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Pure Functional Memoization

There are many computation problems that resonate through the ages. Important Problems. Problems that merit a capital P.

The Halting Problem…

P versus NP…

The Fibonacci sequence…

The Fibonacci sequence is a super-important computation Problem that has vexed mathematicians for generations. It’s so simple!

fibs 0 = 0 fibs 1 = 1 fibs n = (fibs $ n - 1) + (fibs $ n - 2)

Done! Right? Let’s fire up ghci and find out.

*Fibs> fibs 10 55

… looking good…

*Fibs> fibs 40

… still waiting…

102334155

… Well that sure took an unfortunate amount of time. Let’s try 1000!

*Fibs> fibs 1000 *** Exception: <<You died before this returned>>

To this day, science wonders what fibs(1000) is. Well today we solve this!

Memoization

The traditional way to solve this is to use Memoization. In an imperative language, we’d create an array of size n, and prepopulate arr[0] = 0 and arr[1] = 1. Next we’d loop over 2 to n, and for each we’d set arr[i] = arr[i-1] + arr[i-2].

Unfortunately for us, this is Haskell. What to do… Suppose we had a map of the solutions for 0 to i, we could calculate the solution for i + 1 pretty easily right?

fibsImpl _ 0 = 0 fibsImpl _ 1 = 1 fibsImpl m i = (mo + mt) where mo = Map.findWithDefault undefined (i - 1) m mt = Map.findWithDefault undefined (i - 2) m

We return 0 and 1 for i = 0 and i = 1 as usual. Next we lookup n – 1 and n – 2 from the map and return their sum. This is all pretty standard. But where does the map come from?

It turns out that this is one of those times that laziness is our friend. Consider this code:

fibs' n = let m = fmap (fibsImpl m) (Map.fromList (zip [0..n] [0..n])) in Map.findWithDefault undefined n m

When I first saw this pattern (which I call the Wizard Pattern, because it was clearly invented by a wizard), I was completely baffled. We pass the thing we’re creating into the function that’s creating it? Unthinkable!

It turns out that this is just what we need. Because of laziness, the fmap returns immediately, and m points to an unevaluated thunk. So, for i = 0, and i = 1, fibsImpl will return 0 and 1 respectively, and the map will map 0 -> 0 and 1 -> 1. Next for i = 2, Haskell will attempt to lookup from the map. When it does this, it will be forced to evaluate the result of i = 0 and i = 1, and it will add 2 -> 1 to the map. This will continue all the way through i = n. Finally, this function looks up and returns the value of fibs n in linearish time. (As we all know, Map lookup isn’t constant time, but this is a lot better than the exponential time we had before)

So let’s try it out.

*Fibs> fibs' 1 1 *Fibs> fibs' 10 55 *Fibs> fibs' 40 102334155

… so far so good…

*Fibs> fibs' 100 354224848179261915075 *Fibs> fibs' 1000 43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875 *Fibs> fibs' 10000 33644764876431783266621612005107543310302148460680063906564769974680081442166662368155595513633734025582065332680836159373734790483865268263040892463056431887354544369559827491606602099884183933864652731300088830269235673613135117579297437854413752130520504347701602264758318906527890855154366159582987279682987510631200575428783453215515103870818298969791613127856265033195487140214287532698187962046936097879900350962302291026368131493195275630227837628441540360584402572114334961180023091208287046088923962328835461505776583271252546093591128203925285393434620904245248929403901706233888991085841065183173360437470737908552631764325733993712871937587746897479926305837065742830161637408969178426378624212835258112820516370298089332099905707920064367426202389783111470054074998459250360633560933883831923386783056136435351892133279732908133732642652633989763922723407882928177953580570993691049175470808931841056146322338217465637321248226383092103297701648054726243842374862411453093812206564914032751086643394517512161526545361333111314042436854805106765843493523836959653428071768775328348234345557366719731392746273629108210679280784718035329131176778924659089938635459327894523777674406192240337638674004021330343297496902028328145933418826817683893072003634795623117103101291953169794607632737589253530772552375943788434504067715555779056450443016640119462580972216729758615026968443146952034614932291105970676243268515992834709891284706740862008587135016260312071903172086094081298321581077282076353186624611278245537208532365305775956430072517744315051539600905168603220349163222640885248852433158051534849622434848299380905070483482449327453732624567755879089187190803662058009594743150052402532709746995318770724376825907419939632265984147498193609285223945039707165443156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875 *Fibs> fibs' 100000 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Neat. Even that last one only took a few seconds to return!

