Difference between revisions of "The Monadic Way"
m (let's do real output!) 
(Added some text more text) 

Line 1:  Line 1:  
==An evaluation of Philip Wadler's "Monads for functional programming"== 
==An evaluation of Philip Wadler's "Monads for functional programming"== 

−  This tutorial is a "translation" of Philip Wedler's "Monads for 
+  This tutorial is a "translation" of Philip Wedler's [http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf "Monads for functional programming"]. 
−  functional programming". 

(avail. from [http://homepages.inf.ed.ac.uk/wadler/topics/monads.html here]) 
(avail. from [http://homepages.inf.ed.ac.uk/wadler/topics/monads.html here]) 

I'm a Haskell newbie trying to grasp such a difficult concept as the 
I'm a Haskell newbie trying to grasp such a difficult concept as the 

−  +  one of Monad and monadic computation. 

+  
While [http://www.cs.utah.edu/~hal/htut/ "Yet Another Haskell Tutorial"] 
While [http://www.cs.utah.edu/~hal/htut/ "Yet Another Haskell Tutorial"] 

gave me a good understanding of the type system when it 
gave me a good understanding of the type system when it 

Line 14:  Line 14:  
So I decided to "translate it", in order to clarify to myself the 
So I decided to "translate it", in order to clarify to myself the 

−  topic. And I'm now sharing this traslation (not completed yet) 
+  topic. And I'm now sharing this traslation ('''not completed yet'') 
−  the hope it will be useful to someone else. 
+  with the hope it will be useful to someone else. 
Moreover, that's a wiki, so please improve it. And, specifically, 
Moreover, that's a wiki, so please improve it. And, specifically, 

Line 21:  Line 21:  
'''Note: The source of this page can be used as a Literate Haskel 
'''Note: The source of this page can be used as a Literate Haskel 

−  file and can be run with ghci or hugs: so cut paste change 
+  file and can be run with ghci or hugs: so cut paste change and run (in 
emacs for instance) while reading it...''' 
emacs for instance) while reading it...''' 

Line 28:  Line 28:  
Let's start with something simple: suppose we want to implement a new 
Let's start with something simple: suppose we want to implement a new 

programming language. We just finished with 
programming language. We just finished with 

−  [http://swiss.csail.mit.edu/classes/6.001/abelsonsussmanlectures/ Abelson and Sussman's Structure and Interpretation of 
+  [http://swiss.csail.mit.edu/classes/6.001/abelsonsussmanlectures/ Abelson and Sussman's Structure and Interpretation of Computer Programs] 
and we want to test what we have learned. 
and we want to test what we have learned. 

Our programming language will be very simple: it will just compute the 
Our programming language will be very simple: it will just compute the 

−  sum 
+  sum of two terms. 
So we have just one primitive operation (Add) that takes two constants 
So we have just one primitive operation (Add) that takes two constants 

−  and calculates their sum 
+  and calculates their sum. 
+  
+  Moreover we have just one kind of data type: Con a, which is an Int. 

For instance, something like: 
For instance, something like: 

Line 44:  Line 44:  
11 
11 

+  
+  ===The basic evaluator=== 

We will implement our language with the help of a data type 
We will implement our language with the help of a data type 

constructor such as: 
constructor such as: 

−  ===The basic evaluator=== 

<haskell> 
<haskell> 

Line 74:  Line 75:  
*TheMonadicWay> 
*TheMonadicWay> 

−  Very very simple. The evaluator checks if its argument is 
+  Very very simple. The evaluator checks if its argument is of type Con 
−  it is it just returns 
+  Int: if it is it just returns the Int. 
−  If 
+  If the argument is not of type Con, but it is of type Term, it 
−  +  evaluates the first Term and sums the result with the result of the 

+  evaluation of the second Term. 

== Some Output, Please!== 
== Some Output, Please!== 

Line 84:  Line 85:  
Now, that's fine, but we'd like to add some features, like providing 
Now, that's fine, but we'd like to add some features, like providing 

some output, to show how the computation was carried out. 
some output, to show how the computation was carried out. 

+  
Well, but Haskell is a pure functional language, with no side effects, 
Well, but Haskell is a pure functional language, with no side effects, 

we were told. 
we were told. 

Line 89:  Line 91:  
Now we seem to be wanting to create a side effect of the computation, 
Now we seem to be wanting to create a side effect of the computation, 

its output, and be able to stare at it... 
its output, and be able to stare at it... 

+  
If we had some global variable to store the out that would be 
If we had some global variable to store the out that would be 

simple... 
simple... 

Line 94:  Line 97:  
But we can create the output and carry it along the computation, 
But we can create the output and carry it along the computation, 

concatenating it with the old one, and present it at the end of the 
concatenating it with the old one, and present it at the end of the 

−  evaluation together with the evaluation of the expression 
+  evaluation together with the evaluation of the expression given to our 
+  evaluator/interpreter! 

===The basic evaluator with output=== 
===The basic evaluator with output=== 

−  Simple and neat! 

+  
+  Simple and neat: 

<haskell> 
<haskell> 

Line 116:  Line 120:  
Now we have what we want. But we had to change our evaluator quite a 
Now we have what we want. But we had to change our evaluator quite a 

−  bit. First we added a function, that takes a Term (of the expression 

+  bit. 

−  to be evaluated), an Int (the result of the evaluation) and gives back 

−  an output of type Output (that is a synonymous of String). 

−  The evaluator changed quite a lot! Now it has a different type: it 

+  First we added a function, formatLine, that takes an argument of type 

−  takes a Term data type and produces a new type, we called MOut, that 

+  Term (the expression to be evaluated), one of type Int (the result of 

−  is actually a pair of a variable type a (an Int in our evaluator) and 

+  the evaluation of Term) and gives back an output of type Output (that 

−  a type Output, a string. 

+  is a synonymous of String). This is just a helper function to format 

+  the string to output. Not very interesting at all. 

+  
+  The evaluator itself changed quite a lot! Now it has a different type 

+  signature: it takes an argument of type Term and produces a new type, 

+  we called it MOut, that is actually a compound pair of a variable (of 

+  a variable) type a (an Int in our evaluator) and a type Output, a 

+  string. 

So our evaluator, now, will take a Term (the type of the expressions 
So our evaluator, now, will take a Term (the type of the expressions 

Line 165:  Line 174:  
</haskell> 
</haskell> 

−  Why all that? The problem is that we need 
+  Why all that? The problem is that we need: 
−  sum, 
+  * "a" and "b" to calculate their sum (a + b), that will be the first element of the compund pair rapresenting the type (MOut) our evaluator will return 
−  concatenated by the expression x ++ y ++ formatLine 
+  * "x and "y" (the output of each evaluation) to be concatenated with the ourput of formatLine by the expression (x ++ y ++ formatLine(...)): this will be the second element of the compound pair MOut, the string part. 
−  So we need to separate the pairs produced by "evalO t" and " 
+  So we need to separate the pairs produced by "evalO t" and "evalO u". 
−  +  
−  +  We do that within the where clause (remember: evalO now produces a value of type 

+  MOut Int, i.e. a pair of an Int and a String). 

+  
+  Then we use the single element, "extraded" within the where clause, to 

+  return a new MOut composed by 

+  
+  ((a + b),(x ++ y ++ formatLine (Add t u) (a + b))). 

+    

+  Int Output = String 

== Let's Go Monadic== 
== Let's Go Monadic== 

Line 184:  Line 193:  
evaluation). 
evaluation). 

−  The first part of the evaluator does nothing else but creating, from 
+  The first part of the evaluator does nothing else but creating, from a 
−  +  value of type Int, an object of type MOut Int (Int,Output). It does so 

−  by creating a pair with that Int and some text. 
+  by creating a pair with that Int and some text produced by formatLine. 
The second part evaluates the two Term(s) and "stores" the values thus 
The second part evaluates the two Term(s) and "stores" the values thus 

Line 208:  Line 217:  
Our evaluator binds "a" and "x" / "b" and "y" with the evaluation of 
Our evaluator binds "a" and "x" / "b" and "y" with the evaluation of 

−  " 
+  "evalO t" and "evalO u" respectively. 
−  +  
+  Then "a","b","x" and "y" will be then used in the evaluation of 

+  ((a+b),(x++y++formatLine)), the will produce a value of type MOut Int: 

+  
+  <pre> 

+  
+  ((a + b),(x ++ y ++ formatLine (Add t u) (a + b))). 

+    

+  \ / \ / 

+  Int Output = String 

+   

+  \ / 

+  MOut Int 

+  </pre> 

+  
+  The binding happens in the where clause: 

+  <haskell> 

+  where (a, x) = evalO t 

+  (b, y) = evalO u 

+  </haskell> 

We know that there is an ad hoc operator for binding variables to a 
We know that there is an ad hoc operator for binding variables to a 

Line 247:  Line 256:  
resulting from the evaluation of "m" and "f a". 
resulting from the evaluation of "m" and "f a". 

−  So let's write the new version of the evaluator: 

+  As you see, we took the binding part out from evalO and put it in this new function. 

+  
+  So let's write the new version of the evaluator, that we will call evalM_1: 

<haskell> 
<haskell> 

Line 273:  Line 284:  
bind "b". 
bind "b". 

−  So in bindM (evalM_1 u) 
+  So in bindM (evalM_1 u) (\b >) "b" will be bound to the value 
+  returned by evalM_1 u, and this bounded variable will be available in 

+  what comes after ">" as a bounded variable (not free). 

Then the outer part (bindM (evalM_1 t) (\a...) will bind "a" to the 
Then the outer part (bindM (evalM_1 t) (\a...) will bind "a" to the 

−  value needed to evaluate "((a+b), formatLine...) and produce our final 

+  value returned "evalM_1 t", the result of the evaluatuion of the first 

−  MOut Int. 

+  Term. This value is needed to evaluate "((a+b), formatLine...) and 

+  produce our final MOut Int. 

−  +  S we can use lambda notation to write our evaluator in a more convinient way: 

−  what it does: 

<haskell> 
<haskell> 

Line 297:  Line 309:  
</haskell> 
</haskell> 

−  We could use a more general way of creating some output. 
+  We could use a more general way of creating some output. 
+  
+  We can create a function that takes an Int and returns the type MOut 

+  Int. we do that by pairing the received Int with an empty string "". 

+  
+  This will be a general way of creating an object with type MOut Int starting from an Int. 

+  
+  Or, more generaly, a function that takes something of a variable type 

+  a, and return an object of type MOut a, a coumpunt object made up of 

+  an element of type a, and one of type String. 

−  First we need a method for creating someting of type M a, starting from 

+  There it is: 

−  something of type a. This is what <hask>evalM_2 (Con a)</hask> is doing, after all. 

