Conditional Execution

Abstract¶

Conditional execution allows a program to selectively execute different parts of the code based on specific conditions. This capability enables the program to adapt and respond appropriately to various situations it encounters, making it more flexible and functional. By using boolean expressions and conditional statements, students will learn to direct the flow of execution to handle a wide range of inputs and scenarios effectively.

import math

from flowcharts import *
from ipywidgets import interact
from dis import dis

%load_ext divewidgets

if not input('Load JupyterAI? [Y/n]').lower()=='n':
    %reload_ext jupyter_ai

Load JupyterAI? [Y/n]

Motivation¶

Conditional execution means running different pieces of code based on different conditions. Why do programmers need conditional executation?

For instance, when trying to compute a/b, b may be 0 and division by 0 is invalid.

%%optlite -h 450
def multiply_or_divide(a, b):
    print("a:{}, b:{}, a*b:{}, a/b:{}".format(a, b, a * b, a / b))


multiply_or_divide(1, 2)
multiply_or_divide(1, 0)

Although division by 0 is invalid, multiplication remains valid but it is not printed due to the division error. Can we skip only the division but not the multiplication when b is equal to 0?

Solution 1: Use a conditional expression that specifies which code block should be executed under what condition:

... if ... else ...

def multiply_or_divide(a, b):
    q = a / b if b else "undefined"
    print("a:{}, b:{}, a*b:{}, a/b:{}".format(a, b, a * b, q))


multiply_or_divide(1, 2)
multiply_or_divide(1, 0)  # multiplication is valid but not shown

a:1, b:2, a*b:2, a/b:0.5
a:1, b:0, a*b:0, a/b:undefined

Solution 2: Use a boolean expression:

... and ... or ...

def multiply_or_divide(a, b):
    q = b and a / b or "undefined"
    print("a:{}, b:{}, a*b:{}, a/b:{}".format(a, b, a * b, q))


multiply_or_divide(1, 2)
multiply_or_divide(1, 0)  # multiplication is valid but not shown

a:1, b:2, a*b:2, a/b:0.5
a:1, b:0, a*b:0, a/b:undefined

Solution 3: Monitor and catch the error using a try statement:

def multiply_or_divide(a, b):
    try:
        q = a / b
    except ZeroDivisionError:
        q = "undefined"
    print("a:{}, b:{}, a*b:{}, a/b:{}".format(a, b, a * b, q))


multiply_or_divide(1, 2)
multiply_or_divide(1, 0)  # multiplication is valid but not shown

a:1, b:2, a*b:2, a/b:0.5
a:1, b:0, a*b:0, a/b:undefined

In this notebook, we will introduce the first two solutions. The last one is a better way to handle exceptions when the operations get complicated.

%%ai
Explain in one paragraph, using a simple example, the benefits of using a 
try statement over conditional checks for handling exceptions and errors.

Comparison Operators¶

A comparison/relational operators along with its operands form an expression, which evaluates to a boolean value:

True  # if the operands satisfy certain conditions, and

True

False  # otherwise.

False

Unlike many other languages, Python capitalized the keywords for the boolean values to signify that they are constants, just like the keyword None. Hence,

true = False  # is invalid in many languages but not Python
False = true  # is valid in many languages but not Python

Why are boolean values “boolean”?

Boolean expressions are named after George Boole, whose work laid the foundation of digital circuitry. Below are the links to the ebooks, thanks to Project Gutenberg:

An import result derived from Boole’s framework for digital circuits is Boole’s/Shannon expansion theorem.

%%ai
Explain Entscheidungsproblem and its relation to George Boole.

Equality and Inequalties¶

The equality and inequality relationships in mathematics are implemented using the following comparison operators:

Table 1:Comparison operators for equality and inequalities

Expression	True iff
`x == y`^[5]	$x=y$ .
`x < y`^[6]	$x<y$ .
`x <= y`	$x\leq y$ .
`x > y`	$x>y$ .
`x >= y`	$x\geq y$ .
`x != y`	$x\neq y$ .

