languages and finite automata - web2.aabu.edu.jo · fall 2019 costas busch - rpi 3 how can we prove...
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![Page 1: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/1.jpg)
Fall 2019 Costas Busch - RPI 1
Non-regular languages
(Pumping Lemma)
![Page 2: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/2.jpg)
Fall 2019 Costas Busch - RPI 2
Regular languages
ba* acb *
...etc
*)( bacb
Non-regular languages }0:{ nba nn
}*},{:{ bavvvR
![Page 3: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/3.jpg)
Fall 2019 Costas Busch - RPI 3
How can we prove that a language
is not regular?
L
Prove that there is no DFA or NFA or RG
that accepts L
Difficulty: this is not easy to prove
(since there is an infinite number of them)
Solution: use the Pumping Lemma !!!
![Page 4: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/4.jpg)
Fall 2019 Costas Busch - RPI 4
The Pigeonhole Principle
![Page 5: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/5.jpg)
Fall 2019 Costas Busch - RPI 5
pigeons
pigeonholes
4
3
![Page 6: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/6.jpg)
Fall 2019 Costas Busch - RPI 6
A pigeonhole must
contain at least two pigeons
![Page 7: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/7.jpg)
Fall 2019 Costas Busch - RPI 7
...........
...........
pigeons
pigeonholes
n
m mn
![Page 8: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/8.jpg)
Fall 2019 Costas Busch - RPI 8
The Pigeonhole Principle
...........
pigeons
pigeonholes
n
m
mn There is a pigeonhole
with at least 2 pigeons
![Page 9: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/9.jpg)
Fall 2019 Costas Busch - RPI 9
The Pigeonhole Principle
and
DFAs
![Page 10: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/10.jpg)
Fall 2019 Costas Busch - RPI 10
Consider a DFA with states 4
1q 2q 3qa
b
4q
b
a b
b
a a
![Page 11: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/11.jpg)
Fall 2019 Costas Busch - RPI 11
Consider the walk of a “long’’ string:
1q 2q 3qa
b
4q
b
b
b
a a
a
aaaab
1q 2q 3q 2q 3q 4qa a a a b
A state is repeated in the walk of
(length at least 4)
aaaab
![Page 12: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/12.jpg)
Fall 2019 Costas Busch - RPI 12
aaaab
1q 2q 3q 2q 3q 4qa a a a b
1q 2q 3q 4q
Pigeons:
Nests: (Automaton states)
Are more than
Walk of
The state is repeated as a result of
the pigeonhole principle
(walk states)
Repeated
state
![Page 13: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/13.jpg)
Fall 2019 Costas Busch - RPI 13
Consider the walk of a “long’’ string:
1q 2q 3qa
b
4q
b
b
b
a a
a
aabb
1q 2q 3q 4q 4qa a b b
A state is repeated in the walk of
(length at least 4)
Due to the pigeonhole principle:
aabb
![Page 14: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/14.jpg)
Fall 2019 Costas Busch - RPI 14
aabb
1q 2q 3q 4q
Automaton States
Pigeons:
Nests: (Automaton states)
Are more than
Walk of
The state is repeated as a result of
the pigeonhole principle
(walk states) 1q 2q 3q 4q 4qa a b b
Repeated
state
![Page 15: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/15.jpg)
Fall 2019 Costas Busch - RPI 15
iq...... ......
Repeated state
kw 21
1 2 k
Walk of
iq.... iq.... .... 1 2 ki j1i 1j
Arbitrary DFA
DFA of states#|| wIf ,
by the pigeonhole principle,
a state is repeated in the walk
In General:
1qzq
zq1q
w
![Page 16: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/16.jpg)
Fall 2019 Costas Busch - RPI 16
mDFA of states#|| w
1q 2q 1mq mq
Walk of wPigeons:
Nests: (Automaton states)
Are
more
than
(walk states)
iq.... iq.... ....
1q
iq.... ....
zq
A state is
repeated
![Page 17: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/17.jpg)
Fall 2019 Costas Busch - RPI 17
The Pumping Lemma
![Page 18: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/18.jpg)
Fall 2019 Costas Busch - RPI 18
Take an infinite regular language L
There exists a DFA that accepts L
mstates
(contains an infinite number of strings)
![Page 19: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/19.jpg)
Fall 2019 Costas Busch - RPI 19
mw ||(number of
states of DFA)
then, at least one state is repeated
in the walk of w
q...... ...... 1 2 k
Take string with Lw
kw 21
Walk in DFA of
Repeated state in DFA
![Page 20: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/20.jpg)
Fall 2019 Costas Busch - RPI 20
Take to be the first state repeated q
q....
w
There could be many states repeated
q.... ....
Second
occurrence
First
occurrence
Unique states
One dimensional projection of walk :
1 2 ki j1i 1j
![Page 21: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/21.jpg)
Fall 2019 Costas Busch - RPI 21
q.... q.... ....
Second
occurrence
First
occurrence 1 2 ki j
1i 1j
wOne dimensional projection of walk :
ix 1 jiy 1 kjz 1
xyzw We can write
![Page 22: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/22.jpg)
Fall 2019 Costas Busch - RPI 22
zyxw
q... ...
x
y
z
In DFA:
...
...
1 ki
1ij
1j
contains only
first occurrence of q
![Page 23: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/23.jpg)
Fall 2019 Costas Busch - RPI 23
Observation: myx ||length number
of states
of DFA
Since, in no
state is repeated
(except q)
xy
Unique States
q...
x
y
...