−  Very simply: 

<haskell> 
<haskell> 

Line 310:  Line 320:  
</haskell> 
</haskell> 

−  +  Then we need a method of inserting some text in our object of type 

+  MOut. So we will take a string and return it paired with a void 

+  element "()": 

<haskell> 
<haskell> 

Line 322:  Line 332:  
instead of an Int, and the output. 
instead of an Int, and the output. 

−  This way we will be able to define also this firts part in terms of 

+  Now we can rewrite: 

−  bindM, that will take care of concatenating outputs. 

+  <haskell> 

+  evalM_2 (Con a) = (a, formatLine (Con a) a) 

+  </haskell> 

+  using the bindM function: 

+  <haskell> 

+  evalM_2 (Con a) = outPut (formatLine (Con a) a) `bindM` \_ > mkM a 

+  </haskell> 

+  
+  First we create an object of type MOut with the Int part (). As you 

+  see bindM will not use it ("\_"), but will concatenate the String part 

+  with the result of mkM, which in turn is the empry string "". 

+  
+  In other words, first we insert the Output part (a string) in our MOut 

+  Int type, and then we insert the Int. 

−  +  Let's rewrite the evaluator: 

<haskell> 
<haskell> 

> evalM_3 :: Term > MOut Int 
> evalM_3 :: Term > MOut Int 

−  > evalM_3 (Con a) = outPut (formatLine (Con a) a) `bindM` \_ > 
+  > evalM_3 (Con a) = outPut (formatLine (Con a) a) `bindM` \_ > 
+  > mkM a 

> evalM_3 (Add t u) = evalM_3 t `bindM` \a > 
> evalM_3 (Add t u) = evalM_3 t `bindM` \a > 

> evalM_3 u `bindM` \b > 
> evalM_3 u `bindM` \b > 

−  > outPut (formatLine (Add t u) (a + b)) `bindM` \_ > 
+  > outPut (formatLine (Add t u) (a + b)) `bindM` \_ > 
+  > mkM (a + b) 

</haskell> 
</haskell> 

Line 364:  Line 387:  
</haskell> 
</haskell> 

−  Let's put everything together (and change some names 
+  Let's put everything together (and change some names changing M into 
+  MO, so that this file will be still usable as a Literate Haskell 

+  file): 

<haskell> 
<haskell> 

Line 401:  Line 424:  
evalM t >>= \a > evalM u >>= \b > outPut "something" >>= \_ > mkM (a +b) 
evalM t >>= \a > evalM u >>= \b > outPut "something" >>= \_ > mkM (a +b) 

</haskell> 
</haskell> 

−  where >>= is 
+  where >>= is bindMO, obviously. 
−  Let's do some substitution 
+  Let's do some substitution, writing the type of their output of each function: 
−  * 
+  * evalMO t => (a,Out)  where a is Int 
−  * 
+  * evalMO u => (b,Out)  where b is the same of a, an Int, but with a different value 
−  * outMO 
+  * outMO Out = ((),Out) 
−  * mkMO (a+b) = ((a+b),Out) 
+  * mkMO (a+b) => ((a+b),Out)  where (a+b) is the same of a and b, but with a different value from either a and b 
<pre> 
<pre> 

−  +  B  (a,Out) >>= \a > (b,Out) >>= \b > ((),Out) >>= \_ >>= ((a + b), Out)\ 

−  +  i  V V V V V V V V ^ ^ ^ ^ \ 

−  +  n  __________^   ^        MOut Int > ((a+b), Out) 

−  +  d _______(++)___Out_____(++)__V_Out_______(++)__(++)______________/ 

i   ______(b)____________(b)______(b)_____ 
i   ______(b)____________(b)______(b)_____ 

n  _________(a)_________________________(a)__ 
n  _________(a)_________________________(a)__ 

−  +  g  _____()_____ 

</pre> 
</pre> 

Line 422:  Line 445:  
Clear, isn't it? 
Clear, isn't it? 

−  " 
+  "bindMO" is just a function that takes care of gluing together, inside 
a data type, a sequence of computations! 
a data type, a sequence of computations! 

Line 434:  Line 457:  
The process of accumulation and the act of parting the MO Int into its 
The process of accumulation and the act of parting the MO Int into its 

−  component is buried into 
+  component is buried into bindMO, now, that can also preserve some 
−  later uses. 
+  value for later uses. 
So we have: 
So we have: 

* MO a type constructor for a type carrying a pair composed by an Int and a String; 
* MO a type constructor for a type carrying a pair composed by an Int and a String; 

−  * bindMO, that gives a direction to the process of evaluation: it concatenates computations and captures some side effects we created. 
+  * bindMO, that gives a direction to the process of evaluation: it concatenates computations and captures some side effects we created (the direction is given by the changes in the Out part: there's a "before" when Out was something and there's a "later" when Out is something else). 
−  * 
+  * mkMO lets us create an object of type MO Int starting from an Int. 
As you see this is all we need to create a monad. In other words 
As you see this is all we need to create a monad. In other words 

−  monads arise from the type system 
+  monads arise from the type system and the lambda calculus. Everything 
−  sugar. 
+  else is just syntactic sugar. 
−  So, let's have a look 
+  So, let's have a look at that sugar: the famous donotation! 
===Basic monadic evaluator in donotation=== 
===Basic monadic evaluator in donotation=== 

−  We will now rewrite our basic evaluator 
+  We will now rewrite our basic evaluator by using the donotation. 
−  donotation. 

Now we have to crate a new type: this is necessary in order to use 
Now we have to crate a new type: this is necessary in order to use 

specific monadic notation and have at our disposal the more practical 
specific monadic notation and have at our disposal the more practical 

−  donotation. 

+  donotation ('''below we will see the consequences of doing so!'''): 

<haskell> 
<haskell> 

Line 465:  Line 488:  
So, our type will be an instance of the monad class. We will have to 
So, our type will be an instance of the monad class. We will have to 

define the methods of this class (>>= and return), but that will be 
define the methods of this class (>>= and return), but that will be 

−  easy since we already done that 
+  easy since we have already done that when defining bindMO and mkMO: 
<haskell> 
<haskell> 

Line 475:  Line 498:  
</haskell> 
</haskell> 

−  And then we will take the old version of our evaluator and substitute 

+  You can see that return will create, from an argument of a variable 

−  `bindMO` with >>= and `mkMO` with return: 

+  type a (in our case that will be an Int) an object of type Eval Int, 

−  
+  that carries inside just an Int, the result of the evaluation of a 

+  Con. 

+  
+  Bind (>>=) will match for an object of type Eval, extracting what's 

+  inside ("m") and will bind "m" in "f". We know that "f" must return an 

+  object of type Eval with inside an Int resulted by the computations 

+  made by "f" over "m" (that is to say, computations made by "f" where 

+  "f" is a functions with variables, and one of those variables is bound 

+  to the value resulting from the evaluation of "m"). 

+  
+  In exchange for doing so we will now be able to take the old version 

+  of our evaluator and substitute `bindMO` with >>= and `mkMO` with 

+  return: 

+  
<haskell> 
<haskell> 

Line 488:  Line 524:  
</haskell> 
</haskell> 

−  which is, in the donotation: 
+  which is equivalent, in the donotation, to: 
<haskell> 
<haskell> 

Line 500:  Line 536:  
</haskell> 
</haskell> 

−  Simple: do binds the result of "eval_M5 t" to a, binds 
+  Simple: <hask>do</hask> binds the result of "eval_M5 t" to "a", binds 
−  "eval_M5 u" to b and then returns the sum 
+  the result of "eval_M5 u" to "b" and then returns the sum of "a" and 
+  "b". In a very imperative style. 

−  We can now have an image of our monad: it is out type (Eval) that is 

+  ===Monadic evaluator with output in donotation=== 

−  made up of a pair: during evaluation the first member of the pair (the 

+  
−  Int) will get the results of our computation (i.e.: the procedures to 

+  We can now have an image of what our monad should be, if we want it to 

−  calculate the final result). The second part, the String called 

+  produce output: it is out type (Eval) that is made up of a pair, and 

−  Output, will get filled up with the concatenated output of the 

+  Int and a String called Output. 

−  computation. 

+  
+  During evaluation the first member of the pair (the Int) will "store" 

+  the results of our computation (i.e.: the procedures to calculate the 

+  final result). The second part, the String called Output, will get 

+  filled up with the concatenated output of the computation. 

The sequencing done by bindMO (now >>=) will take care of passing to 
The sequencing done by bindMO (now >>=) will take care of passing to 

−  the next evaluation the needed Int and will do 
+  the next evaluation the needed (way to calculate the) Int and will do 
−  calculation to produce the output 
+  some more side calculation to produce the output (concatenating 
−  from computation of the new Int, for instance). 
+  outputs resulting from computation of the new Int, for instance). 
So we can grasp the basic concept of a monad: it is like a label which 
So we can grasp the basic concept of a monad: it is like a label which 

Line 521:  Line 561:  
step bindMO can do some manipulation of it. 
step bindMO can do some manipulation of it. 

We are creating sideeffects and propagating them within our monads. 
We are creating sideeffects and propagating them within our monads. 

−  
−  ===Monadic evaluator with output in donotation=== 

Ok. Let's translate our outputproducing evaluator in monadic 
Ok. Let's translate our outputproducing evaluator in monadic 

Line 612:  Line 650:  
type constructor Eval_IO to the pair that are going to form our monad. 
type constructor Eval_IO to the pair that are going to form our monad. 

−  "return" takes an Int and insert it into our monad. It will also 
+  "return" takes an Int and insert it into our monad. It will also 
−  +  insert an empty String "" that (>>=) or (>>) will then concatenate in 

−  sequence of computations they glue together. 
+  the sequence of computations they glue together. 
The same for (>>=): it will now return something constructed by 
The same for (>>=): it will now return something constructed by 

Line 701:  Line 739:  
==Errare Monadicum Est== 
==Errare Monadicum Est== 

−  '''(Text to be done yet: just a summary)''' 

+  Now that we have a basic understanding of what a monad is, and does, 

+  we will further explore it by making some changes to our evaluator. 

+  
+  In this section we will se how to handle exceptions in our monadic 

+  evaluator. 

+  
+  Suppose that we want to stop the execution of our monad if some 

+  conditions occurs. If our evaluator was to compute divisions, instead 

+  of sums, then we would like to stop the evaluator when a division by 

+  zero occurs, possibly producing some output, instead of the result of 

+  the evaluation of the expression, that explains what happened. 

+  
+  Basic error handling. 

−  In this section we will se how to handle exceptions in our monadic evaluator. 

+  We will do so starting from the beginning once again... 

