Peano’s Axioms Part I: Haskell and Type Theory, and Object Oriented Programming

Well, the title’s a little misleading, this has very little to do with type signatures, I decided to start a little series on what I think is the most powerful aspect of Haskell. Of course, I think that the most part is Math. Specifically in the realm of Type Classes. I’ll give you a little background here.

I’m primarily familiar with Java’s class system. Notably, upon reading about and toying with other class inheritance systems, Java stood out to me as being my favorite. It feels the most algebraic. One thing that should be noted is that typically, the only “real” difference between these systems (I’m generalizing wildly.) is in the world of multiple inheritance. Java, in particular, uses interfaces, which I intend to show as being a kind of type class, in fact, it is about one step away from being a real live type class.

Now, Lets start talking about inheritance.

Assume we have an Object Joe. Joe is a “Reified” (to borrow some language from type theory) object, that is. He’s a finite, usable, object, with real methods, that do real things. Effectively, he’s material, you can “poke” him, as it were.

Lets now say Joe has some Children, Jack and Emily. These are also Reified objects. However, they have a special property, they have a “Parent” object, namely Joe. This “special property” allows them to inherit properties from Joe, (say… devilishly good looks, and powerful intellect?). By now, the female readers are probably moaning, “Those have to come from the mother.” And thats a good point, how do we get the Mom in on this? We need Multiple Inheritence. Before we get there though, lets talk about a different kind of object, an Abstract object.

An Abstract Object describes some underlying property of an object, but it does so through this Parent/Child relationship. An Abstract Object in our setting might be something like “Ancestor” or even “Parent”. Notable things about Abstract objects is that they don’t contain any “Reified” methods, that is, methods containing code. The contain stubs which describe the signature of a method, but not the method itself. So we might have a Abstract class “Parent” which contains a method “getChildren” which has a signature void -> List, its purpose is to return a list of the extending classe’s (like Joe) Children. The important thing is, it doesn’t force that to be the result, it only enforces a type of the content, not the content itself.

But what about Multiple Inheritence. Assume we have another object called “Sarah” which also extends the parent class, and also assume that Jack and Emily are Sarah’s children too. Lets assume further that Sarah implements getChildren differently than Joe does. How then do Jack and Emily access there respective parents getChildren Methods? If they are fundamentally different implementations, then how can they choose which to use? Further, what if we want to have Parents for Joe, but we also want Joe to extend the Ancestor class? Now we have a mess…

How do we fix it? Well, the answer is to create a super version of Abstract classes, lets call them interfaces.

What does an Interface do? Well– in Java, no class is allowed to inherit from more than one parent class. Abstract or Reified. Interfaces exist outside of the usual inheritance tree, and can therefore give information about the tree as a whole. For instance, we can now create our inheritance tree by saying that each ancestor class implements the “hasChildren” interface, which forces the ancestor to have the void -> List method, we can then say that the Children found in that list are instances of the Child class, rather than the class itself, this gives us a solution to the problem by forcing us to access Jack through the Sarah object.

Could we have done this with normal Abstract/Reified classes, sure. Would it work as well? Probably not.

So why are interfaces cool? Well- you can think of an Interface as a way to describe a set of classes. Classes which meet certain properties. Indeed, they use interfaces like this all the time. For instance, Java has a interface called “Comparable” which- as the name suggests, means that any class that implements Comparable can compare instances of itself in a meaninful way. Comparable doesn’t say how to do the actual comparison, it just says that there must be a method compare which has type: (using Haskell-esque syntax)


compare :: Comparable a => a -> a -> {-1,0,+1}

The Javadocs recommends that it should have 0 mean that x == y, but that its not required. The interface only defines the kind of interaction you can have.

Why is this cool? Well, consider the obvious example, what if I wanted to write quicksort over a generic list of elements? How do I ensure that I have a sortable list? Lets say we have an Object Foo, which has no ordering, and another Object Bar, which can. I have a magic method qsort which will actually do the sorting, but how do I write the method such that I know that the list is sortable? That is if I write tests:


assertFail(qsort(List = {foo1, foo2, ...}));
assertPass(qsort(List = {bar1, bar2, ...}));

How can I ensure those both succeed (with the assumption, of course, that those testing constructs exist)? Well, the easy answer is to define qsort as having the method signature:


List qsort(List);

(I think thats the right syntax, it’s been a while since I wrote Java.)

This will prevent (at compile time, no less) the use of qsort on List, because Foo doesn’t implement Comparable. Running it on List is Fine, because its comparable, like we said before.

So now how is this different from Type Classes. Well, principly, there is one difference. Type Classes can not only specify methods the Type must have defined on it, but it can also define other methods derived from those methods. A seemingly trivial extension to Interfaces. However, that one simple change gives you enormous power in creating data abstractions. For instance, given the Haskell equivalent of Comparable, which is the pair of classes Eq and Ord (for Equality and Orderability, resp.), which require only 2 definitions total (== or /=, and compare or <=, /= is not equal). We get for free, about 20-30 functions including: all your comparison and equality operators, the ability to put items implementing the class in a set, the ability to use them as a key in a finite map, sorting of implementing items, some extra list functions, and more. All that, from 2 definitions, in two type classes, along with the Prelude and Haskell’s own polymorphism capabilities.

