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Baby’s First Proof

Unlike many languages that you learn, in Coq, things are truly different. Much like your first functional language after using nothing but imperative languages, you have to re-evaluate things. Instead of just defining functions, you have to prove properties of them. So, let’s take a look at a few basic ways to do that.

Simpl and Reflexivity

Here we have two basic “tactics” that we can use to prove simple properties. Suppose we have some function addition. We’re all familiar with how this works; 2 + 2 = 4, right? Prove it:

Lemma two_plus_two: 2 + 2 = 4. Proof. Admitted.

First, what is this Admitted. thing? Admitted basically tells Coq not to worry about it, and just assume it is true. This is the equivalent of your math professor telling you “don’t worry about it, Aristotle says it’s true, are you calling Aristotle a liar?” and if you let this make it into live code, you are a bad person. We must make this right!

Lemma two_plus_two: 2 + 2 = 4. Proof. simpl. reflexivity. Qed.

That’s better. This is a simple proof; we tell Coq to simplify the expression, then we tell Coq to verify that the left-hand-side is the same as the right-hand-side. One nice feature of Coq is that lets you step through these proofs to see exactly how the evaluation is proceeding. If you’re using Proof General, you can use the buttons Next, Goto, and Undo to accomplish this. If you put the point at Proof. and click Goto, Coq will evaluate the buffer up to that point, and a window should appear at the bottom with the following:

1 subgoals, subgoal 1 (ID 2) ============================ 2 + 2 = 4

This is telling you that Coq has 1 thing left to prove: 2 + 2 = 4. Click next, the bottom should change to:

1 subgoals, subgoal 1 (ID 2) ============================ 4 = 4

Coq processed the simpl tactic and now the thing it needs to prove is that 4 = 4. Obviously this is true, so if we click next…

No more subgoals.

reflexivity should succeed, and it does. If we click next one more time:

two_plus_two is defined

This says that this Lemma has been defined, and we can now refer to it in other proofs, much like we can call a function. Now, you may be wondering “do I really have to simplify 2 + 2?” No, you don’t, reflexivity will simplify on it’s own, this typechecks just fine:

Lemma two_plus_two: 2 + 2 = 4. Proof. reflexivity. Qed.

So, what’s the point of simpl then? Let’s consider a more complicated proof.

Induction

Lemma n_plus_zero_eq_n: forall (n : nat), n + 0 = n.

This lemma state that for any n, n + 0 = n. This is the same as when you’d write in some math. Other bits of new syntax is n : nat, which means that n has the type nat. The idea here is that no matter what natural number n is, n + 0 = n. So how do we prove this? One might be tempted to try:

Lemma n_plus_zero_eq_n: forall (n : nat), n + 0 = n. Proof. reflexivity. Qed.

One would be wrong. What is Coq stupid? Clearly n + 0 = n, Aristotle told me so! Luckily for us, this is a pretty easy proof, we just need to be explicit about it. We can use induction to prove this. Let me show the whole proof, then we’ll walk through it step by step.

Lemma n_plus_zero_eq_n: forall (n : nat), n + 0 = n. Proof. intros n. induction n as [| n']. { reflexivity. } { simpl. rewrite -> IHn'. reflexivity. } Qed.

Place the point at Proof and you’ll see the starting goal:

1 subgoals, subgoal 1 (ID 6) ============================ forall n : nat, n + 0 = n

Click next and step over intros n.

1 subgoals, subgoal 1 (ID 7) n : nat ============================ n + 0 = n

What happened here is intros n introduces the variable n, and names it n. We could have done intros theNumber and the bottom window would instead show:

1 subgoals, subgoal 1 (ID 7) theNumber : nat ============================ theNumber + 0 = theNumber

The intros tactic reads from left to right, so if we had some Lemma foo : forall (n m : nat), [stuff], we could do intros nName mName., and it would read in n, and bind it to nName, and then read in m and bind it to mName. Click next and evaluate induction n as [| n'].

2 subgoals, subgoal 1 (ID 10) ============================ 0 + 0 = 0 subgoal 2 (ID 13) is: S n' + 0 = S n'

The induction tactic implements the standard proof by induction, splitting our goal into two goals: the base case and the n + 1 case. Similarly to intros, this will create subgoals starting with the first constructor of an ADT, and ending with the last.

On Natural Numbers in Coq

Let us take a second to talk about how numbers are represented in Coq. Coq re-implements all types within itself, so nat isn’t a machine integer, it’s an algebraic datatype of the form:

Inductive nat : Set := | O : nat | S : nat -> nat.

O is zero, S O is one, and S (S (S (O))) is three. There is a lot of syntax sugar in place that lets you write 49 instead of S (S ( ... ( S O) ... )), and that’s a good thing.

The point of all of this is that we can pattern match on nat much like we can a list.

More Induction

…anyways, all this brings us back to induction and this mysterious as [| n']. What this is doing is binding names to all the fields of the ADT we are deconstructing. The O constructor takes no parameters, so there is nothing to the left of the |. The S constructor takes a nat, so we give it the name n'. Click next and observe the bottom change:

1 focused subgoals (unfocused: 1) , subgoal 1 (ID 10) ============================ 0 + 0 = 0

The curly braces “focuses” the current subgoal, hiding all irrelevant information. Curly braces are optional, but I find them to be very helpful as the bottom window can become very cluttered in large proofs. Here we see the base case goal being to prove that 0 + 0 = 0. Obviously this is true, and we can have Coq verify this by reflexivity. Click next until the next opening curly brace is evaluated. We see the next subgoal:

1 focused subgoals (unfocused: 0) , subgoal 1 (ID 13) n' : nat IHn' : n' + 0 = n' ============================ S n' + 0 = S n'

So, what do we have here? This is the n + 1 case; here the n’ in S n' is the original n. A particularly bored reader may try to prove forall (n : nat), S n = n + 1 and I’ll leave that as an exercise. However, this follows from the definition of nat.