===The basic evaluator, non monadic, with exception=== 
===The basic evaluator, non monadic, with exception=== 

+  
+  We just take our basic evaluator, without any output, and write a 

+  method to stop execution if a condition occurs: 

<haskell> 
<haskell> 

Line 713:  Line 766:  
> deriving (Show) 
> deriving (Show) 

> type Exception = String 
> type Exception = String 

+  
+  </haskell> 

+  
+  Now, our monad is of datatype "M a" which can either be constructed with the "Raise" constructor, that takes a String (Exception is a synonymous of String), or by the "Return" constructor, that takes a variable type ("a"), an Int in our case. 

+  
+  <haskell> 

> evalE :: Term > M Int 
> evalE :: Term > M Int 

> evalE (Con a) = Return a 
> evalE (Con a) = Return a 

+  
+  </haskell> 

+  
+  If evalE matches a Con it will construct a type Return with, inside, the content of the Con. 

+  
+  <haskell> 

+  
> evalE (Add a b) = 
> evalE (Add a b) = 

> case evalE a of 
> case evalE a of 

Line 728:  Line 794:  
</haskell> 
</haskell> 

+  
+  If evalE matches an Add it will check if evaluating the first part 

+  produces a "Raise" or a "Return": in the first case it will return a 

+  "Raise" whose content is the same received. 

+  
+  If instead the evaluation produces a value of type "Return", the 

+  evaluator will evaluate the second part of the Add type. 

+  
+  If this returns a "Raise", a "Raise" will be returned all the way up the 

+  recursion, otherwise the evaluator will check whether a condition for 

+  raising a "Raise" exists. If not, it will return a "Return" with the sum inside. 

Test it with: 
Test it with: 

evalE (Add (Con 10) (Add (Add (Con 20) (Con 10)) (Con 2))) 
evalE (Add (Con 10) (Add (Add (Con 20) (Con 10)) (Con 2))) 

+  
===The basic evaluator, monadic, with exceptions=== 
===The basic evaluator, monadic, with exceptions=== 

+  
+  In order to produce a monadic version of the previous evaluator, the 

+  one that raises exceptions, we just need to abstract out from the 

+  evaluator all that case analysis. 

<haskell> 
<haskell> 

Line 741:  Line 823:  
> deriving (Show) 
> deriving (Show) 

+  </haskell> 

+  
+  The data type didn't change at all. Well, we changed the name of the Return type constructor (no Ok) so that 

+  this constructor can coexist with the previous one in the same Literate Haskell file. 

+  
+  <haskell> 

+  
> instance Monad M1 where 
> instance Monad M1 where 

> return a = Ok a 
> return a = Ok a 

Line 746:  Line 835:  
> Except e > Except e 
> Except e > Except e 

> Ok a > f a 
> Ok a > f a 

+  
+  </haskell> 

+  
+  Binding operations are now very easy. Basically we check: 

+  * if the result of the evaluation of "m" produces an exception (first matct: Except e >...), in which case we return its content by constructing our M1 Int with the "Raise" constructor". 

+  * if the result of the evaluation of "m" is matched with the "Ok" constructor, we get its content and use it to bind the argument of "f" to its value. 

+  
+  <hask>return a</hask> will just use the Ok type constructor for 

+  inserting "a" (in our case an Int) into M1 Int, the type of our monad. 

+  
+  <haskell> 

> raise :: Exception > M1 a 
> raise :: Exception > M1 a 

> raise e = Except e 
> raise e = Except e 

+  
+  </haskell> 

+  
+  This is just a helper function to construct our "M1 a" type with the 

+  Raise constructor. It takes a string and returns a type (M1 a) to be 

+  matched with the "Raise" constructor. 

+  
+  <haskell> 

> eval_ME :: Term > M1 Int 
> eval_ME :: Term > M1 Int 

Line 759:  Line 867:  
</haskell> 
</haskell> 

+  
+  The evaluator itself is very simple. We bind "a" with the result of 

+  "eval_ME t", "b" with the result of "eval_ME u", and we check for a 

+  condition: 

+  * if the condition is met we raise an exception, that is to say: we return a value constructed with the "Raise" constructor. This value will be matched by ">>=" in the next recursion. And >>= will just return it all the way up the recursion. 

+  * if the condition is not me, we return a value constructed with the "Return" type constructor and go on with the recursion... 

Run with: 
Run with: 

Line 765:  Line 879:  
It is noteworthy the fact that in our datatype definition we used a 
It is noteworthy the fact that in our datatype definition we used a 

−  label field with a label selector (we called it showM) 
+  label field with a label selector (we called it showM), even though it 
+  was not used in our code. We will use this methodology later on. 

−  +  So, just to refresh your memory: 

<haskell> 
<haskell> 

Line 796:  Line 910:  
===Monadic evaluator with output and exceptions=== 
===Monadic evaluator with output and exceptions=== 

+  
+  '''(Text to be done yet: just a summary)''' 