1 i
1ij
![Page 24: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/24.jpg)
Fall 2019 Costas Busch - RPI 24
Observation: 1|| ylength
Since there is at least one transition in loop
q
y
...
1ij
![Page 25: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/25.jpg)
Fall 2019 Costas Busch - RPI 25
We do not care about the form of string z
q...
x
y
z
...
z may actually overlap with the paths of and x y
![Page 26: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/26.jpg)
Fall 2019 Costas Busch - RPI 26
The string
is accepted
zxAdditional string:
q... ...
x z
...
Do not follow loop y
...
1 ki
1ij
1j
![Page 27: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/27.jpg)
Fall 2019 Costas Busch - RPI 27
The string
is accepted
zyyx
q... ... ...
x z
Follow loop
2 times
Additional string:
y
...
1 ki
1ij
1j
![Page 28: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/28.jpg)
Fall 2019 Costas Busch - RPI 28
The string
is accepted
zyyyx
q... ... ...
x z
Follow loop
3 times
Additional string:
y
...
1 ki
1ij
1j
![Page 29: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/29.jpg)
Fall 2019 Costas Busch - RPI 29
The string
is accepted
zyx iIn General:
...,2,1,0i
q... ... ...
x z
Follow loop
times iy
...
1 ki
1ij
1j
![Page 30: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/30.jpg)
Fall 2019 Costas Busch - RPI 30
Lzyx i Therefore: ...,2,1,0i
Language accepted by the DFA
q... ... ...
x z
y
...
1 ki
1ij
1j
![Page 31: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/31.jpg)
Fall 2019 Costas Busch - RPI 31
In other words, we described:
The Pumping Lemma !!!
![Page 32: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/32.jpg)
Fall 2019 Costas Busch - RPI 32
The Pumping Lemma:
• Given a infinite regular language L
• there exists an integer m
• for any string with length Lw mw ||
• we can write zyxw
• with and myx || 1|| y
• such that: Lzyx i ...,2,1,0i
(critical length)
![Page 33: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/33.jpg)
Fall 2019 Costas Busch - RPI 33
In the book:
Critical length = Pumping length m p
![Page 34: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/34.jpg)
Fall 2019 Costas Busch - RPI 34
Applications
of
the Pumping Lemma
![Page 35: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/35.jpg)
Fall 2019 Costas Busch - RPI 35
Observation:
Every language of finite size has to be regular
Therefore, every non-regular language
has to be of infinite size (contains an infinite number of strings)
(we can easily construct an NFA
that accepts every string in the language)
![Page 36: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/36.jpg)
Fall 2019 Costas Busch - RPI 36
Suppose you want to prove that
An infinite language is not regular
1. Assume the opposite: is regular
2. The pumping lemma should hold for
3. Use the pumping lemma to obtain a
contradiction
L
L
L
4. Therefore, is not regular L
![Page 37: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/37.jpg)
Fall 2019 Costas Busch - RPI 37
Explanation of Step 3: How to get a contradiction
2. Choose a particular string which satisfies
the length condition Lw
3. Write xyzw
4. Show that Lzxyw i for some 1i
5. This gives a contradiction, since from
pumping lemma Lzxyw i
mw ||
1. Let be the critical length for m L
![Page 38: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/38.jpg)
Fall 2019 Costas Busch - RPI 38
Note: It suffices to show that
only one string
gives a contradiction
Lw
You don’t need to obtain
contradiction for every Lw
![Page 39: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/39.jpg)
Fall 2019 Costas Busch - RPI 39
Theorem: The language }0:{ nbaL nn
is not regular
Proof: Use the Pumping Lemma
Example of Pumping Lemma application
![Page 40: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/40.jpg)
Fall 2019 Costas Busch - RPI 40
Assume for contradiction
that is a regular language L
Since is infinite
we can apply the Pumping Lemma
L
}0:{ nbaL nn
![Page 41: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/41.jpg)
Fall 2019 Costas Busch - RPI 41
Let be the critical length for
Pick a string such that: w Lw
mw ||and length
mmbaw We pick
m
}0:{ nbaL nn
L
![Page 42: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/42.jpg)
Fall 2019 Costas Busch - RPI 42
with lengths
From the Pumping Lemma:
1||,|| ymyx
babaaaaabaxyz mm ............
mkay k 1,
x y z
m m
we can write zyxbaw mm
Thus:
w
![Page 43: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/43.jpg)
Fall 2019 Costas Busch - RPI 43
From the Pumping Lemma: Lzyx i
...,2,1,0i
Thus:
mmbazyx
Lzyx 2
mkay k 1,
![Page 44: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/44.jpg)
Fall 2019 Costas Busch - RPI 44
From the Pumping Lemma:
Lbabaaaaaaazxy ...............2
x y z
km m
Thus:
Lzyx 2
mmbazyx
y
Lba mkm
mkay k 1,
![Page 45: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/45.jpg)
Fall 2019 Costas Busch - RPI 45
Lba mkm
}0:{ nbaL nnBUT:
Lba mkm
CONTRADICTION!!!
1≥k
![Page 46: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/46.jpg)
Fall 2019 Costas Busch - RPI 46
Our assumption that
is a regular language is not true
L
Conclusion: L is not a regular language
Therefore:
END OF PROOF
![Page 47: Languages and Finite Automata - web2.aabu.edu.jo · Fall 2019 Costas Busch - RPI 3 How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RG that](https://reader030.vdocuments.site/reader030/viewer/2022040913/5e88dc3280701e736f4db315/html5/thumbnails/47.jpg)
Fall 2019 Costas Busch - RPI 47
Regular languages
Non-regular language }0:{ nba nn
)( **baL