Posts tagged "lisp":
Introduction: lists galore
I learned programming backwards, plunging right on into C and, shortly after, C++ and Java from the very beginning. I was knee deep in complex data structures, pointers and abstruse template syntax in no time. And the more complex it all felt, the more i thought i was learning. Of course, i was clueless.
Reading SICP and learning about functional programming changed it all. There were many occasions for revelation and awe, but one of my most vivid recollections of that time is my gradual discovery of the power of simplicity. At about half way into SICP i realised in wonder that that beautiful and powerful world was apparently being constructed out of extremely simple pieces. Basically, everything was a list. Of course there were other important ingredients, like procedures as first-class objects, but the humble list was about the only data structure to be seen. After mulling on it for a little bit, i saw where lists draw their power from: recursion. As you know, lists are data types recursively defined: a list is either the empty list or an element (its head) followed by another list (its tail):
list = 
list = a : list
where i'm borrowing Haskell's notation for the empty list () and the list constructor (:), also known by lispers as () and cons. So that was the trick, i thought: lists have recursion built-in, so to speak, and once you've read a little bit about functional programming you don't need to be sold on the power and beauty of recursive programs.
It is often the case that powerful and beautiful yet simple constructs have a solid mathematical foundation, and only when you grasp it do you really realize how powerful, beautiful and amazingly simple that innocent-looking construct is. Lists, and recursive operations on them, are an excellent case in point. But the path connecting them to their mathematical underpinnings is a long and winding one, which lays in the realm of Category Theory.
I first became acquainted of the relationship between categories and recursive programming reading Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire, by Erik Meijer, Maarten Fokkinga and Ross Paterson. Albeit very enjoyable, this paper presupposes a high degree of mathematical sophistication on the reader's side. I will try in this article to give you a simplified overview of the concepts involved, including Category Theory, its application to programming languages and what funny names like catamorphism, anamorphism or lambda-lifting have to do with your everyday list manipulations. Of course, i'll be only scratching the surface: interspersed links and the Further reading section provide pointers to more in-depth explorations of this wonderland.
Categories are (relatively) simple constructs. A category consists of a set O of objects, and a set A of arrows between elements of O. Arrows are composable: if there's an arrow from a to b, and another one from b to c, there must be an arrow from a to c (where a, b and c are elements of O). Besides, they are associative: if you have arrows from a to b, b to c, and c to d, you can go from a to d via two different paths, namely, first from a to c and then from c to d, or first from a to b and then from b to d. Finally, for each element a in O there's an identity arrow which goes from a to itself (called an identity), such that following this arrow changes nothing. These properties are better visualized with a diagram (or a bit of mathematical notation), as shown in the image on the right.
A category captures a mathematical world of objects and their relationships. The canonical example of a category is Set, which contains, as objects, (finite) sets and, as arrows, (total) functions between them. But categories go far beyond modeling sets. For instance, one can define a category whose objects are natural numbers, and the 'arrows' are provided by the relation "less or equal" (that is, we say that there is an arrow joining two numbers a and b if a is less or equal than b)/./ What we are trying to do with such a definition is to somehow capture the essence of ordered sets: not only integers are ordered but also dates, lemmings on a row, a rock's trajectory or the types of the Smalltalk class hierarchy. In order to abstract what all those categories have in common we need a way to go from one category to another preserving the shared structure in the process. We need what the mathematicians call an isomorphism, which is the technically precise manner of stating that two systems are, in a deep sense, analogous; this searching for commonality amounts to looking for concepts or abstractions, which is what mathematics and (good) programming is all about (and, arguably, intelligence itself, if you are to believe, for instance, Douglas Hofstadter's ideas).
To boot, our definition of a category already contains the concept of isomorphic objects. Think of an arrow from a to b as an operation that transforms a in b. An arrow from b to a will make the inverse transformation. If composing both transformations gives you the identity, you are back to the very same object a, and we say that a and b are isomorphic: you can transform one into the other and back at will. In a deep sense, this concept captures a generic way of expressing equality that pervades all maths: if you're not afraid of a little bit of maths, Barry Mazur's essay When is a thing equal to some other thing? is an excellent introduction to Category Theory with an emphasis in the concept of equality. Among many other things, you will learn how the familiar natural numbers can be understood as a category, or how an object is completely defined by the set of its transformations (and, therefore, how to actually get rid of objects and talk only of transformations; i know this is stretching and mixing metaphors (if not plain silly), but this stress in arrows, as opposed to objects, reminded me of Alan Kay's insistence on focusing on messages rather than objects). Another introductory article with emphasis on categories as a means to capture sameness is R. Brown and T. Porter's Category Theory: an abstract setting for analogy and comparison.