This is the evaluator that produces output, plus exceptions. 
This is the evaluator that produces output, plus exceptions. 
Revision as of 11:42, 1 September 2006
Contents
 1 An evaluation of Philip Wadler's "Monads for functional programming"
 2 A Simple Evaluator
 3 Some Output, Please!
 4 Let's Go Monadic
 5 What Does Bind Bind?
 6 Some Sugar, Please!
 7 What Happened to Our Output??
 8 Errare Monadicum Est
 9 We Need A State
 9.1 The basic evaluator, non monadic, with a counter
 9.2 The evaluator, monadic, with a counter, without donotation
 9.3 The evaluator, monadic, with counter and output, without donotation
 9.4 The monadic evaluator with output and counter in donotation
 9.5 Another version of the monadic evaluator with output and counter, in donotation
 10 If There's A State We Need Some Discipline: Dealing With Complexity
 11 Taking Complexity Out of a Monad: Monadic Transformers
 11.1 The StateT Monad: a Monad Container
 11.2 StateT as a counter, and monadic evaluator with output and exceptions
 11.3 StateT to keep output and counter, and monadic evaluator with (only) exceptions
 11.4 StateT to keep output, counter and debug. The monadic evaluator is only for failable computations
 12 The Final Cut
 13 Next?
 14 Suggested Readings
 15 Acknowledgments
An evaluation of Philip Wadler's "Monads for functional programming"
This tutorial is a "translation" of Philip Wedler's "Monads for functional programming". (avail. from here)
I'm a Haskell newbie trying to grasp such a difficult concept as the one of Monad and monadic computation.
While "Yet Another Haskell Tutorial" gave me a good understanding of the type system when it comes to monads I find it almost unreadable.
But I had also Wedler's paper, and started reading it. Well, just wonderful! It explains how to create a monad!
So I decided to "translate it", in order to clarify to myself the topic. And I'm now sharing this traslation ('not completed yet) with the hope it will be useful to someone else.
Moreover, that's a wiki, so please improve it. And, specifically, correct my poor English. I'm Italian, after all.
Note: The source of this page can be used as a Literate Haskel file and can be run with ghci or hugs: so cut paste change and run (in emacs for instance) while reading it...
A Simple Evaluator
Let's start with something simple: suppose we want to implement a new programming language. We just finished with Abelson and Sussman's Structure and Interpretation of Computer Programs and we want to test what we have learned.
Our programming language will be very simple: it will just compute the sum of two terms.
So we have just one primitive operation (Add) that takes two constants and calculates their sum.
Moreover we have just one kind of data type: Con a, which is an Int.
For instance, something like:
(Add (Con 5) (Con 6))
should yeld:
11
The basic evaluator
We will implement our language with the help of a data type constructor such as:
> module TheMonadicWay where
> data Term = Con Int
>  Add Term Term
> deriving (Show)
After that we build our interpreter:
> eval :: Term > Int
> eval (Con a) = a
> eval (Add a b) = eval a + eval b
That's it. Just an example:
*TheMonadicWay> eval (Add (Con 5) (Con 6)) 11 *TheMonadicWay>
Very very simple. The evaluator checks if its argument is of type Con Int: if it is it just returns the Int.
If the argument is not of type Con, but it is of type Term, it evaluates the first Term and sums the result with the result of the evaluation of the second Term.
Some Output, Please!
Now, that's fine, but we'd like to add some features, like providing some output, to show how the computation was carried out.
Well, but Haskell is a pure functional language, with no side effects, we were told.
Now we seem to be wanting to create a side effect of the computation, its output, and be able to stare at it...
If we had some global variable to store the out that would be simple...
But we can create the output and carry it along the computation, concatenating it with the old one, and present it at the end of the evaluation together with the evaluation of the expression given to our evaluator/interpreter!
The basic evaluator with output
Simple and neat:
> type MOut a = (a, Output)
> type Output = String
>
> formatLine :: Term > Int > Output
> formatLine t a = "eval (" ++ show t ++ ") <= " ++ show a ++ "  "
>
> evalO :: Term > MOut Int
> evalO (Con a) = (a, formatLine (Con a) a)
> evalO (Add t u) = ((a + b),(x ++ y ++ formatLine (Add t u) (a + b)))
> where (a, x) = evalO t
> (b, y) = evalO u
Now we have what we want. But we had to change our evaluator quite a bit.
First we added a function, formatLine, that takes an argument of type Term (the expression to be evaluated), one of type Int (the result of the evaluation of Term) and gives back an output of type Output (that is a synonymous of String). This is just a helper function to format the string to output. Not very interesting at all.
The evaluator itself changed quite a lot! Now it has a different type signature: it takes an argument of type Term and produces a new type, we called it MOut, that is actually a compound pair of a variable (of a variable) type a (an Int in our evaluator) and a type Output, a string.
So our evaluator, now, will take a Term (the type of the expressions in our new programming language) and will produce a pair, composed of the result of the evaluation (an Int) and the Output, a string.
So far so good. But what's happening inside the evaluator?
The first part will just return a pair with the number evaluated and the output formatted by formatLine.
The second part does something more complicated: it returns a pair composed by 1. the result of the evaluation of the right Term summed to the result of the evaluation of the second Term 2. the output: the concatenation of the output produced by the evaluation of the right Term, the output produced by the evaluation of the left Term (each this evaluation returns a pair with the number and the output), and the formatted output of the evaluation.
Let's try it:
*TheMonadicWay> evalO (Add (Con 5) (Con 6)) (11,"eval (Con 5) <= 5  eval (Con 6) <= 6  eval (Add (Con 5) (Con 6)) <= 11  ") *TheMonadicWay>
It works! Let's put the output this way:
eval (Con 5) <= 5  eval (Con 6) <= 6  eval (Add (Con 5) (Con 6)) <= 11 
Great! We are able to produce a side effect of our evaluation and present it at the end of the computation, after all.
Let's have a closer look at this expression:
evalO (Add t u) = ((a + b),(x ++ y ++ formatLine (Add t u) (a + b)))
where (a, x) = evalO t
(b, y) = evalO u
Why all that? The problem is that we need:
 "a" and "b" to calculate their sum (a + b), that will be the first element of the compund pair rapresenting the type (MOut) our evaluator will return
 "x and "y" (the output of each evaluation) to be concatenated with the ourput of formatLine by the expression (x ++ y ++ formatLine(...)): this will be the second element of the compound pair MOut, the string part.
So we need to separate the pairs produced by "evalO t" and "evalO u".
We do that within the where clause (remember: evalO now produces a value of type MOut Int, i.e. a pair of an Int and a String).
Then we use the single element, "extraded" within the where clause, to return a new MOut composed by
((a + b),(x ++ y ++ formatLine (Add t u) (a + b))).   Int Output = String
Let's Go Monadic
Is there a more general way of doing so?
Let's analyze the evaluator from another perspective. From the type perspective.
We solved our problem by creating a new type, a pair of an Int (the result of the evaluation) and a String (the output of the process of evaluation).
The first part of the evaluator does nothing else but creating, from a value of type Int, an object of type MOut Int (Int,Output). It does so by creating a pair with that Int and some text produced by formatLine.
The second part evaluates the two Term(s) and "stores" the values thus produced in some variables to be use later to compute the output.
Let's focus on the "stores" action. The correct term should be "binds".
Take a function:
f x = x + x
"x" appears on both sides of the expression. We say that on the right side "x" is bound to the value of x given on the left side.
So
f 3
binds x to 3 for the evaluation of the expression "x + x".
Our evaluator binds "a" and "x" / "b" and "y" with the evaluation of "evalO t" and "evalO u" respectively.
Then "a","b","x" and "y" will be then used in the evaluation of ((a+b),(x++y++formatLine)), the will produce a value of type MOut Int:
((a + b),(x ++ y ++ formatLine (Add t u) (a + b))).   \ / \ / Int Output = String  \ / MOut Int
The binding happens in the where clause:
where (a, x) = evalO t
(b, y) = evalO u
We know that there is an ad hoc operator for binding variables to a value: lambda, or \.
Indeed f x = x + x is syntactic sugar for:
f = \x > x + x
When we write f 3 we are actually binding "x" to 3 within what's next ">", that will be used (substituted) for evaluating f 3.
So we can try to abstract this phenomenon.
Monadic evaluator with output
What we need is a function that takes our composed type MOut Int and a function in order to produce a new MOut Int, concatenating the output of the computation of the first with the output of the computation of the second.
This is what bindM does:
> bindM :: MOut a > (a > MOut b) > MOut b
> bindM m f = (b, x ++ y)
> where (a, x) = m
> (b, y) = f a
It takes:
 "m": the compound type MOut Int carrying the result of an "eval Term",
 a function "f". This function will take the Int ("a") extracted by the evaluation of "m" ((a,x)=m). This function will produce anew pair: a new Int produced by a new evaluation; some new output.
bindM will return the new Int in pair with the concatenated outputs resulting from the evaluation of "m" and "f a".
As you see, we took the binding part out from evalO and put it in this new function.
So let's write the new version of the evaluator, that we will call evalM_1:
> evalM_1 :: Term > MOut Int
> evalM_1 (Con a) = (a, formatLine (Con a) a)
> evalM_1 (Add t u) = bindM (evalM_1 t) (\a >
> bindM (evalM_1 u) (\b >
> ((a + b), formatLine (Add t u) (a + b))
> )
> )
Ugly, isn't it?
Let's start from the outside:
bindM (evalM_1 u) (\b > ((a + b), formatLine (Add t u) (a + b)))
bindM takes the result of the evaluation "evalM_1 u", a type Mout Int, and a function. It will extract the Int from that type and use it to bind "b".
So in bindM (evalM_1 u) (\b >) "b" will be bound to the value returned by evalM_1 u, and this bounded variable will be available in what comes after ">" as a bounded variable (not free).
Then the outer part (bindM (evalM_1 t) (\a...) will bind "a" to the value returned "evalM_1 t", the result of the evaluatuion of the first Term. This value is needed to evaluate "((a+b), formatLine...) and produce our final MOut Int.