You can explore these operators using the widget below:

comparison_operators = ["==", "<", "<=", ">", ">=", "!="]


@interact(operand1="10", operator=comparison_operators, operand2="3")
def comparison(operand1, operator, operand2):
    expression = f"{operand1} {operator} {operand2}"
    value = eval(expression)
    print(
        f"""{'Expression:':>11} {expression}\n{'Value:':>11} {value}\n{'Type:':>11} {type(value)}"""
    )

%%ai
Explain very briefly how the comparison operators == may be represented 
differently in other programming languages such as maxima, scheme, and bash.

What is the precedence of comparison operators?

All the comparison operators have the same precedence lower than that of + and -.

1 + 2 >= 3  # (1 + 2) >= 3

True

Similar to the assignment operations, comparison operators can be chained together but they are non-associative.

2.0 == 2 > 1, (2.0 == 2) > 1, 2.0 == (2 > 1)

(True, False, False)

1 <= 2 < 3 != 4, (1 <= 2) < (3 != 4)

(True, False)

Solution to Exercise 1 #

The second expression indeed does not involve chained comparison. The expressions inside the parentheses are first evaluated to boolean values, both of which are True. Therefore, the entire expression True < True evaluates to False. You can also try evaluating the following expressions for further understanding:

False < True
- True ** 2 + 1
1 / False

To learn more about the relationship between bool and int, check out PEP 285.

The second expression is not a chained comparison:

The expressions in the parentheses are evaluated to boolean values first to True, and so
the overall expression True < True is evaluated to False. You may also want to try:
- False < True
- - True ** 2 + 1
- 1 / False

To understand how bool is related to int, see the PEP 285.

# Comparisons beyond numbers
@interact(
    expression=[
        "10 == 10.",
        '"A" == "A"',
        '"A" == "A "',
        '"A" != "a"',
        '"A" > "a"',
        '"aBcd" < "abd"',
        '"A" != 64',
        '"A" < 64',
    ]
)
def relational_expression(expression):
    print(eval(expression))

Solution to Exercise 2 #

Checks whether an integer is equal to a floating point number.
Checks whether two characters are the same.
Checks whether two strings are the same. Note the space character.
Checks whether a character is larger than the order character according to their unicodes.
Checks whether a string is lexicographically smaller than the other string.
Checks whether a character is not equal to an integer.
TypeError because there is no implementation that evaluates whether a string is smaller than an integer.

Comparing Floating Point Numbers¶

== can also be used to compare floating point numbers:

x = 10
y = (x ** (1 / 3)) ** 3
x == y

False

Why False? Shouldn’t

(x^{\frac13})^3=x

Floating point numbers have finite precisions and so the computation $x^{\frac13}$ may not be exact.

In practice, a small error in the calculation can be tolerated. Instead of checking equality of floating point numbers, we can check closeness up to a specified tolerance:

abs_tol = 1e-9
y - abs_tol <= x <= y + abs_tol

True

abs_tol, often denoted as $\delta_{\text{abs}}$ , is a positive number called the absolute tolerance.

Why call it absolute tolerance?

Note that the test remains unchanged if we swap x and y:

abs_tol = 1e-9
x - abs_tol <= y <= x + abs_tol

True

Using the absolute function abs, we can also rewrite the comparison as follows:

abs_tol = 1e-9
abs(x - y) <= abs_tol

True

Is an absolute tolerance of 1e-9 good enough?

The same absolute tolerance fails if we set x = 1e10 instead of 10.

x = 1e10
y = (x ** (1 / 3)) ** 3

abs_tol = 1e-9
abs(x - y) <= abs_tol

False

Why absolute tolerance fail?

In floating-point representation, the radix point floats across different scales according to the exponent. The precision error in mantissa is therefore relative to the scale of the number.

We should instead check whether $x$ is within a certain percentage of $y$ :

rel_tol = 1e-9
y * (1 - rel_tol) <= x <= y * (1 + rel_tol)

True

Note that the above test is not symmetric between $x$ and $y$ , i.e., $x$ is relatively close to $y$ does not necessarily mean $y$ is relatively close to $x$ .

x = 1
y = 2
rel_tol = 0.5
y * (1 - rel_tol) <= x <= y * (1 + rel_tol), x * (1 - rel_tol) <= y <= x * (1 + rel_tol)

(True, False)

To make the comparison symmetric, we can use the absolute difference as follows:

Why use

\max\Set{\lvert x\rvert,\lvert y\rvert}

but not

\min\Set{\lvert x\rvert,\lvert y\rvert}

This is to make the test more tolerant, i.e.,

\begin{align} x&=y (1 \pm \delta_{\text{rel}}) && \text{ or } & y&=x (1 \pm \delta_{\text{rel}}) \end{align}

(5)

instead of requiring both.

x = 1e10
y = (x ** (1 / 3)) ** 3

rel_tol = 1e-9
### BEGIN SOLUTION
abs(x - y) <= rel_tol * max(abs(x), abs(y))
### END SOLUTION

True

What if $x$ or $y$ is very close to 0? For instance, can you change rel_tol so that your solution above returns True for the following values of x and y?

x = 1e-15
y = 0

When does relative tolerance fail?