Not only objects inside a category can be transformed into each other. We reveal the common structure of two disjoint categories by means of a functor mapping across two categories. A functor consists of two functions: one that maps each object of the first category to an object in the second, and another one putting in correspondence arrows in one category with arrows in the second. Besides, these functions must preserve arrow composition. Let me spell this mathematically. Consider to categories, C and C' with object sets O and O' and arrow sets A and A'. A functor F mapping C to C' will consist then of two functions (Fo, Fa); the first one taking elements of O to elements of O':
Fo: O -> O'
Fo(a) in O' for every a in O
and the second one taking arrows from A to arrows in A':
Fa: A -> A'
Fa(f) in A' for every f in A
and such that, if f is an arrow from a to b in C, Fa(f) is an arrow from Fo(a) to Fo(b) in C'. Moreover, we want that following arrows in C is 'analogous' to following them in C', i.e., we demand that
Fa(fg) = Fa(f)Fa(g)
In the left hand side above, we are composing two arrows in C and then going to C', while in the right hand side we first take each arrow to C' and, afterwards, compose them in there. If C and C' have the same structure, these two operations must be equivalent. Finally, F must preserve identities: if i is the identity arrow for an element a in O, Fa(i)must be the identity arrow for Fo(a) in O'. The diagram on the left shows a partial graph (i'm not drawing the identity arrows and their mappings) of a simple functor between two categories, and ways of going from an object a in the first category to an object x in the second one which are equivalent thanks to the functor's properties.
As you can see in this simple example, the functor gives us the ability of seeing the first category as a part of the second one. You get a category isomorphism in the same way as between objects, i.e., by demanding the existence of a second functor from C' to C (you can convince yourself that such a functor does not exist in our example, and, therefore, that the two categories in the diagram are not isomorphic).
You have probably guessed by now one nifty property of functors: they let us going meta and define a category whose objects are categories and whose arrows are functors. Actually, Eilenberg and MacLane's seminal paper General theory of natural transformations used functors and categories of categories to introduce for the first time categories (natural transformations are structure-preserving maps between functors: this Wikipedia article gives an excellent overview on them).
But enough maths for now: it is high time to show you how this rather abstract concepts find their place in our main interest, programming languages.
Categories and programming languages
About the only similarity between C and Haskell programming is that one spends a lot of time typing ASCII arrows. But of course, Haskell's are much more interesting: you use them to declare the type of a function, as in
floor:: Real -> Int
The above stanza declares a function that takes an argument of type real and returns an integer. In general, a function taking a single argument is declared in Haskell following the pattern
fun:: a -> b
where a and b are types. Does this ring a bell? Sure it does: if we identify Haskell's arrows with categorical ones, the language types could be the objects of a category. As we have seen, we need identities
id:: a -> a
id x = x
and arrow composition, which in Haskell is denoted by a dot
f:: b -> c
g:: a -> b
fg:: a -> b -> c
fg = f . g
Besides, associativity of arrow composition is ensured by Haskell's referential transparency (no side-effects: if you preserve referential transparency by writing side-effect free functions, it won't matter the order in which you call them): we've got our category. Of course, you don't need Haskell, or a statically typed language for that matter: any strongly typed programming language can be modelled as a category, using as objects its types and as arrows its functions of arity one. It just happens that Haskell's syntax is particularly convenient, but one can define function composition easily in any decent language; for instance in Scheme one would have
(define (compose f g) (lambda (x) (f (g x)))
Functions with more than one arguments can be taken into the picture by means of currying: instead of writing a function of, say, 2 arguments:
(define (add x y) (+ x y))
(add 3 4)
you define a function which takes one argument (x) and returns a function which, in turn, takes one argument (y) and returns the final result:
(define (add x) (lambda (y) (+ x y)))
((add 3) 4)
Again, Haskell offers a pretty convenient syntax. In Haskell, you can
add x y = x + y
which gets assigned, when applied to integers, the following type:
add:: Int -> (Int -> Int)
add is not a function from pairs of integers to integers, but
a function that takes an integer and returns a function of type
Int -> Int.
Finally, we can also deal with functions taking no arguments and
constant values by introducing a special type, called unit or 1 (or
void in C-ish), which has a unique value (spelled
Haskell). Constants of our language (as, e.g.,
True or 43.23) are then
represented by arrows from 1 to the constant's type; for instance,
True is an
1 -> Boolean arrow. The unit type is an example of what in
category theory is known as a terminal object.
