Theory of
Computation
Theory of Computation Peer Instruction Lecture Slides by Dr. Cynthia Lee, UCSD are licensed
under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Based on a work at www.peerinstruction4cs.org.
1
If a language isn’t regular, it just might be one of the…
CONTEXT-FREE LANGUAGES
PDA
Understanding Pushdown Automata
The edge from q0 to q1
does what?
a) Lets you go to q1
without reading any
input (on epsilon)
b) Lets you go to q1 on
epsilon or ‘$’
c) Lets you go to q1 on ‘$’
Understanding Pushdown Automata
In addition to what we said
in the last question, the
edge from q0 to q1 does
what else?
a)
b)
c)
d)
Pushes one entry onto the stack (epsilon)
Pushes one entry onto the stack (‘$’)
Pushes two entries onto the stack (epsilon and ‘$’)
Pops two entries off the stack (epsilon and ‘$’)
Tracing in a Pushdown Automaton
Input “aabb” into this PDA. After “aa” has been read,
what is on the stack?
top of stack
a)
b)
c)
d)
e)
b b
a a
0 a a
a a 0
ϵ ϵ ϵ
bottom of stack 
Language of a Pushdown Automaton
Which is the best description of the language of the
given PDA?
a) { w | number of b’s in w >= number of a’s in w}
b) {w | w = anbn+1 for some n>=0}
c) { w | w = anbn+2 for some n>=0}
d) { w | w = anb2n for some n>=0}
e) {w | w = 0anb2n0 for some n>=0}
Tracing in a Pushdown Automaton
• Which string is NOT
accepted by this PDA?
a)
b)
c)
d)
e)
aabb
abbbc
abbccc
aabcc
None or more than one
of the above
Tracing in a Pushdown Automaton
top of stack
• Which choice depicts a
stack state that occurs at
some point during the
successful* processing of
the string “aaabbbc” on
the given PDA?
bottom of stack 
a) a a a b b b c
b) # a
c) a #
d) # λ
e) None of the above
* Ignore all nondeterministic paths that end in rejecting/getting stuck.
Why did we push ‘#’ onto the stack?
a)
b)
c)
d)
e)
We didn’t have to, because we already “counted” the a’s by pushing
them on the stack
It’s something we do because of convention, but it isn’t necessary to
correctness
We did it to make sure that we didn’t cause a crash/error by trying to
pop something off an empty stack
It is necessary to correctness
None or more than one of the above
CFLs!
THE CLASS OF CONTEXT-FREE
LANGUAGES
Which Venn diagram best represents the
classes of languages we have studied?
(a)
(b)
CFLs
CFLs
RLs
(c)
RLs
(d)
(e) None of the above
CFLs
=
RLs
RLs
CFLs
Note: CFLs = ContextFree Languages, RLs =
Regular Languages
What methods can we use to
prove/disprove each of these?
(a)
(b)
CFLs
CFLs
RLs
(c)
RLs
(d)
CFLs
=
RLs
RLs
CFLs
Note: CFLs = ContextFree Languages, RLs =
Regular Languages
Famous People: Noam Chomsky
• In this class, you know him as the
namesake of “Chomsky Normal Form”
• A linguist who has taught at MIT for 55
years
• Famous for: Developed theories of
conetxt-free grammars for analysis of
human language
– A mathematical model of language
– Discoveries crossed over into Computer
Science
• Perhaps now equally well known for
his outspoken criticism of US foreign
policy and his radical political views
– Anarchist
– Vehement war critic, war on drugs critic
– Co-wrote Manufacturing Consent,
argues that the media in our society is
harming democracy and promoting
corporations and consumerism
Proving that languages that are NOT context-free
CONTEXT-FREE PUMPING LEMMA
Limits of Context-Free Languages
• What are the limitations of
Context-Free languages
• What are the limitations of
PDA?
– Stack size has no limit
(infinite stack)
– ….BUT, there’s only one of
them
– Stack can only be accessed at
the top
• What does that mean
intuitively? When would
you need more than one
stack?