S we can use lambda notation to write our evaluator in a more convinient way:
> evalM_2 :: Term > MOut Int
> evalM_2 (Con a) = (a, formatLine (Con a) a)
> evalM_2 (Add t u) = evalM_2 t `bindM` \a >
> evalM_2 u `bindM` \b >
> ((a + b), (formatLine (Add t u) (a + b)))
Now, look at the first part:
evalM_2 (Con a) = (a, formatLine (Con a) a)
We could use a more general way of creating some output.
We can create a function that takes an Int and returns the type MOut Int. we do that by pairing the received Int with an empty string "".
This will be a general way of creating an object with type MOut Int starting from an Int.
Or, more generaly, a function that takes something of a variable type a, and return an object of type MOut a, a coumpunt object made up of an element of type a, and one of type String.
There it is:
> mkM :: a > MOut a
> mkM a = (a, "")
Then we need a method of inserting some text in our object of type MOut. So we will take a string and return it paired with a void element "()":
> outPut :: Output > MOut ()
> outPut x = ((), x)
Very simple: we have a string "x" (Output) and create a pair with a () instead of an Int, and the output.
Now we can rewrite:
evalM_2 (Con a) = (a, formatLine (Con a) a)
using the bindM function:
evalM_2 (Con a) = outPut (formatLine (Con a) a) `bindM` \_ > mkM a
First we create an object of type MOut with the Int part (). As you see bindM will not use it ("\_"), but will concatenate the String part with the result of mkM, which in turn is the empry string "".
In other words, first we insert the Output part (a string) in our MOut Int type, and then we insert the Int.
Let's rewrite the evaluator:
> evalM_3 :: Term > MOut Int
> evalM_3 (Con a) = outPut (formatLine (Con a) a) `bindM` \_ >
> mkM a
> evalM_3 (Add t u) = evalM_3 t `bindM` \a >
> evalM_3 u `bindM` \b >
> outPut (formatLine (Add t u) (a + b)) `bindM` \_ >
> mkM (a + b)
Well, this is fine, definetly better then before, anyway.
Still we use `bindM` \_ > that binds something we do not use (_). We could write something for this case, when we concatenate computations without the need of binding variables. Let's call it `combineM`:
> combineM :: MOut a > MOut b > MOut b
> combineM m f = m `bindM` \_ > f
So the new evaluator:
> evalM :: Term > MOut Int
> evalM (Con a) = outPut (formatLine (Con a) a) `combineM`
> mkM a
> evalM (Add t u) = evalM t `bindM` \a >
> evalM u `bindM` \b >
> outPut (formatLine (Add t u) (a + b)) `combineM`
> mkM (a + b)
Let's put everything together (and change some names changing M into MO, so that this file will be still usable as a Literate Haskell file):
> type MO a = (a, Out)
> type Out = String
> mkMO :: a > MO a
> mkMO a = (a, "")
> bindMO :: MO a > (a > MO b) > MO b
> bindMO m f = (b, x ++ y)
> where (a, x) = m
> (b, y) = f a
> combineMO :: MO a > MO b > MO b
> combineMO m f = m `bindM` \_ > f
> outMO :: Out > MO ()
> outMO x = ((), x)
> evalMO :: Term > MO Int
> evalMO (Con a) = outMO (formatLine (Con a) a) `combineMO`
> mkMO a
> evalMO (Add t u) = evalMO t `bindMO` \a >
> evalMO u `bindMO` \b >
> outMO (formatLine (Add t u) (a + b)) `combineMO`
> mkMO (a + b)
What Does Bind Bind?
The evaluator looks like:
evalM t >>= \a > evalM u >>= \b > outPut "something" >>= \_ > mkM (a +b)
where >>= is bindMO, obviously.
Let's do some substitution, writing the type of their output of each function:
 evalMO t => (a,Out)  where a is Int
 evalMO u => (b,Out)  where b is the same of a, an Int, but with a different value
 outMO Out = ((),Out)
 mkMO (a+b) => ((a+b),Out)  where (a+b) is the same of a and b, but with a different value from either a and b
B  (a,Out) >>= \a > (b,Out) >>= \b > ((),Out) >>= \_ >>= ((a + b), Out)\ i  V V V V V V V V ^ ^ ^ ^ \ n  __________^   ^        MOut Int > ((a+b), Out) d _______(++)___Out_____(++)__V_Out_______(++)__(++)______________/ i   ______(b)____________(b)______(b)_____ n  _________(a)_________________________(a)__ g  _____()_____
Clear, isn't it?
"bindMO" is just a function that takes care of gluing together, inside a data type, a sequence of computations!
Some Sugar, Please!
Now our evaluator has been completely transformed into a monadic evaluator. That's what it is: a monad.
We have a function that constructs an object of type MO Int, formed by a pair: the result of the evaluation and the accumulated (concatenated) output.
The process of accumulation and the act of parting the MO Int into its component is buried into bindMO, now, that can also preserve some value for later uses.
So we have:
 MO a type constructor for a type carrying a pair composed by an Int and a String;
 bindMO, that gives a direction to the process of evaluation: it concatenates computations and captures some side effects we created (the direction is given by the changes in the Out part: there's a "before" when Out was something and there's a "later" when Out is something else).
 mkMO lets us create an object of type MO Int starting from an Int.
As you see this is all we need to create a monad. In other words monads arise from the type system and the lambda calculus. Everything else is just syntactic sugar.
So, let's have a look at that sugar: the famous donotation!
Basic monadic evaluator in donotation
We will now rewrite our basic evaluator by using the donotation.
Now we have to crate a new type: this is necessary in order to use specific monadic notation and have at our disposal the more practical donotation (below we will see the consequences of doing so!):
> newtype Eval a = Eval a
> deriving (Show)
So, our type will be an instance of the monad class. We will have to define the methods of this class (>>= and return), but that will be easy since we have already done that when defining bindMO and mkMO:
> instance Monad Eval where
> return a = Eval a
> Eval m >>= f = f m
You can see that return will create, from an argument of a variable type a (in our case that will be an Int) an object of type Eval Int, that carries inside just an Int, the result of the evaluation of a Con.
Bind (>>=) will match for an object of type Eval, extracting what's inside ("m") and will bind "m" in "f". We know that "f" must return an object of type Eval with inside an Int resulted by the computations made by "f" over "m" (that is to say, computations made by "f" where "f" is a functions with variables, and one of those variables is bound to the value resulting from the evaluation of "m").
In exchange for doing so we will now be able to take the old version of our evaluator and substitute `bindMO` with >>= and `mkMO` with return:
> evalM_4 :: Term > Eval Int
> evalM_4 (Con a) = return a
> evalM_4 (Add t u) = evalM_4 t >>= \a >
> evalM_4 u >>= \b >
> return (a + b)
which is equivalent, in the donotation, to:
> evalM_5 :: Term > Eval Int
> evalM_5 (Con a) = return a
> evalM_5 (Add t u) = do a < evalM_5 t
> b < evalM_5 u
> return (a + b)
Simple: do
binds the result of "eval_M5 t" to "a", binds
the result of "eval_M5 u" to "b" and then returns the sum of "a" and
"b". In a very imperative style.
Monadic evaluator with output in donotation
We can now have an image of what our monad should be, if we want it to produce output: it is out type (Eval) that is made up of a pair, and Int and a String called Output.
During evaluation the first member of the pair (the Int) will "store" the results of our computation (i.e.: the procedures to calculate the final result). The second part, the String called Output, will get filled up with the concatenated output of the computation.
The sequencing done by bindMO (now >>=) will take care of passing to the next evaluation the needed (way to calculate the) Int and will do some more side calculation to produce the output (concatenating outputs resulting from computation of the new Int, for instance).
So we can grasp the basic concept of a monad: it is like a label which we attach to each step of the evaluation (the String attached to the Int). This label is persistent within the process of computation and at each step bindMO can do some manipulation of it. We are creating sideeffects and propagating them within our monads.
Ok. Let's translate our outputproducing evaluator in monadic notation:
> newtype Eval_IO a = Eval_IO (a, O)
> deriving (Show)
> type O = String
> instance Monad Eval_IO where
> return a = Eval_IO (a, "")
> (>>=) m f = Eval_IO (b, x ++ y)
> where Eval_IO (a, x) = m
> Eval_IO (b, y) = f a
> print_IO :: O > Eval_IO ()
> print_IO x = Eval_IO ((), x)
> eval_IO :: Term > Eval_IO Int
> eval_IO (Con a) = do print_IO (formatLine (Con a) a)
> return a
> eval_IO (Add t u) = do a < eval_IO t
> b < eval_IO u
> print_IO (formatLine (Add t u) (a + b))
> return (a + b)
Let's see the evaluator with output in action:
*TheMonadicWay> eval_IO (Add (Con 6) (Add (Con 16) (Add (Con 20) (Con 12)))) Eval_IO (54,"eval (Con 6) <= 6  eval (Con 16) <= 16  eval (Con 20) <= 20  eval (Con 12) <= 12  \ eval (Add (Con 20) (Con 12)) <= 32  eval (Add (Con 16) (Add (Con 20) (Con 12))) <= 48  \ eval (Add (Con 6) (Add (Con 16) (Add (Con 20) (Con 12)))) <= 54  ") *TheMonadicWay>
Let's format the output part:
eval (Con 6) <= 6 eval (Con 16) <= 16 eval (Con 20) <= 20 eval (Con 12) <= 12 eval (Add (Con 20) (Con 12)) <= 32 eval (Add (Con 16) (Add (Con 20) (Con 12))) <= 48 eval (Add (Con 6) (Add (Con 16) (Add (Con 20) (Con 12)))) <= 54
What Happened to Our Output??
Well, actually something happened to the output. Let's compare the output of evalMO (the monadic evaluator written without the donotation) and eval_IO:
*TheMonadicWay> evalMO (Con 6) (6,"eval (Con 6) <= 6  ") *TheMonadicWay> eval_IO (Con 6) Eval_IO (6,"eval (Con 6) <= 6  ") *TheMonadicWay>
They look almost the same, but they are not the same: the output of eval_IO has the Eval_IO stuff. It must be related to the changes we had to do to our evaluator in order to use the doconation, obviously.
What's changed?
First the type definition. We have now:
newtype Eval_IO a = Eval_IO (a, O)
deriving (Show)
instead of
type MO a = (a, Out)
Moreover our bindMO and mkMO functions changed too, to reflect the change of the type definition:
instance Monad Eval_IO where
return a = Eval_IO (a, "")
(>>=) m f = Eval_IO (b, x ++ y)
where Eval_IO (a, x) = m
Eval_IO (b, y) = f a
Now return a
is the product of the application of the
type constructor Eval_IO to the pair that are going to form our monad.
"return" takes an Int and insert it into our monad. It will also insert an empty String "" that (>>=) or (>>) will then concatenate in the sequence of computations they glue together.
The same for (>>=): it will now return something constructed by
Eval_IO: "b", the result of the application of "f" to "a" (better, the
binding of "a" in "f") and "x" (matched by Eval_IO (a, x)
with
the evaluation of "m") and "y", (matched by "Eval_IO(b,y)" with the
evaluation of "f a".