If $x$ or $y$ is very close to 0, the relative tolerance test becomes ineffective, and so absolute tolerance is a more meaningful test. In particular, with $y=0$ , (4) becomes

\begin{align} \lvert x-0\rvert&\leq \delta_{\text{rel}} \max\Set{\lvert x\rvert, 0}\\ \lvert x\rvert&\leq \delta_{\text{rel}} \lvert x\rvert\\ 1 &\leq \delta_{\text{rel}} \quad\text{or}\quad x=0. \end{align}

(6)

In the nontrivial case $x\neq 0$ , the test $\delta_{\text{rel}}>1$ is independent of the value of $x$ , and therefore cannot check whether $x$ is close to 0. In particular, since $\delta_{\text{rel}}<1$ normally, and the test would always fail unless $x=0$ exactly, which defeats the purpose of allowing a tolerance.

For a more flexible comparison, one can impose both the absolute and relative tolerances:

This is implemented by the function isclose from math module with the default

relative tolerance $\delta_{\text{rel}}$ equal to 1e-9, and
absolute tolerance $\delta_{\text{abs}}$ equal to 0.0.

math.isclose?

True

Signature: math.isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)
Docstring:
Determine whether two floating-point numbers are close in value.

  rel_tol
    maximum difference for being considered "close", relative to the
    magnitude of the input values
  abs_tol
    maximum difference for being considered "close", regardless of the
    magnitude of the input values

Return True if a is close in value to b, and False otherwise.

For the values to be considered close, the difference between them
must be smaller than at least one of the tolerances.

-inf, inf and NaN behave similarly to the IEEE 754 Standard.  That
is, NaN is not close to anything, even itself.  inf and -inf are
only close to themselves.
Type:      builtin_function_or_method

def isclose(x, y, *, rel_tol=1e-09, abs_tol=0.0):
    ### BEGIN SOLUTION
    return x==y or abs(x - y) <= max(rel_tol * max(abs(x), abs(y)), abs_tol)
    ### END SOLUTION

x = 10
y = (x ** (1 / 3)) ** 3
assert isclose(x, y)
x = 1e10
y = (x ** (1 / 3)) ** 3
assert isclose(x, y)
assert not isclose(1e-15, 0)
assert isclose(float('inf'), float('inf'))
assert not isclose(float('nan'), float('nan'))

Conditional Constructs¶

To illustrate how Python can carry out conditional execution, we will consider writing a program that sorts values in ascending order.

If-Then Construct¶

How to sort two values?

Given two values are stored as x and y, we want to

print(x,y) if x <= y, and
print(y,x) if y < x.

Such a program flow is often represented by a flowchart like the following:

sort_two_values_fc1

How to read the flowchart?

A flowchart uses arrows to connects a set of annotated blocks. The rules were first specified by ANSI and later adopted in ISO 5807.

Why use a program flowchart?

A program flowchart is a powerful way of describing an algorithm quickly. Unlike a text-based programming language:

The rules governing the program flow can be shown explicitly by arrows.
The annotated graphical blocks can convey the meaning faster using visual clues.

How to implements the flowchart in Python?

It is often useful to delay detailed implementations until we have written an overall skeleton. To leave a block empty, Python uses the keyword pass.

# write a code skeleton
def sort_two_values(x, y):
    pass
    # print the smaller value first followed by the larger one


sort_two_values(1, 0)
sort_two_values(1, 2)

Python provides the if statement to implement the control flow specified by the diamond boxes in the flowchart.

def sort_two_values(x, y):
    if x <= y:
        pass
        # print x before y
    if y < x: pass  # print y before x


sort_two_values(1, 0)
sort_two_values(1, 2)

To complete the implementations specified by the parallelogram boxes in the flow chart, we fill in the bodies/suites of the if statements as follows:

def sort_two_values(x, y):
    if x <= y:
        print(x, y)
    if y < x: print(y, x)


@interact(x="1", y="0")
def sort_two_values_app(x, y):
    print("Values in ascending order:")
    sort_two_values(eval(x), eval(y))

Test the program by filling in different values of x and y above.

