Stateful computations in F# with update monads
Most discussions about monads, even in F#, start by looking at the wellknown standard monads for handling state from Haskell. The reader monad gives us a way to propagate some readonly state, the writer monad makes it possible to (imperatively) produce values such as logs and the state monad encapsulates state that can be read and changed.
These are no doubt useful in Haskell, but I never found them as important for F#.
The first reason is that F# supports state and mutation and often it is just easier
to use a mutable state. The second reason is that you have to implement three
different computation builders for them. This does not fit very well with the F# style
where you need to name the computation explicitly, e.g. by writing async { ... }
(See also my recent blog about the F# Computation Zoo paper).
When visiting the Tallinn university in December (thanks to James Chapman, Juhan Ernits & Tarmo Uustalu for hosting me!), I discovered the work on update monads by Danel Ahman and Tarmo Uustalu, which elegantly unifies reader, writer and state monads using a single abstraction.
In this article, I implement the idea of update monads in F#. Update monads are
parameterized by acts that capture the operations that can be done on the state.
This means that we can define just a single computation expression update { ... }
and use it for writing computations of all three aforementioned kinds.
Introducing update monads
Before looking at the definition of update monads, it is useful to review the three monads that we want to unify. The update monads paper has more details and you can also find other tutorials that implement these in F#  here, we'll only look at the type definitions:
1: 2: 3: 4: 5: 6: 

If you look at the definitions, it looks like reader and writer are both a versions of the state with some aspect missing  reader does not produce a new state and writer does not take previous state.
How can we define a parameterized computation type that allows leaving one or the other out? The idea of update monads is quite simple. The trick is that we'll take two different types  one representing the state we can read and another representing the updates that specify how to change the state:
1: 2: 3: 4: 

To make the F# implementation a bit nicer, this is not defined as a type alias, but as a
new type labeled with UM
(this makes sure that the inferred types will always use the
name UpdateMonad
for the type, rather than its definition).
To make this work, we also need some operations on the types representing states and updates. In particular, we need:
 Unit update which represents that no update is applied.
 Composition on updates that allows combining multiple updates on the state into a single update.
 Application that takes a state and an update and applies the update on the state, producing new state as the result.
In more formal terms, the type of updates needs to be a monoid (with unit and composition). In the paper, the two types (sets) together with the operations are called act and are defined as \((S, (P,\circ,\oplus), \downarrow)\) where \(\circ\) is the unit, \(\oplus\) is composition and \(\downarrow\) is the application.
Note on naming
In the last case, I'm using different naming than the original paper. In the paper, applying update \(u\) to a state \(s\) is written as \(s \downarrow u\). You can see the "\(\downarrow u\)" part as an action that transforms state and so the authors use the name action. I'm going to use apply instead because I refer more to the operation \(\downarrow\) than to the entire (partiallyapplied action).
Implementing update monads in F#
To implement this idea in F#, we can use static member constraints. If you do not
know about static member constrains in F#, you can think of it as a form of duck typing
(or lightweight type classes). If a type defines certain members, then the code will
compile and run fine, otherwise you'll get a compiletime error. We will require
the user to define two types representing State
and Update
, respectively. The Update
type will need to define the three operations. An abstract definition (not valid F# code)
would look like this:
type State type Update = static Unit : Update static Combine : Update * Update > Update static Apply : State * Update > State
Invocation of members via static member constraints is not entirely easy (the feature is
used mainly by library implementors for things like generic numerical code). But the idea
is that we can define inline functions (unit
, ++
and apply
) that invoke the
corresponding operation on the type (specified either explicitly or via type inference).
If you're not familiar with F#, you can freely skip over this definition, just remember
that we now have functions unit
and apply
and an operator ++
:
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The last thing that we need to do before we can start playing with some
update monads is to implement the monadic operators. In F#, this is done by
defining a computation builder  a type that has Bind
and Return
operations (as well as some others that we'll see later). The compiler then
automatically translates a block update { .. }
using operations update.Return
and update.Bind
.
The computation builder is a normal class with members. Because we are using
static member constraints and inline functions, we need to mark the members
as inline
too:
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The implementation of the Return
operation is quite simple  we return
the specified value and call unit()
to get the unit of the monoid of
updates  as a result, we get a computation that returns the value without
performing any update on the state.
The Bind
member is more interesting  it runs the first computation which
returns a value x
and an update u1
. The second computation needs to be
run in an updated state and so we use apply s u1
to calculate a new state
that reflects the update. After running the second computation, we get the
final resulting value y
and a second update u2
. The result of the
computation combines the two updates using u1 ++ u2
.
How does this actually work? Let's start by looking at reader and writer monads (which are special cases of the update monad.
Implementing the reader monad
The reader monad keeps some state, but it does not give us any way of modifying it.
In terms of update monads, this means that there is some state, but the monoid of
updates is trivial  in principle, we can just use unit
as the type of updates.
You can see that when looking at the type too  the type of reader monad is
'TState > 'T
. To get a type with a structure matching to update monads, we
can use an equivalent type 'TState > unit * 'T
.
Reader state and update
In practice, we still need to define a type for updates, so that we can provide the
required static members. We use a singlecase discriminated union with just one value
NoUpdate
:
1: 2: 3: 4: 5: 6: 7: 8: 

