2. Theory of Computation
• What?
• The theory of computation is a branch of computer science and
mathematics combined
• Deals with how efficiently problems can be solved on a model
of
computation, using an algorithm.
2
6. UNIT I
AUTOMATA FUNDAMENTALS
Introduction to formal proof – Additional forms of Proof –
Inductive Proofs –Finite Automata – Deterministic Finite
Automata – Non-deterministic Finite Automata – Finite Automata
with Epsilon Transitions
6
7. Terminologies
• Alphabet
• Finite, non empty set of symbols
• Basic elements of a language
• Denoted by ∑
• String
• Finite sequence of symbols chosen from some alphabets
• Empty string
• Length of the string
• Power of an alphabet
• Language
• Set of all strings which are chosen from ∑*
7
8. Example
• English
• Alphabet – [a-z]
• String –{hi,hello,…}
• Binary number
• Alphabet – [0,1]
• String –{0,1,00,01,10,11…}
• Hexadecimal
• Alphabet –[0-9][a-e]
• String –[0,1,1A3,…]
8
10. Introduction to Formal Proof
• Deductive Proof
• Reduction to definition
• Other theorem forms
• Theorem that appear not to be if-then statements
10
11. Deductive Proof
• A deductive proof consists of a sequence of statements whose
truth leads us from some initial statement called the
hypothesis or the given statement(s) to a conclusion
statement.
• Each step in the proof must follow by some accepted logical
principle from either the given facts or some of the previous
statement
• The hypothesis may be true or false, depending on the value
of its parameter
• If H then C . C is deducted from H
11
12. • Hypothesis : x ≥ 4
• Conclusion : 2x ≥
x2
• Parameter : x
• Proof:
• If x=3, then 23 ≥ 32
• If x=4 then 24 ≥ 42
8 ≥ 9 which is false
16 ≥ 16 which is true
• For each time x increments by 1, the LHS get incremented by 2
𝑥+
1�
�
2
and the RHS
• If x ≥ 4, the
𝑥+
1�
�
= 1.25, (1.25)2= 1.5625
• 1.5625 is less than 2
• Hence 2x ≥ x2 will be true for x ≥ 4 23
13. If x is the sum of the squares of four
positive integers, then 2x ≥ x2
13
14. Reduction to Definitions
• Convert all the terms in hypothesis to their definitions
• Deductive proof
14
15. Let S be a finite subset of some infinite
set U. Let T be the complement of S with
respect to U. Then T is infinite
• Let us assume T is finite, |T| = m
• As per the given statement, S is finite => |S|=n
• Then |S U T | = n+ m, which is finite which contradicts the
given statement, Hence T should be infinite. 26
16. Other Theorem Forms
• Ways of saying If-Then
• H implies C
• H only if C
• C if H
• Whenever H holds, C follows
• If and only if statement
• A if and only if B
• If part : If B then A
• Only if part: If A then B
16
17. Theorem that appear not to be
If- Then Statement
2
sin 𝛼 ± sin 𝛽 =
sin
𝛼
± 𝛽
co
s
• Example
𝑎2
+ 𝑏2
= 𝑐2
1 1
2
2
𝛼
∓ 𝛽
17
18. Additional Forms of Proof
• Proofs by sets
• Proof by Contrapositive
• Proofs by contradiction
• Proofs by counterexample
18
21. Proof by Contrapositive
• The contrapositive of a statement negates the conclusion as
well as the hypothesis. It is logically equivalent to the original
statement asserted. Often it is easier to prove the
contrapositive than the original statement.
• If H then C is equivalent to If not C then not H
• Example:
• If x ≥ 4 then 2x ≥ x2
• If not 2x ≥ x2 then not x ≥ 4
• If 2x < x2 then x < 4
21
22. Proofs by contradiction
• The method of proof by contradiction is to assume that a
statement is not true and then to show that that assumption
leads to a contradiction.
• To prove if H then C is to prove If H then not C
implies falsehood.
22
23. Proofs by counterexample
• A proof by counterexample is not technically a proof. It is
merely a way of showing that a given statement cannot
possibly be correct by showing an instance that contradicts a
universal statement.
• If integer x is a prime then x is odd
23
25. Induction on integers
• Mathematical Induction is a mathematical technique which is
used to prove a statement, a formula or a theorem is true for
every natural number.