Now that we have successfully modelled our (functional) programming
language as a category (call it C), we can use the tools of the theory
to explore and reason about the language constructs and
properties. For instance, functors will let me recover the original
motivation of this post and explore lists and functions on them from
the point of view of category theory. If our language provides the
ability to create lists, its category will contain objects (types) of
the 'list of' kind; e.g.
[Int] for lists of integers,
lists of Booleans and so on. In fact, we can construct a new
sub-category CL by considering list types as its objects and functions
taking and returning lists as its arrows. For each type a we have a
way of constructing a type, [a] in the sub-category, i.e., we have a
map from objects in C to objects in CL. That's already half a
functor: to complete it we need a map from functions in C to functions
in CL. In other words, we need a way to transform a function acting on
values of a given type to a function acting on lists of values of the
same type. Using the notation of the previous section:
Fo(a) = [a]
Fa(f: a -> b) = f': [a] -> [b]
Fa is better known as
map in most programming languages. We call the
process of going from
f' lifting (not to be confused with a
related, but not identical, process known as lambda lifting), and it's
usually pretty easy to write an operator that lifts a function to a
new one in CL: for instance in Scheme we would write:
(define (lift f) (lambda (lst) (map f lst)))
lift to truly define a functor we need that it behaves well
respect to function composition:
(lift (compose f g)) = (compose (lift f) (lift g))
We can convince ourselves that this property actually holds by means of
a simple example. Consider the function
next which takes an integer to
its successor; its lifting
(lift next) will map a list of integers to
a list of their successors. We can also define
mapping (lists of) integers to (lists of) their predecessors.
(compose next prev) is just the identity, and, therefore,
(lift (compose next prev)) is the identity too (with lifted
signature). But we obtain the same function if we compose
(lift prev) in CL, right? As before, there's nothing specific to
Scheme in this discussion. Haskell even has a
Functor type class
capturing these ideas. The class defines a generic lift operation,
fmap that, actually, generalizes our list lifting to arbitrary
fmap :: (a -> b) -> (f a -> f b)
f a is the new type constructed from
a. In our previous
f a = [a], but if your language gives you a way of
constructing, say, tuples, you can lift functions on given types to
functions on tuples of those types, and repeat the process with any
other type constructor at your disposal. The only condition to name it a
functor, is that identities are mapped to identities and composition is
fmap id = id
fmap (p . q) = (fmap p) . (fmap q)
I won't cover usage of type constructors (and their associated functors) other than lists, but just mention a couple of them: monads, another paradigmatic one beautifully (and categorically) discussed by Stefan Klinger in his Programmer's Guide to the IO Monad - Don't Panic (also discussed at LtU), and the creation of a dance and music library, for those of you looking for practical applications.
To be continued…
Returning to lists, what the lifting and categorical description above buys us is a way to formalize our intuitions about list operations, and to transfer procedures and patterns on simple types to lists. In SICP's section on Sequences as conventional interfaces, you will find a hands-on, non-mathematical dissection of the basic building blocks into which any list operation can be decomposed: enumerations, accumulators, filters and maps. Our next step, following the bananas article i mentioned at the beginning, will be to use the language of category theory to provide a similar decomposition, but this time we will talk about catamorphisms, anamorphisms and similarly funny named mathematical beasts. What the new approach will buy us is the ability to generalize our findings beyond the list domain and onto the so-called algebraic types. But this will be the theme of a forthcoming post. Stay tunned.
The best introductory text on Category Theory i've read is Conceptual Mathematics : A First Introduction to Categories by F. William Lawvere (one of the fathers of Category Theory) and Stephen Hoel Schanuel. It assumes no sophisticated mathematical background, yet it covers lots of ground. If you feel at home with maths, the best option is to learn from the horse's mouth and get a copy of Categories for the Working Mathematician, by Mac Lane.
The books above do not deal with applications to Computer Science, though. For that, the canonical reference is Benjamin Pierce's Basic Category Theory for Computer Scientists, but i find it too short and boring: a far better choice is, in my opinion, Barr and Well's Category Theory and Computer Science. A reduced version of Barr and Well's book is available online in the site for their course Introduction to Category Theory. They are also the authors of the freely available Toposes, Triples and Theories, which will teach you everything about monads, and then more. Marteen Fokkinga is the author of this 80-pages Gentle Introduction to Category Theory, with a stress on the calculational and algorithmical aspects of the theory. Unless you have a good mathematical background, you should probably take gentle with a bit of salt.
Let me close by mentioning a couple of fun applications of Category Theory, for those of you that know a bit about it. Haskell programmers will like this implementation of fold and unfold (as a literate program) using F-(co)algebras and applied to automata creation, while those of you with a soft spot for Physics may be interested in John Baez's musings on Quantum Mechanics and Categories.
