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Module 32

• Pushdown Automata (PDA’s)– definition– Example

• We define configurations and computations of PDA’s

• We define L(M) for PDA’s

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Pushdown Automata

Definition and Motivating Example

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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.

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NFA for {ambn | m,n >= 0}

• What strings end up in each state of the above NFA?– I:

– B:

– C:

• Consider the language {anbn | n >= 0}.

• This NFA can recognize strings which have the correct form, – a’s followed by b’s.

• However, the NFA cannot remember the relative number of a’s and b’s seen at any point in time.

/\ /\

a b

I B C

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PDA for {anbn | n >=0 } *

/\ /\

a b

I B C

NFA

Imagine we now have memory in the form of a stack which we can use to help remember how many a’s 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 if1) The stack is empty.

Finally, when we begin, the stack should be empty.

I B C

b;popa;push a

/\;only if stack is empty

PDA

/\

Initialize stack to empty

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Formal PDA definition

• PDA M = (Q, , , q0, Z, A, )

• Modified elements– is the stack alphabet

• Z is a special character that is initially on the stack

• Often used to represent an empty stack

– 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 PDAQ = {I, B, C} = {a,b} = {Z, a}q0 = IZ is the initial stack characterA = {C}:

S a TopSt NS stack update

I a a I push aaI a Z I push aZI /\ a B push aI /\ Z B push ZB b a B push /\B /\ Z C push Z

I B C

b;a;/\a;a; aaa;Z; aZ

/\;Z;Z/\;a;a /\;Z;Z

Example PDA

Initialize stack toonly contain Z

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Computing with PDA’s *

• 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

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Computation Graph of PDA

Computation graph for this PDA on the input string aabb

(I,aabb,Z)

(C,aabb,Z)

(B,bb,aaZ)

(B,abb,aZ)(I,bb,aaZ)

(I,abb,aZ) (B,aabb,Z)

(B,b,aZ)

(B,/\,Z)

(C,/\,Z)

Q = {I, B, C} = {a,b} = {Z, a}q0 = IZ is the initial stack characterA = {C}:

S a TopSt NS stack updateI a a I push aaI a Z I push aZI /\ a B push aI /\ Z B push ZB b a B push /\B /\ Z C push 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)

I B C

b;a;/\a;a; aaa;Z; aZ

/\;Z;Z/\;a;a /\;Z;Z

(I,aabb,Z)

(C,aabb,Z)

(B,bb,aaZ)

(B,abb,aZ)(I,bb,aaZ)

(I,abb,aZ) (B,aabb,Z)

(B,b,aZ)

(B,/\,Z)

(C,/\,Z)

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Acceptance and Rejection

M accepts string x if one of the configurations reached is an accepting configuration

(q0, x, Z) |-* (f, /\),f in A, in *

Stack contents can be anything

M rejects string x if all configurations reached are either not halting configurations or are rejecting configurations

I B C

b;a;/\a;a; aaa;Z; aZ

/\;Z;Z/\;a;a /\;Z;Z

Input string aabb

(I,aabb,Z)

(C,aabb,Z)

(B,bb,aaZ)

(B,abb,aZ)(I,bb,aaZ)

(I,abb,aZ) (B,aabb,Z)

(B,b,aZ)

(B,/\,Z)

(C,/\,Z)

Not an accepting configuration since input not completely processed

Not an accepting configuration since state is not accepting

An accepting configuration

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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, /\),f in A, in *

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 PDA’s

• A PDA is deterministic if its transition function satisfies both of the following properties– For all q in Q, a in union {/\}, and X in ,

• the set (q,a,X) has at most one element

– For all q in Q and X in G,• if (q, /\, X) < > { }, then (q,a,X) = { } for all a in

• A computation graph is now just a path again• Our default assumption is that PDA’s 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 aZ2 q0 a Z q0 aa

3 q0 /\ Z q0 aZ4 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