NPDA and Context-Free Languages

• Greibach NF
• ?(q0, ?, z) (q1, Sz)
• S ? aSA a ?(q1, a, S) (q1, SA), (q0, ?)
• A ? bB ?(q1, b, A) (q1, B)
• B ? b ?(q1, b, B) (q1, ?)
• ?(q1, ?, z) (q2, ?)

18
Theorem

• For any context-free language L not containing ?,
  there exists
• an NPDA M such that L L(M).

19
Theorem

• Proof
• G (V, T, S, P)
• M (q0, q1, qf, T, V?z, ?, q0, z, qf) z ?
  V
• ?(q0, ?, z) (q1, Sz)
• (q1, u) ? ?(q1, a, A) iff A ? au ? P
• ?(q1, ?, z) (qf, z)

20
Example

• Greibach NF
• ?(q0, ?, z) (q1, Sz)
• S ? aA ?(q1, a, S) (q1, A)
• A ? aABC bB a ?(q1, a, A) (q1, ABC),
  (q1, ?)
• ?(q1, b, A) (q1, B)
• B ? b ?(q1, b, B) (q1, ?)
• C ? c ?(q1, c, C) (q1, ?)
• ?(q1, ?, z) (q2, ?)

21
Context-Free Grammars for NPDA

• M (Q, ?, ?, ?, q0, z, F)
• G (V, T, S, P)
• L(G) L(M)

22
Context-Free Grammars for NPDA

• M (Q, ?, ?, ?, q0, z, F)
• M Single final state
• Final state entered iff the stack is empty
• ?(qi, a, A) c1, c2, ..., cn
• ci (qj, ?) ci (qj, BC)

23
Context-Free Grammars for NPDA

• ?(qi, a, A) (qj, ?), ... ?

24
Context-Free Grammars for NPDA

• ?(qi, a, A) (qj, ?), ... ?
• At qi erase A and move to qj if receiving a

25
Context-Free Grammars for NPDA

• ?(qi, a, A) (qj, BC), ... ?

26
Context-Free Grammars for NPDA

• ?(qi, a, A) (qj, BC), ... ?
• At qi erase A and move to qk if receiving a and
• at qj erase BC and move to qk

27
Context-Free Grammars for NPDA

• ?(qi, a, A) (qj, BC), ... ?
• At qi erase A and move to qk if receiving a and
• at qj erase BC and move to qk
• At qi erase A and move to qk if receiving a and
• at qj erase B and move to qm and
• at qm erase C and move to qk

28
Context-Free Grammars for NPDA

• ?(qi, a, A) (qj, ?), ...
• At qi erase A and move to qj if receiving a
• (qiAqj) ? a

29
Context-Free Grammars for NPDA

• ?(qi, a, A) (qj, BC), ...
• At qi erase A and move to qk if receiving a and
• at qj erase B and move to qm and
• at qm erase C and move to qk
• (qiAqk) ? a(qjBqm)(qmCqk)

30
Context-Free Grammars for NPDA

• Start symbol (q0zqf)

31
Example

• M (Q, ?, ?, ?, q0, z, F)
• Q q0, q1, q2 ? a ? A, z F q2
  
• ?(q0, a, z) (q0, Az)
• ?(q0, a, A) (q0, A)
• ?(q0, b, A) (q1, ?)
• ?(q1, ?, z) (q2, ?)

32
Example

• M (Q, ?, ?, ?, q0, z, F)
• Q q0, q1, q2 , q3 ? a ? A, z F
  q2
• ?(q0, a, z) (q0, Az) ?(q0, a, z) (q0,
  Az)
• ?(q0, a, A) (q0, A) ?(q0, a, A) (q3, ?)
• ?(q0, b, A) (q1, ?) ?(q3, ?, z) (q3,
  Az)
• ?(q1, ?, z) (q2, ?) ?(q0, b, A) (q1, ?)
• ?(q1, ?, z) (q2, ?)

33
Example

• M (Q, ?, ?, ?, q0, z, F)
• Q q0, q1, q2 , q3 ? a ? A, z F
  q2
• ?(q0, a, z) (q0, Az)
• ?(q0, a, A) (q3, ?) (q0Aq3) ? a
• ?(q3, ?, z) (q3, Az)
• ?(q0, b, A) (q1, ?) (q0Aq1) ? b
• ?(q1, ?, z) (q2, ?) (q1zq2) ? ?

34
Example

• M (Q, ?, ?, ?, q0, z, F)
• Q q0, q1, q2 , q3 ? a ? A, z F
  q2
• ?(q0, a, z) (q0, Az) (q0zq0) ? ...
• ?(q0, a, A) (q3, ?) (q0zq1) ? ...
• ?(q3, ?, z) (q3, Az) (q0zq2) ? ...
• ?(q0, b, A) (q1, ?) (q0zq3) ? ...
• ?(q1, ?, z) (q2, ?)

35
Example

• M (Q, ?, ?, ?, q0, z, F)
• Q q0, q1, q2 , q3 ? a ? A, z F
  q2
• ?(q0, a, z) (q0, Az) (q0zq0) ?
  a(q0Aq0)(q0zq0)
• ?(q0, a, A) (q3, ?) a(q0Aq1)(q1zq0)
• ?(q3, ?, z) (q3, Az) a(q0Aq2)(q2zq0)
• ?(q0, b, A) (q1, ?) a(q0Aq3)(q3zq0)
• ?(q1, ?, z) (q2, ?) ...

36
Theorem

• If L L(M) for some NPDA M, then L is a
  context-free
• language.

37
Theorem

• Proof
• M (Q, ?, ?, ?, q0, z, qf)
• G (V, T, S, P)
• T ?
• V (qiAqj) A ? ? S (q0zqf)
• P
• (qiAqj) ? a iff (qj, ?) ? ?(qi, a, A)
• (qiAqm) ? a(qiBqm)(qmCqj) iff (qj, BC) ?
  ?(qi, a, A)

38
Deterministic Pushdown Automata

• A DPDA is a pushdown automaton that never has a
• choice in its move
• ?(q, a, b) contains at most one element.
• if ?(q, ?, b) is not empty, then ?(q, a, b) must
  be empty for
• every a ? ?.

39
Deterministic Context-Free Language

• A language L is said to be a DCFL iff there
  exists a
• DPDA M such that L L(M).

40
Example

• M (Q, ?, ?, ?, q0, z, F)
• Q q0, q1, q2 ?(q0, a, 0) (q1, 10)
• ? a, b ?(q1, a, 1) (q1, 11)
• ? 0, 1 ?(q1, b, 1) (q2, ?)
• z 0 ?(q2, b, 1) (q2, ?)
• F q0 ?(q2, ?, 0) (q0, ?)
• L anbn n ? 0

41
Homework

• Exercises 5, 10, 13 of Section 7.1 - Linzs
  book.
• Exercises 1, 2, 4, 5, 8, 12 of Section 7.2 -
  Linzs book.
• Exercises 1, 2, 3, 6, 7 of Section 7.2 - Linzs
  book.
• Presentations Section 6.3 and Section 7.4.