CS116 - Module 2 - Conditionalscjmorlan/cs116/module02-notes.pdfCS116 - Module 2 - Conditionals...

CS116 - Module 2 - Conditionals

Cameron Morland

Winter 2020

Reminder: if you have not already, ensure you:

Read Think Python, chapters 5, 6.

1 CS116 - Module 2 - Conditionals

Variable assignment

Unlike in Racket, in Python we can change the value of variables as much as we like.Statements run in order, top to bottom.

width = 4

height = 3

area = width * height

# area is now 12

width = 5

# area is still 12

area = width * height

# now area is 15


Basic Conditional Statement

if test:true_block

# scale_positive(x): add half of x to positive x values.

def scale_positive(x):result = x

if x > 0:x = x // 2

result = result + x

return result


Another Conditional Statement

if test:true_block

else:false_block

# scale_unevenly(x): add half of x to positives , third to negatives.

def scale_unevenly(x):if x > 0:

result = x

x = x // 2

result = result + x

return resultelse:

return x + x // 3


Recursion: “Countdown” template in Python

def countdown_fn(n):if n == 0:

return base_answerelse:

return ... n ... countdown_fn(n-1)

Exe

rcise

Use the countdown template to write a function sum_to(n) which sums all the numbers from0 to n.

Exe

rcise

Modify your sum_to(n) function to make a new function sum3(n) which sums the numbersfrom 0 to n which are divisible by 3.


This could get ugly...

If there are multiple conditions to check, we can use nested if statements.

def ticket_cost(age):' ' ' ticket_cost(age) return entryfee for a person age years old.

ticket_cost: Nat -> Float ' ' '

if age < 3: # infants are free.cost = 0.00

else:if age < 18: # cheaper minors

cost = 5.50

else:if age < 65: # full price

cost = 9.25

else: # seniorscost = 8.00

return cost

Racket equivalent:;; (ticket_cost age) return entry fee

;; for a person age years old.

;; ticket_cost: Nat -> Num

(define (ticket-cost age)

(cond [(< age 3) 0.00][else

(cond [(< age 18) 5.50][else

(cond [(< age 65) 9.25][else 8.00])])]))


“Chained” conditionals to the rescue!

def ticket_cost2(age):' ' ' ticket_cost2(age): return entryfee for a person age years old.

ticket_cost: Nat -> Float ' ' '

if age < 3: # infants are free.cost = 0.00

elif age < 18: # cheaper for minorscost = 5.50

elif age < 65: # full pricecost = 9.25

else: # seniorscost = 8.00

return cost

Racket equivalent:;; (ticket_cost2 age) return entry fee

;; for a person age years old.

;; ticket_cost: Nat -> Num

(define (ticket-cost2 age)

(cond [(< age 3) 0.00][(< age 18) 5.50]

[(< age 65) 9.25]

[else 8.00]))

Exe

rcise

Use elif to make sum3(n) easier to read.


Conversion to Boolean

a == b returns True if the values are equal. It works for all types. (Similar to equal? in Racket.)Note that a = b assigns a new value to a and returns nothing. Beware.= return True or False as appropriate.a != b is True if the values are not equal, like a 6= b.


Evaluating if/elif/else

An if statement has one if followed by a code block. It may also have one or more elifstatements. Finally, it may have an else statement.

if conditionA:blockA...

elif conditionB:blockB...

elif conditionC:blockC...

else: # no condition for else.elseBlock...

## This always runs after the if/elif/else,

## unless there was a return. It may be blank.

afterBlock...

You should think of a if/elif/else asa single statement.

If there is an else, exactly one of theblocks will run.

If there is no else, sometimes none ofthe blocks will run.Unlike in Racket, this is not an error.


if/elif/else versus if...if

Learn to keep each if/elif/else block separate in your mind.

x = 10

if x > 5: # This is True...x = 7 # ...so this runs.

elif x > 0: # This is also True...x = 4 # ...but this does not run.

## now x == 7

x = 10

if x > 5: # This is True...x = 7 # ...so this runs.

if x > 0: # This is still True...x = 4 # ...so this does run.

## now x == 4

When you write your own code, I recommendyou leave a blank line any time an if block isfollowed by another if block.if x > 5: # This is True...

x = 7 # ...so this runs.

# Blank for clarity.

if x > 0: # This is still True...x = 4 # ...so this does run


A recursive function

The two-argument Ackermann-Péter function is defined as follows for natural numbers m andn:

A(m, n) =

n + 1 if m = 0

A(m − 1, 1) if m 6= 0 and n = 0A(m − 1,A(m, n − 1)) if m 6= 0 and n > 0.

Exe

rcise Write a Python function ackermann(m,n) which computes this function. Test your function:

check.expect('A(2,3)', ackermann(2,3), 9)check.expect('A(3,3)', ackermann(3,3), 61)


Some limitations to recursion

The Ackermann function blows up really fast! For example, Ackermann(4, 2) = 265536 − 3. Wecan’t do recursion this deep in Python.

Exe

rcise

Revisit your sum_to(n) function. What is sum_to(900) ?What is sum_to(2000) ?

