Prefix, Infix and Postfix Notation

1
Prefix, Infix and Postfix Notation

• Expression Evaluation

We must organize computations in such a way that
our limited powers are sufficient to guarantee
that the computation will establish the desired
effect
- Dijkstra
2
Basic Expressions

• Basic expressions
• Operator , , AND, NOT
• Operands 1,2,3 etc. or, subexpressions
• 3 4 5
• Operations can be written in infix, postfix, or
  prefix order
• Associativity is specified in prefix and postfix
  form

3
Operators and Functions

• Some languages distinguish between operators
  (built-in) and functions (user defined)
• Typically, built-ins are specified as infix,
  functions as prefix
• 34 5 add(3, mul(4,5))
• Operators and functions are essentially
  equivalent concepts
• Why differentiate?

4
Operator Overload

• Some languages allow the user to overload
  built-in operators
• Some require prefix or postfix expressions
• Postscript and FORTH use postfix
• LISP and Scheme use prefix (and fully
  parenthesized)
• Each language has rules for evaluating
  expressions
• Recall that compilers build parse trees, so, the
  order of expression evaluation matters in order
  to get the desired result

5
Expression Evaluation

• Applicative order evaluation
• All operands are evaluated first then operators
  are applied to them
• Also called strict evaluation
• (314) 3 (82-6) 14
• 3 Mul(1add(3,4), 2sub(8,6))
• In what order are the subexpressions computed?
• Natural order is left to right
• Many languages leave this undefined

6
Infix Precedence

• Issue Is the value of the expression
• 2 3 4
• (2 3) 4 20
• Or,
• 2 (3 4) 14?
• Operator precedence governs evaluation order.
• has higher precedence than , so it is applied
  first, making the answer 14.

7
Operator Precedence

• ( )
  HIGH
• (unary) , - (unary) , ! (NOT)
• , /,
• (addition), - (subtraction)
• lt, lt, gt, gt
• , !
• 
  LOW

8
Associativity

• Does the expression 8 - 4 - 2
• evaluate (8 - 4) - 2 2,
• or 8 - (4 - 2) 6?
• Precedence doesnt help as it is the same
  operator appearing multiple times in an
  expression
• Use Associativity. Since - is left-associative,
  the left - is evaluated first, giving us 2.
• Can you think of a right-associative expression,
  meaning, right hand side is evaluated first?

- Assignment Operation
9
Order of Evaluation

• Different machines may have different
  requirements for function calls
• Might want to make the code more efficient
• (34)(34)
• Evaluation order only matters in the presence of
  side effects
• If the order is unspecified, any program that
  relies on a particular order could give incorrect
  final result

10
Expressions

• C is an expression oriented language, written in
  Infix form
• x (f - z) (5/9)
• Y X
• z a
• Some expressions can be evaluated without
  evaluating all subexpressions
• Short-circuit boolean evaluation

11
Short Circuit Evaluation

• Programming languages specify evaluation of
  boolean expressions
• Left to right until truth value is known
• Order of evaluation is important!
• Many different ways of writing boolean operators

12
Short Circuit Boolean Evaluation
exp1 exp2 . . . expn Evaluate left to
right, stop and return true when first exp is
true short circuit evaluation. if all are
false, return false. exp1 exp2 . . .
expn Evaluate left to right, stop and return
false when first exp is false short circuit
evaluation. if all are true, return true.
.
13
Short Circuit Evaluation of Boolean Expressions

• Example in Java
• if ( !(obj NULL) (obj.num 0))
• obj.num obj.num 1
• is a binary expression, so can we just do
• Let t1 be (obj NULL)
• t2 be !t1
• t3 be (obj.num 0)
• t4 be (t2 t3)
• Simple isnt it? Wrong!!
• - If obj is NULL, then evaluating obj.num will
  lead to Segmentation Fault.

14
Expression Notations

• Arity of an operator
• unary one operand, binary two operands, etc.
• Notation infix, prefix, postfix
• constant or variable is the constant or variable
  itself
• the application of op to E1 and E2 op E1 E2
• more general, opk E1 EK

15
Infix and Prefix Notation

• Infix notation Operator appears between
  operands
• 2 3 ? 5
• 3 6 ? 9
• Implied precedence 2 3 4 ? 2 (3 4 ),
• not (2 3 ) 4 See slide 7 for Order
  of precedence
• Prefix notation Operator precedes operands
• 2 3 ? 5
• 3 5 2 ? ( ( 3 5 ) 2) ? 2 15 ? 17
• Exercise (prefix)
• 20 30 60 ( ( 20 30) 60)
• ( 50 60)
• 3000

16
Postfix Notation

• Postfix notation Operator follows operands
• 2 3 ? 5
• 5 2 3 ? (5 ( 2 3 ) ) ? 5 6 ? 11
• Called Polish postfix since few could pronounce
  Polish mathematician Lukasiewicz, who invented
  it.
• An interesting, but unimportant mathematical
  curiosity when presented in 1920s. Only became
  important in 1950s when Burroughs rediscovered it
  for their ALGOL compiler.
• Prefix Notation is referred to as Reverse Polish
  Notation