That is to say: in the "where" clause, we are matching for the elements paired in a type Eval_IO: this is indeed the type of "m" (corresponding to "eval_IO t" in the body of the evaluator) and "f a" (where "f" correspond to another application of "eval_IO" to the result of the previous application of "m").
And so, "Eval_IO (a,x) = m" means: match "a" and "x", paired in a type Eval_IO, and that are produced by the evaluation of "m" (that is to say: "eval_IO t"). The same for Eval_IO (b,y): match "b" and "y" produced by the evaluation of "f a".
So the output of the evaluator is now not simply a pair made of and Int and a String. It is a specific type (Eval_IO) that happens to carry a pair of an Int and a String. But, if we want the Int and the string, we have to extract them from the Eval_IO type, as we do in the "where" clause: we unpack our type object (let's call it with its name: our monad!) and take out the Int and the String to feed the next function application and the output generation.
The same to insert something in our monad: if we want to create a pair of an Int and a String, pair of type Eval_IO, we now have to pack the together by using our type constructor, feeding it with pair composed by and Int and a String. This is what we do with the "return" method of out monad and with "print_IO" function, where:
 return insert into the monad an Int;
 print_IO insert into the monad a String.
Notice that "combineM" disappeared. This is because it comes for free by just defining our type Eval_IO as an instance of the Monad class.
Indeed, if we look at the definition of the Monad class in the Prelude we read:
class Monad m where
return :: a > m a
(>>=) :: m a > (a > m b) > m b
(>>) :: m a > m b > m b
fail :: String > m a
 Minimal complete definition: (>>=), return
p >> q = p >>= \ _ > q
fail s = error s
You can see that the "combineM"" method (or (>>)) is automatically derived by
the "bindMO" (or >>=) method:
p >> q = p >>= \ _ > q
So, what the hell is the old type MO a = (a, Out)
that
did not required all this additional work (apart the need to
specifically define (>>)?
Thanks the help of some nice guy of the haskellcafe mailing list (look at the thread started by this silly question of mine) we can answer.
Type MO is just a synonymous for (a,Out): the two can be substituted one for the other. That's it.
We did not have to pack "a" and "Out" together with a type constructor to have a new type MO.
As a consequence, we cannot use MO as an instance of Monad, and so, we cannot use with it the syntactic sugar we needed: the donotation.
That is to say: a type created with the "type" keyword cannot be an instance of a class, and cannot inherits its methods (in our case (>>=, >> and return). And without those methods the donotation is not usable.
Anyway we will better understand all the far reaching consequences of this new approach later on.
Errare Monadicum Est
Now that we have a basic understanding of what a monad is, and does, we will further explore it by making some changes to our evaluator.
In this section we will se how to handle exceptions in our monadic evaluator.
Suppose that we want to stop the execution of our monad if some conditions occurs. If our evaluator was to compute divisions, instead of sums, then we would like to stop the evaluator when a division by zero occurs, possibly producing some output, instead of the result of the evaluation of the expression, that explains what happened.
Basic error handling.
We will do so starting from the beginning once again...
The basic evaluator, non monadic, with exception
We just take our basic evaluator, without any output, and write a method to stop execution if a condition occurs:
> data M a = Raise Exception
>  Return a
> deriving (Show)
> type Exception = String
Now, our monad is of datatype "M a" which can either be constructed with the "Raise" constructor, that takes a String (Exception is a synonymous of String), or by the "Return" constructor, that takes a variable type ("a"), an Int in our case.
> evalE :: Term > M Int
> evalE (Con a) = Return a
If evalE matches a Con it will construct a type Return with, inside, the content of the Con.
> evalE (Add a b) =
> case evalE a of
> Raise e > Raise e
> Return a >
> case evalE b of
> Raise e > Raise e
> Return b >
> if (a+b) == 42
> then Raise "The Ultimate Answer Has Been Computed!! Now I'm tired!"
> else Return (a+b)
If evalE matches an Add it will check if evaluating the first part produces a "Raise" or a "Return": in the first case it will return a "Raise" whose content is the same received.
If instead the evaluation produces a value of type "Return", the evaluator will evaluate the second part of the Add type.
If this returns a "Raise", a "Raise" will be returned all the way up the recursion, otherwise the evaluator will check whether a condition for raising a "Raise" exists. If not, it will return a "Return" with the sum inside.
Test it with:
evalE (Add (Con 10) (Add (Add (Con 20) (Con 10)) (Con 2)))
The basic evaluator, monadic, with exceptions
In order to produce a monadic version of the previous evaluator, the one that raises exceptions, we just need to abstract out from the evaluator all that case analysis.
> data M1 a = Except Exception
>  Ok {showM :: a }
> deriving (Show)
The data type didn't change at all. Well, we changed the name of the Return type constructor (no Ok) so that this constructor can coexist with the previous one in the same Literate Haskell file.
> instance Monad M1 where
> return a = Ok a
> m >>= f = case m of
> Except e > Except e
> Ok a > f a
Binding operations are now very easy. Basically we check:
 if the result of the evaluation of "m" produces an exception (first matct: Except e >...), in which case we return its content by constructing our M1 Int with the "Raise" constructor".
 if the result of the evaluation of "m" is matched with the "Ok" constructor, we get its content and use it to bind the argument of "f" to its value.
return a
will just use the Ok type constructor for
inserting "a" (in our case an Int) into M1 Int, the type of our monad.
> raise :: Exception > M1 a
> raise e = Except e
This is just a helper function to construct our "M1 a" type with the Raise constructor. It takes a string and returns a type (M1 a) to be matched with the "Raise" constructor.
> eval_ME :: Term > M1 Int
> eval_ME (Con a) = do return a
> eval_ME (Add t u) = do a < eval_ME t
> b < eval_ME u
> if (a+b) == 42
> then raise "The Ultimate Answer Has Been Computed!! Now I'm tired!"
> else return (a + b)
The evaluator itself is very simple. We bind "a" with the result of "eval_ME t", "b" with the result of "eval_ME u", and we check for a condition:
 if the condition is met we raise an exception, that is to say: we return a value constructed with the "Raise" constructor. This value will be matched by ">>=" in the next recursion. And >>= will just return it all the way up the recursion.
 if the condition is not me, we return a value constructed with the "Return" type constructor and go on with the recursion...
Run with:
eval_ME (Add (Con 10) (Add (Add (Con 20) (Con 10)) (Con 2)))
It is noteworthy the fact that in our datatype definition we used a label field with a label selector (we called it showM), even though it was not used in our code. We will use this methodology later on.
So, just to refresh your memory:
> data Person = Person {name :: String,
> age :: Int,
> hobby :: String
> } deriving (Show)
> andreaRossato = Person "Andrea" 37 "Haskell The Monadic Way"
> personName (Person a b c) = a
will produce:
*TheMonadicWay> andreaRossato Person {name = "Andrea", age = 37, hobby = "Haskell The Monadic Way"} *TheMonadicWay> personName andreaRossato "Andrea" *TheMonadicWay> name andreaRossato "Andrea" *TheMonadicWay> age andreaRossato 37 *TheMonadicWay> hobby andreaRossato "Haskell The Monadic Way" *TheMonadicWay>
Monadic evaluator with output and exceptions
(Text to be done yet: just a summary)
This is the evaluator that produces output, plus exceptions.
> data M2 a = Ex Exception
>  Done {unpack :: (a,O) }
> deriving (Show)
> instance Monad M2 where
> return a = Done (a, "")
> m >>= f = case m of
> Ex e > Ex e
> Done (a, x) > case (f a) of
> Ex e1 > Ex e1
> Done (b, y) > Done (b, x ++ y)
Since we have to concatenate output we must check that also the next run of the evaluator will not raise an exception.
> raise_IOE :: Exception > M2 a
> raise_IOE e = Ex e
> print_IOE :: O > M2 ()
> print_IOE x = Done ((), x)
> eval_IOE :: Term > M2 Int
> eval_IOE (Con a) = do print_IOE (formatLine (Con a) a)
> return a
> eval_IOE (Add t u) = do a < eval_IOE t
> b < eval_IOE u
> let out = formatLine (Add t u) (a + b)
> print_IOE out
> if (a+b) == 42
> then raise_IOE $ out ++ "The Ultimate Answer Has Been Computed!! Now I'm tired!"
> else return (a + b)
Run with
eval_IOE (Add (Con 10) (Add (Add (Con 20) (Con 10)) (Con 2)))
Look at the let clause within the do notation: we do not need let since all variable bound within a do procedure will be available all the way down.
Remember m >>= \a > f >>= \ > ...
We Need A State
We start adding complexity to our monadic evaluator. But in order to
add a counter we will start over again (to review out knowledeg).
The basic evaluator, non monadic, with a counter
The basic evaluator plus a counter: evalNM takes now the expression to
be evaluated plus an initial state (0) to start counting from.
>  non monadic
> evalNMS :: Term > MS Int
> evalNMS (Con a) x = (a, x + 1)
> evalNMS (Add t u) x = let (a, y) = evalNMS t x in
> let (b, z) = evalNMS u y in
> (a + b, z +1)
The evaluator, monadic, with a counter, without donotation
The moadic version without do notation.
>  monadic
> type MS a = State > (a, State)
> type State = Int
> mkMS :: a > MS a
> mkMS a = \x > (a, x)
> bindMS :: MS a > (a > MS b) > MS b
> bindMS m f = \x >
> let (a, y) = m x in
> let (b, z) = f a y in
> (b, z)
> combineMS :: MS a > MS b > MS b
> combineMS m f = m `bindMS` \_ > f
> incState :: MS ()
> incState = \s > ((), s + 1)
> evalMS :: Term > MS Int
> evalMS (Con a) = incState `combineMS` mkMS a
> evalMS (Add t u) = evalMS t `bindMS` \a >
> evalMS u `bindMS` \b >
> incState `combineMS` mkMS (a + b)
> evalMS (Add (Con 6) (Add (Con 16) (Add (Con 20) (Con 12)))) 0
The evaluator, monadic, with counter and output, without donotation
Now we'll add Output to the stateful evaluator:
>  state and output
> type MSO a = State > (a, State, Output)
> mkMSO :: a > MSO a
> mkMSO a = \s > (a, s, "")
> bindMSO :: MSO a > (a > MSO b) > MSO b
> bindMSO m f = \x >
> let (a, y, s1) = m x in
> let (b, z, s2) = f a y in
> (b, z, s1 ++ s2)
> combineMSO :: MSO a > MSO b > MSO b
> combineMSO m f = m `bindMSO` \_ > f
> incMSOstate :: MSO ()
> incMSOstate = \s > ((), s + 1, "")
> outMSO :: Output > MSO ()
> outMSO = \x s > ((),s, x)
> evalMSO :: Term > MSO Int
> evalMSO (Con a) = incMSOstate `combineMSO`
> outMSO (formatLine (Con a) a) `combineMSO`
> mkMSO a
