Module 32

1
Module 32

• Pushdown Automata (PDAs)
• definition
• Example
• We define configurations and computations of
  PDAs
• We define L(M) for PDAs

2
Pushdown Automata

• Definition and Motivating Example

3
Pushdown Automata (PDA)

• In this presentation we introduce the PDA model
  of computation (programming language).
• The key addition to a PDA (from an NFA-/\) is the
  addition of external memory in the form of an
  infinite capacity stack
• The word pushdown comes from the stacks of
  trays in cafeterias where you have to pushdown on
  the stack to add a tray to it.

4
NFA for ambn m,n gt 0

• Consider the language anbn n gt 0.
• This NFA can recognize strings which have the
  correct form,
• as followed by bs.
• However, the NFA cannot remember the relative
  number of as and bs seen at any point in time.

• What strings end up in each state of the above
  NFA?
• I
• B
• C

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PDA for anbn n gt0
Imagine we now have memory in the form of a stack
which we can use to help remember how many as we
have seen by pushing onto and popping from the
stack When we see an a in state I, we do the
following two actions 1) We push an a on the
stack. 2) We stay in state I. When we see a b
in state B, we do the following two actions 1)
We pop an a from the stack. 2) We stay in state
B. From state B, we allow a /\-transition to
state C only if 1) The stack is empty. Finally,
when we begin, the stack should be empty.

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Formal PDA definition

• PDA M (Q, S, G, q0, Z, A, d)
• Modified elements
• G is the stack alphabet
• Z is a special character that is initially on the
  stack
• Often used to represent an empty stack
• d is modified as follows
• Pop to read the top character on the stack
• Stack update action
• What to push back on the stack
• If we push /\, then the net result of the action
  is a pop

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Example PDA

• Q I, B, C
• S a,b
• G Z, a
• q0 I
• Z is the initial stack character
• A C
• d
• S a TopSt NS stack update
• I a a I push aa
• I a Z I push aZ
• I /\ a B push a
• I /\ Z B push Z
• B b a B push /\
• B /\ Z C push Z

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Computing with PDAs

• Configurations change compared with NFA-/\s
• Configuration components
• current state
• remaining input to be processed
• stack contents
• Computations are essentially the same as with
  NFA-/\s given the modified configurations
• Determining which transitions of a PDA can be
  applied to a given configuration is more
  complicated though

9
Computation Graph of PDA
Computation graph for this PDA on the input
string aabb
Q I, B, C S a,b G Z, a q0 I Z is
the initial stack character A C d S
a TopSt NS stack update I a a I push
aa I a Z I push aZ I /\ a B push a I /\ Z B push
Z B b a B push /\ B /\ Z C push Z
(I,aabb,Z)
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Definition of -
Input string aabb
(I, aabb, Z) - (I,abb,aZ) (I, aabb, Z) - (B,
aabb, Z) (I, aabb, Z) -2 (C, aabb, Z) (I, aabb,
Z) -3 (B, bb, aaZ) (I, aabb, Z) - (B, abb,
aZ) (I, aabb, Z) - (B, /\, Z) (I, aabb, Z) -
(C, /\, Z)
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Acceptance and Rejection
Input string aabb
M accepts string x if one of the configurations
reached is an accepting configuration (q0, x,
Z) - (f, /\, a),f in A, a in G Stack
contents can be anything M rejects string x if
all configurations reached are either not halting
configurations or are rejecting configurations
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Defining L(M) and LPDA

• L(M) (or Y(M))
• The set of strings ?
• N(M)
• The set of strings ?
• LPDA
• Language L is in language class LPDA iff ?

M accepts string x if one of the configurations
reached is an accepting configuration (q0, x,
Z) - (f, /\, a),f in A, a in G Stack
contents can be anything M rejects string x if
all configurations reached are either not halting
configurations or are rejecting configurations
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Deterministic PDAs

• A PDA is deterministic if its transition function
  satisfies both of the following properties
• For all q in Q, a in S union /\, and X in G,
• the set d(q,a,X) has at most one element
• For all q in Q and X in G,
• if d(q, /\, X) lt gt , then d(q,a,X) for
  all a in S
• A computation graph is now just a path again
• Our default assumption is that PDAs are
  nondeterministic

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Two forms of nondeterminism
Trans Current Input Top of Next Stack
State Char. Stack State
Update -------------------------------------------
------------ 1 q0 a
Z q0 aZ 2 q0
a Z q0 aa 3 q0
/\ Z q0 aZ 4
q0 a Z q0
aa
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LPDA and DCFL

• A language L is in language class LPDA if and
  only if there exists a PDA M such that L(M) L
• A language L is in language class DCFL
  (Deterministic Context-Free Languages) if and
  only if there exists a deterministic PDA M such
  that L(M) L
• To be proven
• LPDA CFL
• CFL is a proper superset of DCFL

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PDA Comments

• Note, we can use the stack for much more than
  just a counter
• See examples in chapter 7 for some details