How does that work? Well, lets dissect some code, here’s the Ord class:

ignore the ‘.”s, they’re just for spacing.

class (Eq a) => Ord a where
........compare :: a -> a -> Ordering
........(<), (=),(>) :: a -> a -> Bool
........max, min :: a -> a -> a
................--minimal complete definition:
................-- (<=) or compare
................-- using compare can be more efficient for complex types.
........compare x y
.............| x == y = EQ
.............| x <= y = LT
.............| otherwise = GT
........x <= y = compare x y /= GT
........x < y = compare x y == LT
........x >= y = compare x y /= LT
........x > y = compare x y == GT
........-- note that (min x y, max x y) = (x,y) or (y,x)
........max x y
.............| x <= y = y
.............| otherwise = x
........min x y
.............| x <= y = x
.............| otherwise = y

So, lets dissect.

First, we declare what the class implements, that is, what methods are required to exist if the class is implemented. However, this is not the same as saying that all those methods must be defined over the structure before we can say it implements the class. We instead only need the “MRD/MCD” or “Minimal Requisite/Complete Definition” that is, the minimal number of functions needed to define all the other functions the class provides. How do we know what the MRD is? Well- thats up to you to decide. We do that by defining each function of the class in terms of other functions in the class, even cyclicly if necessary. I like to call this set of cyclic definitions the “internal definitions” of the set. These internal definitions provide the real power of the type class. Also note how, in the class definition, we have an inheritance bit, in that of Eq a => Ord a. This allows us to assume that we have basic equality functions provided by Eq.

In the next few posts in this series, I’ll be implementing Peano Arithmetic in Haskell, principly by pushing one simple data type:


Nat = Z | S Nat

Into as many type classes as I can, as of this writing, I have two more posts planned, and have pushed Nat into Eq, Ord, Num, Integral, Enum, and Real, I hope to get it into Rational and Fractional which will turn it into Nat-QQ (Natural Rationals (that is, all the Positive Rationals)), and maybe then refactor Nat into ZZ (the integers), and get the Regular Rationals out of that.

Advertisements
Published in: on July 9, 2007 at 2:52 am  Comments (2)  

Human Error Bugs

I fixed comments not coming up, and such. Thanks for pointing them out, folks.

Published in: on July 6, 2007 at 3:10 am  Comments (1)  

An Analogy for Functional versus Imperative programming.

I was thinking the other day, about many things, as we drove back from my Aunt and Uncles in NY. I had been discussing with my father about C vs Haskell and, more generally, about functional versus imperative programming. I had been trying to explain why Functional programming is useful, particularly in areas relating to parallelism and concurrent code. To put this in perspective, my father has been writing low level system verification code for a very long time. (he started way back in the 70s) so he’s pretty situated in the imperative world, with a (vast) knowledge of C and languages like Verilog and stuff. Me, I’ve only been writing code since I was 14-15. I have far more knowledge in languages like Scheme, Haskell, and to an extent, the ML family. I also regularly use Java and friends, So it’s safe to say that trying to cross our respective expertises is most often quite difficult. Anyway, I came up with a pretty clever analogy, I think, for how Functional and Imperative programs relate.

Factorys

I have no idea if this analogy has been used before, but if it has, kudos to whoever came up with it, basically, it goes like this.

an imperative language is monolithic. it effectively can be modelled as one giant state machine that gets permuted. Like a Rube Goldberg machine, you don’t design an imperative program to manipulate inputs, you design it for its side effects.

Here in Massachusetts, we have the Boston science museum, in which, there is a rube goldberg machine (RGM). It is, (next to the math room), by far one of my favorite exhibits. But what does an RGM do? Put simply, its just a while(1) loop. It ferries some bowling balls or whatnot to the top, and then drops then. The interesting part is the side effects. The bangs and whizzes and clanking and ratcheting of the chain as various balls drop and gears spin and cogs do whatever they do, cogulate, I guess. The point of the RGM is to manipulate the world indirectly. Someone, at some point, “said” to the machine, “start.” From thence, it has worked to make noise and spin and do all that nifty stuff.

So, RGM’s, you say, are mildly useless, right? Well. We’ll come back to that in a minute, but suffice to say, like everything else in the world, they have there place.

So if an Imperative language is Like a RGM, whats a functional language?

