2. BOOKS
Theory of computer Science: K.L.P.Mishra &
N.Chandrasekharan
Intro to Automata theory, Formal languages and
computation: Ullman,Hopcroft
Motwani
Elements of theory of computation Lewis &
papadimitrou
3. Syllabus
Introduction
Deterministic and non deterministic Finite
Automata, Regular Expression,Two way
finite automata,Finite automata with
output,properties of regular sets,pumping
lemma, closure properties,Myhill nerode
theorem
4. Context free Grammar: Derivation trees,
Simplification forms
Pushdown automata: Def, Relationship
between PDA and context free
language,Properties, decision
algorithms
Turing Machines: Turing machine
model,Modification of turing
machines,Church’s
thesis,Undecidability,Recursive and
recursively enumerable languages Post
correspondence problems recursive
functions
5. Chomsky Hierarchy: Regular grammars,
unrestricted grammar, context sensitive
language, relationship among
languages
10. Different Kinds of Automata
Automata are distinguished by the temporary memory
• Finite Automata: no temporary memory
• Pushdown Automata: stack
• Turing Machines: random access memory
13. Power sets
A power set is a set of sets
S = { a, b, c }
Powerset of S = the set of all the subsets of S
2S = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }
Observation: | 2S | = 2|S| ( 8 = 23 )
14. Cartesian Product
A = { 2, 4 } B = { 2, 3, 5 }
A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 4) }
|A X B| = |A| |B|
Generalizes to more than two sets
AXBX…XZ
15. RELATIONS
R = {(x1, y1), (x2, y2), (x3, y3), …}
xi R yi
e. g. if R = ‘>’: 2 > 1, 3 > 2, 3 > 1
In relations xi can be repeated
16. Equivalence Relations
• Reflexive: xRx
• Symmetric: xRy yRx
• Transitive: x R Y and y R z xRz
Example: R = ‘=‘
•x=x
•x=y y=x
• x = y and y = z x=z
17. Equivalence Classes
For equivalence relation R
equivalence class of x = {y : x R y}
Example:
R = { (1, 1), (2, 2), (1, 2), (2, 1),
(3, 3), (4, 4), (3, 4), (4, 3) }
Equivalence class of 1 = {1, 2}
Equivalence class of 3 = {3, 4}
18. Example of Equivalence relation
Let Z = set of integers
R be defined on it as:
R= {(x,y)| x ∈Z, y ∈ Z and
(x - y)is divisible by
5}
This relation is known as
” congruent modulo 5”
21. Induction
We have statements P1, P2, P3, …
If we know
• for some k that P1, P2, …, Pk are true
• for any n >= k that
P1, P2, …, Pn imply Pn+1
Then
Every Pi is true
22. Trees
root
parent
leaf
child
Trees have no cycles
23. Proof by Induction
• Inductive basis
Find P1, P2, …, Pk which are true
• Inductive hypothesis
Let’s assume P1, P2, …, Pn are true,
for any n >= k
• Inductive step
Show that Pn+1 is true
28. We want to show: l(i) <= 2i
• Inductive basis
l(0) = 1 (the root node)
• Inductive hypothesis
Let’s assume l(i) <= 2i for all i = 0, 1, …, n
• Induction step
we need to show that l(n + 1) <= 2n+1
30. Proof by Contradiction
We want to prove that a statement P is true
• we assume that P is false
• then we arrive at an incorrect conclusion
• therefore, statement P must be true
31. Example
Theorem: 2 is not rational
Proof:
Assume by contradiction that it is rational
2 = n/m
n and m have no common factors
We will show that this is impossible
32. 2 = n/m 2 m2 = n2
n is even
Therefore, n2 is even
n=2k
m is even
2 m2 = 4k2 m2 = 2k2
m=2p
Thus, m and n have common factor 2
Contradiction!
