This is the content discussed for Theory of Computation, on Day1 of Girlscript Education Outreach Program, Batch 5

1. Theory of Computation
By Rushabh Wadkar

2. Topics to be covered
Introduction to TOC
i. Solvability of a problem
ii. Order of an
algorithm(asymptotic
notations)
iii. Set theory
iv. Graphs and trees
Formal languages
Day 1

3. Introduction to TOC
In theoretical computer science and mathematics, the theory of computation is
the branch that deals with how efficiently problems can be solved on a model of
computation, using an algorithm.
The theory of computation can be considered the creation of models of all kinds
in the field of computer science. Therefore, mathematics and logic are used.

4. Solvability of a problem
Solvable Problems
● The problem has a definite solution.
● The problem will be solved in finite number
of steps.
● Example: Shortest Path problems
Unsolvable Problems
● The problem has no definite solution.
● The solution doesn’t exist yet. No finite
steps can provide u with a solution.
● Example: Division by Zero
All problems are divided into 2 categories, Solvable and Unsolvable

5. Königsberg
bridge problem
The Königsberg bridge problem
asks if the seven bridges of the city
of Königsberg (formerly in Germany
but now known as Kaliningrad and part of
Russia) over the river Preger can all
be traversed in a single trip without
doubling back, with the additional
requirement that the trip ends in the
same place it began.

6. Königsberg
bridge problem
Let us consider each land like a
node of a graph. And the bridges
are the edges connecting the
nodes.
The degree of each node is odd,
hence it is not possible to start from
a particular land and come back
there without traversing a bridge
more than once.

7. PROBLEM
SOLVABLE UNSOLVABLE
DECIDABLE UNDECIDABLE
Can be moved to
● Algorithm
+
Procedure
exists
● Only
Procedure
exists

8. Asymptotic Notations
● f(n)=O( g(n))
"Big O(micron)" – upper bound => worst case
● f(n) = Ω(g(n))
"Big Omega" – lower bound => best case
● f(n) = θ(g(n))
"Big Theta" – upper & lower bound => "average" case

9. Set theory
● (d1) A∩B = df {x: x∈A & x∈B} [simple intersection]
● (d2) A–B = df {x: x∈A & x≠B} [set-difference]
● (d3) A∪B = df {x: x∈A ∨ x∈B} [simple union]
● (d4) •(f) = df {x: ∃Y(Y∈f & x∈Y)} [general union]
● (d5) €(f) = df {x: ∀Y(Y∈f → x∈Y)} [general intersection]

10. Relations
● R is a relation => R is a set of ordered pair
(a,b)∈AxB
● R is a relation from A to B iff it satisfies the following restrictions
dom(R) ⊆ A
ran(R) ⊆ B

11. Functions
● A function is, by definition, a relation R satisfying the following
restriction.
∀xyz(xRy & xRz .→ y=z)
No two images will have same preimage.

12. DeMorgan’s Law
● (A∪B)’= A’∩ B’
● (A∩B)’= A’∪ B’

13. Important Set Properties
● Disjoint Sets: (A∩B) = ∅
● Size of Set= |S|, No. of elements in a set

14. ● Graph consists of edges and
nodes.
● A non-cyclic graph is called a
tree.
● There exist directed and
undirected graphs.

15. Formal Languages & its comparison
Language: English Formal Language
(For a fan)
Symbols: (A . . . Z)
(a . . . z)
0 & 1
On and off states
Alphabet: Set of all symbols
(A, B . . . Y, Z, a, b . . . y, z)
Set of all symbols
(0 , 1)
Strings: Any set of words
(including, these, themselves) (0 , 1, 01, 10, 001, . . . )

16. End of Day 1
www.linkedin.com/in/wadkar-rushabh
@RushabhWadkar
Thank you...