> evalMSO (Add t u) = evalMSO t `bindMSO` \a >
> evalMSO u `bindMSO` \b >
> incMSOstate `combineMSO`
> outMSO (formatLine (Add t u) (a + b)) `combineMSO`
> mkMSO (a + b)
> evalMSO (Add (Con 6) (Add (Con 16) (Add (Con 20) (Con 12)))) 0
The monadic evaluator with output and counter in donotation
State, Output in donotation. Look at how much the complexity of our (>>=) founction is increasing:
>  thanks to Brian Hulley
> newtype MSIO a = MSIO (State > (a, State, Output))
> instance Monad MSIO where
> return a = MSIO (\s > (a, s, ""))
> (MSIO m) >>= f = MSIO $ \x >
> let (a, y, s1) = m x in
> let MSIO runNextStep = f a in
> let (b, z, s2) = runNextStep y in
> (b, z, s1 ++ s2)
> incMSOIstate :: MSIO ()
> incMSOIstate = MSIO (\s > ((), s + 1, ""))
> print_MSOI :: Output > MSIO ()
> print_MSOI x = MSIO (\s > ((),s, x))
> eval_MSOI :: Term > MSIO Int
> eval_MSOI (Con a) = do incMSOIstate
> print_MSOI (formatLine (Con a) a)
> return a
> eval_MSOI (Add t u) = do a < eval_MSOI t
> b < eval_MSOI u
> incMSOIstate
> print_MSOI (formatLine (Add t u) (a + b))
> return (a + b)
> run_MSOI :: MSIO a > State > (a, State, Output)
> run_MSOI (MSIO f) s = f s
> run_MSOI (eval_MSOI (Add (Con 6) (Add (Con 16) (Add (Con 20) (Con 12))))) 0
Another version of the monadic evaluator with output and counter, in donotation
This is e second version that exploit label fields in datatype to decrease the complexity of the binding operations.
>  Thanks Udo Stenzel
> newtype Eval_SIO a = Eval_SIO { unPackMSIOandRun :: State > (a, State, Output) }
> instance Monad Eval_SIO where
> return a = Eval_SIO (\s > (a, s, ""))
> (>>=) m f = Eval_SIO (\x >
> let (a, y, s1) = unPackMSIOandRun m x in
> let (b, z, s2) = unPackMSIOandRun (f a) y in
> (b, z, s1 ++ s2))
> incSIOstate :: Eval_SIO ()
> incSIOstate = Eval_SIO (\s > ((), s + 1, ""))
> print_SIO :: Output > Eval_SIO ()
> print_SIO x = Eval_SIO (\s > ((),s, x))
> eval_SIO :: Term > Eval_SIO Int
> eval_SIO (Con a) = do incSIOstate
> print_SIO (formatLine (Con a) a)
> return a
> eval_SIO (Add t u) = do a < eval_SIO t
> b < eval_SIO u
> incSIOstate
> print_SIO (formatLine (Add t u) (a + b))
> return (a + b)
> unPackMSIOandRun (eval_SIO (Add (Con 6) (Add (Con 16) (Add (Con 20) (Con 12))))) 0
If There's A State We Need Some Discipline: Dealing With Complexity
In order to increase the complexity of our monad now we will try to mix State (counter), Exceptions and Output.
This is an email I send to the haskellcafe mailing list:
Now I'm trying to create a statefull evaluator, with output and exception, but I'm facing a problem I seem not to be able to conceptually solve. Take the code below. Now, in order to get it run (and try to debug) the Eval_SOI type has a Raise constructor that produces the same type of SOIE. Suppose instead it should be constructing something like Raise "something". Moreover, I wrote a second version of >>=, commented out. This is just to help me illustrate to problem I'm facing. Now, >>= is suppose to return Raise if "m" is matched against Raise (second version commented out). If "m" matches SOIE it must return a SOIE only if "f a" does not returns a Raise (output must be concatenated). I seem not to be able to find a way out. Moreover, I cannot understand if a way out can be possibly found. Something suggests me it could be related to that Raise "something". But my feeling is that functional programming could be something out of the reach of my mind... by the way, I teach Law, so perhaps you'll forgive me...;) If you can help me to understand this problem all I can promise is that I'll mention your help in the tutorial I'm trying to write on "the monadic way"... that seems to lead me nowhere. Thanks for your kind attention. Andrea
This was the code:
data Eval_SOI a = Raise { unPackMSOIandRun :: State > (a, State, Output) }
 SOIE { unPackMSOIandRun :: State > (a, State, Output) }
instance Monad Eval_SOI where
return a = SOIE (\s > (a, s, ""))
m >>= f = SOIE (\x >
let (a, y, s1) = unPackMSOIandRun m x in
case f a of
SOIE nextRun > let (b, z, s2) = nextRun y in
(b, z, s1 ++ s2)
Raise e1 > e1 y only this happens
)
 (>>=) m f = case m of
 Raise e > error "ciao"  why this is not going to happen?
 SOIE a > SOIE (\x >
 let (a, y, s1) = unPackMSOIandRun m x in
 let (b, z, s2) = unPackMSOIandRun (f a) y in
 (b, z, s1 ++ s2))
incSOIstate :: Eval_SOI ()
incSOIstate = SOIE (\s > ((), s + 1, ""))
print_SOI :: Output > Eval_SOI ()
print_SOI x = SOIE (\s > ((),s, x))
raise x e = Raise (\s > (x,s,e))
eval_SOI :: Term > Eval_SOI Int
eval_SOI (Con a) = do incSOIstate
print_SOI (formatLine (Con a) a)
return a
eval_SOI (Add t u) = do a < eval_SOI t
b < eval_SOI u
incSOIstate
print_SOI (formatLine (Add t u) (a + b))
if (a + b) == 42
then raise (a+b) " = The Ultimate Answer!!"
else return (a + b)
runEval exp = case eval_SOI exp of
Raise a > a 0
SOIE p > let (result, state, output) = p 0 in
(result,state,output)
runEval (Add (Con 10) (Add (Con 28) (Add (Con 40) (Con 2))))
This code will produce
eval (Con 10) <= 10  eval (Con 28) <= 28  eval (Con 40) <= 40  eval (Con 2) <= 2  = The Ultimate Answer!! eval (Add (Con 28) (Add (Con 40) (Con 2))) <= 70  eval (Add (Con 10) (Add (Con 28) (Add (Con 40) (Con 2)))) <= 80 
The exception appears in the output, but executioon is not stopped.
Monadic evaluator with output, counter and exception, in donotation
Brian Hulley came up with this solution:
>  thanks to Brian Hulley
> data Result a
> = Good a State Output
>  Bad State Output Exception
> deriving Show
> newtype Eval_SIOE a = SIOE {runSIOE :: State > Result a}
> instance Monad Eval_SIOE where
> return a = SIOE (\s > Good a s "")
> m >>= f = SIOE $ \x >
> case runSIOE m x of
> Good a y o1 >
> case runSIOE (f a) y of
> Good b z o2 > Good b z (o1 ++ o2)
> Bad z o2 e > Bad z (o1 ++ o2) e
> Bad z o2 e > Bad z o2 e
> raise_SIOE e = SIOE (\s > Bad s "" e)
> incSIOEstate :: Eval_SIOE ()
> incSIOEstate = SIOE (\s > Good () (s + 1) "")
> print_SIOE :: Output > Eval_SIOE ()
> print_SIOE x = SIOE (\s > Good () s x)
> eval_SIOE :: Term > Eval_SIOE Int
> eval_SIOE (Con a) = do incSIOEstate
> print_SIOE (formatLine (Con a) a)
> return a
> eval_SIOE (Add t u) = do a < eval_SIOE t
> b < eval_SIOE u
> incSIOEstate
> let out = formatLine (Add t u) (a + b)
> print_SIOE out
> if (a+b) == 42
> then raise_SIOE $ out ++ "The Ultimate Answer Has Been Computed!! Now I'm tired!"
> else return (a + b)
> runEval exp = case runSIOE (eval_SIOE exp) 0 of
> Bad s o e > "Error at iteration n. " ++ show s ++
> "  Output stack = " ++ o ++
> "  Exception = " ++ e
> Good a s o > "Result = " ++ show a ++
> "  Iterations = " ++ show s ++ "  Output = " ++ o
Run with runEval (Add (Con 18) (Add (Con 12) (Add (Con 10) (Con 2))))
Taking Complexity Out of a Monad: Monadic Transformers
We have seen how the complexity of (>>=) was growing by adding operations to be done. N We will do the opposite: we will implement a state transformer (I copied StateT).
We will embed our monad in the StateT monad and we will start moving state and output from the inner monad (our one) to the outer monad (StateT).
The StateT Monad: a Monad Container
Let me introduce StateT with some useful functions:
> newtype StateT s m a = StateT {runStateT :: s > m (a,s) } StateT (s > m (a,s))
> instance Monad m => Monad (StateT s m) where
> return a = StateT (\s > return (a,s))
> StateT m1 >>= k = StateT (\s > do ~(a,s1) < m1 s
> let StateT m2 = k a
> m2 s1)
>   Execute a stateful computation, as a result we get
>  the result of the computation, and the final state.
> runState :: s > StateT s m a > m (a,s)
> runState s (StateT m) = m s
>   Execute a stateful computation, ignoring the final state.
> evalState :: Functor m => s > StateT s m a > m a
> evalState s m = fmap fst (runState s m)
>   Execute a stateful computation, just for the side effect.
> execState :: Functor m => s > StateT s m a > m s
> execState s m = fmap snd (runState s m)
> lift :: (Monad m) => m a > StateT s m a
> lift m = StateT (\s > do x < m
> return (x,s))
StateT is pleased to meet you!.
StateT as a counter, and monadic evaluator with output and exceptions
And now out monad, with state out from it:
> data MTa a = FailTa Exception
>  DoneTa {unpackDoneTa :: (a,O) }
> deriving (Show)
> instance Monad MTa where
> return a = DoneTa (a, "")
> m >>= f = case m of
> FailTa e > FailTa e
> DoneTa (a, x) > case (f a) of
> FailTa e1 > FailTa e1
> DoneTa (b, y) > DoneTa (b, x ++ y)
> instance Functor MTa where
> fmap _ (FailTa e) = FailTa e
> fmap f (DoneTa (r,o)) = DoneTa ((f r),o)
> raiseTa_SIOE :: O > StateT Int MTa a
> raiseTa_SIOE e = lift (FailTa e)
> printTa_SIOE :: O > StateT Int MTa ()
> printTa_SIOE x = lift (DoneTa ((), x))
> incTaState :: StateT Int MTa ()
> incTaState = StateT (\s > return ((), s + 1))
> evalTa_SIOE :: Term > StateT Int MTa Int
> evalTa_SIOE (Con a) = do incTaState
> printTa_SIOE (formatLine (Con a) a)
> return a
> evalTa_SIOE (Add t u) = do a < evalTa_SIOE t
> b < evalTa_SIOE u
> incTaState
> let out = formatLine (Add t u) (a + b)
> printTa_SIOE out
> if (a+b) == 42
> then raiseTa_SIOE $
> out ++ "The Ultimate Answer Has Been Computed!! Now I'm tired!"
> else return (a + b)
> runEvalTa :: Term > String
> runEvalTa exp = case runStateT (evalTa_SIOE exp) 0 of
> FailTa e > e
> DoneTa (~(r,s),o)> "Result = " ++ show r ++
> "; Iteration = " ++ show s ++
> "; Output = " ++ o
> runEvalTa1 :: Term > String
> runEvalTa1 exp = case runState 0 (evalTa_SIOE exp) of
> FailTa e > e
> DoneTa ((r,s),o) > "Result = " ++ show r ++
> "; Iteration = " ++ show s ++
> "; Output = " ++ o
> runEvalTa2 :: Term > String
> runEvalTa2 exp = case evalState 0 (evalTa_SIOE exp) of
> FailTa e > e
> DoneTa (r,o) > "Result = " ++ show r ++ "; Output = " ++ o
> runEvalTa3 :: Term > String
> runEvalTa3 exp = case execState 0 (evalTa_SIOE exp) of
> FailTa e > e
> DoneTa (s,o) > "Iterations = " ++ show s ++ "; Output = " ++ o
StateT to keep output and counter, and monadic evaluator with (only) exceptions
Now we take output away from the inner monad and place it in the outer one (StateT):
> data MTb a = FailTb Exception
>  DoneTb {unpackDoneTb :: a }
> deriving (Show)
> type StateIO = (O,Int)
> instance Monad MTb where
> return a = DoneTb a
> m >>= f = case m of
> FailTb e > FailTb e
> DoneTb a > f a
> instance Functor MTb where
> fmap _ (FailTb e) = FailTb e
> fmap f (DoneTb b) = DoneTb (f b)
> raiseTb_SIOE :: O > StateT StateIO MTb a
> raiseTb_SIOE e = lift (FailTb e)
> printTb_SIOE :: O > StateT StateIO MTb ()
> printTb_SIOE x = StateT (\(o,s) > return ((), (o ++ x,s)))
> incTbStateIO :: StateT StateIO MTb ()
> incTbStateIO = StateT (\(o,s) > return ((), (o,s + 1)))
> evalTb_SIOE :: Term > StateT StateIO MTb Int
> evalTb_SIOE (Con a) = do incTbStateIO
> printTb_SIOE (formatLine (Con a) a)
> return a
> evalTb_SIOE (Add t u) = do a < evalTb_SIOE t
> b < evalTb_SIOE u
> incTbStateIO
> let out = formatLine (Add t u) (a + b)