34. ∑
• String: A sequence of letters
– Examples: “cat”, “dog”, “house”, …
– Defined over an alphabet:
Σ={a, b, c,, z}
Language: It is a set of strings on some
alphabet
35. Alphabets and Strings
• We will use small alphabets:
• Strings Σ = { a, b}
a
u = ab
ab
v = bbbaaa
abba
w = abba
baba
aaabbbaabab
36. String Operations
abba
w = a1a2 an
bbbaaa
v = b1b2 bm
Concatenation
wv = a1a2 anb1b2 bm abbabbbaaa
37. w = a1a2 an ababaaabbb
Reverse
R
w = an a2 a1 bbbaaababa
38. String Length
w = a1a2 an
w =n
• Length:
abba = 4
• Examples: aa = 2
a =1
39. Recursive Definition of Length
a =1
For any letter:
wa wa = w + 1
For any string : abba = abb + 1
= ab + 1 + 1
Example:
= a +1+1+1
= 1+1+1+1
=4
40. Length of Concatenation
uv = u + v
u = aab, u = 3
• Example: = abaab,
v v =5
uv = aababaab = 8
uv = u + v = 3 + 5 = 8
41. Proof of Concatenation Length
uv = u + v
• Claim:
v
• Proof: By induction on the length
v =1
– Induction basis:
– From definition of length:
uv = u +1 = u + v
42. uv = u + v
– Inductive hypothesis:
v = 1,2,, n
• for
– Inductive step: we will prove uv = u + v
–
– for v = n +1
43. Inductive Step
v = wa w = n, a = 1
• Write , where
uv = uwa = uw + 1
• From definition of length:
wa = w + 1
• From inductive hypothesis: uw = u + w
• Thus:
uv = u + w + 1 = u + wa = u + v
44. Empty String
λ
• A string with no letters:
λ =0
• Observations:
λw = wλ = w
λabba = abbaλ = abba
45. Substring
• Substring of string:
– a subsequence of consecutive
characters
abbab ab
• String Substring
abbab abba
abbab b
abbab bbab
46. Prefix and Suffix
• Prefixes Suffixes
abbab w = uv
λ abbab
a bbab prefix
ab bab suffix
abb ab
abba b
abbab λ
47. Another Operation
wn = ww w
n
( abba ) = abbaabba
2
• Example:
0
• Definition:
w =λ
–
( abba ) = λ
0
48. The * Operation
• Σ * : the set of all possible strings from
• Σ alphabet
Σ = { a, b}
Σ* = { λ, a, b, aa, ab, ba, bb, aaa, aab,}
•
49. The + Operation
+ : the set of all possible strings from
Σ
alphabet Σ except λ
Σ = { a, b}
Σ* = { λ , a, b, aa, ab, ba, bb, aaa, aab,}
+
Σ = Σ * −λ
+
Σ = { a, b, aa, ab, ba, bb, aaa, aab,}
50. Language
• A language is any subset of Σ*
Σ = { a, b}
• Example: Σ* = { λ , a, b, aa, ab, ba, bb, aaa,}
{ λ}
{ a, aa, aab}
• Languages:
{λ , abba, baba, aa, ab, aaaaaa}
51. Another Example
n n
L = {a b : n ≥ 0}
• An infinite language
λ
ab
∈L abb ∉ L
aabb
aaaaabbbbb
52. Operations on Languages
• The usual set operations
{ a, ab, aaaa} { bb, ab} = {a, ab, bb, aaaa}
{ a, ab, aaaa} { bb, ab} = {ab}
{ a, ab, aaaa} − { bb, ab} = { a, aaaa}
• Complement:
L = Σ * −L
{ a, ba} = { λ , b, aa, ab, bb, aaa,}
53. Reverse
R R
L = {w : w ∈ L}
• Definition:
•
{ ab, aab, baba} = { ba, baa, abab}
Examples:
R
n n
L = {a b : n ≥ 0}
R n n
L = {b a : n ≥ 0}