• The technique involves two steps to prove a statement, as
stated below −
• Step 1(Base step) − It proves that a statement is true for
the
initial value.
• Step 2(Inductive step) − It proves that if the statement is true
for the nth iteration (or number n), then it is also true for
(n+1)th iteration ( or number n+1).
25
26. 1 + 2 + ... + n = n(n+1)/2
Proof. (Proof by Mathematical Induction)
Let's let P(n) be the statement "1 + 2 + ... + n = (n (n+1)/2."
Basis Step.
P(1) asserts "1 = 1(2)/2", which is clearly true. So we are done
with the initial step.
26
27. Inductive Step.
Induction Hypothesis/ Inductive assumption:
Assume, 1 + 2 + ... + k = k (k+1)/2 is true
Prove for k+1,
(i.e) 1 + 2 + ... + k + (k+1) = (k+1)(k+2)/2.
1 + 2 + ... + k + (k+1)
= k(k+1)/2 + (k+1)
= (k(k+1) + 2 (k+1))/2
= (k+1)(k+2)/2.
27
30. Structural Induction
• Structural induction is a proof methodology similar to
mathematical induction, only instead of working in the
domain of positive integers (N) it works in the domain of such
recursively defined structures!
• It is terrifically useful for proving properties of such
structures.
• Its structure is sometimes “looser” than that of mathematical
induction.
30
31. Every tree has one more node than its edges
• If T is a tree and T has n nodes and e edges then n=e+1
• Basis:
• T is single node tree, then n=1, e=0, so n=e+1 holds true
• Inductive Hypothesis:
• Assume the statement S(Ti ) hold for i=1,2,3,…,K and Ti have ni
nodes and ei edges then ni = ei + 1
• Induction
31
33. Every expression has an equal number of left and
right parentheses
• Let G is an expression
• Basis :
• G is a number or variable, so the number of left and right
parenthesis is 0
• Inductive Hypothesis:
• Assume E and F are two expressions which has equal
number of
left and right parentheses.
• Induction :
33
38. Finite Automata
• Finite automata are used to recognize patterns.
• It takes the string of symbol as input and changes its state
accordingly. When the desired symbol is found, then the
transition occurs.
• At the time of transition, the automata can either move to the
next state or stay in the same state.
• Finite automata have two states, Accept state or Reject state.
When the input string is processed successfully, and the
automata reached its final state, then it will accept.
38
39. • Input Tape
• It is a linear tape having some number of cells. Each input symbol is
placed in each cell.
• Finite control
• It decides the next state on receiving particular input from input
tape.
• Tape reader
• It reads the cells one by one from left to right, and at a time only
one input symbol is read.
39
40. • A finite automaton consists of:
• a finite set S of N states
• a special start state
• a set of final (or accepting) states
• a set of transitions T from one state to another, labelled with
chars in C
40
41. • Execution of FA on an input sequence as follows:
• Begin in the start state
• If the next input char matches the label on a transition from the
current state to a new state, go to that new state
• Continue making transitions on each input char
• If no move is possible, then stop
• If in accepting state, then accept
41
42. Deterministic Finite Automata
• Deterministic refers to the uniqueness of the computation.
• On each input there is one and only one state to which the
automaton can transition from its current state
• DFA does not accept the null move.
42
43. Formal Definition of DFA
• A deterministic finite automaton (DFA) is a 5-tuple
(Q,Σ,δ,q0,F),where
• Q is a finite set called the states,
• Σ is a finite set called the alphabet,
• δ:Q×Σ→Q is the transition function,
• q0 ∈ Q is the start state, and
• F ⊆ Q is the set of accepting states.
43
44. Transition Table
• A transition table is a tabular representation of the transition
function that takes two arguments and returns a state.
• The column contains the state in which the automaton will be
on the input represented by that column.
• The row corresponds to the state the finite control unit can
be
in.
• The entry for one row corresponding to state q and the
column corresponds to input a is the state δ(q, a).
44
45. Transition Diagram
• Transition graph can be interpreted as a flowchart for an
algorithm recognizing a language.
• A transition graph consists of three things:
• A finite set of states, at least one of which is designated the start
state and some of which are designated as final states.
• An alphabet Σ of possible input symbols from which the input
strings are formed.
• A finite set of transitions that show the change of state from the
given state on a given input.