Also of note here is IHn'. IH stands for induction hypothesis, and this is that n’ + 0 = n’. So, how do we proceed? Click next and observe how the subgoal changes:

1 focused subgoals (unfocused: 0) , subgoal 1 (ID 15) n' : nat IHn' : n' + 0 = n' ============================ S (n' + 0) = S n'

It brought the + 0 inside the S constructor. Notice that now there is n' + 0 on the left hand side. Click next and watch closely what happens:

1 focused subgoals (unfocused: 0) , subgoal 1 (ID 16) n' : nat IHn' : n' + 0 = n' ============================ S n' = S n'

Here we use the induction hypothesis to rewrite all occurrences of n' + 0, which was the left hand side of the induction hypothesis as n', which was the right hand side of the induction hypothesis. This is what the rewrite tactic does. Notice now that the subgoal is S n' = S n' which reflexivity will surely find to be true. So, what would happen if we had done rewrite <- IHn'.?

1 focused subgoals (unfocused: 0) , subgoal 1 (ID 16) n' : nat IHn' : n' + 0 = n' ============================ S (n' + 0 + 0) = S (n' + 0)

It rewrote all instances of n', which was the right hand side of the induction hypothesis with n' + 0 which was the left hand side of the induction hypothesis. Obviously, this isn’t what we want. I should note that you can undo this by rewriting to the right twice…

{ simpl. rewrite <- IHn'. rewrite -> IHn'. rewrite -> IHn'. reflexivity. }

…and it will technically work. But don’t do this, it’s silly and there’s no room for silliness in a rigorous mathematical proof.

Personally, I have a hard time keeping it straight what the left and right rewrites do. I sometimes find myself just trying one, and then the other if I guessed wrong. Think of it like this: rewrite -> foo rewrites the current goal, replacing all occurrences of the thing on the left hand side of the equation of foo with the thing on the right hand side of the equation. It changes from the left to the right. And vice-versa for rewrite <-, which changes from the right to the left.

Functional Doubly Linked Lists

It’s often been said that functional programming just isn’t cut out for certain tasks. File IO? Please… Databases? Forget about it!

I’ve always figured that the humble Doubly Linked list was on this list. After all, how do we implement these in C? An implementation would probably look something like this:

struct _DList { struct DList * next; struct DList * prev; void * element; } DList;

In this case, the element pointed to by prev and next have pointers to this element, if they aren’t NULL. The Doubly Linked list isn’t exactly a complicated structure, this is basically the way to do it. So, how would we do this in Haskell?

In the past, I’ve thought there were three answers to this question:

1) Use C pointers. This would involve use of unsafePerformIO, and you’d be a monster.

2) Use a singly linked list, and pretend it’s doubly linked. This would involve a “prev” function that just walks from the beginning of the list to the element before the current one. You’d be an even bigger monster than the guy who did option 1.

3) Don’t use a doubly linked list. To me, this is actually reasonable. In a world where arrays are basically always better, the only reason to use a list is because they’re very easy to deal with. If you need the level of complexity of a doubly linked list, you’re probably just better off with a different data structure.

But recently, I thought of a way to implement a doubly linked list in Haskell without being a bad person.

Preserved Here For Posterity

The implementation is actually quite simple. Here’s our type:

data DoublyLinkedList a = DList [a] [a] | Nil

Obviously, we have our Nil type for the empty list. We also have DList, which has two lists. Why two? On the left, we have the previous elements. This starts empty. On the right, we have our next elements. As we walk the list, we pop elements from the head of the right list to the head of the left list. The head of the right list is our current element. Let’s see some functions:

get :: DoublyLinkedList a -> a get (DList _ (x:xs)) = x

This function returns the “current” element; the element at the current position in the list.

next :: DoublyLinkedList a -> DoublyLinkedList a next (DList _ []) = undefined next (DList prev (c:next)) = DList (c:prev) next prev :: DoublyLinkedList a -> DoublyLinkedList a prev (DList [] _) = undefined prev (DList (c:prev) next) = DList prev (c:next)

These two functions move the cursor forward or backward. As you can see, when next is called, the current element is popped from the next list, and pushed onto the previous list. The opposite happens when prev is called. I’ve left the edge cases undefined, but a real implementation should do something sane here. (return a Maybe, throw an exception, etc…)

insert :: DoublyLinkedList a -> a -> DoublyLinkedList a insert Nil e = DList [] [e] insert (DList prev next) e = DList prev (e:next)

Elements are inserted in front of the current element.

makeDouble :: [a] -> DoublyLinkedList a makeDouble [] = Nil makeDouble l = DList [] l

…and it is trivial to convert a singly linked list into a doubly linked list. We can even add a pretty show instance!

instance (Show a) => Show (DoublyLinkedList a) where show Nil = "Nil" show (DList prev (c:next)) = (show $ reverse prev) ++ "*[" ++ (show c) ++ "]*" ++ show next

Obviously, we could go crazy and make a Monad instance and other things, but you get the idea. The final solution was very simple. One could even say simpler than the C version as you don’t have to worry about updating the next and prev pointers when adding elements. Sure, it took a bit of thought, but like many things in Haskell, the result was simple and elegant.

Checking Our Heads With Liquid Haskell

Consider this function:

headInTail :: [a] -> [a] headInTail l = (tail l) ++ [head l]

Pretty straightforward, right? It takes a list, extracts the head and sticks it in the tail. Surely you’ve written something like this before. It should be fine, right?

*Main> headInTail [1,2,3] [2,3,1]

…checks out. Let’s try a few more:

*Main> headInTail "hello" "elloh" *Main> headInTail ["cat"] ["cat"]

…good. And the moment we’ve all been waiting for:

*Main> headInTail [] *** Exception: Prelude.tail: empty list

Oh yeah, head and tail don’t work for empty lists… Normally, we have some choices on how to proceed here. We could wrap the function in a Maybe:

maybeHeadInTail :: [a] -> Maybe [a] maybeHeadInTail [] = Nothing maybeHeadInTail l = Just $ headInTail l

…which introduces an annoying Maybe to deal with just to stick our heads in our tails. Or, we could just do something with the empty list:

headInTail :: [a] -> [a] headInTail [] = [] headInTail l = (tail l) ++ [head l]

…but what if returning the empty list isn’t the correct thing to do?