Well, lets realize that effectively, all a program does is turn some set of inputs, to some set of outputs. Kind of like how a factory may take in some raw material, (steel, plastics, etc.) and create a new, better “material” from those outputs, (eg, a car). A language does the same thing, a user “inputs” a query to a database server (technically, he, or someone else, have given the program the database, too. Kind of like currying, here. hmm). Then after your query, you get back a database entry or list thereof which match your search parameters. Now, a RGM-type machine, or more accurately, a monolithic program, which are typically written in imperative languages (though you can write monolithic functional programs) take an input, and, completely internally, turn that input to an output. Kind of like a whole factory in a box. useful, yes, reusable not necessarily. A functional approach, on the other hand, is like the components that make up a factory. When I write a program in Haskell, I have a series of functions which map input data to intermediate data, and functions which map intermediate data to intermediate data, and then finally functions which take intermediate data and map it to output data. For instance, if I want to create a function which takes a list and returns it along with its reversal, in Haskell, I’d write:


myReverse :: [a] -> [a]
retPair :: [a] -> ([a],[a])

myReverse [] = []
myReverse (x:xs) = myReverse xs : x

retPair ls = (ls , myReverse ls)

So you can see how retPair starts the chain by taking input. copies ls and sends it to an output, and then sends the other copy to another “machine” in the factory, which turns a list of anything ‘[a]’ to a list of anything. The result is then sent to output with the original as a pair ‘([a],[a])’

You can see this in the diagram:


..........................myReverse..............................
................/---------[a]->[a]-------\...they get............
input...........|........................|...rejoined............
>---------------|split ls................|---([a],[a])-->output..
................|........................|.......................
................\---------[a]->[a]-------/.......................
..........................identity...............................
..........................function...............................

So what does this “individual machine method” give us? For one, its free to reuse, its very easy to pull apart this “factory” of “machines” and reuse any given “machine” in some other “factory”. It would not be as easy to do if we had written it procedurally, as in C/C++. I can hear the screams of imperative programmers now, “We would have written exactly the same thing, more or less!” and I know this, and don’t get me wrong, you _can_ write this “factory style” code in C/C++, but what about less trivial examples? In Haskell, I can only write pure functional code (barring monads, which are borderline non-functional). Whereas in C/C++, writing this kind of reusable code is often hard. In a functional language, writing this kind of code is almost implicit to the nature of how you think about code. The point I’m trying to make is simply this, FP-style languages force you to write (more or less) reusable code. Imperative languages in many cases force you to write once-off code you’ll never see again. I’m not saying this makes FP better, in fact, in a huge number of cases, I, as an FP programmer, have to write one-off imperative-esque state mangling RGM’s to get things done. The point is Haskell helps me avoid those things, which makes for more reusable code.

Another thing, FP is famous for being “good” at concurrency. This analogy works wonders at explaining why. Think about the factory example, when I split ls into two copies, I split the program into two “threads”. I effectively set up an assembly line, when I send done the copy of ls to the myReverse function, you can imagine a little factory worker turning the list around pi/2 radians so that it was backwards, and sending it down the line… You can even imagine the type constrictions as another little worker who hits a siren when you send down the wrong material. Imagine, however, trying to parallelize an RGM. RGM’s are often dependent on the inability to be made concurrent, even if that wasn’t the programmers intention. Imperative programs fight the programmer with things like deadlocks (two balls in the RGM get stuck in the same spot) and race conditions (two balls in the RGM racing towards the conveyor belt, with no way of determining who will win, how do you handle that?) whereas FP implicitly allows multiple “workers” to manipulate there own personal materials to make there own personal products at there station. In a purely functional program, each function is implicitly a process, you could even go so far as to give it its own thread. Each machine’s thread would just yield until it got something to work on, at which point it would do its work, and go back to waiting, it doesn’t matter which information gets to the next machine first, because it will just wait till it has all the information it needs to execute. Bottlenecking (a common problem in all code) is easier to see in this “factory” style view, since a bottle neck will (*gasp*) look like a bottle neck, all the functions will output to a single function. Thats a sign that its time to break it up, or have two copies of it running in parallel. FP makes this stuff simple, which makes FP powerful. Because for a language to have true power, it must make it so the programmer has to only think about what he wants to do, and not how to do it.

On the other hand, Imperative programming world. You have a number of excellent things going for you. Imperative code is remarkably good at clever manipulations of the machine itself. It is, in some ways, “closer” to the machine than a Functional language could ever be. So even though you have your share of problems, (parallelism and code reuse are the two I think are the biggest.) you have benefits to. Code in C is well know to be very fast, Object Oriented Languages are, I think, best described imperatively, and OO is a wonderfully intuitive way to think about programming. Imperative Programming is also makes it infinitely easier to deal with state, which can be a good thing, if use properly. Don’t get me wrong, I love monads, but even the cleverness of monads can’t compare to the ease of I/O in Perl, C, Java, Bash, or any other imperative language.

Hopefully this was enlightening, again, I want to say, Imperative programming isn’t Bad, just different. I like FP because it solves some big problems in Imperative Programming. Other people are of course allowed to disagree. It’s not like I’m Jake or anything.

Published in: on July 5, 2007 at 5:21 pm  Leave a Comment  

Programming Lanugages Part II: Beginner Friendly

Title says it all, really, I’m not asking for much.

Recently, my girlfriend asked about what languages were good for a beginner to learn. Her sister is interested in computers and is going to be going to community college soon, and wants (or at least, is wanted) to prepare for possible classes in programming by learning a simple language which will teach her the fundamentals.