Context-Free
Regular
Not Context-Free
PDA Language is 0n12n
Can we change this so it is 0n12n0n?
Classic Not-Context-Free Language
• anbncn for some n>=0
• INTUITIVELY, the problem is that a PDA recognizing this
language would try to:
– Push on the stack to count the ‘a’ section, then
– Pop off the stack to match the ‘b’ section, then
– You’ve “forgotton” n, now you can’t count the ‘c’ section
• Aside: what could you do with a second stack?
• THIS IS NOT A PROOF!!
– Maybe there is a completely different way to approach this
problem using PDA, which we just didn’t think of yet
– But, turns out, there is no way to do this, and we can actually
prove that
Aside: Recognizing anbncn for some
n>=0 using a second stack
CFL Pumping Lemma
Proving {aibjck | 0<=i<=j<=k} is not
Context-Free
• s = apbpcp
• i = ???
– Can’t solve it with just one i!
– For the case analysis in this proof, we need to use
i=0 for some cases of how the string is sliced up,
and i=2 for other cases of how the string is sliced
up
– Is that legal???
• Yes.
• The Pumping Lemma Game shows us why…
The REGULAR LANGUAGES
Pumping Lemma Game
Your Script
Pumping Lemma’s Script
• “I’m giving you a language L that
I’m assuming is regular.”
• “Thanks. For the language L that
you’ve given me, I pick this nice
pumping length I call p.”
• “Excellent. I’m giving you this
string s that I made using your p.
It is in L and |s| >= p. I think you’ll
really like it.”
• “Great string, thanks. I’ve cut s up
into parts xyz for you. I won’t tell
you what they are exactly, but I will
say this: |y| > 0 and |xy| <= p.
Also, you can remove y, or copy it
as many times as you like, and the
new string will still be in L, I
promise!”
• “Hm. I followed your directions
for xyz, but when I [copy y N
times or delete y], the new string
is NOT is L! What happened?”
• “Well, then L wasn’t a regular
language. Thanks for playing.”
The CONTEXT-FREE LANGUAGES
Pumping Lemma Game
Your Script
Pumping Lemma’s Script
• “I’m giving you a language L that
I’m assuming is context-free.”
• “Thanks. For the language L that
you’ve given me, I pick this nice
pumping length I call p.”
• “Excellent. I’m giving you this
string s that I made using your p.
It is in L and |s| >= p. I think you’ll
really like it.”
• “Great string, thanks. I’ve cut s up
into parts uvxyz for you. I won’t tell
you what they are exactly, but I will
say this: |vy| > 0 and |vxy| <= p.
Also, you can remove v and y, or
copy them as many times as you
like (as in uvixyiz), and the new
string will still be in L, I promise!”
• “Hm. I followed your directions
for uvxyz, but when I [copy vy N
times or delete vy], the new string
is NOT is L! What happened?”
• “Well, then L wasn’t a context-free
language. Thanks for playing.”
Review review review…
MIDTERM REVIEW
From last year’s midterm:
5. To prove a language is not regular, which method can
be used?
a) Show several DFAs or NFAs that almost, but not quite,
recognize the language, and then conclude that no
DFA or NFA can recognize the language.
b) Use the Pumping Lemma for regular languages.
c) Show a CFG that recognizes the language, and then
conclude that the language is context-free, not
regular.
d) (b) and (c)
e) None of the above.
(a) TRUE or (b) FALSE
The following proof is valid:
• A = {w | w=an for n>=0} is a regular language,
and B = {w | w=bn for n>=0} is a regular
language because we can produce DFA’s MA
and MB for A and B, respectively (see
drawings). Regular languages are closed under
concatenation, therefore language AB = {w |
w=anbn for n>=0} is a regular language.
(valid in this case means the flow of logic is sound, even though may not do a
great job of stating GIVEN/WANT TO SHOW, etc)