Another choice is to document this behavior (as head and tail do), and just never call headInTail []. But how can we guarantee that we never attempt to call this function on an empty list? Shouldn’t this be the type system’s problem?

Unfortunately, not all is roses and puppies. Sometimes the type system cannot help us. Sometimes somebody thought it’d be a good idea to use Haskell’s Evil exception system. Whatever the case, there are tools to help us.

Liquid Haskell

Liquid Haskell is a static code analysis tool that is used to catch just these sorts of problems. Liquid Haskell allows us to define invariants which will be enforced by the tool. Liquid Haskell is a research project that is still in development. As such, it has some rough spots, however it’s still very much capable of helping us with our problem here. But before we begin, we need to get the tool installed.

To install the tool, execute:

cabal install liquidhaskell

…simple right? Unfortunately, we’re not quite done. We need to install an SMT solver. This tool is used by Liquid Haskell. Currently, the tool defaults to Z3. I’m not sure how to use a different solver (and Z3 works just fine), so I suggest you you Z3. You’ll have to build Z3, and place the binary somewhere on the PATH. After this is done, and assuming your .cabal/bin directory is also on the PATH, testing your source file is a simple as:

liquid [FILE].hs

Let’s Have A Look

Create a haskell source file that contains the following:

headInTail :: [a] -> [a] headInTail l = (tail l) ++ [head l]

After that’s done, let Liquid Haskell analyze your file:

liquid [YOUR_FILE].hs

A bunch of stuff should scroll by, then in a second you’ll see something similar to the following:

**** UNSAFE ********************************************* Doop.hs:5:22: Error: Liquid Type Mismatch Inferred type VV : [a] | VV == l && len VV >= 0 not a subtype of Required type VV : [a] | len VV > 0 In Context VV : [a] | VV == l && len VV >= 0 l : [a] | len l >= 0

If you go to the line and column indicated, you’ll find the argument to tail. Conveniently, it seems that Liquid Haskell comes pre-loaded with definitions for some library functions. Normally, you’ll have to define those yourself. In fact, let’s do just that.

The next logical step here is to write a specification for our function. This specification is a statement about what sorts of values the function can take. Add the following to your source file, in the line above the signature for headInTail:

{-@ headInTail :: {l:[a] | len l > 0} -> [a] @-}

If you re-run liquid on your source file, you’ll see that the warning went away, and the program now indicates that your source is “SAFE”. So, what does this refinement mean?

Basically, these refinements are machine-checked comments. They have no impact on the program, they exist for Liquid Haskell. Think of it as being like an enhanced type signature. Like a normal type signature, we start with the name of the function, then two colons. This part, however:

{l:[a] | len l > 0}

…should be new. Basically, this part says that the list should not be empty. You should read it as “l is a [a] such that len l is greater than zero.” A lot of the notation used by Liquid Haskell comes from formal logic. Let’s break this down some more:

l:[a]

Here we bind a symbol, l, to the first list argument. At any point to the right of this symbol until the end of the scope defined by {}, we can reference this symbol.

{... | ...}

The pipe symbol indicates that we are going to make some statement about the type on the left hand side.

len l > 0

Here we state that the length of l must be greater than 0. It looks like we are calling a function, and we sort of are; len is a measure which is a special function that is used in specifications. However, the subject of measures is a post for another day.

You may now be thinking: “Well this is all well and good, but what’s to stop me from calling this function on an empty list?” To answer that, let’s implement main:

main = do i <- getLine putStrLn $ headInTail i

Add this to your source file, and then run liquid [YOUR_FILE].hs and you’ll notice that Liquid Haskell has a problem with your attempt to call headInTail:

**** UNSAFE ********************************************* Doop.hs:3:29: Error: Liquid Type Mismatch Inferred type VV : [Char] | VV == i && len VV >= 0 not a subtype of Required type VV : [Char] | len VV > 0 In Context VV : [Char] | VV == i && len VV >= 0 i : [Char] | len i >= 0

Liquid Haskell is telling you that it can’t prove that the length of i is greater than 0. If you execute your main function, you should see that it works as expected. Type in a string, and it’ll do the right thing. Push enter right away and you’ll get an exception.

*Main> main hello elloh *Main> main *** Exception: Prelude.tail: empty list

…ick… Let’s fix this:

main = do i <- getLine case i of [] -> putStrLn "Get your head checked." _ -> putStrLn $ headInTail i

Now if you analyze your file, Liquid Haskell should be happy. Honestly, you should be happy too: the tool caught this problem for you. It didn’t go to testing messed up, and the bug certainly didn’t escape testing unfound. Your program is now objectively better:

*Main> main isn't that neat? sn't that neat?i *Main> main Get your head checked.

My Kingdom For a File Browser

As you may know, lately I’ve been trying to get used to using emacs. This has been going well, but there are some things I find myself missing from my old editors. One of them is the files list on the left side:

file_browser

Sure, I can use dired, Speedbar, or any number of other solutions. But none of these work exactly like my old file browser. In despair, I got used to the C-x C-f Do [tab] c [tab] [tab] d [tab] p [tab] [tab] ... workflow that autocompletion brings us. However, it never quite settled with me.

Lately, I’ve found myself using org-mode in a way that has been quieting the proverbial fire. I’ve been able to use org-mode as a project browser, let me show you how!

Some Background

As you may know, org-mode is kind of a catch-all tool for organizing, note taking, and keeping todo lists. You can create hierarchical trees of headings. Hierarchical trees… Sound like anything we know?