Myself, being born of a number of languages, Common Lisp and VB6 as well as others, immediately thought, “Scheme.” I soon realized though that, if this girl is going to community college, chances are they don’t really want to teach her to be a brilliant, deep thinking, professorial type of person, but rather a run-of the mill, decent, get the job frakking done coder. Now I want to say, run-of-the-mill coders are not run-of-the-mill intelligent people, they are often orders of magnitude smarter than most. Maybe I’m bias, but to give some perspective, I consider my father, who has a Bachelors degree from Northeastern University in Boston, and about 25 years of experience in the field, to be a run-of-the-mill coder. There is noone on the planet who I think is smarter than my father, not even me. Now that you have that nice perspective thing. Realize that though Scheme is a wonderful language for learning about CS as Theory, and even Math to some extent. It presents an unfortunately distorted world view. In the real world, we write code in an imperative style (though thats slowly changing, and I’m quite happy of that). In the real world, we write code in C, Java, or similar languages. In the real world, we write mostly object oriented code. In the real world, we generally solve problems iteratively using arrays as a principal data-structure– not recursively with lists as a principal data-structure. So I said to my girlfriend, “Well, you have a few options.” I continued to think about what makes a good languages, heres my list:

  • Easy to Setup

Chance are, if this is your first programming language ever, you might not understand all the intricacies of setting up a compiler/editor chain, or an IDE, or whatever, so this is obviously critical. You can’t use a language you can’t set up. This is why languages like VB are so popular, in my day, Visual Studio was trivial to install and use, and thats why I used it. I could have just as easily learned C or C++, my brain was plenty big enough for them, but I couldn’t grasp all the arcane mysticism of the GCC compiler at that time, so how was I supposed to do anything? In this area, I think Scheme beats Java, Notably, mzscheme’s DrScheme, which I still use when I write scheme code. It is an excellent, intuitive IDE for Scheme, It just works. It’s as easy (if not easier) than VS was to install, and I really wish I had found it before Dad’s copy of VS. Java, though I think a better “real world” option, is a little tougher to set up, obviously a editor/compiler chain, though a wonderful, no-frills way to write code, is not necessarily as intuitive to a complete beginner as a nice IDE. Since Eclipse (admittedly, the only Java IDE I’ve ever used) is not designed (like DrScheme is) to be used as a learning tool, there are alot of superfluous things that I, as a seasoned Java programmer, might use, but to a beginner, these things are just clutter. Clutter, as far as I’m concerned, means confusion.

  • Good Errors

I’ve known many languages in my time, and many– many of them had the worst, most inconceivably bad error messages in the world. It’s getting better, but even now — with languages designed for teaching, like haskell– the error messages are archaic and often unreadable. Now mind you, they are only such to the uninitiated haskellers, and as such, I don’t have a problem with trying to decipher “ambiguous type variables” and type errors and other such things. But to a beginner, all these kind of errors perpetrate is the myth that writing code is hard, and something that should be left to the realm of the ubergeek. So my opinion here is pretty standard, Good Errors => Good Beginner language.

  • Results Oriented

By “Results Oriented” I mean that it should be easy to see the results of your work. The Idea is simple, when someone is learning a language, they want to see there helloworld program just work. The don’t want to go through the work of writing line after line of code and then 12 different commands to compile the code, and then finally get to type that brilliant ./a.out command and see that you misspelt hello as hewwo. The point is, a beginner programmer, more than anything, needs encouragement. If a language can’t give a beginner a positive boost every time they do something right– then its not a good language for a beginner. HTML/Javascript are both wonderful examples of 100% results oriented languages. If I write an html file, and look at it in a browser, then I know immediately whether I did it correctly or not. I know exactly if it looks and reads the way I think its supposed to. Similarly with Javascript, its trivial for me to see whether my script works or not. This kind of language allows the newbie programmer to just get results, and thats the best thing a newbie programmer can have.

So, by now, you probably want to know what I thought was the best language to learn, well- I didn’t pick just one, but for what its worth, here is my list of the top few good beginner languages:

  1. HTML/Javascript
  2. Python
  3. Ruby
  4. Scheme or Java
  5. Other Web Languages (PHP, ASP, etc)
  6. C
  7. C++
  8. Perl
  9. Haskell/Erlang/ML et al
  10. Assembler?

Those rate from one being the absolute best language I think a beginner should learn to 10 being the absolute worst language for a beginner, I split up C/C++ because the object oriented stuff in C++ makes it even more complicated than just C alone with its pointers. Between gcc’s mild, jovial inanity, and pointers, makes C just a little to tough to make me think it’s a good option for a beginner. I want to mention that I am not judging these languages absolutely, I know most of them (though I have considerably less experience than I would like in most of them) and think that they are all quite wonderful. I’d especially like to learn Python soon. It’s been a while since I picked up an imperative language.

Oh, by that way– I only tossed Assembler on the list to make the list an even ten. Assembler is a terribly confusing subject to the uninitiated, and makes a good +infinity on the list. I suppose the list should also have a 0, which would be the metamagical mythical languages of “JustRight” where no matter what you type, there are never any bugs, it compiles and runs in optimal space and time, all NP problems become P, and magical unicorns prance in fields of cotton candy and happiness outside your cube.