Operator Overloading in C++

My thoughts on Operator Overloading are hardly a secret. Some languages, such as Haskell, support it in a sane manner. Some languages, such as C++ don’t. Either way though, I’m not a fan.

That said, there is one case where I support it; when what you are trying to overload an operator to do makes sense for your type. In other words, if it makes sense to multiply an Employee object, feel free to overload the multiplication operator. Since we don’t usually model mating when we implement an employee object, I don’t usually approve.

Multiplying a Fraction

However, if we have a fraction class, that has a numerator and denominator component, overloading the multiplication operator suddenly becomes reasonable:

frac<T> operator *(const frac<T> & rhs) const { return frac<T>(numer * rhs.numer, denom * rhs.denom); };

In C++, many of the built in operators are available to be overloaded. You can do basically anything in an operator overload, but if you do, that makes you a bad person. Here we have a straightforward implementation, that I’m going to assume you don’t need to have explained to you. But let’s talk const correctness.

In this signature, the const appears twice. The right hand side argument is a const reference. Since this is a reference, no copy occurs, but with references can come mutability. The const here prevents any mutation from occurring. There is also a const at the end of the signature. That means that this function cannot modify the this pointer.

This function returns a new frac<t>, so no need for const there.

Convert a Fraction to a Double

Next, you can overload the casting operator, and if you are a good person you’ll use this for type conversions. Let’s implement (double) for our fraction:

operator double() const { return (double) numer / denom; }

This code is pretty straight forward. We divide the numerator by the denominator, and cast it to double in case integer division occurs. Cast operator overloads don’t have a return type, which is a bit strange but something we can work with.

Streaming operators

Finally, I’d like to talk about streaming operator overloads. For any overloaded operator, there are two forms: member and non-member. Member overloads take n - 1 arguments, where the minus 1 is the current object (and the left hand side). Non member overloads can’t access the this pointer, so they need to take the left hand side as a parameter.

Member overloads obviously have access to private class members, which makes them more powerful than non-member. Which leads us to the streaming operators. Streaming operators need to return std::ostream &, so they must be non-member. The problem is that we want to print the private numerator and denominator fields. The solution? make it a friend:

friend std::ostream & operator <<(std::ostream & lhs, const frac<T> & rhs) { lhs << rhs.numer << '/' << rhs.denom; return lhs; };

With that problem solved, this function becomes pretty straight forward. We stream the fields, formatted correctly, into the left hand side std::ostream &, then we return the left hand side argument for further use by the next part of the streaming chain.

Back in the Saddle with C++

Next quarter, I’ll be taking a graphics programming class. It was a year ago to the day that I last wrote about getting a makefile set up for OpenGL development. As you can imagine nothing came of it. Why? The obvious answer is that OpenGL is hard.

Secondary answers include the fact that almost all of the literature is assuming C++, and I was swimming against the tide trying to use C. Well, I won’t have a choice this time, the class is in C++, so I’m stuck with my old nemesis again.

What follows in the next few posts is an attempt to refamiliarize myself with C++. It’s been a while since I’ve done any significant work in C++ and a lot has happened. I’ll be going over a lot of basic stuff, so that I have a centralized crash course for myself when I’m stuck trying to remember if it’s const &, & const, & const &, or co&n&st (stranger things have happened).

Namespaces

First up: namespaces. Namespaces are a formalization of the c convention of writing functions called stuff like dmp_doStuff. First we can declare our namespace:

namespace dmp { void doStuff() }

…and we can call this function by doing dmp::doStuff(). This may involve an extra keystroke, but we can also use it like so:

using namespace dmp; doStuff(); doStuff(); doStuff(); while (stuffUndone()) doStuff;

<Template> Classes

Now that we got that out of the way, let’s get into the meat of it: template classes. First up, due to some incredibly unfortunate reason, a C++ template class has to go in the header file. This sad fact is one of C++’s many transgressions against society, but we’ll not worry about that for now.