I’ve been trying to use org-mode primarily as a TODO list to keep track of all the stuff I have going on. For projects that I’m working on, I’ve been creating TODO lists for the various source files I’m writing. A few links later, and we have the beginning of a project browser.

* TODO programming project

One of the nice things about tracking your own project is that you can decide what the tree looks like. An embedded project might group C and assembly source files separately, where a Haskell project might group source files by module nesting level.

Let’s talk C. We can create a project.org file in the root of our project directory (where the Makefile is). Next, let’s create some headers:

* C * ASM * Unit Tests * Misc

Here, I’ve created 4 groups. C source goes in the C group, Assembly source goes in ASM, Unit test source files go in Unit Tests, and miscellaneous stuff like Makefiles, .gitignores, and READMEs go in Misc.

Next, let’s add some source files:

* C ** [[./main.c]] * ASM * Unit Tests * Misc

Here, I’ve added a link to main.c to the C group. At this point, this file doesn’t exist. If you click on main.c with org-mode enabled, the file will be opened in emacs, creating it if doesn’t exist! Sure, it’s manual, but I find it is not hard to keep up. If you commit your project.org to your version control system, all members of the project can keep it updated and in sync.

However, what the manual updating buys you is the freedom to organize it how you like! We can add tasks to the tree:

* C ** [[./main.c]] *** TODO implement main *** TODO file header *** TODO add to git * ASM * Unit Tests * Misc

We can add deadlines:

*C ** [[./main.c]] *** TODO implement main DEADLINE: <2015-02-13 Fri> *** TODO file header *** TODO add to git * ASM * Unit Tests * Misc

We can add commentary:

* C ** [[./main.c]] *** TODO implement main DEADLINE: <2015-02-13 Fri> - I tried to implement main, but printf is such a hard function to use! I'll revisit this after I've had four beers... *** TODO file header *** TODO add to git * ASM * Unit Tests * Misc

…and this can lead to more tasks!

* C ** [[./main.c]] *** TODO implement main DEADLINE: <2015-02-13 Fri> - I tried to implement main, but printf is such a hard function to use! I'll revisit this after I've had four beers... **** TODO drink four beers *** TODO file header *** TODO add to git * ASM * Unit Tests * Misc

Let’s see your silly little file browser window do that!

Fun With Undo-Tree-Mode

No more. This ends now. This will not stand!

What’s my problem, you ask? I’ll tell you what. Emacs’ stupid undo system is my problem. Humor me for a second; fire up Emacs, and type the following:

foo bar baz

Still with me? Good, now press

C-x u

…baz disappeared. “Well, duh! If you’re gonna whine about the fact that C-u doesn’t undo, then maybe you shouldn’t be using Emacs…” you say. Were that my issue, I’d be inclined to agree. Unfortunately, the problem is much worse. Enter the following into your buffer:

oof

…then…

C-x u C-x u

See that? Yeah… Not OK. The buffer should be:

foo

with the entry of oof and bar having been undone. Instead, the buffer is:

foo bar baz

with the entry of oof and the undoing of the undoing of the entry of baz having been undone.

Apparently, an undo operation itself goes onto the undo stack in Emacs. Meaning that if you undo enough times, it will redo all the undone operations, then undo them again, then undo the operations before. This lets us get the buffer back into any state it has ever been in, which sounds nice on paper. In practice, it’s yet another way that Emacs goes its own way, the rest of the world be darned. And I gotta say, its pretty annoying. In a normal program, you can just hold Ctrl+u, and watch the undoes fly. In Emacs, you get to press C-x, release, press u, and release to undo one thing. Then do it again n times.

Luckily for us, this being Emacs, there is likely a better way out there.

undo-tree-mode

Undo tree mode is a minor mode for Emacs that helps us deal with this undo behavior. In the short amount of time that I’ve played with it, I’d say it makes the on-paper benefit of Emacs’ undo behavior a reality. But first, some installation.

Undo-tree-mode is in Melpa stable, however, I was unable to get it to work. To install undo-tree-mode, you can install it via Melpa, then add the following to your .emacs file:

(global-undo-tree-mode)

Unfortunately, when I restarted Emacs, I got the following error:

Symbol's function definition is void: global-undo-tree-mode

Adding:

(require 'undo-tree)

does nothing to alleviate this situation, neither does opening undo-tree.el and entering:

M-x byte-compile-file

However, I was able to install it manually. First, I deleted all the files for undo-tree-mode from my .emacs.d/ directory. Then I downloaded the current version of the plugin from the author's website, and placed it in my ~/.emacs.d/, naming it undo-tree.el. Finally, I added the following lines to my .emacs file:

(add-to-list 'load-path "~/.emacs.d/") (require 'undo-tree) (global-undo-tree-mode t)

That first line is only required if you haven’t already added it elsewhere. Restart Emacs, and you should see that the Undo-Tree minor mode is active.

Climbing The Tree

Now, let’s try that previous sequence of events.

foo bar baz C-x u

A new window should appear with the following in it:

s | | o | | o | | o | | o | | x

Neat. This represents the operations that you did. Try clicking through them (you can also use the arrow keys to browse). Each node is a state of the document, x is the current state, and s is the initial state. Select the node two nodes up from the leaf to undo entry of baz. Lets continue our sequence:

oof C-x u

You should see a new tree that looks something like this:

s | | o | | o | | o | | o | / \ x o | | o

If you click around the tree, you’ll see the various states of your buffer. You’ll see that it is simple to restore any previous state of the buffer.

Now this is something I can get used to!

Again, This Time In Reverse!

Today we’ll be doing Exercise 5 from The C Programming Language. Today’s exercises aren’t terribly challenging, but there is some good stuff in here. The problem is:

Modify the temperature conversion program to print the table in reverse order, that is, from 300 degrees to 0.