Then again, you could argue JustRight would present a distorted realworld view too.

~~Joe

Published in: on June 2, 2007 at 2:24 am  Comments (3)  

God, get the frak out of my Barne’s and Nobel section.

Scientists, please, I beg you, stop with this God vs Science bull already.

I went into my local Barne’s and Nobel today, and started to casually browse through my favorite section of the bookstore–

The Science Section.

Last time I checked, Books about God fell under the “Religion” category. Furthermore, I am pretty sure that Science, in fact, is not religion. Now, I like reading philosophical books, religious texts, etc, just as much as the next guy — heck, I even like reading things like the Bible or the Koran. But why the hell does my community feel it necessary to invade on religion’s ground? Leave God alone, I don’t want to hear about him, I don’t want books about him in my section of the bookstore.

Scientists reading this right now are screaming something like, “But Jooooe! They do it to us all the time!!!” Here’s my reply, Let them. If they feel that they, in order to be validated, need to shove their beliefs down other peoples throats, treat them like they act. You don’t get a toddler to stop whining about not getting what they want. Religious types want you to believe what they believe. The way you teach a toddler to stop crying about not getting their way is by letting them cry themselves out. If you continuously argue with them, they will lose. The people slamming science don’t take time to be unbiased about science, and unfortunately, I think that most of the people slamming Religion back are not taking the time to be unbiased about religion either. Aren’t we, as scientists, supposed to be unbiased about all ideas?

I’m annoyed that people who call themselves scientists are wasting there time on this whole God business, I’m annoyed that the religious community isn’t reigning in the idiots who are stepping outside there bounds. It’s a community’s responsibility to regulate itself. So, Science community, consider yourselves regulated. Religious community, take a hint.

~~Joe

Published in: on May 21, 2007 at 10:25 pm  Comments (4)  

Annoyances with Blogger.

I’m Mildly annoyed that blogger does not seem to recognized that when I put in some indented code, it replaces things with &nbsp’s and other gibberish. The same thing occurs with other punctuation. It’s bothersome.



C’est la vie.





~~Joe

Published in: on May 20, 2007 at 8:30 pm  Comments (2)  

Programming Languages Part I: Syntactic Similarity

I like languages. when I was younger, I remember reading The Hobbit and spending more time reading the first two preface pages about the Moon Runes, A gussied up version of the Futhark, than actually reading the book. For a good bit of time after that, I wanted to be a Linguist, and not a mathematician.

But Alas, over time my interests went from Natural Language to Formal, from Formal Language to Abstract Language, and from there to the wide world of Algebra and Logic. A good transition, I think. Nevertheless I still love Languages, and that love has now been turned to specifically Programming languages. I like these languages because they are first and foremost utilitarian. They are designed from the start to do one thing, get a point across to an idiot. Lets face it, Programmers and Computers alike are pretty damn dumb. The language is the smart thing. A Programmer has no capability to tell a computer what to do without the help of a good language, and a Computer can’t do anything it isn’t told to do. So the most fundamental piece of Computer Technology is that of the Language, the layer that binds Programmer with Programmee.

I love languages, but I often hate them too. Take, for instance, Java. Java is an exceptionally pretty language, but it’s also ugly as your Aunt Anita Maykovar (May – ko – var). Java effectively boils down to one syntactic structure, the class. Every Java file revolves around it, and in some ways, this is really good. The fundamental commonality this structure brings allows Java code to be easier to learn to read. Your brain is good at processing things that are similar, the pathways it has to draw are all about the same– so it’s easier to optimize up there. The issue I have with Java actually is that, sometimes its too good at looking the same. To the point where I forget where I am. I get lost in my own neural pathways while I try to figure out whether I’m looking at an Abstract class or an Interface, or if I’m looking at a dirty hack of a databearing class or if I’m looking at something more legitimate. C/C++ is great at making this distinction, but it’s also, IMO, ugly as shit, uglier even. I like C often for its ability to compartmentalize things, but I think it takes it to far, nothing looks alike, even if it should. One of my peeves with C vs Java is they’re taking extreme views on something which should be easily decided. I’d like to sum it up as a fundamental rule I want to see in all programming languages (though that will probably never happen). Here it is:

Syntacticly similar things should be Semantically similar things, and vice versa, according to the proportion of similarity.

That is, If I want to create an object called “List” which implements a linked list, and then I want to create an interface (which is really just a Type Class, I’ve come to realize, but thats a story for another day.) called “Listable” which, when implemented, forces a object to have list-like properties. These things should have some similar structure. However, this is not to say we should copy Java. Java takes this too far, I think. In that, Java follows the rule: “If things are Semantically similar, they are Structurally almost Identictal.” This is bad, Interfaces should look different than Classes, but only in a minor way. I’d like Java Interfaces, heck, Java in general if I could specify type signatures ala Haskell. I think Haskell has got it right when it comes to how types should work syntactically. The brilliance of this comes in when you try to write a Java Method with this Type Sig ala Haskell type syntax, here’s Fibonacci:

fib :: public, static :: int -&gt; int

fib(nth) {
(nth == 1) ? 1 : fib(nth-1) + fib(nth-2);
}

(I think that’ll work, but it’s been a while, so I might have it wrong.)

Granted, there are issues with this. Notably, Java has things like side effects, but these could be built into the type signatures. I think that the ultimate benefit of this kind of type signaturing is a separation of concerns syntactually. I think that overall, this would make the language as a whole a lot cleaner. As interfaces would no longer have stubs like:

public static int fib(int nth);

which, though nice, doesn’t carry the same amount of information that could be held in something like:

fib :: public, static :: int -> int

Syntactually, the latter structure is more extensible, it could allow for the incorporation things like side effect tracking, or thread signatures, which might look like:

fib :: public, static, threaded :: int -> (int, int, …) -> int

which says that effectively fib is a method with takes a int to a unspecified number of threads, to an integer.

I’m really just spitballing at the end here, with some neat ideas I think that a small change in Java’s syntactic structure could bring.

Just my thoughts.

PS: I don’t know if the Syntactic/Semantic Similarity rule is well known, but either way, its a damn good idea.

Published in: on May 18, 2007 at 4:09 am  Leave a Comment  

Haskell: The Good, Bad, and Ugliness of Types

I’ve started to learn Haskell, for those who don’t know, Haskell is a wonderful little language which is based on Lazy Evaluation, Pure Functional Programming, and Type Calculus.

Effectively, this means that, like Erlang and other sister languages, If I write a function foo in Haskell, and evaluate it at line 10 in my program. Then I evaluate it again at Line 10000, or 10000000, or any other point in my code. It will– given the same input– always return the same value. Furthermore, if I write a function to generate an arbitrary list of one’s, like this:

listOfOnes = 1 : listOfOnes

Haskell just accepts it as valid. No Questions asked. Schemer’s and ML’ers of the world are probably cowering in fear. Recursive data types are scary in an Eager language, But Haskell is lazy. Where the equivalent definition in scheme:

(define list-of-ones
(cons 1 list-of-ones))

would explode and probably crash your computer, (that is, if the interpreter didn’t catch it first.) in Haskell, its not evaluated till its needed, so until I ask Haskell to start working on the listOfOnes structure, it won’t. I like Languages like that, IMO, if a language is at least as lazy as I am, its good.

The third really neat thing about Haskell, and what really drew me to it in the first place, is the Type Checker. I’ve used Scheme for a while now, and I love it to death. Sometimes, though- Scheme annoys me. For instance, I was working on a function like this once:

;count-when-true : [bool] x [num] -&gt; num
;supposed to be a helper for filter, I want do a conditional sum. So I pass in (filter foo some-list-of-numbers) and some-list-of-numbers,
; and I should get out a sum of the elements
(define (count-when-true list-of-bools list-of-numbers)
;or-list : ([a] -> bool) x [[a]] -> bool
;applys a filter across a list of lists and ors the results
(cond [(or-list nil? (list-of-bools list-of-numbers)) 0]
[(car x) (+ (car list-of-number) (count-when-true (cdr list-of-bools) (cdr list-of-numbers)))]
[else (count-when-true (cdr list-of-bools) (cdr list-of-numbers))]))

This probably has bugs in it, doesn’t work right, etc. but the idea is to return a conditional sum, now. I want to use this on lists, thats how its defined, but sometimes the calling function would try to call it on atoms, instead of lists. Big problem? not really, pain in the ass to find, you bet. The issue was, when I was trying to figure out what was wrong, Scheme didn’t realize that the type of the inputs were wrong. This would have made the error obvious, but Scheme doesn’t care about types, thats it’s principle strength, until it starts making bugs hard to find. I HATE it when its hard to find bugs.

Lets face it, as programmers, we suck, we write lots of buggy functions, things are generally done wrong the first (two or three… thousand) times. Programming is a recursive process, we write some code, run it, check for bugs, fix bugs, run it, check, fix, etc. Until we get tired of finding bugs/the program doesn’t come up with any. IMO, languages should not be designed to force programmers to write bug-free code, which seems to be the consensus today. At least, thats what I gather from the interweb and such. The goal should be to make all bugs so blatently obvious, that when the programmer sits down and trys to debug his program, he can’t help but to smack himself in the face and proclaim, “!@#$, I missed that!” This is where Haskell Shines.

When I write Scheme, I typically don’t want to be burdened by knowing which types go where. Scheme is great at this, however, it takes things to far, I think, in that it forces you to never have types. Sure, typed schemes exist, but most of them suck, because scheme isn’t designed for types. Don’t get me wrong, typed schemes are wicked cool, I’ve used types in CL too, and they’re great, especially when you want to compile. So to solve the problem of not having types, we invented contracts, which are cool. For the unenlightened: a contract is a specification of what the given datastructure or function does in terms of its arguments. eg:

+ : num * num -> num
toASCII : string -> num
toCHAR : num ->; string

etc.

These can be read as follows:

literally:

+ is num cross num to num
etc

in english

+ is a function which takes two numbers and returns another number.