None of the operations on the ReaderUpdate
type does anything interesting.
Both unit and combine simply returns the only possible value and the apply
operation returns the state without a change.
Reader monad primitives
Next, we need a primitive that allows us to read the state and a run operation that executes a computation implemented using the reader monad (given the value of the readonly state). The operations look as follows:
1: 2: 3: 4: 5: 

When you look at the type of computations (hover the mouse pointer over the
read
identifier), you can see a parameterized update monad type. The read
primitive has a type UpdateMonad<ReaderState, ReaderUpdate, ReaderState>
. This
means that we have an update monad that uses ReaderState
and ReaderUpdate
as
the act (specifying the computation details) and, when executed, produces a
value of ReaderState
.
Sample reader computations
Now we can use the update { .. }
block together with the read
primitive
to write computations that can read an immutable state. The following
basic example reads the state and adds one (in demo1
), and then adds
1 again in demo2
:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 

If you run the code, you'll see that the result is 42. The interesting thing
about this approach is that we only had to define two types. The update { .. }
computation works for all update monads and so we get the computation builder
"for free". However, thanks to the parameterization, the computation really represents
an immutable state  there is no way to mutate it.
Implementing the writer monad
Similarly to the reader monad, the writer monad is just a simple special case of the
update monad. This time, the state is trivial and all the interesting things are
happening in the updates. The type of the usual writer monad is 'TState * 'T
and
so if we want to make this a special case of update monad, we can define the type
as unit > 'TState * 'T
.
Writer state and update
The state needs to be a monoid (with unit and composition) so that we can compose
the states of multiple subcomputations. The following example uses a list as a
concrete example. We define a (writer) monad that keeps a list of 'TLog
values
and returns that as the result (more generally, we could use an arbitrary monoid
instead of a list):
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 

The writer monad appears (in some informal sense) dual to the earlier reader monad.
The state (that can be read) is always empty and is represented by the NoState
value,
while all the interesting aspects are captured by the WriterUpdate
type  which is a
list of values produced by the computation. The updates of a writer monad have to form
a monoid  here, we use a list that concatenates all logged values. You could easily
change the definition to implement other monoids (e.g. to keep the last produced value).
Writer monad primitives
Similarly to the previous example, we now need two primitives  one to add a new element
to the log (write
of the writer monad) and one to run a computation and extract the
result and the log:
1: 2: 3: 4: 5: 

The write
function creates a singleton list containing the specified value Log [v]
as
the update and returns the unit value as the result of the computation. When combined with
other computations, the updates are concatenated and so this will become a part of the
list l
in the result (Log l, v)
that is made accessible in the writerRun
function.
Sample writer computations
Let's have a look at a sample computation using the new definitions  the remarkable thing
(from the practical F# programming perspective) is that we wrap the computation
in the update { .. }
block just like in the previous example. But this time, we'll use
the write
primitive to write 20 and then 10 to the log and the F# compiler correctly
infers that we are using WriterState
and WriterUpdate
types:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 

