Computation involves any type of information processing that can be represented as an algorithm. An algorithm is a finite sequence of simple instructions that is guaranteed to halt in a finite amount of time. However, this definition is abstract as it does not specify the nature of the instructions or the entity executing them. Some problems cannot be solved algorithmically as there is no program that can provide answers for all possible instances of the problem within a finite time period; these are called undecidable problems. Finding integral solutions to collections of polynomial equations is an example of an undecidable problem.

1. ComputationA very fundamental and traditional branch of
computation.
Computation is a general term for any type of
information processing that can be represented as
an algorithm precisely .
Computation is a general term for any type of
information processing that can be represented as an
algorithm precisely.

2. Use In Algorithm
A finite sequence of simple instructions that is guaranteed to halt in a
finite amount of time.
This is a very abstract definition, since:
We didn’t specify the nature of this simple instructions.
We didn’t specify the entity which can execute these instructions.
If it is a computer what is the processor type? How much memory does it
have? …. ?
To make a more solid definition of algorithm we need to define an
abstract (general) machine which can perform any algorithm that can
be executed by any computer.

3. Decidable ProblemsProblems, for which we can’t find an algorithm that
answer all possible instances of the problem.
That is there is no TM program which answer all
possible instances of the problem in a finite amount
of time.
For a decidable problem there is a program such that
if an instance of the problem has solution, the
program eventually halts with answer. But if there is
no solution for that instance, the program will not
ever halt.

4. Decidable Problems
For a decidable problem there is a program such that
if an instance of the problem has solution, the
program eventually halts with answer. But if there is
no solution for that instance, the program will not
ever halt.
Can we consider such programs as algorithms?

5. An Un-decidable ProblemThe problem of finding an integral solution for a
collection of multi-variable polynomial equations, is not
decidable.
For example consider the following two instances of
problem:

6. An Un-decidable ProblemThe problem of finding an integral solution for a
collection of multi-variable polynomial equations, is not
decidable.
For example consider the following two instances of
problem: