ChatGPT Tutoring Session: Haskell Monads
ChatGPT Logs
I talk to ChatGPT way too much these days. But some of my chats are incredibly instructive, and more helpful than related Google searches had been.
When I have a particularly good chat, I’ll post it on here. It’s easy content, and an easy way for me to track things I don’t want to forget.
Monads in Haskell
I’m sure we’ve all asked ourselves at some point: “What the fuck is a monad?”
I spent a brief amount of time trying to answer that question in the past, but never found explanations that worked for how my brain works. So now that I’m trying to learn some functional languages, I wanted to make sure I really had a good answer. The convo is below
The Logs
Yes, knowing either OCaml or Haskell will make learning the other easier. Here’s why:

Functional Programming Paradigm: Both OCaml and Haskell are primarily functional programming languages. If you’re familiar with one, you’ve already been exposed to core functional programming concepts such as immutability, firstclass functions, higherorder functions, pattern matching, and recursion. These concepts transfer directly from one language to the other.

Type Systems: Both languages have strong, static type systems with type inference. While there are differences in the specifics of the type systems (e.g., Haskell’s type classes vs. OCaml’s modules and functors), the general mindset of working with and thinking in terms of types will carry over.

Pattern Matching: Both languages heavily utilize pattern matching, a powerful tool for deconstructing and processing data structures. The syntax and usage are somewhat similar between the two languages.

Functional Data Structures: Lists, trees, and other functional data structures, as well as the associated standard library functions (like
map
,fold
,filter
), are common in both languages. 
Algebraic Data Types: Both OCaml and Haskell support algebraic data types, allowing you to define sum and product types. This is a foundational concept in both languages.
However, there are also significant differences between the two languages:

Purity and Laziness: Haskell is a purely functional language and uses lazy evaluation by default. This means that side effects are isolated in monads, and expressions are only evaluated when needed. OCaml, on the other hand, is strict (eager evaluation) and allows for imperative programming constructs.

Syntax: While both languages share some syntactic similarities due to their functional nature, they also have distinct syntaxes. For example, Haskell uses
=
for function and pattern definitions, while OCaml useslet ... = ... in ...
ormatch ... with ...
. 
Module Systems: Haskell uses type classes and modules in a way that’s different from OCaml’s module and functor system.

Tooling and Ecosystem: The tooling, libraries, and package management systems differ between the two languages.
In conclusion, while there are differences between OCaml and Haskell, a background in one will provide a solid foundation for learning the other. The functional programming mindset, in particular, is the most valuable transferable skill between the two.
you said “This means that side effects are isolated in monads, and expressions are only evaluated when needed.”
Can you explain this? What is a monad?
Certainly! Let’s break down the statement and then dive into what a monad is.

Side Effects: In programming, a side effect refers to any change in the system that is observable outside the called function. This can include modifying global variables, writing to files, reading from or writing to a database, sending data over the network, etc. In purely functional languages, functions don’t have side effects; they only compute and return values based on their inputs.

Lazy Evaluation: Haskell uses lazy evaluation, which means that expressions are not evaluated immediately when they’re defined. Instead, they’re evaluated only when their results are actually needed. This can lead to certain efficiencies, but it also means that the order of operations might not be as straightforward as in languages with eager evaluation.
Now, let’s discuss monads:
Monad:
A monad is a design pattern in Haskell (and other functional languages) that allows for the sequencing of operations. It’s a way to structure computations in terms of values and the operations that produce them. Monads become particularly useful when dealing with side effects in a pure functional setting.
At its core, a monad is a type class with specific laws and operations. The two primary operations are:
return
: Wraps a value into the monadic context.>>=
(bind): Chains operations together, passing the result of one operation as the input to another.
Here’s a simple analogy: Imagine you have a series of conveyor belts (operations) and boxes (values). Each conveyor belt can modify the box in some way. The monad is the mechanism that ensures the boxes move from one conveyor belt to the next in a structured manner.
Some common monads in Haskell:

Maybe Monad: Represents computations that might fail. It can have a value (
Just x
) or no value (Nothing
). 
List Monad: Represents computations that return multiple possible results.

IO Monad: This is where Haskell’s purity meets the real world. The IO monad allows for side effects (like reading from the keyboard or writing to a file) in a controlled manner. When you see a function with a type like
IO String
, it’s a computation that, when executed, will perform some IO and produce aString
.
The beauty of monads is that they allow Haskell to maintain its pure functional nature while still interacting with the outside world and handling side effects. Instead of executing side effects directly, Haskell functions produce a description of the side effects to be executed, encapsulated within a monad. The Haskell runtime system then executes these side effects.
Understanding monads can be challenging, and many developers find them to be one of the more complex aspects of Haskell. However, once grasped, they provide a powerful way to structure and sequence computations.
Maybe
monad just seems like Rust’s Option
, which is just a type?You’re right in observing that Haskell’s Maybe
monad is similar to Rust’s Option
type. Both can represent a value (Just
in Haskell, Some
in Rust) or the absence of a value (Nothing
in Haskell, None
in Rust). However, what makes Maybe
a monad in Haskell is not just its ability to represent a value or its absence, but the way it adheres to the monadic interface and laws.
A monad, in the context of Haskell and functional programming, is defined by:

Type Constructors: A way to take a basic type and produce a monadic type. For
Maybe
, this isJust
andNothing
. 
Monad Operations: Specifically, two primary operations:
return
: Takes a value and puts it into a monadic context. ForMaybe
, this is equivalent toJust
.>>=
(bind): A way to chain operations together. It takes a monadic value and a function that can transform the inner value into another monadic value.

Monad Laws: Any valid monad must satisfy three laws:
 Left Identity:
return a >>= f
is the same asf a
.  Right Identity:
m >>= return
is the same asm
.  Associativity:
(m >>= f) >>= g
is the same asm >>= (\x > f x >>= g)
.
 Left Identity:
For Maybe
to be a monad, it must provide implementations for return
and >>=
that satisfy these laws.
Here’s a brief example with Maybe
:


Rust’s Option
does have monadic behavior, especially when you consider methods like map
and and_then
, which are analogous to monadic operations. However, Rust doesn’t explicitly use the term “monad” or enforce the monadic laws in the same way Haskell does.
In summary, while many types across different languages exhibit monadic behavior, in Haskell, a monad is a specific design pattern with a defined interface (type constructors and operations) and laws. The power of monads in Haskell comes from this consistent interface, which allows for generic operations and abstractions across different monadic types.
Option
not satisfy?Rust’s Option
type exhibits behavior that aligns with the monadic interface, especially when considering methods like map
and and_then
. Let’s examine how Option
aligns with the monadic properties:

Type Constructors:
 Haskell’s
Maybe
:Just
andNothing
 Rust’s
Option
:Some
andNone
 Haskell’s

Monad Operations:

Haskell’s
return
: In the context ofMaybe
, this is equivalent toJust
. 
Rust’s equivalent: Directly using
Some
. 
Haskell’s
>>=
(bind): Chains operations. 
Rust’s equivalent:
and_then
method onOption
.


Monad Laws:

Left Identity:
return a >>= f
should be the same asf a
. In Rust:Some(a).and_then(f)
is equivalent tof(a)
. 
Right Identity:
m >>= return
should be the same asm
. In Rust:option.and_then(Some)
is equivalent tooption
. 
Associativity:
(m >>= f) >>= g
should be the same asm >>= (\x > f x >>= g)
. In Rust: This translates to chainingand_then
calls. If you have two functionsf
andg
that both transform anOption
, then chaining them withand_then
should be associative.

Given the above, Rust’s Option
does satisfy the monadic laws and exhibits monadic behavior. However, there are a few nuances:

Rust doesn’t explicitly use the term “monad” in its documentation or design. The language doesn’t enforce monadic laws or provide a generic monadic interface in the same way Haskell does.

The monadic behavior in Rust is more of a consequence of its design rather than an intentional adherence to the monadic pattern.

While
Option
in Rust andMaybe
in Haskell are conceptually similar and both exhibit monadic behavior, the surrounding ecosystem and language features make the experience of working with them different.
In summary, Rust’s Option
does align with the monadic properties, but the language doesn’t emphasize or require monadic design in the same way Haskell does.
Exactly! Rust’s Option
type behaves like a monad and satisfies the monadic laws as defined in Haskell. However, the Rust community and documentation don’t explicitly label or discuss it using the “monad” terminology.
The monadic behavior is a natural outcome of the design and methods provided for the Option
type, especially methods like map
and and_then
. Rustaceans benefit from the monadic properties when using these methods, even if they don’t refer to them as such.
In contrast, the Haskell community places a strong emphasis on the theoretical foundations and terminology, so monads are a central and explicitly discussed concept.
So, while both languages have types with monadic behavior, the cultural and pedagogical approaches to them differ.
Certainly! Haskell has several monads, but the following are arguably the five most commonly used:

Maybe Monad:
 Represents computations that might fail or produce no result.
 Data constructors:
Just a
(represents a value) andNothing
(represents the absence of a value).  Commonly used for functions that might not return a valid result for all inputs.

List Monad:
 Represents computations that return multiple possible results.
 Essentially, the list itself
[]
is the monad.  Commonly used for nondeterministic computations where there are multiple possible outcomes.

IO Monad:
 Represents sideeffecting computations.
 Type signature:
IO a
.  Used for operations that interact with the outside world, such as reading from or writing to the console, files, network, etc. It allows Haskell to maintain its pure functional nature while still being able to perform side effects in a controlled manner.

Either Monad:
 Represents computations that might result in an error.
 Data constructors:
Left a
(usually represents an error) andRight b
(usually represents a successful result).  Unlike
Maybe
, which just indicates the absence of a result,Either
can provide more information about an error.

State Monad:
 Represents computations that carry some state along.
 Type signature:
State s a
, wheres
is the type of the state anda
is the type of the result.  Used for functions that need to maintain and modify some state during their execution.
Each of these monads provides a way to handle specific kinds of effects or computational contexts in a pure functional setting. They come with their own set of operations and utilities that make working within those contexts more straightforward and more abstracted.
Certainly! The concept of “encapsulating side effects in monads” in Haskell is a foundational idea that allows the language to maintain its purely functional nature while still being able to interact with the outside world and perform computations that inherently seem “impure.” Let’s break this down:

Pure Functions: In a purely functional language, functions are expected to be pure. This means that for the same input, they always produce the same output and have no side effects. Side effects include things like modifying global state, reading/writing to files, interacting with databases, etc.

The Problem with Side Effects: If you were to directly introduce side effects into a purely functional language, you’d break the fundamental guarantee of referential transparency (i.e., you can replace a function call with its result without changing the program’s behavior). This would make reasoning about code, optimizing it, and testing it much more difficult.

Monads to the Rescue: Monads provide a solution by not actually executing side effects when you “write” them. Instead, they build up a computation that, when executed, will produce those side effects. This computation is represented as a value (the monadic value). This means you’re still working with values and functions that produce values, keeping everything pure.
For example, when you use the
IO
monad in Haskell, you’re not directly performing I/O. Instead, you’re building anIO
value that describes the I/O operations you want to perform. The actual I/O happens when the Haskell runtime system interprets thisIO
value. 
Sequencing with Monads: One of the challenges with side effects is the order in which they occur. Monads, with their
bind
operation (>>=
in Haskell), allow you to sequence operations. This ensures that effects occur in a specific order, even though the effects themselves are encapsulated. 
Isolation of Effects: Different monads encapsulate different kinds of effects. For instance, the
Maybe
monad encapsulates computations that might not return a value, while theState
monad encapsulates computations that carry some state. By using specific monads, you can be explicit about the kinds of effects a function might have.
In essence, monads allow Haskell to “simulate” impure operations in a pure environment. They provide a structured way to build up computations with effects without actually causing those effects until the very end. This approach retains the benefits of pure functional programming—like referential transparency and easier reasoning about code—while still allowing for practical, realworld operations.
Maybe
monad encapsulating computations that might not return a value  my question is, how is that a side effect? In my rust way of thinking, it’s all just a value either way. It’s just a specific variant of the Option (or Maybe in Haskell) type. Why does it need to be encapsulated?You’re right in pointing out that the Maybe
monad doesn’t encapsulate side effects in the same way that, say, the IO
monad does. The term “side effect” is perhaps a bit overloaded, and in the context of the Maybe
monad, it’s not about side effects in the traditional sense (like I/O operations). Instead, it’s about modeling the possibility of failure or absence of a value in computations.
Let’s clarify a few things:

Maybe and Option: Both Haskell’s
Maybe
and Rust’sOption
types are used to represent a value that might be absent. They’re not about side effects but about the potential absence of a result. In both languages, this is a pure operation. 
Why Call It a Monad?: In Haskell, the term “monad” isn’t just about side effects. It’s a broader concept that represents a certain kind of abstraction that can model computations. The
Maybe
monad models computations that can fail, and it provides a way to chain these computations together. The monadic aspect ofMaybe
comes into play when you want to perform a series of operations that might fail, and you want to shortcircuit the computation as soon as one of them does. Thebind
operation (>>=
) forMaybe
allows for this kind of chaining. 
Encapsulation in Haskell’s Context: When we talk about “encapsulation” in the context of Haskell’s monads, it’s about encapsulating a certain kind of computation or behavior. For
IO
, it’s about side effects. ForMaybe
, it’s about potential failure. ForList
, it’s about nondeterministic multiple results. The encapsulation is about providing a consistent interface (likereturn
andbind
) to work with these different kinds of computations. 
Rust’s Option: Rust’s
Option
provides methods likemap
andand_then
that allow for similar chaining of operations, much like the monadic operations in Haskell. However, Rust doesn’t label this as “monadic behavior” in its official documentation or discussions, even though it aligns with the concept.
In summary, you’re correct that the Maybe
type (or Option
in Rust) by itself is just a value representing the presence or absence of another value. The monadic behavior is about how you work with these types, especially when chaining operations. The term “encapsulation” in the context of monads in Haskell is broader than just side effects; it’s about encapsulating a pattern of computation.
Can you go into more detail about the IO monad, then, since it does encapsulate side effects? What exactly is it doing?
use putStrLn
as an example, since that is a simple case of IO
Certainly! The IO
monad in Haskell is a fascinating construct because it allows a purely functional language to interact with the outside world, which is inherently impure. Let’s delve into it using putStrLn
as an example.
The IO Monad

What is the IO Monad?
At a high level, you can think of values of typeIO a
as descriptions of computations that, when executed, will interact with the outside world and produce a value of typea
. These descriptions are pure values in Haskell, and they don’t do anything by themselves. The Haskell runtime system is responsible for actually executing them. 
Why is this Useful?
By representing side effects as values, Haskell can remain purely functional. You can build, combine, and transform theseIO
values using pure functions. The actual side effects only occur when the Haskell runtime system interprets these values.
The putStrLn
Function
Consider the function putStrLn
:



Function Signature:
The type signature tells us thatputStrLn
takes aString
and returns anIO
action that, when executed, will produce a value of type()
, which is the unit type in Haskell (similar tovoid
in some other languages). 
What does it do?
When you useputStrLn
in your code, you’re not immediately printing a string to the console. Instead, you’re creating a value that describes the action of printing a string. This is a crucial distinction. The action doesn’t happen when you create the value; it happens later when the runtime system decides to execute it. 
Chaining IO Actions:
Using the monadic interface (specifically, thebind
operation>>=
), you can chain multipleIO
actions together. For example:


In this do
notation (which is syntactic sugar for chaining monadic operations), multiple IO
actions are sequenced. The actual side effects (printing to the console, reading from the console) will occur in the order specified when the main
function is executed by the Haskell runtime.
How Does Haskell Remain Pure?
The genius of the IO
monad is that it allows Haskell to model impure operations (like reading from or writing to the console) without actually being impure. The functions themselves remain pure; they just produce values that describe impure actions. The language itself doesn’t perform the impure actions; the runtime system does.
This separation allows Haskell code to enjoy the benefits of purity (like referential transparency, easier reasoning, and potential for certain optimizations) while still being able to perform necessary side effects. The IO
monad essentially provides a controlled environment in which these side effects can be sequenced and combined in a predictable manner.
Yes, in a sense, the types of impure actions you can perform in Haskell are determined by the capabilities provided by the Haskell runtime system and its standard libraries. The IO
monad and associated functions define a set of operations that the runtime knows how to execute. When you create an IO
action in Haskell, you’re essentially creating a value that the runtime can interpret and execute according to its predefined semantics.
However, there are a few things to keep in mind:

Extensive Standard Library: Haskell’s standard library, including the
base
package, provides a wide range ofIO
operations that cover most common needs, such as file I/O, network I/O, concurrency, and more. These operations are sufficient for a vast majority of applications. 
FFI (Foreign Function Interface): If you need to perform actions not covered by the standard library or if you want to interface with libraries written in other languages (like C), Haskell provides a Foreign Function Interface (FFI). Using the FFI, you can call functions from other languages and have them return
IO
actions to Haskell. This greatly extends the range of impure operations you can perform, as you can leverage the capabilities of other languages and libraries. 
Extensibility: Over time, as needs evolve, the Haskell ecosystem can (and does) extend its libraries, including those related to
IO
. New libraries can be created to support new kinds of operations or to interface with new external systems. 
Purity and Safety: While the set of
IO
operations is extensive, Haskell’s design ensures that these operations are clearly marked as impure by their types. This makes it evident when a function has side effects and when it doesn’t, leading to safer and more predictable code.
In summary, while the Haskell runtime and its libraries define the set of impure operations you can perform directly in Haskell, the language provides mechanisms (like the FFI) to extend these capabilities and interface with external systems. This design strikes a balance between purity and practicality, allowing developers to write realworld applications while benefiting from the advantages of functional programming.
do
syntax sugar you mentioned? I’m not yet familiar with that syntax as I just started learningOf course! The do
notation is syntactic sugar that makes monadic code more readable by giving it a more imperative feel. However, under the hood, it’s all translated to uses of the monadic bind operation (>>=
) and the return
function.
Here’s the do
notation version again for reference:


Without the do
notation, the above code would look like this:


Here’s a breakdown of the translation:

The Bind Operation (
>>=
): This operation takes anIO
action on its left and a function on its right. The function takes the result of theIO
action and returns anotherIO
action. The bind operation sequences these actions, executing the left one first and then the right one. 
Lambda Functions (
\... > ...
): These are anonymous functions in Haskell. In the desugared version, we use lambda functions to capture the results ofIO
actions and use them in subsequent actions. 
The Sequence Operation (
>>
): This is a special case of bind where the result of the left action is ignored. It’s equivalent toa >>= \_ > b
. In the example, we use it afterputStrLn
since we don’t care about the result of printing to the console (which is()
).
The do
notation is essentially a more readable way to write this chain of bind operations, especially when there are many sequential steps. As you delve deeper into Haskell, you’ll likely find the do
notation more intuitive for writing monadic code, but it’s beneficial to understand the underlying operations it represents.
Certainly! For something to be considered a monad in Haskell, it must satisfy the following prerequisites:

Type Constructors: A way to take a basic type and produce a monadic type. For example, with the
Maybe
monad, the type constructors areJust
andNothing
. 
Monad Operations: There are two primary operations that a monad must define:
return
: This operation takes a value and puts it into a monadic context. For theMaybe
monad,return
is equivalent toJust
.>>=
(bind): This operation is used to chain or sequence monadic operations. It takes a monadic value and a function that can transform the inner value into another monadic value.

Monad Laws: Any valid monad must satisfy three fundamental laws. These laws ensure consistent behavior across different monads and allow for generic operations and reasoning:
 Left Identity: For any value
a
and monadic functionf
,return a >>= f
should be the same asf a
.  Right Identity: For any monadic value
m
,m >>= return
should be the same asm
.  Associativity: For any monadic value
m
and monadic functionsf
andg
,(m >>= f) >>= g
should be the same asm >>= (\x > f x >>= g)
.
 Left Identity: For any value
These prerequisites ensure that monads provide a consistent interface and behavior, allowing for a unified way to work with different types of computations in Haskell. The monad laws, in particular, are crucial for reasoning about monadic code and for certain compiler optimizations.
why is it called return
? I guess the verbiage is what’s confusing me (it often does when learning new languages / styles). Just x
putting x
into the monad makes sense, but I don’t get why this is called return
, as return
in all other languages I know means.. you know.. returning something from a function
>>=
(bind) is just a method on the (monad) type that takes an instance of the monad and returns another one? Can you give any simple example of something that is almost
a monad but fails to have a bind operation? It seems like it would be trivial to implement for anything I can think of
You’ve touched on some points that often trip up newcomers to Haskell and functional programming, especially those coming from imperative backgrounds.

Why is it called
return
?
The namereturn
in the context of monads in Haskell is indeed a bit misleading, especially for those familiar with imperative languages. In Haskell’s monadic context,return
doesn’t mean “exit a function and give back a value” as it does in many other languages. Instead, it means “take a value and lift it into a monadic context.” The naming is historical and, in hindsight, might not have been the best choice given its potential for confusion. 
What is
>>=
(bind)?
Yes,>>=
(often called “bind”) is a method that takes an instance of a monad and a function that transforms the inner value of that monad into another monadic value. It’s a way to chain or sequence monadic computations. 
Example of something that’s
almost
a monad:
Let’s consider a hypothetical type calledAlmostMonad
that represents optional values, similar toMaybe
. We’ll define areturn
like function for it, but we’ll intentionally make the bind operation (>>=
) nonassociative, violating the monad laws.1 2 3 4 5 6 7 8 9 10 11
data AlmostMonad a = SomeValue a  NoValue ourReturn :: a > AlmostMonad a ourReturn = SomeValue ourBind :: AlmostMonad a > (a > AlmostMonad b) > AlmostMonad b ourBind NoValue _ = NoValue ourBind (SomeValue x) f = case f x of SomeValue y > if y == "specialValue" then NoValue else SomeValue y NoValue > NoValue
Here,
ourBind
behaves differently for a “specialValue” string, making it nonassociative. ThisAlmostMonad
fails to be a true monad because it doesn’t satisfy the associativity law of monads.While it’s easy to define the bind operation for many structures, ensuring that the monad laws hold is the tricky part. The laws are crucial for reasoning about monadic code and ensuring consistent behavior.
In summary, the terminology in Haskell’s monadic context can indeed be confusing, especially when compared to similar terms in imperative languages. However, once you get past the initial terminology hurdle and start thinking in terms of the underlying concepts, monads become a powerful and consistent way to work with various computational contexts in Haskell.
Exactly! The challenge with monads isn’t so much defining the bind
operation (or method, as you’re referring to it) in terms of its function signature or basic behavior. Instead, the difficulty lies in ensuring that the behavior of bind
(and return
) satisfies the monad laws. These laws are what give monads their consistent and predictable behavior across different computational contexts.
The monad laws ensure that you can reason about monadic code in a generic way, regardless of the specific monad you’re working with. They also enable certain compiler optimizations and transformations that rely on the predictable behavior guaranteed by the laws.
So, while many structures can have a bind
like operation, not all of them will satisfy the monad laws, and thus, not all of them can be considered true monads in the Haskell sense.
Option
/ Maybe
now, and my language server’s Goto Definition
features doesn’t seem to be working for haskell right now :)Certainly! Let’s explore the List
monad, which is another common monad in Haskell. The List
monad represents nondeterministic computations, where a computation can have multiple possible results.
List Monad
In Haskell, lists themselves are monads. Here’s how the monad operations work for lists:

return: This takes a value and puts it into a list context. For lists, this means creating a singleelement list.
1
return x = [x]

bind (»=): For lists, the bind operation can be thought of as a “flattening map.” It maps a function over the list and then flattens the result.
1
xs >>= f = concat (map f xs)
Left Identity Law with Lists
The left identity law states:


Let’s see this in action with the List
monad:

Using return and bind:
1
return 3 >>= (\x > [x, x+1, x+2])
Breaking it down:
return 3
produces the list[3]
. The function
(\x > [x, x+1, x+2])
takes an integer and produces a list of that integer and the next two integers.  Using bind, we map this function over the list
[3]
and then flatten the result.
The result is
[3, 4, 5]
. 
Directly applying the function:
1
(\x > [x, x+1, x+2]) 3
This directly applies the function to the value
3
, producing the list[3, 4, 5]
.
As you can see, both approaches produce the same result, [3, 4, 5]
, demonstrating that the left identity law holds for the List
monad.
The List
monad provides a different perspective on monadic behavior compared to the Maybe
monad. While Maybe
models computations that can fail, List
models computations with multiple possible outcomes. The monad laws ensure that, despite these different semantics, you can reason about and work with these monads in a consistent way.
List
’s bind operator to make it not satisfy the left identity?The left identity law for monads states that for any value a
and monadic function f
:


To make the List
monad’s bind operation (>>=
) violate the left identity law, we can introduce a behavior that changes the result of return a >>= f
so that it’s not equivalent to f a
.
Here’s a simple alteration to the bind operation for lists that violates the left identity:


In this altered version, if the list xs
is not empty, we append its first element to the result. This means that when we use return
(which produces a singleelement list) with this altered bind, the result will always have the original value appended, making it different from just applying f
to the value.
Let’s see this in action:

Using our altered bind with return:
1
return 3 `alteredBind` (\x > [x, x+1, x+2])
This produces the list
[3, 4, 5, 3]
. 
Directly applying the function:
1
(\x > [x, x+1, x+2]) 3
This produces the list
[3, 4, 5]
.
Clearly, the two results are not the same, so our altered bind operation violates the left identity law for monads.
This example demonstrates how subtle changes in the behavior of monadic operations can lead to violations of the monad laws, emphasizing the importance of these laws in ensuring consistent and predictable behavior for monads.
I asked for a bad example, this feels very similar to the Maybe
monad.
How does this work for the IO
monad? I can’t really picture what the monadic value
even looks like for it. What’s actually happening?
The IO
monad is indeed a special case in Haskell and is somewhat more abstract than other monads like Maybe
or List
. The IO
monad represents sideeffecting computations, and its inner workings are deliberately obscured to maintain the purity of the language. Let’s delve into it.
The IO
Monad

What’s Inside an
IO
Value?
The exact nature of anIO
value is not exposed to the programmer. You can think of anIO a
value as a “recipe” or a “description” of a computation that, when executed, will perform some side effects and produce a value of typea
. The specifics of this “recipe” are hidden, and Haskell provides no way to “peek inside” or directly manipulate it. 
How Do We Work with
IO
Values?
Even though we can’t see inside anIO
value, we can still transform it, chain it with otherIO
values, and produce newIO
values. This is done using the monadic operations (return
and>>=
) and other provided functions. 
Execution of
IO
Values:
The actual execution ofIO
actions is handled by the Haskell runtime system. When you run a Haskell program, the main action (of typeIO ()
) is executed, and anyIO
actions it depends on are executed in turn.
Left Identity with IO
The left identity law for the IO
monad would look like this:


For the IO
monad:

return:
Thereturn
function forIO
creates anIO
action that does nothing and produces the given value. 
bind (»=):
The bind operation forIO
sequences twoIO
actions. It first executes the action on the left, then takes its result and feeds it into the function on the right to produce a newIO
action, which it then executes.
Given the abstract nature of IO
, it’s hard to provide a concrete example of violating the left identity without delving into the internals of the Haskell runtime or making hypothetical changes to the language itself. The IO
monad is designed to always satisfy the monad laws, and any violation would break many assumptions and guarantees provided by the language.
What’s Actually Happening with IO
?
When you create an IO
action in Haskell, you’re not causing side effects right then and there. Instead, you’re building up a “recipe” of actions to be performed. These actions are only actually executed when the Haskell runtime system interprets the main IO
action of your program.
This design allows Haskell to maintain its pure functional nature while still being able to interact with the outside world. The IO
monad provides a controlled environment in which side effects can be sequenced and combined in a predictable manner, without exposing the underlying details or allowing for arbitrary side effects throughout the code.
Ok. Maybe I’m getting too caught up in stuff that’s handled by the runtime then.
In the List
monad example, what is the inner type a
.
My first thought was that since Lists are singly linked lists, a
is really just the first node of the list
But then I’m confused, because does the function f
that takes a
and returns List b
supposed to operate just on the value of the first node a
or follow the nodes to the end?
The fact that bind
was given as concat (map f xs)
means that return a >>= f
will transform the whole list, but if our bind
is doing the concat
and map
, wouldn’t f
also have to operate on all the elements of the list? but surely it can’t ifbind
is doing it?
Your questions touch on the heart of how the List
monad works. Let’s break it down step by step.
The Inner Type a
in the List
Monad
In the context of the List
monad, the inner type a
refers to the type of elements contained within the list. For a list of integers ([Int]
in Haskell), a
would be Int
. For a list of strings ([String]
), a
would be String
, and so on.
The Function f
in the Bind Operation
The function f
that you provide to the bind operation (>>=
) should take a single value of type a
and produce a list of type b
. It operates on individual elements of the list, not on entire lists.
When you use the bind operation with a list, the bind operation will apply f
to each element of the list in turn. This is why the implementation of bind for lists involves map
, which applies a function to each element of a list.
How Bind Works with Lists
Let’s break down the bind operation for lists:



Mapping with
f
:
Themap f xs
part applies the functionf
to each element of the listxs
. Sincef
produces a list for each element, the result ofmap f xs
is a list of lists. 
Concatenation:
The outerconcat
then flattens this list of lists into a single list.
Here’s an example to illustrate:
Suppose we have a list of integers and a function that produces lists:


Using bind:


This is equivalent to:


Which becomes:


Resulting in:


In this example, the function f
operates on individual elements of the list xs
, and the bind operation sequences these operations and combines the results into a single list.
In summary, in the context of the List
monad, the function f
in the bind operation works on individual elements, and the bind operation itself takes care of applying f
to each element and combining the results.
Ok, I think this is starting to make sense.
For the Maybe
monad, we’re *encapsulating the possibility of failure. There are thus two distinct variants, Just or Nothing, where Just a
says that the operation was successful, and we have a value of type a.
List
on the other hand, doesn’t have multiple variants, it’s just a list. It can have 0 or more items in it, of type a. The thing you’re encapsulating is not knowing how many of type a. So f a
is a function that takes a single one of type a
, and the monad is there to deal with the fact that you have no idea how many.
Ok I feel like I basically just repeated what you said back to you but for whatever reason it actually means something to me now. Something clicked I guess
Exactly! You’ve captured the essence of it. Monads in Haskell are a way to encapsulate and manage different kinds of computational contexts:

For the
Maybe
monad, the context is the possibility of failure. It can either have a value (Just a
) or represent the absence of a value (Nothing
). 
For the
List
monad, the context is nondeterminism or multiple possibilities. A function that returns a list can have many potential results, and theList
monad provides a way to work with these multiple outcomes in a structured manner.
Your understanding is spot on. Each monad encapsulates a specific kind of computational effect or context, and the monadic operations (return
and >>=
) provide a consistent interface to work within that context. The power of monads in Haskell is that they allow you to work with these different contexts using a unified set of principles, making it easier to reason about and combine different kinds of computations.
And don’t worry about repeating things back; sometimes, rephrasing or restating a concept in your own words is a great way to solidify understanding. I’m glad it clicked for you!
The right identity is much more intuitive I think. It’s just.. if I take the inner value of my monadic value and throw it back into the same monad, I have the same monadic value.
in rust, Some(Some(x).unwrap())
is the same thing all the time
Exactly! The right identity law for monads captures the idea that if you take a monadic value, extract its inner value, and then immediately put it back into the monadic context using return
, you should end up with the original monadic value.
In Haskell notation, the right identity law is:


Using your Rust example with the Option
type:
Some(x).unwrap()
extracts the inner valuex
from theSome
variant.Some(...)
then wraps it back into theOption
context.
So, Some(Some(x).unwrap())
is indeed equivalent to Some(x)
, which aligns with the right identity law.
This law ensures that the return
function for a monad behaves as expected: it simply wraps a value in the monadic context without introducing any additional effects or behaviors. It’s a fundamental property that ensures the consistency and predictability of monadic operations.
for associativity, again, i get what it is, but i’m struggling to think of an example again
Using the List
monad, can you give me an example of a way you could break associativity while maintaining the validity of the left and right identity
Certainly! Associativity for monads ensures that when chaining operations together, the order in which you group them doesn’t matter. In Haskell notation, the associativity law is:


To break associativity for the List
monad while maintaining left and right identity, we’d need to introduce some inconsistency in the way we chain operations together.
Here’s a contrived example:
Let’s say we have a modified bind operation for lists (brokenBind
) that behaves normally most of the time, but if it encounters the number 42
in the list, it reverses the result of the next bind operation:


Now, let’s consider two functions:


Using our brokenBind
:

Grouping operations one way:
1
([41, 42] `brokenBind` f) `brokenBind` g
This first applies
f
to each element of[41, 42]
, producing[41, 42, 42, 43]
. Since42
is in the list, the result of the next bind operation (withg
) will be reversed. So, the final result is[42, 41, 43, 42, 44, 43]
. 
Grouping operations the other way:
1
[41, 42] `brokenBind` (\x > f x `brokenBind` g)
This chains the operations differently. For each element in
[41, 42]
, it appliesf
and theng
. The presence of42
will reverse the result of the bind withg
for each element. The final result is[41, 40, 42, 41, 42, 41, 43, 42]
.
Clearly, the two results are different, so our brokenBind
violates the associativity law. However, it still satisfies the left and right identity laws, as the special behavior around 42
doesn’t affect those properties.
This example is contrived and wouldn’t make sense in a realworld scenario, but it illustrates how associativity can be broken while other properties remain intact.
That’s a good observation, and in general, it’s a useful guideline. The bind operation (>>=
) should be consistent in its behavior regardless of the specific content of the monadic value. It should depend only on the structure or context that the monad represents, not on the specific inner values.
However, there are a few nuances:

Consistency with Structure: While the bind operation should be consistent, it can and often does depend on the structure or “shape” of the monadic value. For instance, in the
Maybe
monad, the behavior of bind is different forJust
andNothing
values. Similarly, for theList
monad, an empty list might be treated differently than a nonempty list. 
Avoiding Arbitrary Behavior: The bind operation shouldn’t introduce arbitrary or unexpected behaviors based on the specific inner values of the monad. In our previous example with the
brokenBind
operation, the arbitrary behavior introduced when encountering the number42
is what broke associativity. 
State Monad: There’s a specific monad called the
State
monad where the bind operation does depend on some internal state. However, this is a controlled and expected behavior that’s central to what theState
monad represents. The key is that even in theState
monad, the behavior of bind is consistent and predictable, and it satisfies the monad laws.
In summary, while the bind operation can depend on the structure or context that the monad represents, it should avoid arbitrary behaviors based on specific inner values. Doing so ensures that the monad remains consistent, predictable, and adheres to the monad laws.
that’s a good point  I overgeneralized.
Is it closer, then, to say that the bind operation shouldn’t depend on the state of the specific inner values of type a
, but only the variant (e.g. Just
/ Nothing
) or structure (e.g. List
length)? Does that guideline still break for the State
monad?
Your refined statement is closer to the mark. In many monads, the behavior of the bind operation depends on the structure or variant of the monadic value but not on the specific details of the inner value of type a
. This ensures that the behavior is consistent and predictable across different inner values.
However, the State
monad is a special case, and it does introduce a nuance to this guideline.
The State
Monad
The State
monad is used to model computations that carry along some mutable state. A value of the State
monad represents a function that takes a state and returns a value along with a new state.
The type of a State
monad value is:


Where s
is the type of the state and a
is the type of the value being produced.
In the State
monad:
 The
return
function produces a computation that returns the given value without changing the state.  The bind operation (
>>=
) sequences two stateful computations, passing the state from one to the next.
Because the State
monad inherently deals with state transformation, the bind operation does depend on the specific state value (of type s
). However, it doesn’t arbitrarily change behavior based on the state; it consistently passes the state from one computation to the next.
Conclusion
Your refined guideline holds for many common monads: the bind operation should depend on the structure or variant of the monadic value but not on the specific inner value of type a
.
However, the State
monad is an exception in that its bind operation does depend on the specific state value. But this behavior is consistent, predictable, and central to the purpose of the State
monad. It’s not an arbitrary dependency but a fundamental aspect of stateful computations.
" the bind operation does depend on the specific state value (of type s). However, it doesn’t arbitrarily change behavior based on the state; it consistently passes the state from one computation to the next."
Can I say then that the bind operation shouldn’t ever change it’s behavior based on specific inner values of type a
, but only the variant (e.g. Just
/ Nothing
) or structure (e.g. List
length)?
Yes, that’s a more accurate and generalizable statement. The bind operation, for most monads, should not change its behavior based on the specific inner values of type a
. Instead, its behavior can depend on the variant or structure of the monadic value.
For the State
monad, the behavior of the bind operation depends on the state (of type s
), but this is consistent with its purpose of chaining stateful computations. The specific value within the monad (of type a
) doesn’t dictate the behavior of the bind operation.
Your refined statement captures the essence of how the bind operation should behave across different monads. It ensures that the monadic operations remain consistent and predictable, which is crucial for reasoning about and working with monads effectively.