If you run the code, the demo3
computation first writes 20 to the log,
which is then combined (using the ++
operator that invokes WriterUpdate.Combine
)
with the value 10 written in demo4
.
Building richer computations
One of the key things about F# computation expressions that I emphasized in my
previous blog post and in the PADL 2014 paper is that computation
expressions provide rich syntax that includes resource management (the use
keyword),
exception handling or loops (for
and while
)  in simple words, it mirrors the
normal syntax of F#.
So far, we have not used any of these for update monads. All these additional constructs have to be provided in the computation builder (so that the author can define them in the most suitable way). The great thing about update monads (for F#) is that we have just a single computation builder and so we can define a number of additional operations to enable richer syntax.
The following snippet extends UpdateBuilder
defined earlier with more operations. If you're
not interested in the details, feel free to skip to the next section. The key idea is
that this only has to be written once!
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 

You can find more details about these operations in the F# Computation Zoo paper
or in the F# language specification. In fact, the definitions mostly follow
the samples from the F# specification. It is worth noting that all the members are
marked as inline
, which allows us to use static member constrains and to write
code that will work for any update monad (as defined by a pair of update and
state types).
Let's look at a trivial example using the writer computation:
1: 2: 3: 4: 

As expected, when we run the computation using writeRun
, the result is a tuple containing
a list with numbers from 1 to 10 and a unit value. The computation does not explicitly
return and so the Zero
member is automatically used.
Implementing the state monad
Interestingly, the standard state monad is not a special case of update monads. However, we can define a computation that implements the same functionality  a computation with state that we can read and write.
States and updates
In this final example, both the type representing state and the type representing update will have a useful role. We make both of the types generic over the value they carry. State is simply a wrapper containing the value (current state). Update can be of two kinds  we have an empty update (do nothing) and an update to set the state:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 

This definition is a bit more interesting than the previous two, because there is some
interaction between the states and updates. In particular, when the update is Set v
(we want to replace the current state with a new one), the Apply
member returns a new
state instead of the original. For the Unit
member, we need an update SetNop
which
simply means that we want to keep the original state (and so Apply
just returns the
original value in this case).
Another notable thing is the Combine
operation  it takes two updates (which may be
either empty updates or set updates) and produces a single one. If you read a composition
a1 ++ a2 ++ .. ++ an
as a sequence of state updates (either Set
or SetNop
), then the
Combine
operation returns the last Set
update in the sequence (or SetNop
if there are
no Set
updates). In other words, it builds an update that sets the last state that was
set during the whole sequence.
State monad primitives
Now that we have the type definitions, it is quite easy to add the usual primitives:
1: 2: 3: 4: 5: 6: 

The set
operation is a bit different than the usual one for state monad. It ignores the
state and it builds an update that tells the computation to set the new state.
The get
operation reads the state and returns it  but as it does not intend to change it,
it returns SetNop
as the update.
Sample stateful computation
If you made it this far in the article, you can already expect how the example will look!
We'll again use the update { .. }
computation. This time, we define a computation
demo5
that increments the state and call it from a loop in demo6
:
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Running the code yields 10 as expected  we start with zero and then increment the
state ten times. Since we extended the definition of the UpdateBuilder
(in the
previous section), we now got a few nice things for free  we can use the for
loop
and write computations (like demo5
) without explicit return
if they just need to
modify the state.
Conclusions
People coming to F# from the Haskell background often dislike the fact that
F# does not let you write code polymorphic over monads and that computation
expressions always explicitly state the type of computations such as
async { .. }
. I think there are good reasons for this and tried to explain some
of them in a recent blog post and PADL paper.
As a result, using reader, writer and state monads in F# was always a bit cumbersome. In this blog post, I looked at an F# implementation of the recent idea called update monads (see the original paper (PDF)), which unifies the three staterelated monads into a single type. This works very nicely with F#  we can define just a single computation builder for all staterelated computations and then define a concrete staterelated monad by defining two simple types. I used the approach to define a reader monad, writer monad useful for logging and a state monad (that keeps a state and allows changing it).
I guess that making update monads part of standard library and standard programming style in Haskell will be tricky because of historical reasons. However, for F# libraries that try to make purely functional programming easier, I think that update monads are the way to go.
Full name: Updatemonads_.Reader<_,_>
Given a readonly state, produces a value
Full name: Updatemonads_.Writer<_,_>
Produces a value together with additional state
Full name: Updatemonads_.State<_,_>
Given state, produces new state & a value
Full name: Updatemonads_.UpdateMonad<_,_,_>
Represents an update monad  given a state, produce
value and an update that can be applied to the state
val unit : unit > 'S (requires member get_Unit)
Full name: Updatemonads_.unit