17
Summary Associativity and Precedence

• Infix notation operators appear between operands
  a b c
• sum of a and b c? ? a (bc)
• product of a b and c ? ? (a b) c
• Precedence of operators
• and / have higher precedence than and -
• except Smalltalk
• Associativity
• usually from left to right 4 - 2 - 1 ?
• are you sure it is not 3?
• operator is left associative if subexpressions
  containing multiple occurrences of the operator
  are grouped left to right
• Parentheses

18
Summary Prefix Notation

• ltExpressiongt (ltoperatorgt ltoperandsgt)
• Example
• 3 5 ( ltgt lt3 5gt)
• 15
• 3 5 5 (lt lt 3 5gt 5gt)
• 20
• 3. / 2 3 5 6 (lt/ lt 2 3 5gt 6gt)
• (lt/ lt lt 2 3gt 5gt 6gt)
• (lt/ lt 6 5gt 6gt
• (lt/ 30 6gt)
• 5

19
Prefix Notation and Scheme
Scheme uses prefix notation, where the operation
(or function) call occurs first, followed by the
data or arguments.
( ltoperationgt ltdatagt )
Examples ( 2 2) ( 3 1) (/ 10 0) ( 7 (
4 5))
Which of these would cause problems if entered
into a simple calculator?
20
Exercises Prefix Notation

• Write the following expression in prefix
  notation
• 1) subtract 3 from 5
• 2) subtract 10 from the quantity 5 minus 8
• 3) divide 3 into 42
• 4) sum the squares of both 3 and 4
• Evaluate the following prefix expressions
• (/ ( 2 4) (/ ( 6 4) ( 12 4)))
• / 2 6 14 2

21
Documentation for self-paced learning using
Dr.Scheme
http//www.htdp.org
22
You will use the DrScheme program as your editor
and development environment. It is what is known
as an IDE integrated development environment.
23
Definitions Window
Interactions Window
24
Execute button loads the Definitions into
the Interactions window
25
Saving your work
26
Important
Be sure to set the language level in Dr. Scheme
to Beginning Scheme
27
Recall Scheme is Functional Interpreted

• Scheme can execute a single line of code as soon
  as it is typed into the interactions window
• ( 2 2)
• 4

28
Simple Evaluation
Lets start with something simple. Suppose we
wanted to solve the following expression 3
4 5 This looks easy, but what might go
wrong?
29
Simple Evaluation
It might be that by mistake someone could read
this as 7 5 by adding before multiplying. A
clear mistake. 3 4 5 How can we be
sure we dont make this mistake? ? paranthesize
30
Simple Evaluation
We can add unnecessary but helpful parentheses
around part of the expression. 3 (4
5) We all know that parenthetical expressions
are evaluated first So this will make it
totally clear.
31
Simple Evaluation
We can go further, and add parentheses around the
entire expression. (3 (4 5)) This
doesnt change the meaning of our original short
notation. It makes things explicit.
32
Recall the notations for writing arithmetic
expressions ?
As we saw earlier, the choice of representing
operators (, ) between the operands (numbers)
is completely arbitrary. (1 2)
Convention could be to write this as (
1 2) or even 1 2
Assuming binary operands, because of the way
postfix notation works, theres no need for a
set of parentheses
33
Using DrScheme
In class activity Boot up your machines and
find Dr.Scheme and launch the application now.
34
Learning Index
Desired level of learning in CS160
Actions You Will Take This Evening
Try numerous examples in DrScheme AND the
Stepper Run DrScheme, but not the
Stepper Merely read notes/slides/online
docs Blow it off until somethings due
A
B
C
D/F
35
Using the Stepper
DrScheme includes a remarkable stepping tool that
lets yu see the order of evaluation. Lets
evaluate a complex expression with this tool we
did by hand earlier (/ ( 2 4) (/ ( 6 4) (
12 4))) (In class activity all students try
this on your terminal)
36
Scheme Numbers
Elements
Integers, e.g. 4, -6, 0 Real Numbers, e.g.,
3.14159 Ratios, e.g., 1/2, 4/3 Symbols, e.g.,
x, y, foo, bar -- often used for the name of a
parameter, or a function name, or a variable
37
Scheme
Predefined symbols. As it turns out, there are
about 100 predefined symbols in scheme, including
the arithmetic operators
addition - subtraction
multiplication / division
38
Using a pre-defined function in Scheme

• Let us use the sqrt function pre-defined in
  Scheme, which computes the square root of a given
  number.
• We are used to function call as sqrt(4)
• In Scheme, it is written as (sqrt 4)
• Try sqrt and other pre-defined functions in
  Scheme after viewing the next slide.