I bet you thought we were done with temperature conversion, didn’t you? I’m right there with you, but luckily for us, this is also the section that introduces the for loop, so this is much less tedious. The new program for modification:

#include <stdio.h> int main (int argc, char ** argv) { for (int fahr = 0; fahr <= 300; fahr = fahr + 20) { printf("%3d %6.1f\n", fahr, (5.0/9.0)*(fahr-32)); } }

I made a few minor modifications. I gave main the correct signature to keep gcc from complaining, and I also moved the initialization of fahr to the loop. As you may know this is not valid C, and the compiler will throw an error. You might also know that this was made to be valid C in the C 99 revision. To compile this, you just need to add the -std=c99 flag to your gcc call.

To complete this exercise, you only need to modify the loop declaration. Change this line…

for (int fahr = 0; fahr <= 300; fahr = fahr + 20)

…to this…

for (int fahr = 300; fahr >= 0; fahr = fahr - 20)

Pretty straight forward stuff here. fahr starts at 300 instead of 0, and the loop continues so long as it remains greater than 0. Instead of adding 20 each iteration, we subtract it. This can be compiled like so:

gcc -std=c99 -Wall ex5.c -o ex5

In Haskell

Unfortunately, the authors of The C Programming Language did not see fit to include a starting Haskell program for us to modify. This seems like a pretty serious oversight to me, but we’ll just have to make due. Luckily for us, we can use the temperature conversion program I wrote for the last entry in this series. This program handles conversions, and we can use it to produce a table of all conversions between 0 and 300 degrees.

While I’ve established that Haskell does in fact have loops, we won’t be using them here. Instead we’ll be using ranges, functors, and sequencing to solve this problem in a more functional way.

First, we need a list of every 20th number between 0 and 300:

[300.0, 280.0 .. 0.0]

The .. there basically says “You get the idea, do that.” If you put enough information into it so that the compiler can figure out the pattern, and start and end points, the compiler will fill the rest in. In this case, I gave the first two values, telling the compiler I want increments of 20, and I gave it the last value so it knows where to stop.

Unfortunately, as you recall, my conversion program needs members of the Temperature typeclass. Surely I don’t plan to make Double a member, right? Right. We need to map a function over this list to produce a list of Temperatures. But what function should we use?

Something that beginning Haskellers may not realize is that constructors are in fact functions!

Prelude> :t Just Just :: a -> Maybe a Prelude> :t Left Left :: a -> Either a b

…and of course…

*Main> :t Fahrenheit Fahrenheit :: Double -> Fahrenheit

That’s right, we can map Fahrenheit over a functor just like any other function!

map Fahrenheit [300.0, 280.0 .. 0.0]

This will produce a list of every 20th Fahrenheit temperature between 300 and 0. However, we aren’t done yet. We actually need a list of Conversions, because this type is required for our insertConv function. To get this we can map toConversion over this list:

map toConversion $ map Fahrenheit [300.0, 280.0 .. 0.0]

…but that’s starting to get a bit ugly, there must be a better way. Normally, I’m not terribly fond of function composition, in this case it will make our life easier.

Function Composition

Haskell provides an operator . that is used for function composition. Let’s take a look at its type:

Prelude> :t (.) (.) :: (b -> c) -> (a -> b) -> a -> c

The composition operator takes a function that takes some type b as an argument, and that returns some type c. It takes a second function that takes some type a as an argument, and that returns some type b. Finally, it returns some type c.

What does this do for us? Basically, it takes a function a -> b, and a function b -> c, and turns them into a function a -> c. Let’s see an example:

Prelude> :t show show :: Show a => a -> String

The function show takes a member of the Show typeclass and returns a String.

Prelude> :t putStr putStr :: String -> IO ()

The function putStr takes a String, and returns an IO action.

Prelude> :t putStr . show putStr . show :: Show a => a -> IO ()

When we compose the two functions, we get a function that takes a member of the Show typeclass, and returns an IO action.

Logically, calling putStr . show someShowable is the same as calling putStr $ show someShowable or putStr (show someShowable). However, putStr . show is a valid function, where putStr $ show and putStr (show) are compiler errors if you don’t give show an argument.

How does this help us?

*Main> :t Fahrenheit Fahrenheit :: Double -> Fahrenheit *Main> :t toConversion toConversion :: (Temperature a, Temperature b) => a -> (a, b) *Main> :t toConversion . Fahrenheit toConversion . Fahrenheit :: Temperature b => Double -> (Fahrenheit, b)

By composing toConversion and Fahrenheit, we end up with a function that takes a Double as an argument and returns a Conversion from Fahrenheit. We can then map this function over our list of Doubles:

map (toConversion . Fahrenheit) [300.0, 280.0 .. 0.0]

Next, we need to turn these Conversions into TableBuilders. This is a simple matter of composing insertConv:

map (insertConv . toConversion . Fahrenheit) [300.0, 280.0 .. 0.0]

The type of this new function is:

:t (insertConv . toConversion . Fahrenheit) (insertConv . toConversion . Fahrenheit) :: Double -> TableBuilder ()

…or at least it would be if the type inferrer could figure out the type of toConversion. Unfortunately for us, it can’t because the implementation is purposely ambiguous. We need to add a type signature to it:

(insertConv . (toConversion :: Fahrenheit -> (Fahrenheit, Celsius)) . Fahrenheit)

Sequencing

Now we are ready to create our table. This part is actually very easy. Thanks to the library function sequence_ we won’t even need a do block. What does this function do? Let’s look at its signature:

Prelude> :t sequence_ sequence_ :: Monad m => [m a] -> m () Prelude> :t sequence sequence :: Monad m => [m a] -> m [a]

There are two variations of this function. The first, sequence_ evaluates each monadic action, and ignores the results. Basically, suppose we have some function: func :: a -> Monad b

let mList = map func ["foo", "bar", "baz"] in sequence_ mList

…is equivalent to…

do func "foo" func "bar" func "baz" return ()

The second variation evaluates each monadic action, and returns a list of all the results. Suppose we have our same function: func :: a -> Monad b

let mList = map func ["foo", "bar", "baz"] in sequence mList

…is equivalent to…

do foo <- func "foo" bar <- func "bar" baz <- func "baz" return [foo, bar, baz]

If you have a list of monads, you can use sequence or sequence_ to have them all evaluated. Use sequence if you care about the resulting values, use sequence_ if you only care about the monad’s side effect. Which do we want? Let’s look at the signature of insertConv:

*Main> :t insertConv insertConv :: (Temperature a, Temperature b) => Conversion a b -> TableBuilder ()

If we were to use sequence, we’d get a resulting value with the type TableBuilder [()]. Since nobody has any use for a list of empty tuples, we’ll be using sequence_.