type unit = Unit
Full name: Microsoft.FSharp.Core.unit
Full name: Updatemonads_.apply
type UpdateBuilder =
new : unit > UpdateBuilder
member Bind : UpdateMonad<'S,'U,'T> * f:('T > UpdateMonad<'S,'U,'R>) > UpdateMonad<'S,'U,'R> (requires member Combine and member Apply)
member Combine : c1:UpdateMonad<'a,'b,unit> * c2:UpdateMonad<'a,'b,'c> > UpdateMonad<'a,'b,'c> (requires member Combine and member Apply)
member Delay : f:(unit > UpdateMonad<'a,'b,'c>) > UpdateMonad<'a,'b,'c> (requires member get_Unit and member Combine and member Apply)
member For : sq:seq<'V> * f:('V > UpdateMonad<'S,'P,unit>) > UpdateMonad<'S,'P,unit> (requires member Combine and member Apply and member get_Unit)
member Return : v:'T > UpdateMonad<'S,'U,'T> (requires member get_Unit)
member ReturnFrom : m:UpdateMonad<'S,'P,'T> > UpdateMonad<'S,'P,'T>
member Using : r:'a * f:('a > UpdateMonad<'b,'c,'d>) > UpdateMonad<'b,'c,'d> (requires 'a :> IDisposable)
member While : t:(unit > bool) * f:(unit > UpdateMonad<'S,'P,unit>) > UpdateMonad<'S,'P,unit> (requires member Combine and member Apply and member get_Unit)
member Zero : unit > UpdateMonad<'a,'b,unit> (requires member get_Unit)
Full name: Updatemonads_.UpdateBuilder

new : unit > UpdateBuilder
Full name: Updatemonads_.UpdateBuilder.Return
Returns the specified value, together
with empty update obtained using 'unit'
Full name: Updatemonads_.UpdateBuilder.Bind
Compose two update monad computations
Full name: Updatemonads_.update
Instance of the computation builder
that defines the update { .. } block
Full name: Updatemonads_.ReaderState
The state of the reader is 'int'
val int : value:'T > int (requires member op_Explicit)
Full name: Microsoft.FSharp.Core.Operators.int

type int = int32
Full name: Microsoft.FSharp.Core.int

type int<'Measure> = int
Full name: Microsoft.FSharp.Core.int<_>
 NoUpdate
static member Apply : s:'a * NoUpdate:ReaderUpdate > 'a
static member Combine : NoUpdate:ReaderUpdate * NoUpdate:ReaderUpdate > ReaderUpdate
static member Unit : ReaderUpdate
Full name: Updatemonads_.ReaderUpdate
Trivial monoid of updates
Full name: Updatemonads_.ReaderUpdate.Unit
Full name: Updatemonads_.ReaderUpdate.Combine
Full name: Updatemonads_.ReaderUpdate.Apply
Full name: Updatemonads_.read
Read the current state (int) and return it as 'int'
Full name: Updatemonads_.readRun
Run computation and return the result
Full name: Microsoft.FSharp.Core.Operators.snd
Full name: Updatemonads_.demo1
Returns state + 1
Full name: Updatemonads_.demo2
Returns the result of demo1 + 1
Full name: Updatemonads_.WriterState
Writer monad has no readable state
 Log of 'TLog list
static member Apply : NoState:WriterState * 'a > WriterState
static member Combine : WriterUpdate<'a> * WriterUpdate<'a> > WriterUpdate<'a>
static member Unit : WriterUpdate<int>
Full name: Updatemonads_.WriterUpdate<_>
Updates of writer monad form a list
Full name: Microsoft.FSharp.Collections.list<_>
Full name: Updatemonads_.WriterUpdate`1.Unit
Returns the empty log (monoid unit)
Full name: Updatemonads_.WriterUpdate`1.Combine
Combines two logs (operation of the monoid)
module List
from Microsoft.FSharp.Collections