45
46. Example DFA
states A b
q0 q1 q2
q1 q1 q3
q2 q2 q3
*q3 q3 q3
46
A=({q0,q1,q2,q3},{a,b}δ,q0,{q3})
δ is given by
δ(q0,a)=q1
δ(q0,b)=q2
δ(q1,a)=q1
δ(q2,b)=q2
δ(q1,b)=q3
δ(q2,a)=q3
δ(q3,a)=q3
δ(q3,b)=q3
47. • Design a DFA with ∑ = {0, 1} accepts those string which starts
with 1 and ends with 0.
• Design a DFA with ∑ = {0, 1} accepts the only input 101.
47
48. • Design a DFA with ∑ = {0, 1} accepts the strings with an even
number of 0's end by single 1.
48
49. Extended transition function
δ^
• The DFA define a language: the set of all strings that result in a
sequence of state transitions from the start state to an accepting
state
• Extended Transition Function
• Describes what happens when we start in any state and follow any
sequence of inputs
• If δ is our transition function, then the extended transition function is
denoted by δ^
• The extended transition function is a function that takes a state q and
a string w and returns a state p (the state that the automaton
reaches when starting in state q and processing the sequence of
inputs w)
• Let w=va then
60
δ(q, va) = δ(δ^ (q, v),
a).
50. Language accepted by DFA
• The language of a DFA A = (Q, Σ, δ, q0, F), denoted L(A) is
defined by
L(A) = { w | δ^(q0, w) is in F }
• The language of A is the set of strings w that take the start
state q0 to one of the accepting states
• If L is a L(A) from some DFA, then L is a regular language
50
51. Nondeterministic Finite
Automata
• An NFA is like a DFA, except that it can be in several states at
once.
• This can be seen as the ability to guess something about the
input.
• Useful for searching texts
51
52. Formal Definition of NFA
• A nondeterministic finite automaton (NFA) is a 5-tuple
(Q,Σ,δ,q0,F),where
• Q is a finite set called the states,
• Σ is a finite set called the alphabet,
• δ:Q×Σ→P(Q) is the transition function,
• q0 ∈ Q is the start state, and
• F ⊆ Q is the set of accepting states.
52
53. Extended transition function
δ^
• The extended transition function is a function that takes a
state q and a string w and returns a set of states P (The set of
possible state that the automaton reaches when starting in
state q and processing the sequence of inputs w)
• Let w=va then
δ^
𝑞0, 𝑣𝑎
= ∪ ′ ^
𝑞 ∈ δ
𝑞0,𝑣
δ( 𝑞
′,a)
53
54. Language accepted by NFA
• The language L(A) accepted by the NFA A is defined as follows:
L(A) = {w | δ^(q0, w) ∩ F ≠ ∅}
54
56. • Design an NFA with ∑ = {0, 1} in which double '1' is followed by
double '0'.
• Design an NFA with ∑ = {0, 1} accepts all string in which the
third symbol from the right end is always 0.
56
57. Epsilon Nondeterministic Finite
Automata
• Formal Definition
A ε-NFA is a quintuple A=(Q,Σ,δ,q0,F) where
• Q is a set of states
• Σ is the alphabet of input symbols
• ε is never a member of Σ. Σε is defined to be (Σ ∪ ε)
• δ: Q × Σε → P(Q) is the transition function
• q0 ∈ Q is the initial state
• F ⊆ Q is the set of final states
57
59. -
ε closure
• Epsilon means present state can goto other state without any
input. This can happen only if the present state have epsilon
transition to other state.
• Epsilon closure is finding all the states which can be reached
from the present state on one or more epsilon transitions.
ε- closure (0)={0,1,2}
ε- closure(1)={1,2}
ε- closure(2)={2}
59
66. • Every DFA is NFA but not vice versa.
• Both NFA and DFA have same power and each NFA can be
translated into a DFA.
• There can be multiple final states in both DFA and NFA.
• NFA is more of a theoretical concept.
• DFA is used in Lexical Analysis in Compiler.
• Every NFA is ε-NFA but not vice-versa
66
67. Conversion of NFA to DFA
• Subset Construction Method
1. Create state table from the given NFA.
2. Create a blank state table under possible input alphabets for the
equivalent DFA.
3. Mark the start state of the DFA by q0 (Same as the NFA).
4. Find out the combination of States {Q0, Q1,... , Qn} for each
possible input alphabet.
