# Discussion 2: Environment Diagrams, Higher-Order Functions

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# Call Expressions

**Call expressions**, such as `square(2)`

, apply functions to arguments. When
executing call expressions, we create a new frame in our diagram to keep
track of local variables:

- Evaluate the operator, which should evaluate to a function.
- Evaluate the operands from left to right.
Draw a new frame, labelling it with the following:

- A unique index (
`f1`

,`f2`

,`f3`

, ...). - The
**intrinsic name**of the function, which is the name of the function object itself. For example, if the function object is`func square(x) [parent=Global]`

, the intrinsic name is`square`

. - The parent frame ([
`parent=Global`

]).

- A unique index (
- Bind the formal parameters to the argument values obtained in step 2 (e.g.
bind
`x`

to 3). - Evaluate the body of the function in this new frame until a return value is obtained. Write down the return value in the frame.

If a function does not have a return value, it implicitly returns `None`

. In that case,
the “Return value” box should contain `None`

.

**Note:**
Since we do not know how built-in functions like `min(...)`

or imported
functions like `add(...)`

are implemented, we do not draw a new frame when we
call them, since we would not be able to fill it out accurately.

### Q1: Call Diagram

Let’s put it all together! Draw an environment diagram for the following code. You may not have to use all of the blanks provided to you.

```
def double(x):
return x * 2
hmmm = double
wow = double(3)
hmmm(wow)
```

Return value |

Return value |

### Q2: Nested Calls Diagrams

Draw the environment diagram that results from executing the code below. You may not need to use all of the frames and blanks provided to you.

```
def f(x):
return x
def g(x, y):
if x(y):
return not y
return y
x = 3
x = g(f, x)
f = g(f, 0)
```

Return value |

Return value |

Return value |

Return value |

# Lambda Expressions

A lambda expression evaluates to a function, called a lambda function. For
example, `lambda y: x + y`

is a lambda expression, and can be read as "a
function that takes in one parameter `y`

and returns `x + y`

."

A lambda expression by itself evaluates to a function but does not bind it to a name. Also note that the return expression of this function is not evaluated until the lambda is called. This is similar to how defining a new function using a def statement does not execute the function’s body until it is later called.

```
>>> what = lambda x : x + 5
>>> what
<function <lambda> at 0xf3f490>
```

Unlike `def`

statements, lambda expressions can be used as an operator or an
operand to a call expression. This is because they are simply one-line
expressions that evaluate to functions. In the example below,
`(lambda y: y + 5)`

is the operator and `4`

is the operand.

```
>>> (lambda y: y + 5)(4)
9
>>> (lambda f, x: f(x))(lambda y: y + 1, 10)
11
```

### Q3: Lambda the Environment Diagram

Draw the environment diagram for the following code and predict what Python will output.

```
a = lambda x: x * 2 + 1
def b(b, x):
return b(x + a(x))
x = 3
x = b(a, x)
```

Return value |

Return value |

Return value |

# Higher Order Functions

A **higher order function** (HOF) is a function that manipulates other
functions by taking in functions as arguments, returning a function, or both.
For example, the function `compose`

below takes in two functions as arguments
and returns a function that is the composition of the two arguments.

```
def composer(func1, func2):
"""Return a function f, such that f(x) = func1(func2(x))."""
def f(x):
return func1(func2(x))
return f
```

HOFs are powerful abstraction tools that allow us to express certain general patterns as named concepts in our programs.

# HOFs in Environment Diagrams

An **environment diagram** keeps track of all the variables that
have been defined and the values they are bound to. However, values are not
necessarily only integers and strings. Environment diagrams can model more
complex programs that utilize higher order functions.

Lambdas are represented similarly to functions in environment diagrams, but since they lack instrinsic names, the lambda symbol (λ) is used instead.

The parent of any function (including lambdas) is always the frame in which
the function is defined. It is useful to include the parent in environment
diagrams in order to find variables that are not defined in the current
frame. In the previous example, when we call `add_two`

(which is really the
lambda function), we need to know what `x`

is in order to compute `x + y`

.
Since `x`

is not in the frame `f2`

, we look at the frame’s parent, which is
`f1`

. There, we find `x`

is bound to 2.