To declare a class (inside of a namespace if you’re a good citizen):

namespace dmp { template <typename T> class frac { public: /* public stuff goes here */ private: T numer; T denom; /* and your privates go here */ }; }

template <typename T> would be omitted if this weren’t a template class. Here we declare a class frac with one generic type T. Our class has two private fields of type T. As we all know, class members are private by default, but I don’t like to rely on “default behaviors”, so it doesn’t hurt to make it explicit. Let’s add some public constructors:

frac(T inNumer, T inDenom) : numer(inNumer), denom(inDenom) {}; frac(const frac<T> & toCopy) : numer(toCopy.numer), denom(toCopy.denom) {};

Here we use initializer lists to populate the numerator and denominator. The first constructor takes two arguments of the same type (T), and the second is the copy constructor.

According to Microsoft, initializer lists are more efficient than assignment in the function body, which is why we prefer them here. There are certain cases where you must use an initializer list, and cases where you cannot use them. Your compiler will tell you if you mess this up.

Finally, let’s add a destructor:

~frac() {};

Here we have a marvel of software engineering. Since we really don’t have anything to destroy, we can let it do nothing. It should be noted that this destructor is not virtual, so it’s not really safe to extend this class (more on this another day).

Conclusion

Honestly, namespaces and constructors/destructors are two of my favorite features of C++. Simple, but helpful in eliminating boilerplate. I almost might be convinced to compile “C” with a C++ compiler to get these features. Coming up will be some features that I’m a bit more iffy about, such as operator overloading. however, when used appropriately, these things can also be good, more to come!

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.

Everything Is An Integer

Today I found myself plumbing the murky depths of C++. My latest task: file IO.

C++ provides a type fstream. Being a proper object-oriented system, the class hierarchy is completely indecipherable. However, the actual use of this type is fairly simple. An instance of fstream has a “get pointer” and a “put pointer”. To write to the file, you must seek the put pointer to a location, then write. To read, you must seek the get pointer, then read. Fairly simple, right?

The seek methods: seekp and seekg are overloaded:

ostream& seek[pg] (streampos pos); ostream& seek[pg] (streamoff off, ios_base::seekdir way);

Should be no problem, right? Let’s de-typedef-ify that:

ostream& seek[pg] (int pos); ostream& seek[pg] (int off, int way);

Ok, still no big deal. This is C++ after all, enums are integers. So, I carry on my merry way. I implemented the following functions:

template <class to_write> static void write_out(fstream * fio, offset location, const to_write & value) { fio->seekp(location, ios_base::end); write_out<to_write>(fio, value); } template <class to_write> static void write_out(fstream * fio, const to_write & value, streampos position) { fio->seekp(position); write_out<to_write>(fio, value); } template <class to_read> static void read_in(fstream * fio, to_read & into_location) { ...stuff... }

The idea: wrap the seeking and writing into one operation, so my high level code need not concern itself with it. The behavior:

write_out<long>(foo, 8, ios_base::beg);

…and…

offset bar = 0; write_out<long>(foo, bar, 0);

…both result in the first function being called! Were this a language with a proper type system, all those types would be distinct and the function overloading would work as advertised. Unfortunately, this is C++, where there is only 1 type: byte.

Let this serve as a warning, so you don’t spend half your day troubleshooting such a stupid problem.

DMP Photobooth: On the Road to 2.0

It’s been a busy few months for me, and things have been quiet here. Quiet, but not forgotten; big things are in the works! What big things you ask?

Back in June, I reflected on DMP Photobooth. The completion of the photobooth was a major accomplishment for me: it worked perfectly (dead camera batteries and jammed printers aside) for a good six hours, and was a hit with the guests! However, like many things in life, it has problems. On my list of things to fix are a wonky UI, and an aggressively imperative code style.

A few months of reflection later, and I’ve had time to consider these issues. I thought about why I wrote DMP Photobooth. Aside from saving ~ $1,000 on renting a photobooth, it was really an opportunity for me to truly explore the C programming language.

While I had some experience with it, it was mostly in an academic capacity. Writing linked lists and trees and such is one thing, creating a real program is another entirely. DMP Photobooth gave me a chance to truly do something with it. Having implemented a non-trivial application in it has given me an appreciation of the strengths and weaknesses of the language. I’ve seen what’s easy, and what’s hard. I’ve made questionable design choices, and attempted to fix them. I’ve made good design choices that made my life easier. I can truly say that I’ve been there and done that.