How to indent code in Python?

As Python uses indentation to indicate code blocks or suites:

print(x, y) (Line 3) is indented to the right of if x <= y: (Line 2) to indicate it is the body of the if statement.
For convenience, if y < x: print(y, x) (Line 4) is a one-liner for an if statement that only has one line in its block.
Both if statements (Line 2-4) are indented to the right of def sort_two_values(x,y): (Line 1) to indicate that they are part of the body of the function sort_two_values.

The style guide recommends using 4 spaces for each indentation. In JupyterLab, you can simply type the tab key.

We can visualize the execution as follows. Step through the execution to

see which lines are skipped, and
understand why they are skipped.

%%optlite -h 450
def sort_two_values(x, y):
    if x <= y:
        print(x, y)
    if y < x: print(y, x)


sort_two_values(1, 0)
sort_two_values(1, 2)

If-Then-Else Construct¶

Can the sorting algorithm be improved further?

Hint

not (x <= y) and (y < x) is a tautology, i.e., always true.

Consider the following modified flowchart:

sort_two_values_fc2

This can implemented by the else clause of the if statement as follows:

%%optlite -h 450
def sort_two_values(x, y):
    if x <= y:
        print(x, y)
    else:
        print(y, x)


sort_two_values(1, 0)
sort_two_values(1, 2)

Can we shorten the code to one line? This is possible with the conditional expression.

def sort_two_values(x, y):
    print(("{0} {1}" if x <= y else "{1} {0}").format(x, y))


@interact(x="1", y="0")
def sort_two_values_app(x, y):
    print("Values in ascending order:")
    sort_two_values(eval(x), eval(y))

Solution to Exercise 5

A conditional expression must be an expression:

It must give a value under all cases. To enforce that, else keyword must be provided.
An assignment statement does not return any value and therefore cannot be used for the conditional expression.
x = 1 if True else 0 is valid because x = is not part of the conditional expression.

Nested Conditionals¶

Now, consider a slight more challenging problem of sorting three values instead of two. A feasible algorithm is as follows:

sort_three_values_fc

To implement flowchart, we can use nested conditional constructs, where one conditional statement is placed within another, allowing for multiple layers of condition checks:

def sort_three_values(x, y, z):
    if x <= y <= z:
        print(x, y, z)
    else:
        if x <= z <= y:
            print(x, z, y)
        else:
            if y <= x <= z:
                print(y, x, z)
            else:
                if y <= z <= x:
                    print(y, z, x)
                else:
                    if z <= x <= y:
                        print(z, x, y)
                    else:
                        print(z, y, x)


def test_sort_three_values():
    sort_three_values(0, 1, 2)
    sort_three_values(0, 2, 1)
    sort_three_values(1, 0, 2)
    sort_three_values(1, 2, 0)
    sort_three_values(2, 0, 1)
    sort_three_values(2, 1, 0)


test_sort_three_values()

Imagine what would happen if we have to sort many values. The program will not only be long, but also fat due to the indentation. To avoid an excessively long line due to the indentation, Python provides the elif keyword that combines else and if.

def sort_three_values(x, y, z):
    if x <= y <= z:
        print(x, y, z)
    elif x <= z <= y:
        print(x, z, y)
    elif y <= x <= z:
        print(y, x, z)
    elif y <= z <= x:
        print(y, z, x)
    elif z <= x <= y:
        print(z, x, y)
    else:
        print(z, y, x)


test_sort_three_values()

Exercise 6

The above sorting algorithm is inefficient because some comparisons may be checked more than once. Improve the program by:

Eliminating redundunt checks so it can run fast.
Eliminate redundunt variables so it does not require too much memory.
Making it short and easy to read.