type List<'T> =
 ( [] )
 ( :: ) of Head: 'T * Tail: 'T list
interface IEnumerable
interface IEnumerable<'T>
member Head : 'T
member IsEmpty : bool
member Item : index:int > 'T with get
member Length : int
member Tail : 'T list
static member Cons : head:'T * tail:'T list > 'T list
static member Empty : 'T list
Full name: Microsoft.FSharp.Collections.List<_>
Full name: Microsoft.FSharp.Collections.List.append
Full name: Updatemonads_.WriterUpdate`1.Apply
Applying updates to state does not affect the state
Full name: Updatemonads_.write
Writes the specified value to the log
Full name: Updatemonads_.writeRun
Runs a "writer monad computation" and returns
the log, together with the final result
Full name: Updatemonads_.demo3
Writes '20' to the log and returns "world"
Full name: Updatemonads_.demo4
Calls 'demo3' and then writes 10 to the log
Full name: Updatemonads_.UpdateBuilder.Zero
Represents monadic computation that returns unit
(e.g. we can now omit 'else' branch in 'if' computation)
Returns the specified value, together
with empty update obtained using 'unit'
Full name: Updatemonads_.UpdateBuilder.Delay
Delays a computation with (uncontrolled) side effects
Compose two update monad computations
Represents monadic computation that returns unit
(e.g. we can now omit 'else' branch in 'if' computation)
Full name: Updatemonads_.UpdateBuilder.Combine
Sequential composition of two computations where the
first one has no result (returns a unit value)
Full name: Updatemonads_.UpdateBuilder.ReturnFrom
Enable the 'return!' keyword to return another computation
Full name: Updatemonads_.UpdateBuilder.Using
Ensure that resource 'r' is disposed of at the end of the
computation specified by the function 'f'
Full name: Updatemonads_.UpdateBuilder.For
Support 'for' loop  runs body 'f' for each element in 'sq'
val seq : sequence:seq<'T> > seq<'T>
Full name: Microsoft.FSharp.Core.Operators.seq

type seq<'T> = System.Collections.Generic.IEnumerable<'T>
Full name: Microsoft.FSharp.Collections.seq<_>
member Current : 'T
Full name: System.Collections.Generic.IEnumerator<_>
Ensure that resource 'r' is disposed of at the end of the
computation specified by the function 'f'
Full name: Updatemonads_.UpdateBuilder.While
Supports 'while' loop  run body 'f' until condition 't' holds
Full name: Updatemonads_.logNumbers
Logs numbers from 1 to 10
Full name: Updatemonads_.StateState<_>
Wraps a state of type 'T
union case StateState.State: 'T > StateState<'T>

type State<'TState,'T> = 'TState > 'TState * 'T
Full name: Updatemonads_.State<_,_>
Given state, produces new state & a value
 Set of 'T
 SetNop
static member Apply : s:StateState<'a> * p:StateUpdate<'a> > StateState<'a>
static member Combine : a:StateUpdate<'a> * b:StateUpdate<'a> > StateUpdate<'a>
static member Unit : StateUpdate<int>
Full name: Updatemonads_.StateUpdate<_>
Represents updates on state of type 'T
union case StateUpdate.Set: 'T > StateUpdate<'T>

module Set
from Microsoft.FSharp.Collections

type Set<'T (requires comparison)> =
interface IComparable
interface IEnumerable
interface IEnumerable<'T>
interface ICollection<'T>
new : elements:seq<'T> > Set<'T>
member Add : value:'T > Set<'T>
member Contains : value:'T > bool
override Equals : obj > bool
member IsProperSubsetOf : otherSet:Set<'T> > bool
member IsProperSupersetOf : otherSet:Set<'T> > bool
...
Full name: Microsoft.FSharp.Collections.Set<_>

new : elements:seq<'T> > Set<'T>
Full name: Updatemonads_.StateUpdate`1.Unit
Empty update  do not change the state
Full name: Updatemonads_.StateUpdate`1.Combine
Combine updates  return the latest (rightmost) 'Set' update
Full name: Updatemonads_.StateUpdate`1.Apply
Apply update to a state  the 'Set' update changes the state
Full name: Updatemonads_.set
Set the state to the specified value
Full name: Updatemonads_.get
Get the current state
Full name: Updatemonads_.setRun
Run a computation using a specified initial state
Full name: Updatemonads_.demo5
Increments the state by one
Full name: Updatemonads_.demo6
Call 'demo5' repeatedly in a loop
and then return the final state
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