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Calculators employing reverse Polish notation (also known as postfix notation )use a stack structure to hold values.

**EXPRESSION
EVALUATION AND SYNTAX PARSING**

Calculators employing reverse Polish notation (also known as
postfix notation )use a stack structure to hold values.

Expressions can be represented in
prefix, postfix or infix notations. Conversion from one form of the expression
to another form needs a stack. Many compilers use a stack for parsing the
syntax of expressions, program blocks etc. before translating into low level
code. Most of the programming languages are
context-free languages allowing them to be parsed with stack based machines.
Note that natural languages are context
sensitive languages and stacks alone are not enough to interpret their meaning.

**Infix,
Prefix and Postfix Notation**

We are accustomed to write arithmetic expressions
with the operation between the two operands: **a+b **or** c/d**. If we
write** a+b*c**, however, we have to apply precedence rules to avoid the** **ambiguous
evaluation (add first or multiply first?).

There's no real reason to put the operation between
the variables or values. They can just as well precede or follow the operands.
You should note the advantage of prefix and postfix: the need for precedence
rules and parentheses are eliminated.

**Examples
of use: ( application of stack )**

**Arithmetic
Expressions: Polish Notation**

An arithmetic expression will have
operands and operators. Operator precedence listed below:

Highest : ($)

Next Highest : (*) and (/)

Lowest : (+) and (-)

For most common arithmetic operations, the operator
symbol is placed in between its two operands. This is called ** infix
notation**.

*Example:
A + B , E * F *

Parentheses can be used to group the operations.

*Example:
(A + B) * C*

Accordingly, the order of the operators and operands
in an arithmetic expression does not uniquely determine the order in which the
operations are to be performed.

Polish notation refers to the notation in which the
operator symbol is placed before its two operands. This is called ** prefix
notation**.

*Example: +AB, *EF *

The fundamental property of polish notation is that
the order in which the operations are to be performed is completely determined
by the positions of the operators and operands in the expression.

Accordingly, one never needs parentheses when
writing expressions in Polish notation.

** Reverse Polish Notation **refers
to the analogous notation in which the operator symbol is

*Example:
AB+, EF**

Here also the parentheses are not needed to
determine the order of the operations.

*The computer usually evaluates an
arithmetic expression written in infix notation in two steps, *

*1.
**It converts the expression to postfix
notation. *

*2.
**It evaluates the postfix
expression. *

*In each step, the stack is the main tool
that is used to accomplish the given task. *

**(1)Question
: ( Postfix evaluation )**

**How to evaluate a
mathematical expression using a stack The algorithm for Evaluating a postfix
expression ?**

•
Initialise an
empty stack

•
While
token remain in
the input stream

–Read
next token

–If
token is a number, push it into the stack

–Else, if token is an operator, pop top two tokens
off the stack, apply the operator, and push the answer back into the stack

• Pop
the answer off
the stack.

**Algorithm postfixexpression**

Initialize a stack, opndstk to be empty.

{scan the input string
reading one element at a time into symb } While ( not end of input string )

{

Symb
:= next input character;

If
symb is an operand Then

push
(opndstk,symb)

Else

[symbol
is an operator]

{

Opnd1:=pop(opndstk);

Opnd2:=pop(opndnstk);

Value := result of applying symb to opnd1 &
opnd2 Push(opndstk,value);

}

Result := pop (opndstk);

Example:

6 2 3 + - 3 8 2 / + * 2 $ 3 +

The
Final value in the STACK is 52. This is the answer for the given expression.

**(2) run time stack for function calls (
write factorial number calculation procedure) **push
local data and return address onto stack

return by popping off local data and then popping
off address and returning to it return value can be pushed onto stack before
returning, popped off by caller

**(3)
expression parsing**

e.g. matching brackets: [ ... ( ... (
... ) [ ...( ... ) ...] ... ) ... ] push left ones, pop off and compare with
right ones

**4)
INFIX TO POSTFIX CONVERSION**

Infix expressions are often translated into postfix
form in which the operators appear after their operands.

**Steps:**

1. Initialize
an empty stack.

2. Scan
the Infix Expression from left to right.

3. If
the scannned character is an operand, add it to the Postfix Expression.

4. If
the scanned character is an operator and if the stack is empty, then push the
character to stack.

5. If
the scanned character is an operator and
the stack is not empty, Then

(a) Compare
the precedence of the character with the operator on the top of the stack.

(b) While
operator at top of stack has higher precedence over the scanned character &
stack is not empty.

(i) POP
the stack.

(ii) Add
the Popped character to Postfix String.

(
c ) Push the scanned character to stack.

6. Repeat the
steps 3-5 till all the characters

7. While
stack is not empty,

(a) Add
operator in top of stack

(b) Pop
the stack.

8. Return
the Postfix string.

**Algorithm
Infix to Postfix conversion ( without parenthesis)**

1. Opstk
= the empty stack;

2. while
( not end of input )

{

symb = next input character;

3. if
( symb is an operand )

add
symb to the Postfix String

4. else

{

5.
While( ! empty (opstk) && prec (
stacktop ( opstk), symb) )

{

topsymb = pop ( opstk )

add topsymb to the Postfix String; } / * end of
while */

Push(opstk,
symb);

} /* end else
*/

6.
} /*
end while * /

7. While(
! empty ( opstk ) )

{

topsymb
= pop (opstk)

add
topsymb to the Postfix String

} /* end of
while */

8. Return
the Postfix String.

**OTHER
APPLICATIONS OF STACK**

Some
of the applications of stack are :

(i) Evaluating
arithmetic expression

(ii) Balancing
the symbols

(iii) Towers
of Hannoi

(iv) Function
Calls.

(v)8
Queen Problem.

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