So, what does our main look like?

main = do temps <- return (map (insertConv . (toConversion :: Fahrenheit -> (Fahrenheit, Celsius)) . Fahrenheit) [300.0, 280.0 .. 0.0]) putStrLn $ prettyPrint $ buildTable $ do sequence_ temps

This produces the following output:

------------------------------ || Fahrenheit || Celsius || ------------------------------ || 300.0 |> 148.9 || || 280.0 |> 137.8 || || 260.0 |> 126.7 || || 240.0 |> 115.6 || || 220.0 |> 104.4 || || 200.0 |> 93.3 || || 180.0 |> 82.2 || || 160.0 |> 71.1 || || 140.0 |> 60.0 || || 120.0 |> 48.9 || || 100.0 |> 37.8 || || 80.0 |> 26.7 || || 60.0 |> 15.6 || || 40.0 |> 4.4 || || 20.0 |> -6.7 || || 0.0 |> -17.8 || ------------------------------

K&R Challenge 3 and 4: Functional Temperature Conversion

The other day, I implemented the C solution to exercises 3 and 4 in The C Programming Language, today I’ll be implementing the Haskell solution. As a reminder, the requirements are:

Modify the temperature conversion program to print a heading above the table

… and …

Write a program to print the corresponding Celsius to Fahrenheit table.

I could take the easy way out, and implement the Haskell solution almost identically to the C solution, replacing the for loop with a call to map, but that’s neither interesting to do, nor is it interesting to read about. I’ll be taking this problem to the next level.

Requirements

For my temperature program, I’d like it to be able to convert between any arbitrary temperature unit. For the purposes of this post, I will be implementing Celsius, Fahrenheit, Kelvin (equivalent to Celsius, except 0 degrees is absolute zero, not the freezing point of water), and Rankine (the Fahrenheit version of Kelvin).

That said, nothing about my solution should rely on the fact that these four temperature scales are being implemented. A programmer should be able to implement a new temperature unit with minimal changes to the program. Ideally, just by implementing a new type.

Additionally, given a bunch of conversions, the program should be able to output a pretty table showing any number of temperature types and indicating what converts to what.

First, Conversion

This problem is clearly broken up into two sub-problems: conversion, and the table. First, we need to handle conversion. This being Haskell, the right answer is likely to start by defining some types. Let’s create types for our units:

newtype Celsius = Celsius Double deriving (Show) newtype Fahrenheit = Fahrenheit Double deriving (Show) newtype Kelvin = Kelvin Double deriving (Show) newtype Rankine = Rankine Double deriving (Show)

I’ve chosen to make a type for each unit, and since they only contain one constructor with one field, I use a newtype instead of a data. Now, how to convert these? A straightforward solution to this would be to define functions that convert each type to each other type. Functions that look like:

celsiusToFahrenheit :: Celsius -> Fahrenheit celsiusToKelvin :: Celsius -> Kelvin ... rankineToKelvin :: Rankine -> Kelvin rankineToCelsius :: Rankine -> Celsius

A diagram for these conversions looks like this:

many-to-many

That’s a lot of conversions! One might argue that it’s manageable, but it certainly doesn’t meet requirement #1 that implementing a new unit would require minimal work; to implement a new conversion, you’d need to define many conversion functions as well! There must be a better way.

Let’s think back to chemistry class. You’re tasked with converting litres to hours or somesuch. Did you do that in one operation? No, you used a bunch of intermediate conversion to get to what you needed. If you know that X litres are Y dollars, and Z dollars is one 1 hour, then you know how many litres are in 1 hour! These are called conversion factors.

Luckily for us, our conversions are much simpler. For any temperature unit, if we can convert it to and from celsius, then we can convert it to and from any other unit! Let’s define a typeclass for Temperature:

class Temperature a where toCelsius :: a -> Celsius fromCelsius :: Celsius -> a value :: a -> Double scaleName :: a -> String convert :: (Temperature b) => a -> b convert f = fromCelsius $ toCelsius f

Our Temperature typeclass has five functions: functions to convert to and from celsius, a function to get the value from a unit, a function to get the name of a unit, and a final function convert. This final function has a default implementation that converts a unit to celsius, then from celsius. Using the type inferrer, this will convert any unit to any other unit!

convert Rankine 811 :: Kelvin convert Celsius 123 :: Fahrenheit convert Kelvin 10000 :: RelativeToHeck

Now to implement a new temperature, you only need to implement four functions, as convert is a sane solution for all cases. This arrangement gives us a conversion diagram that looks like:

many-to-one

Much better. Let’s go through our Temperature implementations for our types:

instance Temperature Celsius where toCelsius c = c fromCelsius c = c value (Celsius c) = c scaleName _ = "Celsius"

Of course Celsius itself has to implement Temperature. It’s implementation is trivial though; no work needs to be done.

instance Temperature Fahrenheit where toCelsius (Fahrenheit f) = Celsius ((5 / 9) * (f - 32)) fromCelsius (Celsius c) = Fahrenheit ((9 / 5) * c + 32) value (Fahrenheit f) = f scaleName _ = "Fahrenheit"

Now things are heating up. The conversion functions are identical to the C implementation.

instance Temperature Kelvin where toCelsius (Kelvin k) = Celsius (k - 273.15) fromCelsius (Celsius c) = Kelvin (c + 273.15) value (Kelvin k) = k scaleName _ = "Kelvin"

The Kelvin implementation looks much like the Fahrenheit one.

instance Temperature Rankine where toCelsius (Rankine r) = toCelsius $ Fahrenheit (r - 459.67) fromCelsius c = Rankine $ 459.67 + value (fromCelsius c :: Fahrenheit) value (Rankine r) = r scaleName _ = "Rankine"

The conversion between Fahrenheit and Rankine is much simpler than the conversion between Celsius and Rankine; therefore I will do just that. After converting to and from Fahrenheit, it’s a simple matter of calling toCelsius and fromCelsius.

