In refinement, each statement is repeatedly expanded into more detailed statements in the subsequent levels.

Algorithm	Program
Algorithm is for human readers to understand.	Program is for the computers to execute directly.
Knowledge of English is needed.	Knowledge of programming language is required.
Easy to understand.	It is difficult to understand.

1. Recursion is a method of solving problems that involves breaking a problem down into smaller and smaller sub problems until user gets in to a small problem that it can be solved trivially.
2. Usually recursion involves a function calling itself. While it may not seem like much on the surface, recursion allows us to write elegant solutions to problems that may otherwise be very difficult to program.

An expression involving variables, which remains unchanged by an assignment to one of these variables is called as an invariant of the assignment.

1. --minimum (a,b)
2. -- inputs: a, b are real numbers.
3. --output: result: minimum (a,b)

An algorithm is a step by step sequence of statements intended to solve a problem. An algorithm starts execution with the input data, executes the statements and finishes execution with the output data.

1. Enter A,B
2. Initialise Q-0, r-A
3. If r ≥ B, then do Q-Q+1; r-r-B else r,q
4. Exit.

(1) double (n)
--inputs: n is a real number or an integer, n > 0
--outputs: y is a real number or an integer such that y= n x h.
(2) double (n)
--inputs: n is a real number or an integer, n > 0
--outputs: y is a real number or an integer such that y = 2 x n.

1. Case analysis statement generalizes it to multiple cases.
2. Case analysis splits the problem into an exhaustive set of disjoint cases.

Let u - No. of tumblers right side up
v - No. of tumblers up side down
Initial stage: u = 0, v=7 (All tumblers upside down)
Final stage output: u=7, v=0 (All tumblers right side up)
Possible Iterations:
(i) Turning both up side down tumblers to right side up
u=u+ 2,   v=y-2   [u is even]
(ii) Turning both right side up tumblers to upside down.
u=u-2,  v=v+2  [u is even]
(iii) Turning one right side up tumblers to upside down and other tumbler from upside down to right side up.
u=u+1-1=u, v=y+1-1=v [u is even]
Initially u = 0 and continuous to be even in all the three cases. Therefore u is always even.
Invariant: u is even (i. e. No. of right side up tumblers are always even)
But in the final stage (Goal), u = 7 and v=0 i. e. u is odd.
Therefore it is not possible to reach a situation where all the tumblers are right side up.

Assignment statement is used to assign values to the variables.
The variable on the left side of the assignment operator and a value on the right side.
Variable := Value

Abstraction :
It is the facility to define objects. It also involves removal of unnecessary attributes and defining only essential attributes.
For example, when we represent the state of a process we select only the variables essential to the problem and ignore inessential details.

Let P be the required property of the inputs and Q the property of the desired outputs.
Format of specification of an algorithm:
(a) Algorithm name (Inputs)
(b) Inputs - P
(c) Outputs - Q

Flowchart is a diagrammatic notation for representing algorithms. They show the control flow of algorithms using diagrams in a visual manner. In flowcharts, rectangular boxes represent simple statements, diamond-shaped boxes represent conditions, and arrows describe how the control flows during the execution of the algorithm. A flowchart is a collection of boxes containing statements and conditions which are connected by arrows showing the order in which the boxes are to be executed.
(i) A statement is contained in a rectangular box with a single outgoing arrow, which points to the box to be executed next.

(ii) A condition is contained in a diamond-shaped box with two outgoing arrows, labelled true and false. The true arrow points to the box to be executed next if the condition is true, and the false arrow points to the box to be executed next if the condition is false.

(iii) Parallelogram boxes represent inputs given and outputs produced.

(iv) Special boxes marked Start and End are used to indicate the start and end of execution:

(i) f=1 i=1 f = f * i i = i+1
f = 1 x 1 = 1 i = 2
(ii) f = 1 x 2 i = 3 = 2
(iii) f = 2 × 3 i = 4 = 6
(iv) f = 6 x 4 i = 5 (loop terminates) = 24

Let the variables a, b, c represent Glass A, Glass B and Glass C and a, b, c can store values APPLE, GRAPE or EMPTY.
Specification:
1 exchange (a, b)
2-inputs: a, b: APPLE, GRAPE
3-outputs: a, b: = GRAPE, APPLE
Algorithm:
1. exchange (a, b)
2. a, b, c :- APPLE, GRAPE, EMPTY.
3. c:=b
4. -- a, b, c :- APPLE, EMPTY, GRAPE,
5. b:= a
6. --a, b, c = EMPTY, APPLE, GRAPE
7. a: = c
8. --a, b, c = GRAPE, APPLE, EMPTY

There are three important control flow .statements to alter the control flow depending on the state.
(i) In sequential control flow, a sequence of statements are executed one after another in the same order as they are written.
(ii) In alternative control flow, a condition of the state is tested, and if the condition is true, one statement is executed; if the condition is false an alternative statement is executed.
(iii) In iterative control flow, a condition of the state is tested, and if the condition is true, a statement is executed. The two steps of testing the condition and executing the statement are repeated until the condition becomes false.

Let the variables a, b, c represent the glass A, glass B and Glass C respectively. Variables A, B, C can store values APPLE, GRAPE or EMPTY.
Initial State:
1.-- a, b, c : = APPLE, GRAPE, EMPTY
2. C : = b
3.-- a, b, c : = APPLE, EMPTY, GRAPE
4. b : = a
5.-- a, b, c : = EMPTY, APPLE, GRAPE
6. a : = C
7.-- a, b, c : = GRAPE, APPLE, EMPTY
Final State:
Specification:
1. Exchange ()
2. -- inputs: a, b, c :- APPLE, GRAPE, EMPTY
3.--outputs: a, b, c:- GRAPE, APPLE, EMPTY

1. hypotenuse (a, b)
2. -- inputs: a, b are real numbers, a > 0, b > 0
3 -- outputs: c²=a² + b² where c is real number, c > 0