39
What is i
When we next type in (sqrt (sqrt 4)) we
see i1.4142135623730951 That looks like the
right value, but whats with the i in front?
This is schemes way of indicating an
approximation.
40
Mathematical operations

• Numerous built-in functions for common
  operations
• (sqrt A) ? ?A
• (expt A B) ? AB
• (remainder A B) ? remainder of integer division
  A/B (remember modulus operator?)
• e.g. (remainder 23 8) ? 7
• (log A) ? natural log of A
• (sin A) ? sine of A radians

41
In-class Exercise 30 minutes

• Try several numerical expression evaluations as
  well as pre-defined functions evaluation in
  DrScheme now.
• Visit
• http//download.plt-scheme.org/doc/drscheme/
• Use the handout as your reference for checking
  your progress.
• Visit
• http//www.htdp.org/2002-09-22/Companion/drscheme-
  Z-H-6.html
• When satisfied with using arithmetic expressions
  evaluation proceed to learn about functions
  online at
• http//www.htdp.org/2002-09-22/Companion/drscheme-
  Z-H-7.html_chap_4
• Or, wait till we discuss it in class.

42
Writing our Own Mathematical Functions
Lets start with a simple function from
mathematics f(x) x x This function
squares a number, thus f(2) 2 2
4 f(5) 5 5 25


43
Functions
Recall that a function maps a set of values,
called the domain, onto a set of return
values, called the range, so that one of the
domain values is paired with one and only one of
the range values.
1
2
3
44
f(X)
f(X) XX
range
X
domain
45
Functional Vocabulary
Consider again
f(x) x x
46
Functional Vocabulary
f(x) x x f(2)
The function body is applied to the value 2
47
Observations
f(x) x x f(2) 4
Predictable, reproducible. Each time we apply
the function to the value 2, we get the same
result. Nothing changed about the parameter.
Each time we call f(2), we do not change the
value of 2. The function doesnt alter anything
in the world.
48
So let's write a Scheme function!
Remember square f(x) x x We start by naming
the function (define (sq x) and continue by
adding the body (define (sq x) ( x x)) And
we test (sq 3) 9
49
Remember Pythagoras?
50
Pythagoras
c
b
a
c2 a2 b2 c sqrt(a2 b2)
51
Pythagoras...
(sqrt ( ( 3 3) ( 4 4))) 5
c
3
4
52
Note on Prefix Notation in Scheme
We use prefix notation, instead of infix ( 3
3) rather than ( 3 3) because scheme
expects the name of the function first (after the
opening paren). "" is a actually the name of a
function that multiplies numbers together
53
Many Calculations of Hypotenuse
(sqrt ( ( 3 3) ( 4 4)))
(sqrt ( ( 2 2) ( 8 8)))
c
2
8
If we had many triangles to work with, typing the
same expression over and over gets tedious.
54
Assignment ? Binding values
In mathematics, wed recognize the function at
play hypotenuse(a, b) sqrt ( aa bb
) Remember what that "" sign meant? It binds
the body to the function/parameter list. In
Scheme, we can do this binding by defining a
function. It begins with ( define
55
Binding the name and the expression
At this point, the Scheme interpreter would
expect you to name the function you hope to
define. We need to come up with a symbol
name. The symbol/name hypotenuse is
good (define (hypotenuse side1 side2)
Here side1 and side2 are "parameters" or
"arguments basically placeholders for the side
lengths. In our pseudocode (infix) notation, this
is equivalent to Procedure hypotenuse
(side1, side 2)
56
Still More

• At this point, we have
• a symbol for the name of the function, and
• the names of the parameters to the function.

(define (hypotenuse side1 side2)
The body of the function is easy
(define (hypotenuse side1 side2) (sqrt ( (
side1 side1) ( side2 side2))))
57
Scheme and Pseudocode

• (define (hypotenuse side1 side2)
• (sqrt ( ( side1 side1) ( side2 side2))))

Pseudocode for the same, assuming sqrt(x) is
pre-defined procedure hypotenuse (side1, side2)
print sqrt((side1side1)
(side2side2))
58
Style matters!(not to the scheme interpreter,
but)
Even though the interpreter doesnt care, as good
programming practice, we will always seek good
style! As a matter of style, we will use indents
and returns to better format the statement
(define (hypotenuse side1 side2)
(sqrt ( (
side1 side1) (
side2 side2)))) As opposed to (define
(hypotenuse side1 side2) (sqrt ( ( side1
side1) ( side2 side2))))
59
Abstraction and Re-use of functions
Note we square side1 and then square side2, then
add the result and find the sqrt of the
result. We defined a square function sq earlier.
We should use it. Its better abstraction.
(define (hypotenuse side1 side2) (sqrt (
( side1 side1)
( side2 side2))))
(define (hypotenuse side1 side2) (sqrt (
(sq side1)
(sq side2))))
60
In-class Exercise Functions

• Write a Scheme function for converting from
  Fahrenheit to Celsius as done for Quiz 5 using
  the formula
• C (5/9) (F-32)
• Write a Scheme function for computing the volume
  of a box as done for Quiz 5
• Write a Scheme function which when given the area
  of a square, calculates the perimeter of the
  square.

61
Self-paced Learning

• Bookmark the following documentation pages and
  work through them on your own.
• http//download.plt-scheme.org/doc/drscheme/
• TOC for self-paced learning
• http//www.htdp.org/2002-09-22/Companion/
• drscheme-Z-H-1.html_toc_start