Python can’t recurse deeper than about 1000 function calls.We’ll see new approaches for bigger problems.


Global and Local variables

Variables defined inside a function are called local variables.Variables not defined inside a function are called global variables.This code works fine:foo = 3 # foo is a global variable

def func(x):baz = x + foo # baz and x are both local variables

return baz + foo

Inside a function you cannot change a global variable. So we will call them global constants.This code will not work:foo = 3 # foo is a global variable

def func(x):foo = x + foo #

Boolean arithmetic

In addition to Int and Float, Python has a Bool type. It can only be True or False.

v1 and v2 ⇒ True only if both values are True.v1 or v2 ⇒ True if at least one value is True.not v1 ⇒ True if v1 if False, otherwise False.

A Truth Table can be useful to describe combinations:

def baz(a, b):return a and not b

a b baz(a, b)

False False False

False True False

True False True

True True False

Exe

rcise

Write a Python function quux(a, b) which has thefollowing truth table:

a b quux(a, b)

False False False

False True True

True False True

True True False


Short Circuit Evaluation – Just like in RacketExe

rcise

Type in this code, and crash it:import math

def wrecktify(x):' ' ' do some calculation , badly.wrecktify: Float -> Float ' ' 'if math.cos(1/x) > 0:

return math.cos(1/x)else:

return 0.0

A problem when x == 0. To fix it, we could add anextra if, or use short circuit evaluation:

A and B: B is evaluated only if A is True

A or B: B is evaluated only if A is False

One solution:def wrecktify2(x):

if x == 0:return 0.0

elif math.cos(1/x) > 0:return math.cos(1/x)

else:return 0.0

Another solution:def wrecktify3(x):

if x != 0 and math.cos(1/x) > 0:return math.cos(1/x)

else:return 0.0


Interpreting numbers as Booleans

0 and 0.0 are treated as False, while any other numeric value is treated as True.Generally it’s not a good idea to write code that does this, but you might see code that does.

def sum_to(n):if n:

return n + sum_to(n-1)else:

return 0

This program behaves the same as it would if I wrote if n != 0: ...


Sum of a seriesExe

rcise Use recursion to write a Python function sum_powers(b, n) which consumes two Nat and

returns the sumS = 1 + b + b2 + b3 + · · ·+ bn−1 + bn

Hint:S = 1 + b (1 + b (1 + b (1 + . . . )))


Recursion PracticeExe

rcise

Complete digit_sum(n).def digit_sum(n):

' ' ' digit_sum(n) return the sum of the digits of n.digit_sum: Nat -> Nat

Examples:

digit_sum(245) => 11

digit_sum(11000) => 2 ' ' '

The nth Fibonacci number is the sum of the two previous Fibonacci numbers:

fn = fn−1 + fn−2

where f0 = 0, f1 = 1.

Exe

rcise Write a recursive function fib(n) that returns the nth Fibonacci number.

fib(1) => 1

fib(6) => 8

Recursion PracticeExe

rcise Write a function sum_squares_between(low, high) that returns the sum of the squares of all

the numbers between low and high, inclusive.sum_squares_between(3,5) => 3*3 + 4*4 + 5*5 => 50

Exe

rcise

Write a function sum_multiples(n, d1, d2) that adds up all the numbers up to n that aremultiples of d1 or d2.

Exe

rcise

Write a function count_digits(n) that consumes a Nat and returns the number of digits init.count_digits(9) => 1

count_digits(245) => 3

count_digits(12345678901234567890) => 20


Recursion Practice

If r =√n, then r2 = n, so r = nr . If r 6=

√n, then one of r , nr is greater than

√n, and the

other is smaller.So if r is a “guess” of the square root of n, the average of r and nr is a better guess.

Exe

rcise Use recursion to write a function sqrt(n, guess, threshold) which computes

√n with an

error less than threshold.check.within('2 +- 0.1', sqrt(4.0, 1.0, 0.01), 2.0, 0.01)


Recursion Practice

The greatest common divisor (GCD) of two natural numbers is the largest natural number thatdivides evenly into both.

Exe

rcise

Complete my_gcd_under.def my_gcd_under(a, b, t):

' ' ' return largest value t or less that divides a & b.my_gcd_under: Nat Nat Nat -> Nat

Example:

my_gcd_under(60,40,100) => 20

my_gcd_under(60,40,18) => 10 ' ' '

Exe

rcise

Using my_gcd_under as a helper function, write my_gcd(a,b) that returns the GCD of two Natvalues.


Goals of Module 2

Become comfortable with changing variables.

Understand local and global variables.

Write programs that use conditionals.

Get used to writing recursive functions in Python.

Before we begin the next module, please

Read Think Python, chapters 8, 10.


CS116 - Module 2 - Conditionalscjmorlan/cs116/module02-notes.pdfCS116 - Module 2 - Conditionals...

Documents

Transcript of CS116 - Module 2 - Conditionalscjmorlan/cs116/module02-notes.pdfCS116 - Module 2 - Conditionals...