Bringing The Monads

Now that the easy part is done, we get to create the table. Our table should have as many columns as it needs to display an arbitrary number of conversions. To that end, let’s define a data structure or two:

data ConversionTable = ConversionTable [String] [[TableRowElem]] deriving (Show) data TableRowElem = From Double | To Double | NullConv Double | Empty deriving (Show)

The ConverstionTable, like the name suggests, is our table. The list of strings is the header, and the list of lists of TableRowElem are our conversions. Why not just have a list of Double? We need to have our cells contain information on what they mean.

To that end, I created a TableRowElem type. From is an original value, To is a converted value, NullConv represents the case were we convert from some type to the same type, and Empty is an empty cell. The problem of how to place elements into this data structure still remains however. To solve that, things are going to get a bit monadic. Let’s define some intermediate builder types:

type Conversion a b = (a, b) toConversion :: (Temperature a, Temperature b) => a -> (a, b) toConversion a = (a, convert a)

First we have Conversion, and the corresponding toConversion function. This simply takes a unit, and places it in a tuple with its corresponding conversion. Next, we have the TableBuilder:

type TableBuilder a = WriterT [[TableRowElem]] (State [String]) a

Here we have a WriterT stacked on top of a State monad. The writer transformer contains the list of table rows, and the state monad contains the header. The idea is that as rows are “logged” into the writer, the header is checked to make sure no new units were introduced. To this end, if only two units are introduced, the table will have two columns. If 100 units are used, then the table will have 100 columns.

NOTE: I realize that WriterT and State are not in the standard library. I only promised to limit the usage of libraries for Haskell solutions. This means avoiding the use of things like Parsec or Happstack. Frameworks and libraries that vastly simplify some problem or change the way you approach it. To this end, if I feel a monad transformer or anything along these lines are appropriate to a problem, I will use them. I’ll try to point out when I do though. Besides, I could have just re-implemented these things, but in the interest of not being a bad person and re-inventing the wheel, I’ve decided to use a wheel off the shelf.

So, how do we use this TableBuilder? I’ve defined a function for use with this monad:

insertConv :: (Temperature a, Temperature b) => Conversion a b -> TableBuilder () insertConv (a, b) = do oldHeader <- lift $ get workingHeader <- ensureElem a oldHeader finalHeader <- ensureElem b workingHeader lift $ put finalHeader tell [buildRow a b finalHeader []] where ensureElem a h = return $ case ((scaleName a) `elem` h) of True -> h False -> h ++ [(scaleName a)] buildRow _ _ [] r = r buildRow a b (h:xs) r | (scaleName a) == (scaleName b) && (scaleName a) == h = r ++ [NullConv $ value a] | (scaleName a) == h = buildRow a b xs (r ++ [From $ value a]) | (scaleName b) == h = buildRow a b xs (r ++ [To $ value b]) | otherwise = buildRow a b xs (r ++ [Empty])

Yeah, that one is kind of a doosey. Let me walk you through it. This function takes a Conversion, and returns a TableBuilder.

In the first four lines of the do block, we update the header. We lift the State monad, then get we call ensureElem with the first and second units, then we put the new updated header back into the State monad.

The ensureElem function checks the header list to see if the current unit is a member. If it is, the header list is returned unchanged, if it’s not the unit is appended to the end and the new list is returned. In this way, whenever a conversion is added to the table, the header is updated.

After updating the header, we call tell with the result of buildRow, “writing” the row into the Writer monad. The buildRow function recursively adds TableRowElems to the result list depending on the current heading. In this way, conversions are placed in the appropriate column.

In addition to that function, I’ve defined a function to simplify working with the TableBuilder:

buildTable :: TableBuilder a -> ConversionTable buildTable b = let result = runState (runWriterT b) [] in ConversionTable (snd result) (snd $ fst result)

Working with some of these MTL monads can be confusing for people coming from imperative backgrounds. I’ve been working with Haskell for almost a year now and I still get extremely confused by them. It can take some muddling through haddoc pages to work them out, but the good news is that you mainly just need to define one function that takes a monad (in the form of a do block), and returns a whatever. The buildTable function takes a TableBuilder, and returns a ConversionTable. It handles calls to runState and runWriterT, and then unwraps the resulting tuple and builds the ConversionTable.

This function can be called like this:

buildTable $ do insertConv someConversion insertConv someOtherConversion

… and so on. The only thing to remember is that the final value of a for the do block must be (). Conveniently, insertConv return a value of type TableBuilder (), so if the last call is to this function, then you are good. You can also always end it with return () if you like.