Hint

Do not use chained comparison operators or compound boolean expressions. E.g.,

1
2
3
4
5
6
7
8
9
10
11
12
13
def sort_three_values(x, y, z):
    if x <= y <= z:
        print(x, y, z)
    elif x <= z <= y:
        print(x, z, y)
    elif y <= x <= z:
        print(y, x, z)
    elif y <= z <= x:
        print(y, z, x)
    elif x <= y: # z <= x is redundant, why?
        print(z, x, y)
    else:
        print(z, y, x)

def sort_three_values(x, y, z):
    ### BEGIN SOLUTION
    # How many comparisons are needed in a program to sort n numbers?
    if x > y:
        x, y = y, x
    if y > z:
        y, z = z, y
    if x > y:
        x, y = y, x
    print(x, y, z)
    ### END SOLUTION


sort_three_values(10, 17, 14)

10 14 17

Sorting algorithms

A sorting algorithm refers to the procedure of sorting any number of values in order. The performance of a sorting algorithm is measured primarily by its time complexity (the number of comparisons and swaps it makes) and its space complexity (the amount of additional memory it requires). It would be even better if the program is short and easy to read, but that depends partly on the design of the programming language. Python implemented its default sorted function using the Timsort algorithm. See if you can understand how it works and how well it works.

%%ai
Explain in one paragraph what the minimum possible number of comparisons needed
to sort n numbers is and why it is not the same as the maximum number of 
inversions.

Boolean Operations¶

Since chained comparisons are non-associative, it follows a different evaluation rule than arithmetic operators.

E.g., 1 <= 2 < 3 != 4 is equivalent to:

1 <= 2 and 2 < 3 and 3 != 4

True

The above is called a compound boolean expression, which is formed using the boolean/logical operator and.

Why use boolean operators?

What if we want to check whether a number is either $< 0$ or $\geq 100$ ?

The following program checks whether a number is either $< 0$ or $\geq 100$ :

# Check if a number is outside a range.
@interact(x="15")
def check_out_of_range(x):
    x_ = float(x)
    is_out_of_range = x_ < 0 or x_ >= 100
    print("Out of range [0,100):", is_out_of_range)

Can you rewrite the logical expression to use only the and instead of or?

Turns out this is mission impossible because and alone is not functionally complete, i.e., it is not enough to give all possible boolean functions. We can add or and not operators to make it functionally complete.

%%ai
Explain in a paragraph why the set of operators including only "and" is not 
functionally complete.

%%ai
Explain in a paragraph why the set of operators "and", "or", and "not" is 
functionally complete.

The following table is called a truth table. It enumerates all possible input and output combinations for each boolean operator available in Python:

Table 2:Truth table for different logical operators

`x`	`y`	`x and y`	`x or y`	`not x`
`True`	`True`	`True`	`True`	`False`
`True`	`False`	`False`	`True`	`False`
`False`	`True`	`False`	`True`	`True`
`False`	`False`	`False`	`False`	`True`

Solution to Exercise 7

Expression A evaluates to True because and has higher precedence and so the expression has the same value as True or (False and True).
Expression B evaluates to False because and is left associative and so the expression has the same value as (True and False) and True.
Expression C evaluates to True because and has a higher precedence and so the expression has the same value as True or (True and False). Note that (True or True) and False evaluates to something False instead, so precedence matters.

A compound boolean expression actually uses a short-circuit evaluation.

To understand this, we will use the following function to evaluate a boolean expression verbosely.

def verbose(id, expr):
    """Identify evaluated boolean expressions."""
    print(id, "evaluated:", expr)
    return expr

For instance:

verbose("1st expression", True)
verbose("2nd expression", False)

1st expression evaluated: True
2nd expression evaluated: False

False

Visualization by Thonny IDE

You may also use the debugger of Thonny to visualize the step-by-step evaluations of the operands:

In JupyterLab, click File $\to$ New Launcher $\to$ Desktop to launch a remote desktop for your server.^[1]
Start Thonny in the remote desktop by clicking Applications $\to$ Development $\to$ Thonny.^[2]
In the Thonny editor, write some Python code such as True or False and True and then save it to a file, say test.py.^[3]
Debug the code by clicking the Thonny menu item Run $\to$ Debug current script (nicer).^[4]
Click the step into button to step through the executions.

Short-circuit `or`¶

Consider evaluating True or False and True in a verbose manner:

verbose("A", verbose(1, True) or verbose(2, False) and verbose(3, True))

1 evaluated: True
A evaluated: True

True

Why the second and third expressions are not evaluated?

