1. Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death
Nathan Brunelle Department of Computer Science University of Virginia

2. Hierarchy: So Far anbn bn abba wwR Regular Languages
Violates Pumping Lemma
For CFLs
anbn bn
Violates Pumping Lemma For RLs
abba
Described by DFA, NFA, RegExp,
wwR
Described by CFG, NDPDA
Described by DPDA
finite Languages
Regular Languages
Deterministic Context-Free
Context-Free Languages

3. Hierarchy: Next step Languages Recognized by any computing machine
Regular Languages
Context-Free Languages
finite Languages
Deterministic Context-Free
Problems solvable by humans?

4. PDA Enhancements c b c b $ $ a a b b c c
Theorem: 2-stack PDAs are more powerful than 1-stack PDAs.
Hint: Find an example non-CFL accepted by a 2-stack PDA.
a, ↓b, ↓c
b, ↑b,-
c, -,↑c c b
ε, ↓$, ↓$
b, ↑b, -
c,-,↑c
ε, ↑$, ↑$
start
a‘s
b‘s
c‘s
end c b $
Input $ a a b b c c
Stack 2
Stack 1

5. PDA Enhancements a b b b b a $ $ a b b a b b
Theorem: 2-stack PDAs are more powerful than 1-stack PDAs.
Hint: Find an example non-CFL accepted by a 2-stack PDA.
a, ↓a,-
ε, ↑a, ↓a a
a, -, ↑a b
b, ↓b,-
ε, ↑b, ↓b
b, -, ↑b b b
ε, ↓$, -
ε, -, ↓$
ε, ↑$,-
ε, -, ↑$
start w1
flip w2
end b a $
Input $ a b b a b b
Stack 2
Stack 1

6. PDA Enhancements a b b $ a $ b b a b b b $ a $ $ b b b a b a $
Is there anything we can’t simulate with a 2-PDA?
How about a 3-PDA?
To pop from stack 1:
Do: σϵ{a,b}, ωϵ{a,b,$} a b
ε, ↑b, -, - x y $ b
Stack 2 a
Stack 1
Stack 3 b $ a $ b b b b x
ε, ↑σ, ↓σ
ε, ↑$, ↓$
ε, ↑b, - y ε
ε, ↓ω, ↑ω $ a b $ b b
Move stack 3
Move stack 2
Pop
reset a b $ a
Stack 1
Stack 2

7. PDA Enhancements a b a b $ $
Is there anything we can’t simulate with a 2-PDA?
2-PDA=Turing Machine $ a b $ b a
Current Symbol
Stack 1
Stack 2

8. Turing Machine … FSM Infinite tape: Γ*
Tape head: read current square on tape,
write into current square,
move one square left or right
FSM: like PDA, except:
transitions also include direction (left/right)
final accepting and rejecting states

9. Turing Machine … FSM 7-tuple: (Q, , Γ, δ, q0, qaccept, qreject)
Q: finite set of states
: input alphabet (cannot include blank symbol, _)
Γ: tape alphabet, includes  and _ (blank)
δ: transition function: Q  Γ  Q  Γ  {L, R} (in FSM)
q0: start state, q0  Q
qaccept: accepting state, qaccept  Q
qreject: rejecting state, qreject  Q

10. Turing Machine Configurationq …
FSM u v
(u, q, v)ϵΓ* × Q × Γ*
tape contents
left of head
head and right
current
FSM state

11. Initial Configurationb _ …
FSM
input
blanks
Configuration: (a, start, bba)

12. Operation … δ*: Γ*  Q  Γ*  Γ*  Q  Γ*
Transition from configuration to configuration …
FSM u v q
The qaccept and qreject states are final (you never leave):
δ*(u, qaccept, v)  (u, qaccept, v)
δ*(u, qreject, v)  (u, qreject, v)

13. Operation … a c b δ*(ua, q, bv) = (uac, qr, v) if δ(q, b) = (qr, c, R)
FSM q qr
b,c,R … a c b u v
δ*(ua, q, bv) = (uac, qr, v) if δ(q, b) = (qr, c, R)
δ*(ua, q, bv) = (u, qr, acv) if δ(q, b) = (qr, c, L)

14. Operation … a c b δ*(ua, q, bv) = (uac, qr, v) if δ(q, b) = (qr, c, R)
FSM q qr
b,c,L … a c b u v
δ*(ua, q, bv) = (uac, qr, v) if δ(q, b) = (qr, c, R)
δ*(ua, q, bv) = (u, qr, acv) if δ(q, b) = (qr, c, L)

15. ≡ Church Turing Thesis Read notes Ponder Scribble notes Repeat
Read from tape
Change state
Write to tape
Repeat