5. Each time we generate a new DFA state under the input
alphabet columns, we have to apply step 4 again, otherwise go
to step 6.
6. The states which contain any of the final states of the NFA are
the final states of the equivalent DFA.
67
73. Now we will obtain δ' transition for state q0.
δ'([q0], 0) = [q0]
δ'([q0], 1) = [q1]
The δ' transition for state q1 is obtained as:
δ'([q1], 0) = [q1, q2] (new state generated)
δ'([q1], 1) = [q1]
73
74. Now we will obtain δ' transition on [q1, q2].
δ'([q1, q2], 0) = δ(q1, 0) ∪ δ(q2, 0)
= {q1, q2} ∪ {q2}
= [q1, q2]
δ'([q1, q2], 1) = δ(q1, 1) ∪ δ(q2,
1)
= {q1} ∪ {q1, q2}
= {q1, q2}
= [q1, q2]
The state [q1, q2] is the final state as well because
it contains a final state q2. 85
75. The transition table for the constructed DFA will
be:
State 0 1
→[q0] [q0] [q1]
[q1] [q1, q2] [q1]
*[q1, q2] [q1, q2] [q1, q2]
86
76. Conversion of -
ε NFA to DFA
• Modified subset construction
1. Find the ε-closure for the starting state of ε - NFA as a starting
state of DFA.
2. Find the states for each input symbol that can be traversed from
the present. That means the union of transition value and their
closures for each state of NFA present in the current state of
DFA.
3. If a new state is found, take it as current state and repeat step
2.
4. Repeat Step 2 and Step 3 until there is no new state present in
the transition table of DFA.
5. Mark the states of DFA as a final state which contains the final
state of ε-NFA. 87
77. Example
• Let us obtain ε-closure of each state.
ε-closure {q0} = {q0, q1, q2}
ε-closure {q1} = {q1}
ε-closure {q2} = {q2}
ε-closure {q3} = {q3}
ε-closure {q4} = {q4}
77
78. •let ε-closure {q0} = {q0, q1, q2} be state
A δ'(A, 0) = ε-closure {δ((q0, q1, q2), 0) }
= ε-closure {δ(q0, 0) ∪ δ(q1, 0) ∪
δ(q2, 0) }
= ε-closure {q3}
= {q3} call it as state B. δ'(A,
1) = ε-closure {δ((q0, q1, q2), 1) }
= ε-closure {δ((q0, 1)
∪ δ(q1, 1) ∪ δ(q2, 1)}
= ε-closure {q3}
= {q3} = B. 78
![Example
• English
• Alphabet – [a-z]
• String –{hi,hello,…}
• Binary number
• Alphabet – [0,1]
• String –{0,1,00,01,10,11…}
• Hexadecimal
• Alphabet –[0-9][a-e]
• String –[0,1,1A3,…]
8](https://image.slidesharecdn.com/unit1-250311050745-24f11123/85/unit-1-pptx-theory-of-computation-complete-notes-8-320.jpg)
![Now we will obtain δ' transition for state q0.
δ'([q0], 0) = [q0]
δ'([q0], 1) = [q1]
The δ' transition for state q1 is obtained as:
δ'([q1], 0) = [q1, q2] (new state generated)
δ'([q1], 1) = [q1]
73](https://image.slidesharecdn.com/unit1-250311050745-24f11123/85/unit-1-pptx-theory-of-computation-complete-notes-73-320.jpg)
![Now we will obtain δ' transition on [q1, q2].
δ'([q1, q2], 0) = δ(q1, 0) ∪ δ(q2, 0)
= {q1, q2} ∪ {q2}
= [q1, q2]
δ'([q1, q2], 1) = δ(q1, 1) ∪ δ(q2,
1)
= {q1} ∪ {q1, q2}
= {q1, q2}
= [q1, q2]
The state [q1, q2] is the final state as well because
it contains a final state q2. 85](https://image.slidesharecdn.com/unit1-250311050745-24f11123/85/unit-1-pptx-theory-of-computation-complete-notes-74-320.jpg)
![The transition table for the constructed DFA will
be:
State 0 1
→[q0] [q0] [q1]
[q1] [q1, q2] [q1]
*[q1, q2] [q1, q2] [q1, q2]
86](https://image.slidesharecdn.com/unit1-250311050745-24f11123/85/unit-1-pptx-theory-of-computation-complete-notes-75-320.jpg)