As illustrated above, higher order functions that return a function have their return value represented with a pointer to the function object.

### Q4: Make Adder

Draw the environment diagram for the following code:

```
n = 9
def make_adder(n):
return lambda k: k + n
add_ten = make_adder(n+1)
result = add_ten(n)
```

Return value |

Return value |

There are 3 frames total (including the Global frame). In addition, consider the following questions:

- In the Global frame, the name
`add_ten`

points to a function object. What is the intrinsic name of that function object, and what frame is its parent? - What name is frame
`f2`

labeled with (`add_ten`

or λ)? Which frame is the parent of`f2`

? - What value is the variable
`result`

bound to in the Global frame?

### Q5: Make Keeper

Write a function that takes in a number `n`

and returns a function
that can take in a single parameter `cond`

. When we pass in some condition
function `cond`

into this returned function, it will print out numbers from
1 to `n`

where calling `cond`

on that number returns `True`

.

# HOFs and Lambdas

### Q6: Make Your Own Lambdas

For each of the following expressions, write functions `f1`

, `f2`

,
`f3`

, and `f4`

such that the evaluation of each expression
succeeds, without causing an error. Be sure to use lambdas in your
function definition instead of nested `def`

statements. Each function
should have a one line solution.

# Currying

One important application of HOFs is converting a function that takes
multiple arguments into a chain of functions that each take a single
argument. This is known as **currying**. For example, the function below
converts the `pow`

function into its curried form:

```
>>> def curried_pow(x):
def h(y):
return pow(x, y)
return h
>>> curried_pow(2)(3)
8
```

This is useful if, say, you needed to calculate a lot of powers of 2.
Using the normal `pow`

function, you would have to put in `2`

as the first
argument for every function call:

```
>>> pow(2, 3)
8
>>> pow(2, 4)
16
>>> pow(2, 10)
1024
```

With `curried_pow`

, however, you can create a one-argument function specialized for taking powers of 2 one time, and then keep using that function for taking powers of 2:

```
>>> pow_2 = curried_pow(2)
>>> pow_2(3)
8
>>> pow_2(4)
16
>>> pow_2(10)
1024
```

This way, you don't have to put `2`

in as an argument for every call! If instead you wanted to take powers
of 3, you could quickly make a similar function specialized in taking powers of 3 using `curried_pow(3)`

.

Another point that will be relevant once you learn about sequences: Currying is also helpful in contexts where only one-argument functions are allowed, such as with the `map`

function. The `map`

applies a one-argument function to every term in a sequence. If we wanted to take the power of 2 for every number in a sequence using `map`

, we would be forced to use the one-argument function `pow_2`

(as defined in the example above) instead of using `pow`

with 2 as the first argument.

### Q7: Curry2 Diagram

Draw the environment diagram that results from executing the code below.

```
def curry2(h):
def f(x):
def g(y):
return h(x, y)
return g
return f
make_adder = curry2(lambda x, y: x + y)
add_three = make_adder(3)
add_four = make_adder(4)
five = add_three(2)
```

Return value |

Return value |

Return value |

Return value |

Return value |

# Extra Practice

This question is particuarly challenging, so it is recommended to attempt if you are feeling confident on the previous questions or are studying for the exam.

### Q8: Match Maker

Implement `match_k`

, which takes in an integer `k`

and returns a function
that takes in a variable `x`

and returns `True`

if all the digits in `x`

that
are `k`

apart are the same.

For example, `match_k(2)`

returns a one argument function that takes in `x`

and checks if digits that are 2 away in `x`

are the same.

`match_k(2)(1010)`

has the value of `x = 1010`

and digits 1, 0, 1, 0 going
from left to right. `1 == 1`

and `0 == 0`

, so the `match_k(2)(1010)`

results
in `True`

.

`match_k(2)(2010)`

has the value of `x = 2010`

and digits 2, 0, 1, 0 going
from left to right. `2 != 1`

and `0 == 0`

, so the `match_k(2)(2010)`

results
in `False`

.

**Important:** You may not use strings or indexing for this problem.
You do not have to use all the lines, one staff solution does not use the
line directly above the while loop.

**Hint:** Floor dividing by powers of 10 gets rid of the rightmost digits.