In Scheme, these contracts are basically comments, so Type checking is left to the programmer. This is all well and good, but I find it often leads to the practice of what I like to call single-typing. In which the programmer attempts to force all of his data to have the same type, or lists of the same type, or lists of lists, or etc. Typically, this results in convoluted datastructures which give FP in general a bad name. I’ve seen some horrible code written by single-typers, its bad, horrific even, It makes me want to gauge out my eyes with a pencil and tear my brain out… Okay, maybe its not that bad. Still, single-typing is most often bad. So how does Haskell fix it?

By not changing a thing.

Contracts are a wonderful Idea, they work, they just don’t work in Scheme. Because it was designed that way. Haskell has type inference, you don’t ever need to touch the Type Calculus capabilities of Haskell, You can– more or less– literally translate Scheme to Haskell with minimal difficulty. (Though, it may be easier just to write scheme in haskell.) But the brilliance of haskell is this:

Heres the Standard Factorial function in Scheme:

;Fac : int -> int

(define (fac x)
(cond [(= 0 x) 1]
[else (* n (fac (-n 1)))]))

Here it is in Haskell:

fac :: Int -> Int
fac(x)
| (x == 0) 1
| otherwise x * fac(x – 1)

(I used a ML style to make things look the same.)

The only real difference (besides some syntax changes) is the lack of the semicolon in front of the contract.

But what does all this do? Well, the difference comes during evaluation, watch this:

In Scheme:

(fac 1.414)

we have an infinite recursion, because:

(fac 1.414) -> 1 * fac(0.414) -> 1 * 0.414 * (fac -0.586) …

In Haskell:

fac 1.414

is a type error, and the whole thing kersplodes. Over, Evaluation Done, Haskell has Denied your function the right to evaluate.

In short, you have been rejected.Enough about the wonderfulness of the Type system. My title says the Good -> Bad -> Ugliness, obviously we’ve seen the good. How about the Bad?

Type Errors in Haskell:

Type errors in haskell suck, easy as that. They’re hard to understand, and in general, not very helpful. Further, alot of the differences between types are very subtle. For instance, consider the factorial function again, (just the type contracts for succinctness)

fac0 :: Int -> Int
fac1 :: Num -> Num

The look equivalent, right? Wrong. Num != Int, it includes Reals too.* So no lovely type errors here. These things are unfortunate, yes, but nothings really perfect. I could deal with this, but what I can’t deal with is exactly the problem I hoped to solve with Haskell, My bugs are hard to find. Not only that, they’re not hard to locate, I know exactly where they are, I just can’t decipher the cryptic text from the Haskell error stream to know exactly what the bug is. So I have to resort to piecing through the code bit by bit, trying to figure it out.

Silly.

Type Signatures are Ugly:

I Like Contracts, but Haskell doesn’t technically use them. Haskell has type signatures. Which are different.

So far, I’ve written contracts like this:

F : S * T * U * … -> D

I could also have:

F : S * T * … -> (D1, D2, …)

or if I wanted HOF’s

F : (G : X -> Y) * … -> (D, …)

these are all pretty easy to understand, (if you know how to read the shorthand). We know exactly what the arguments should be, elements of the set of elements of type S, or T etc. We also know exactly what the return types are, elements of the typed-set D, or ordered k-tuples of elements of typesets D1 through Dn, etc. Equivalent signatures in Haskell are:

(assuming f = F, and any capital letter is a valid type, and that …’s would be replaced with types in the end result.)**

f :: S -> T -> U -> … -> D
f :: S -> T -> … -> (D1, D2, …)
f :: (X -> Y) -> … -> (D, …)

Now, I understand that, since Haskell is Lazily evaluated, we want the type signatures to be heavily curried, hence the load of arrows. Honestly though, how hard is it to convert all that to a form Haskell can use? I’m not saying get rid of the arrow version, maybe just add an option to provide a “normal form” version, I shouldn’t have to add these in my code as comments, solely so I can understand whats going on. I understand that the implication method more accurately reflects what the compiler is doing, but as a programmer, I don’t really give a rats ass what the compiler is doing. As a mathematician,

foo :: Int -> String -> Num -> Bool

looks ugly, do I know what it means? Yes. Do I like the way it looks? No. I grasp that, as a Haskell Compiler, reading these type of signatures in makes things easier, and further, that these definitions make things easier to prove correct*** but damnit Haskell, I’m a mathematician, not a miracle worker, I want to be able to read those definitions intuitively, and not have to muddle around trying to figure out exactly what that signature represents. It’s ugly, fix it.

On that note, I am beginning to work on some Haskell Code which will convert a Type Signature of the form:

f :: S^n1 * T^n2 * … -> (D1,D2, … Dn)

to the form:

f :: S -> S -> .. n1 times .. -> T -> T -> ..n2 times.. -> (D1, D2, … Dn)

and hopefully, given some user input, the latter to the former as well. (This is not harder, sortof, but I can’t know what the normal form of the type signature should be without some user input about the in-arity (arity) and out-arity (ority) of the function.

Anywho, Haskell is awesome, go play with it.