Pretty Printing

Finally, we have the matter of printing a nice pretty table. For that, we need yet another function:

prettyPrint :: ConversionTable -> String prettyPrint (ConversionTable h r) = let widestCol = last $ sort $ map length h columnCount = length h doubleCell = printf ("%-" ++ (show widestCol) ++ ".1f") stringCell = printf ("| %-" ++ (show widestCol) ++ "s |") emptyCell = replicate widestCol ' ' horizontalR = (replicate (((widestCol + 4) * columnCount) + 2) '-') ++ "\n" formatRow row = "|" ++ (concat $ map formatCell row) ++ "|\n" formatCell (From from) = "| " ++ (doubleCell from) ++ " |" formatCell (To to) = "> " ++ (doubleCell to) ++ " |" formatCell Empty = "| " ++ emptyCell ++ " |" formatCell (NullConv nc) = "| " ++ (doubleCell nc) ++ " |" in horizontalR ++ ("|" ++(concat $ map stringCell h) ++ "|\n") ++ horizontalR ++ (concat $ map formatRow (normalizeRowLen (columnCount) r)) ++ horizontalR where normalizeRowLen len rows = map (nRL' len) rows where nRL' len' row | (length row) < len' = nRL' len' (row ++ [Empty]) | otherwise = row

Yeah… Sometimes the littlest things take the most work. You’d think all this plumbing we’ve been doing would be the most complecated bit, but you’d be wrong. Let’s try to make sense of this mess function by function:

widestCol = last $ sort $ map length h

This function determines the widest column based on the header. Typically, this is going to be “Fahrenheit”, but it doesn’t have to be. It should be noted that if a data cell is wider than this, then the pretty printer will mess up. Like most things in life, there is room for improvement here. That said, unless you’re converting the temperature of the core of the sun, you probably won’t have an issue here.

columnCount = length h

Returns the number of columns in the table. Used by the horizontal rule function.

doubleCell = printf ("%-" ++ (show widestCol) ++ ".1f")

Ahh, our old friend printf. It exists in Haskell and works in much the same way as it did in C. The doubleCell function converts a temperature value to a string, left aligns it, pads it by widestCol, and has it show one decimal place.

stringCell = printf ("| %-" ++ (show widestCol) ++ "s |")

Much like with doubleCell, this function pads, and left-aligns a string. This is used by the header.

emptyCell = replicate widestCol ' '

This one is pretty self-explanatory. It prints an empty cell of the appropriate width.

horizontalR = (replicate (((widestCol + 4) * columnCount) + 2) '-') ++ "\n"

This function prints a horizontal rule. This will be a solid line of “-” across the width of the table.

formatRow row = "|" ++ (concat $ map formatCell row) ++ "|\n"

This function formats a table data row. It maps formatCell over the list of cells, flattens it, then adds a pretty border around it.

formatCell (From from) = "| " ++ (doubleCell from) ++ " |" formatCell (To to) = "> " ++ (doubleCell to) ++ " |" formatCell Empty = "| " ++ emptyCell ++ " |" formatCell (NullConv nc) = "| " ++ (doubleCell nc) ++ " |"

In this function, much of the work is done. It formats the cell using doubleCell or emptyCell, the applies a border to the cell. It denotes a cell containing a To by adding a > on the left.

Now that we’ve covered the let-bound functions, let’s talk about the actual function body:

horizontalR concat $ map stringCell h) ++ "|\n") horizontalR concat $ map formatRow (normalizeRowLen (columnCount) r)) horizontalR

This bit is prett straightforward. First, it prints a horizontal line. Second, it maps stringCell over the header list, flattens it, and gives it a border. Third it prints another horizontal line. Fourth is maps formatRow over the normalized row list, then flattens it. Finally, one last horizontal line. After this is all said and done, it concats it all together.

You may be wondering about that normalizeRowLen function. If you were paying particularly close attention to the insertConv function, you may have noticed an issue. Let’s walk through it in ghci:

*Main> let fc = toConversion (Fahrenheit 100) :: (Fahrenheit, Celsius) *Main> buildTable $ do insertConv fc ConversionTable ["Fahrenheit","Celsius"] [[From 100.0,To 37.77777777777778]]

We add one conversion, we get two columns. Everything seems to be in order here, but let’s add another conversion and see what happens:

*Main> let fc = toConversion (Fahrenheit 100) :: (Fahrenheit, Celsius) *Main> let cr = toConversion (Celsius 100) :: (Celsius, Rankine) *Main> buildTable $ do {insertConv fc; insertConv cr;} ConversionTable ["Fahrenheit","Celsius","Rankine"] [[From 100.0,To 37.77777777777778],[Empty,From 100.0,To 671.6700000000001]]

See the problem? Let’s add some newlines to make it clearer:

ConversionTable ["Fahrenheit","Celsius","Rankine"] [[From 100.0,To 37.77777777777778], [Empty,From 100.0,To 671.6700000000001]]

As we add more columns, the rows with less columns are never updated to have the new column count. Logically, this is fine, since the extra entries would just be Empty anyways, but our pretty printer would print this table like so:

-------------------------------------------- || Fahrenheit || Celsius || Rankine || -------------------------------------------- || 100.0 |> 37.8 || || || 100.0 |> 671.7 || --------------------------------------------

As you add more and more columns, the problem gets worse and worse. Enter our normalizeRowLen function:

normalizeRowLen len rows = map (nRL' len) rows where nRL' len' row | (length row) < len' = nRL' len' (row ++ [Empty]) | otherwise = row

This is another fairly straightforward function. If the row has the same number of columns as the header, it is returned unchanged. If it doesn’t, Empty is added to the end until it does.

With that, our program is complete. Let’s try it out:

main = do k <- return (toConversion $ Kelvin 100 :: (Kelvin, Rankine)) f <- return (toConversion $ Fahrenheit 451 :: (Fahrenheit, Kelvin)) r <- return (toConversion $ Rankine 234 :: (Rankine, Celsius)) c <- return (toConversion $ Celsius 9 :: (Celsius, Fahrenheit)) nc <- return (toConversion $ Rankine 123 :: (Rankine, Rankine)) putStrLn $ prettyPrint $ buildTable $ do insertConv k insertConv f insertConv r insertConv c insertConv nc

In our main, we create a bunch of conversions. Then we prettyPrint them and putStrLn the result. The following will be printed to the console:

---------------------------------------------------------- || Kelvin || Rankine || Fahrenheit || Celsius || ---------------------------------------------------------- || 100.0 |> 180.0 || || || |> 505.9 || || 451.0 || || || || 234.0 || |> -143.2 || || || |> 48.2 || 9.0 || || || 123.0 || || || ----------------------------------------------------------

Any type that implements Temperature can be put into a table this way. To add a new unit to the program, it’s as easy as implementing four one-line functions!

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