But let’s be honest with ourselves: C wasn’t really an appropriate choice for this sort of program. Java, Python, or something along those lines really would have been the correct choice. But where’s the fun in that? However, now that the project is “complete” its time to find a way forward. C no longer has the allure for me that it once did, and since my customers include me and my wife, I feel confident that a re-write won’t lose me any business.

That’s right: DMP Photobooth is being completely re-written. Version 2.0 will be re-implemented in Haskell. However, this doesn’t mean that I have to completely walk away from all the work I did. Early versions will likely use the original Camera, Printer, and Trigger modules via FFI.

Why re-write? I thought about it for a while, I’ve wanted to do something with Haskell to cement my knowledge as I did with C, but I was lacking inspiration. Since DMP Photobooth was really a playground to explore C, why not do the same with Haskell? It seems like a logical choice, implementing a non-trivial application in Haskell will cement my knowledge in the same way it did in C. Additionally, trying to solve the same problem in a different way will give me an appreciation of the pros and cons of each approach.

Change in the Wind

So, surely this isn’t going to be a one-to-one port of DMP Photobooth to Haskell, right?

With this new version, I am making some fundamental changes to the program. Probably the biggest change is the elimination of runtime loading of modules. In 2.0, all modules will be set at compile time. I feel that the machinery of loading and operating modules at runtime was a bit wonky. Too much effort was put into “not restricting a module developer’s freedom”. In 2.0, modules will in place at compile time, and they will be treated as a first-class part of the program.

This isn’t to say that the module concept is going away. In fact, it is being expanded. There are now six total modules. The Camera, Trigger, and Printer modules return in 2.0; but the Photostrip, Persistence, and Interface modules are new in this version. The original modules will continue to work as they did, but let’s discuss the newcomers.

Interface

The interface is fairly straightforward. This module is responsible for the user’s interaction with the program through the computer. Triggering the capture process will still be the trigger module’s domain, but anything that appears on the computer monitor is the domain of the Interface module. All log messages generated by the program will be sent to the Interface module for presentation to the user. Additionally, the Core will provide callbacks to the Interface that the interface will be responsible for enabling the user to call these. There will no longer be a concept of “starting” or “stopping” the photobooth; the photobooth will start when the program starts, and it will stop when the program exits. However, the interface will likely be responsible for allowing a user to print a previously processed strip, alter the configuration, and close the program.

Photostrip

Another fairly self-explanatory module. This module encompasses all the functions that were previously defined in photo_strip.c. In 2.0, the use of temporary files will be limited, therefore all images will be passed in ByteStrings.

Persistence

Of the three new modules, this one is probably the most novel. This module’s responsibility is persisting data between invocations of the program. This will include at the very least program configuration, but it can also include things like log history or even photo sessions. All modules will be given hooks into the persistence module, so they can persist and restore anything that they see fit.

The functionality here closely resembles that of configuration.c. It will work in a similar way. I have defined the following data type:

data Property = IntProperty {propName :: String, intValue :: Integer} | StringProperty {propName :: String, stringValue :: String} | DoubleProperty {propName :: String, doubleValue :: Double} | BoolProperty {propName :: String, boolValue :: Bool} | ListProperty {propName :: String, listValue :: [Property]} deriving (Eq, Show)

The persistence module will be able to persist and restore this data type. The modules can store anything in this, including arbitrary types if they are instances of Show and Read. This list of types will probably change over time, but its a good start.

The Way Ahead

Its going to be a long road to 2.0, but as the project progresses, expect to see updates here. The codebase for DMP Photobooth has been uploaded to GitHub, and can be found here. Version 1.0’s repository has been renamed to dmp-photo-booth-legacy. I have no plans to take this version down, so it should remain available for the foreseeable future.

Object Oriented Haskell

You may recall, earlier this year I wrote about object orientation in C. The basic Idea being that “Object Oriented Programming” is more a mindset, then a language feature. You can do object orientation and access control in C using free-floating functions and opaque structs. Well, guess what? You can do Object Oriented Programming in Haskell as well!

As a quick recap, if you didn’t read "C With Classes", for our purposes there are three components that need to be present to be considered “an object”: data consolidation, access control, and method calling. Inheritance is also important to real “Object Oriented Programming” (or OOP, as I’ll start calling it), but is really just gravy. Inheritance is largely supported by typeclasses in Haskell, so I won’t be going into it today.

Functional OOP

What would OOP look like in Haskell? Thanks to the magic of higher-order functions, we can actually very closely approximate what you’d see in a traditional OOP language, such as Java or C++.