Because True or ... must be True (why?) so Python does not look further. From the documentation:

Short-circuit evaluation of or

The expression x or y

first evaluates x;
if x is true, its value is returned;
otherwise, y is evaluated and the resulting value is returned.

Put it another way, (x or y) translate to the following

_ if (_ := x) else y

Program 1:Translation of (x or y)

except for the extra variable _. For example, True or False and True translates to _ if (_ := True) else False and True:

verbose("A", (_ if (_ := verbose(1, True)) else verbose(2, False) and verbose(3, True)))

1 evaluated: True
A evaluated: True

True

Short-circuit `and`¶

Now, consider evaluating False or False and True in a verbose manner:

False or False and True

False

verbose("B", verbose(4, False) or verbose(5, False) and verbose(6, True))

4 evaluated: False
5 evaluated: False
B evaluated: False

False

Why expression 6 is not evaluated?

False or False and ... must be False so Python does not look further.

Short-circuit evaluation of and

The expression x and y first evaluates x;
if x is false, its value is returned;
otherwise, y is evaluated and the resulting value is returned.

Put it another way, (x and y) translate to the following

_ if not (_ := x) else y

Program 2:Translation of (x and y)

except for the extra variable _. For example, False or False and True translates to False or (_ if not (_ := False) else True):

verbose(
    "B", verbose(4, False) or (_ if not (_ := verbose(5, False)) else verbose(6, True))
)

4 evaluated: False
5 evaluated: False
B evaluated: False

False

Non-Boolean Values for Boolean Operations and Control Flow Statements¶

Interestingly, Python also allows logical operations to have non-boolean operands and return values. From the documentation:

Boolean interpretation

In the context of Boolean operations, and also when expressions are used by control flow statements, the following values are interpreted as false:

False
None
Numeric zero of all types
Empty strings and containers (including strings, tuples, lists, dictionaries, sets and frozensets)

All other values are interpreted as true.

The following is an application of this rule.

print("You have entered", input() or "nothing")

You have entered nothing

Solution to Exercise 8 #

The code replaces an empty user input by the default string nothing because an empty string is regarded as False in a boolean operation.
If user input is non-empty, it is regarded as True in the boolean expression and returned immediately as the value of the boolean operation.

Exercise 9 (chain)

Let’s play a game to test your understanding of boolean expressions. Run the following program and choose your input so that 1, 2, 3 are each printed in a separate line as follows:

1
2
3

input(1) or input(2) and input(3)

Solution to Exercise 9

Enter ’ ', ‘’ and then ‘’.

### BEGIN SOLUTION
_ if (_ := input(1)) else (_ := input(3)) if (_ := input(2)) else _
### END SOLUTION

Is an empty string "" equal to False?

Try "" == False.

An empty string is regarded as false in a boolean operation but
a comparison operation is not a boolean operation, even though it forms a boolean expression.

Should the following return False?

0 == False and 1 == True

Try and explain what you see.

%%ai
In Python, explain in one paragraph what is the value of the following expression:
0 == False and 1 == True

Footnotes¶

The equality operator == consists of two equal signs, different from the assignment operator =.
↩
Different from C or javascript:
- We can write 1 != 2 as not 1 == 2 but not !(1 == 2) because
- ! is not a logical operator. It is used to call a system shell command in IPython.
↩
Thonny is an IDE based on TKinter, which requires a proper desktop environment to run.
↩
You can also launch a terminal in the remote desktop and run the command thonny.
↩
You can copy and paste code into the remote desktop by clicking Remote Clipboard button at the top right of the browser page and pasting your code into the textbox that appears below.
↩
Short-cut keys for Thonny may not work as the keys may be captured by the browser or your OS before they can reach Thonny.
↩

Lab 2

Symbolic Calculators

Conditional Execution

Abstract¶

Motivation¶

Comparison Operators¶

Equality and Inequalties¶

Comparing Floating Point Numbers¶

Conditional Constructs¶

If-Then Construct¶

If-Then-Else Construct¶

Nested Conditionals¶

Boolean Operations¶

Short-circuit or¶

Short-circuit and¶

Non-Boolean Values for Boolean Operations and Control Flow Statements¶

Short-circuit `or`¶

Short-circuit `and`¶