~~Joe

*= Aside: I’m quite glad Haskell calls them Reals and not
something silly like Float (though that is allowed) or Double. Us
Mathematicians have had these names for years, IEEE can call the format
Double precision floating point of w/e the hell they want, they’re
reals, not doubles. Silly computer scientists…

Edit: Note that in fact I understand that floats != reals, but its about state of mind. I know I’m working on a computer, and so I’m not going to treat things as reals, but I want to be thinking as if I’m not limited, so that when I work with my code, I’m not tuning the algorithm to work with the computer, I’m tuning the computer to work with my algorithm. In this way, the problem becomes a problem of making the compiler better, rather than hacking my algorithm to work.

**= Haskell doesn’t really like capitalized function names.

***= Proofs of correctness are done through the Curry-Howard Isomorphism, which effectively states that if the contract of a given function is a valid statement of Logic, then the function is correct, otherwise its not. Note that this requires the Signature to be correctly written, ie:

concatString :: String -> String -. String as a signature for a function which zipped two strings together would be “correct” but only in the sense that the contract would be satisfied. A Proof of correctness means that the function of that type can exist, there are other methods related to this Isomorphism which allow for a better proof of the semantic correctness, as opposed to the more syntactual flare of Curry-Howard

Published in: on May 1, 2007 at 9:14 pm  Comments (5)  

The Second Stupidest thing on the planet.

This is only marginally brighter than TimeCube



The Earth Is Not Moving



Somehow, despite all scientific evidence, this man seems to believe that the earth is in fact the center of the universe, I’m not entirely sure why, but he says that the Bible, of all things, supports him.



Now, don’t get me wrong, if you want to believe in the bible, thats your right, I won’t judge. Hell, I believe in the bible, to an extent. But Come on, Can we really deny the evidence around us like this?



My Sister says that technically, I’m not agnostic, I’m a “comtemplative” (con – TEM – plah – tihv). I’m not sure what that means, but my belief is that- I, like the rest of humanity, am fallible. I’m fundamentally capable of making mistakes. And as such, I choose to put complete stock in nothing, I choose instead to believe that all things are innocent till proven guilty. Right until proven wrong, by a contradiction of pure fact, pure logic, or otherwise. I’m sure that everyone can agree- thats not a radical point of view, thats just common sense. As such, since I cannot prove that Evolution or Creation or any such nonsense is true, then they are both equally correct.



Hold on, aren’t they mutually exclusive? Isn’t it true that you can’t believe in both? Bullshit. Of course you can believe in both. The only important thing is that these things must make sense. They must fit the theory. Here’s my theory.



God Created some rules, laws of math. From these he derived the laws of physics, and then he sat his ass down, and watched the 100 billion year show. We Evolved, isn’t it amazing? That simple rules that God came up with led to such complex beauty as Sentience? Look at us, how did we get here by chance? Against all odds we evolved from nothing, doesn’t that sound like a concept that God would want us to learn from, Perseverence? Why the hell does it matter where we came from anyway? Why can’t we just agree to disagree that we got here- and so now we should figure out where we’re going? It’s ridiculous, we fight, we kill, maim, murder, yell, cheat, steal. Whats the Point? We’re supposed to be enlightened, but we behave like animals. I don’t seem to understand, why can’t we realize that we have different Ideas, that people are different, and that they’re entitled to be different. Maybe I’m getting a little Laodecian, but maybe thats what we need, some Lukewarm politics to get the status quo thrown the fuck up.



Just remember folks. Apparently, The Earth Is Not Moving







Published in: on April 18, 2007 at 4:54 pm  Leave a Comment  

Compilers and Orchestra

When I started this silly blog thing, I had hoped to be able to post it in on a regular basis about current events and happenings in my many (yah, right) mathematical (sometimes) travels (sittings) .



Needless to say, it didn’t work out that way.



Heres what I’m working on, and what I’m not working on anymore:



Orchestra, A CAMaCS:



Orchestra is designed to be a Composer Assistant, something that helps you get out of a rut by learning from previous compositions and applying rules you supply it to suggest the next few notes of whatever your working on. I hope to have the Paper I’m writing about it as well as the actual source up on my site soon. It’ll be pretty bare, I don’t think I’m going to get to the GUI any time soon, but it’ll be workable, more or less, if you don’t mind hand editing code to make it work… 🙂



Joe, a mini java compiler:

I’m taking CS41 something or other, a Senior level CS course,

I’m a Sophmore level Math major.

I suck.

I don’t think I’m going to be able to finish it, fortunately, I only took this course as an elective, I don’t need it to graduate, so I should be fine when I fail it miserably.



The material is interesting, I just don’t have the code-writing skill to handle the requirements.





Anywho, I intend to take PLT next year (what I should have taken this year) which I do want for my “Major” (which is Math + Algebra + AI + Functional Programming Languages + Logic = Math w/ concentration in Logic and Algebra, and a healthy smattering of AI and Formal Languages) I hope to end up working with Automated Theorem Proving/Proof Assistant systems, (hence my inspiration for Orchestra, and Automated Music Writing system, more or less.) with all that, but who knows.



Anyway,





~~Joe

Published in: on March 31, 2007 at 5:38 am  Leave a Comment