Data

This part is trivial, there is literally a data keyword for this purpose. You should have this down by now.

Access Control

Access Control can be accomplished by way of the module keyword. Simply expose what you’d like to be public, and don’t expose your private members. Obviously, if you have private fields in your “Class”, you should make a factory function for your class instead of exposing its constructor. This is all pretty standard stuff.

Method Calls

This is an area that Haskell shines, and C shows its ugly side. In Haskell, we can actually create a method call operator, and create something that looks very much like the traditional Class.method calling convention that we’re used to!

The Method Call Operator

First, we need some contrived classes. Let’s go with everybody’s favorite: the Employee!

data Employee = Employee {name :: String, employeeId :: Integer, title :: String}

Nothing fancy or non-standard here. We created an Employee “Class” in the traditional Haskell way. Due to our use of record syntax, we already have three getters: name, employeeId, and title.

Let’s make another function, so that we can have a method that takes arguments:

isSeniorTo :: Employee -> Employee -> Bool isSeniorTo s f = (employeeId s) > (employeeId f)

There is something extremely important to note about this function: the last argument is the object, not the first as it was in “C With Classes”. The reason for this is to allow us to partially apply this function, this will be crucial.

Now, let’s give it a shot to make sure everything is working:

*ghci> let boss = newEmployee "Chris" 1 "Author" *ghci> let notBoss = newEmployee "Geoffrey" 2 "Associate" *ghci> name boss "Chris" *ghci> isSeniorTo notBoss boss True

All looks well, we create two Employee objects. Since Chris‘s employee ID is lower than Geoffrey‘s, the isSeniorTo method returns True. But this is all very wonky still, let’s create that method call operator already!

(>-) :: a -> (a -> b) -> b (>-) o f = f o

Since . and -> are already spoken for, I’ve gone with >- for my method call operator. The method call operator takes an arbitrary type a, and a function that takes the same type a, and returns a type b. Finally a type b is returned. The definition of this function couldn’t be simpler: we call the function on the passed-in object! Let’s see this in action:

*ghci> notBoss >- title "Associate" *ghci> boss >- isSeniorTo notBoss True

This is just about as elegant as it gets; using the method call operator we just defined, we call a method on an object! In the case of isSeniorTo, we partially apply the function with all but the last argument, then the method call operator is able to use it just as any other function.

But something doesn’t sit right. Isn’t the definition of isSeniorTo a bit smelly? It calls methods on objects, shouldn’t it use the method call operator? Now that we have our method call operator, let’s fix that function:

isSeniorTo :: Employee -> Employee -> Bool isSeniorTo s f = (s >- employeeId) > (f >- employeeId)

There, that’s better. isSeniorTo now properly calls methods on s and f, and all is again well in the world.

Here’s The Kicker

You may be experiencing some deja vu from all of this. It feels like we’ve done this sort of thing before. But where? You may recall this little operator:

*ghci> :t (>>=) (>>=) :: Monad m => m a -> (a -> m b) -> m b

That’s right, the monadic bind operator looks suspiciously like our method call operator. But does it behave the same?

*ghci> 1 >- (+) 1 2 *ghci> Just 1 >>= (\a -> return $ a + 1) Just 2

Let’s ignore for a second that we had to use a monad for the bind operator, and see the fact that the two basically did the same thing. In fact, to further illustrate the point, let’s make a Class monad:

data Class a = Class a deriving (Show) instance Monad Class where (Class a) >>= f = f a return = Class

Unlike most monads, which do important work, our Class monad does nothing but prove a point. However, you should note that the implementation of the bind operator is essentially the same as the implementation of the method call operator. Now, let’s see our monad in action!

let boss = Employee "Chris" 1 "Author" *ghci> boss >- name "Chris" *ghci> Class boss >>= return . name Class "Chris"

As you can see, they essentially do the same thing. It turns out that Haskell had classes all along!

Of course, this isn’t exactly true. I’d say that the State monad serves many of the same purposes as objects in OOP languages. However, the semantics of things like Reader, Maybe, and IO have little in common with objects. But much like we implemented Objects in Haskell, the various OOP languages are implementing monads into their standard libraries.

Indeed, OOP and functional programming are not supported by languages, they are supported by programmers. Some languages may make one or the other easier, but the differences are small, and get smaller as time passes.

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