> printTb_SIOE out
> if (a+b) == 42
> then raiseTb_SIOE $
> out ++ "The Ultimate Answer Has Been Computed!! Now I'm tired!"
> else return (a + b)
We take away complexity from >>= and put it in the function we need to use to manipulate content in our StateT monad.
These are some wrapper to the evaluator to get the result and the sideeffects produced by evaluation:
> runEvalTb :: Term > String
> runEvalTb exp = case runStateT (evalTb_SIOE exp) ("",0) of
> FailTb e > e
> DoneTb (r,~(o,s)) > "Result = " ++ show r ++
> "; Iteration = " ++ show s ++
> "; Output = " ++ o
> runEvalTb1 :: Term > String
> runEvalTb1 exp = case runState ("",0) (evalTb_SIOE exp) of
> FailTb e > e
> DoneTb (r,~(o,s)) > "Result = " ++ show r ++
> "; Iteration = " ++ show s ++
> "; Output = " ++ o
> runEvalTb2 :: Term > String
> runEvalTb2 exp = case evalState ("",0) (evalTb_SIOE exp) of
> FailTb e > e
> DoneTb r > "Result = " ++ show r
> runEvalTb3 :: Term > String
> runEvalTb3 exp = case execState ("",0) (evalTb_SIOE exp) of
> FailTb e > e
> DoneTb (o,s) > "Iterations = " ++ show s ++
> "  Output = " ++ o
StateT to keep output, counter and debug. The monadic evaluator is only for failable computations
And now we will keep in the inner monad only the result of the evaluation.
> data MT a = FailT Exc
>  DoneT {unpackDoneT :: a }
> deriving (Show)
> type Exc = String
> type IOstack = [Output]
> newtype StateTIO = StateTIO {unPackStateTIO :: (IOstack,Exc,Int)}
> deriving(Show)
> instance Monad MT where
> return a = DoneT a
> m >>= f = case m of
> FailT e > FailT e
> DoneT a > f a
> instance Functor MT where
> fmap _ (FailT a) = FailT a
> fmap f (DoneT a) = DoneT (f a)
Simple isn't it?
The complexity is now below:
> stopExecT_SIOE :: Output > StateT StateTIO MT Int
> stopExecT_SIOE exc = StateT (\s > do x < FailT exc
> return (x, s))
> catchT_SIOE exc = StateT (\(StateTIO (o,e,s)) >
> return ((), StateTIO (o ,"Exception at Iteration " ++
> show s ++ ": " ++ exc ++ "  " ++ e,s)))
> printT_SIOE :: Output > StateT StateTIO MT ()
> printT_SIOE x = StateT (\(StateTIO (o,e,s)) > return ((), StateTIO (x:o,e,s)))
> incTstateIO :: StateT StateTIO MT ()
> incTstateIO = StateT (\(StateTIO (o,e,s)) > return ((),StateTIO (o,e,s + 1)))
> evalT_SIOE :: Term > StateT StateTIO MT Int
> evalT_SIOE (Con a) = do incTstateIO
> printT_SIOE (formatLine (Con a) a)
> return a
> evalT_SIOE (Add t u) = do a < evalT_SIOE t
> b < evalT_SIOE u
> incTstateIO
> let out = formatLine (Add t u) (a + b)
> printT_SIOE out
> case (a+b) of
> 42 > do catchT_SIOE "The Ultimate Answer Has Been Computed!! Now I'm tired!"
> return (a+b)
> 11 > stopExecT_SIOE "11.... I do not like this number!"
> otherwise > return (a + b)
But now we have exceptions to stop execution and debugging output.
Some helper functions:
> runEvalT :: Term > String
> runEvalT exp = case runStateT (evalT_SIOE exp) (StateTIO ([],"",0)) of
> FailT e > e
> DoneT (r,StateTIO (o,e,s)) > "Result = " ++ show r ++ "; Iteration = " ++ show s ++
> "; Output = " ++ show o ++ "  Exceptions = " ++ e
> runEvalT1 :: Term > String
> runEvalT1 exp = case runState (StateTIO ([],"",0)) (evalT_SIOE exp) of
> FailT e > e
> DoneT (r,StateTIO(o,e,s)) > "Result = " ++ show r ++ "; Iteration = " ++ show s
> ++ "; Output = " ++ show o ++ "  Exceptions = " ++ e
> runEvalT2 :: Term > String
> runEvalT2 exp = case evalState (StateTIO ([],"",0)) (evalT_SIOE exp) of
> FailT e > e
> DoneT r > "Result = " ++ show r
> runEvalT3 :: Term > String
> runEvalT3 exp = case execState (StateTIO ([],"",0)) (evalT_SIOE exp) of
> FailT e > e
> DoneT (StateTIO (o,e,s)) > "Iterations = " ++ show s ++
> "  Output = " ++ show o ++ "  Exceptions = " ++ e
> showOut :: [String] > IO ()
> showOut [] = return ()
> showOut (a:xs) = do print a
> showOut xs
> runMyEval :: Term > IO ()
> runMyEval exp = let StateTIO (a,b,c) = unpackDoneT $ execState (StateTIO ([],"",0)) (evalT_SIOE exp) in
> showOut $ reverse a
Some tests:
*TheMonadicWay> runEvalT (Add (Con 18) (Add (Con 12) (Add (Con 10) (Con 2)))) "Result = 42; Iteration = 7; Output = [\"eval (Add (Con 18) (Add (Con 12) (Add (Con 10) (Con 2)))) <= 42  \", \"eval (Add (Con 12) (Add (Con 10) (Con 2))) <= 24  \", \"eval (Add (Con 10) (Con 2)) <= 12  \", \"eval (Con 2) <= 2  \", \"eval (Con 10) <= 10  \", \"eval (Con 12) <= 12  \", \"eval (Con 18) <= 18  \"]  Exceptions = Exception at Iteration 7: The Ultimate Answer Has Been Computed!! Now I'm tired!  " *TheMonadicWay> runEvalT2 (Add (Con 18) (Add (Con 12) (Add (Con 10) (Con 2)))) "Result = 42" *TheMonadicWay> runEvalT3 (Add (Con 18) (Add (Con 12) (Add (Con 10) (Con 2)))) "Iterations = 7  Output = [\"eval (Add (Con 18) (Add (Con 12) (Add (Con 10) (Con 2)))) <= 42  \", \"eval (Add (Con 12) (Add (Con 10) (Con 2))) <= 24  \", \"eval (Add (Con 10) (Con 2)) <= 12  \", \"eval (Con 2) <= 2  \", \"eval (Con 10) <= 10  \", \"eval (Con 12) <= 12  \", \"eval (Con 18) <= 18  \"]  Exceptions = Exception at Iteration 7: The Ultimate Answer Has Been Computed!! Now I'm tired!  " *TheMonadicWay> runEvalT3 (Add (Con 1) (Add (Con 7) (Add (Con 1) (Con 2)))) "Iterations = 7  Output = [\"eval (Add (Con 1) (Add (Con 5) (Add (Con 1) (Con 2)))) <= 9  \", \"eval (Add (Con 5) (Add (Con 1) (Con 2))) <= 8  \", \"eval (Add (Con 1) (Con 2)) <= 3  \", \"eval (Con 2) <= 2  \", \"eval (Con 1) <= 1  \", \"eval (Con 5) <= 5  \", \"eval (Con 1) <= 1  \"]  Exceptions = " *TheMonadicWay> runEvalT3 (Add (Con 1) (Add (Con 7) (Add (Con 1) (Con 2)))) "11.... I do not like this number!" *TheMonadicWay> runMyEval (Add (Add (Add (Add (Con 10) (Con 2)) (Add (Con 12) (Con 3))) (Con 3)) (Con 10)) "eval (Con 10) <= 10  " "eval (Con 2) <= 2  " "eval (Add (Con 10) (Con 2)) <= 12  " "eval (Con 12) <= 12  " "eval (Con 3) <= 3  " "eval (Add (Con 12) (Con 3)) <= 15  " "eval (Add (Add (Con 10) (Con 2)) (Add (Con 12) (Con 3))) <= 27  " "eval (Con 3) <= 3  " "eval (Add (Add (Add (Con 10) (Con 2)) (Add (Con 12) (Con 3))) (Con 3)) <= 30  " "eval (Con 10) <= 10  " "eval (Add (Add (Add (Add (Con 10) (Con 2)) (Add (Con 12) (Con 3))) (Con 3)) (Con 10)) <= 40  " *TheMonadicWay>
The Final Cut
StateT for output, counter, debug, using the Standard Library
module MyStateT where
import Control.Monad.State hiding (State)
data Term = Con Int
 Add Term Term
deriving (Show)
type IOStack = [Output]
type Output = String
type Debug = [String]
data EvalST = State {getIOS :: IOStack, getDebug :: Debug, getCount:: Int}
deriving(Show)
type Exception = String
data MT a = Fail Exception
 Done {unpackDone :: a }
deriving (Show)
type Eval s a = StateT s MT a
instance Monad MT where
return a = Done a
m >>= f = case m of
Fail e > Fail e
Done a > f a
instance Functor MT where
fmap _ (Fail a) = Fail a
fmap f (Done a) = Done (f a)
emptyState = State [] [] 0
stopExecT exc = lift $ Fail exc
catchT e = do st < get
let s = getCount st
let es = getDebug st
let o = getIOS st
let exc = "Debug msg at Iteration " ++ show s ++ ": " ++ e
put $ State o (exc:es) s
printT :: Output > Eval EvalST ()
printT o = do st < get
let s = getCount st
let e = getDebug st
let os = getIOS st
let out = show s ++ "  " ++ o
put $ State (out:os) e s
incTcounter :: Eval EvalST ()
incTcounter = do st < get
let s = getCount st
let e = getDebug st
let o = getIOS st
put $ State o e (s+1)
evalT :: Term > Eval EvalST Int
evalT (Con a) = do incTcounter
printT (formatLine (Con a) a)
return a
evalT (Add t u) = do a < evalT t
b < evalT u
incTcounter
let out = formatLine (Add t u) (a + b)
printT out
case (a+b) of
42 > do catchT "The Ultimate Answer Has Been Computed!! Now I'm tired!"
return (a+b)
11 > stopExecT "11.... I do not like this number!"
otherwise > return (a + b)
formatLine :: Term > Int > Output
formatLine t a = "eval (" ++ show t ++ ") <= " ++ show a
printAll :: [String] > IO ()
printAll [] = return ()
printAll (a:xs) = do putStrLn a
printAll xs
eval :: Term > IO ()
eval exp = case execStateT (evalT exp) emptyState of
Fail e > putStrLn e
Done (State a b c )
> do printAll $ reverse a
putStrLn $ show $ unpackDone $
evalStateT (evalT exp) emptyState
case b of
[] > putStrLn $ "Iterations: " ++ show c
_ > do printAll $ reverse b
putStrLn $ "Iterations: " ++ show c
 testing functions 
runEvalT :: Term > String
runEvalT exp = case runStateT (evalT exp) emptyState of
Fail e > e
Done (r,State o e s) > "Result = " ++ show r ++
"; Iteration = " ++ show s ++
"; Output = " ++ show o ++
"  Exceptions = " ++ show e
getEvalResult :: Term > String
getEvalResult exp = case evalStateT (evalT exp) emptyState of
Fail e > e
Done r > "Result = " ++ show r
getSideEffects :: Term > String
getSideEffects exp = case execStateT (evalT exp) emptyState of
Fail e > e
Done (State o e s) > "Iterations = " ++ show s ++
"  Output = " ++ show o ++
"  Exceptions = " ++ show e
{
Some runs:
*MyStateT> eval (Add (Add (Add (Add (Con 40) (Con 2)) (Add (Con 12) (Con 30))) (Con 3)) (Con 10))
1  eval (Con 40) <= 40
2  eval (Con 2) <= 2
3  eval (Add (Con 40) (Con 2)) <= 42
4  eval (Con 12) <= 12
5  eval (Con 30) <= 30
6  eval (Add (Con 12) (Con 30)) <= 42
7  eval (Add (Add (Con 40) (Con 2)) (Add (Con 12) (Con 30))) <= 84
8  eval (Con 3) <= 3
9  eval (Add (Add (Add (Con 40) (Con 2)) (Add (Con 12) (Con 30))) (Con 3)) <= 87
10  eval (Con 10) <= 10
11  eval (Add (Add (Add (Add (Con 40) (Con 2)) (Add (Con 12) (Con 30))) (Con 3)) (Con 10)) <= 97
97
Debug msg at Iteration 3: The Ultimate Answer Has Been Computed!! Now I'm tired!
Debug msg at Iteration 6: The Ultimate Answer Has Been Computed!! Now I'm tired!
Iterations: 11
*MyStateT>
*MyStateT> eval (Add (Add (Add (Add (Con 10) (Con 2)) (Add (Con 12) (Con 3))) (Add (Con 5) (Con 2))) (Con 2))
1  eval (Con 10) <= 10
2  eval (Con 2) <= 2
3  eval (Add (Con 10) (Con 2)) <= 12
4  eval (Con 12) <= 12
5  eval (Con 3) <= 3
6  eval (Add (Con 12) (Con 3)) <= 15
7  eval (Add (Add (Con 10) (Con 2)) (Add (Con 12) (Con 3))) <= 27
8  eval (Con 5) <= 5
9  eval (Con 2) <= 2
10  eval (Add (Con 5) (Con 2)) <= 7
11  eval (Add (Add (Add (Con 10) (Con 2)) (Add (Con 12) (Con 3))) (Add (Con 5) (Con 2))) <= 34
12  eval (Con 2) <= 2
13  eval (Add (Add (Add (Add (Con 10) (Con 2)) (Add (Con 12) (Con 3))) (Add (Con 5) (Con 2))) (Con 2)) <= 36
36
Iterations: 13
*MyStateT>
*MyStateT> eval (Add (Con 5) (Con 6))
11.... I do not like this number!
*MyStateT>
}
Next?
We need a parser to get a string from input and turn into something of type Term!
Let's see if we'll time for it... Fist we must complete the text above!!
Suggested Readings
Cale Gibbard, Monads as Containers
Jeff Newbern, All About Monads
You Could Have Invented Monads! (And Maybe You Already Have.) by sigfpe
Acknowledgments
Thanks to Neil Mitchell, Daniel Fisher, Bulat Ziganzhin, Brian Hulley and Udo Stenzel for the invaluable help they gave, in the haskellcafe mailing list, in understanding this topic.
I couldn't do it without their help.
Obviously errors are totally mine. But this is a wiki so, please, correct them!