After a few scheming years, i had come to view objects as little more than poor-man closures. Rolling a simple (or not so simple) object system in scheme is almost a textbook exercise. Once you've got statically scoped, first-order procedures, you don't need no built-in objects. That said, it is not that object-oriented programming is not useful; at least in my case, i find myself often implementing applications in terms of a collection of procedures acting on requisite data structures. But, if we restrict ourselves to single-dispatch object oriented languages, i saw little reason to use any of them instead of my beloved Scheme.
Things started to change recently due to my discovering the pleasures of Smalltalk. First and foremost, it offers a truly empowering integrated ambient to live and develop in. Second, if you're going to use objects, using the simplest, cleanest syntax will not hurt. Add to that some reading on the beautiful design principles underlying Smalltalk, and one begins to wonder if closures aren't, in fact, poor-man objects–or at least i do, whenever i fall in an object-oriented mood (i guess i'm yet not ready to reach satori).
But Scheme is not precisely an ugly or bad designed language, so i needed some other reason to switch language gears for my OO programming. I knew there's more than encapsulation or subtype polymorphism in object-land from my readings on CLOS (the Common Lisp Object System), or on Haskell's type classes (and its built-in parametric polymorphism), but i was after something retaining Smalltalk's elegance. And then i remembered that, when i was a regular lurker in the Tunes project's mailing lists and IRC channel, a couple of smart guys were implementing an OO language whose syntax was smalltalkish. That language (which, if memory servers, started life with the fun name who me?) has evolved during the last few years into a quite usable programming environment named Slate, started by Lee Salzman and currently developed and maintained by Brian Rice.
I've been reading about Slate during the last few days, and decided to learn it. What motivated me was discovering how Slate goes beyond mainstream object-oriented programming by incorporating well-known (but hardly used) and really powerful paradigms. In short, Slate improves Smalltalk's single-dispatch model by introducing and combining two apparently incompatible technologies: multiple dispatch and prototype-based programming. To understand the whys and hows of Slate, there's hardly a better way than reading Lee Salzman's Prototypes with Multiple Dispatch. The following discussion is, basically, an elaboration of Lee's explanation on the limitations of mainstream OO languages, and how to avoid them with the aid of PMD.
Note: the images are thumbnails of this PDF file, with clickable links inside.
Fishes and sharks
Let's start by showing why on earth would you need anything beyond Smalltalk's object system (or any of its modern copycats). Consider a simple oceanographic ecosystem analyser, which deals with (aquatic) Animals, Fishes and Sharks. These are excellent candidates for class definitions, related by inheritance. Moreover, we are after modeling those beasts' behaviours and, in particular, their reactions when they encounter each other: each time a Shark meets a Fish of other species, the Shark will swallow the other Fish, while when a Shark meets Shark they will fight. As a result of such fighting, Sharks get unhealthy, which regrettably complicates matters: wound sharks won't try to eat other fishes, and will swim away other sharks instead of fighting them. The image on the left provides a sketchy representation of the code we need to model our zoo. Waters are quickly getting muddled implementation-wise.
On the one hand, subtype polymorphism based just on the object receiving the encounter message: we need, in addition, to take into account the argument's concrete type to implement the desired behaviour. This is a well-known issue in single-dispatch languages, whose cure is, of course, going to multiple dispatching (see below). In particular, we want to avoid the need to modify existing classes whenever our hierarchy is extended.
On the second hand, varying state (exemplified here by the Shark's isHealthy instance variable complicates the implementation logic. As we will see, prototype-based languages offer a way to factor out this additional complexity.
The need to adjust behaviour on the basis of the type of both a message receiver and its arguments arises frequently in practice. So frequently, that a standard way of dealing with it has been christened as the Visitor design pattern. The technique, also known as double-dispatch, is well known: you can see, for instance, how it's applied to arithmetic expressions in Smalltalk, or read about a generic implementation of multimethods in Python (which also includes a basically language-independent discussion on the issues at hand). If you happen to be a C++ programmer, you may be tempted to think that global functions and overloading solve the problem in that language. Well, think twice: a proper implementation of multiple dispatch in C++ needs of RTTI and templates, as shown in this article.
CLOS and Dylan are two examples of languages solving the issue from the onset by including support for multi-methods. The idea is to separate methods from classes (which only contain data slots). As shown in the pseudo-code of the accompanying figure, methods are defined as independent functions with the same name, but differing in their arguments' types (in CLOS, a set of such methods is called a generic function). When a generic function is called, the system selects the actual method to be invoked using the types of all the arguments used in the invocation. The encounter generic function in our running example provides a typical example, as shown in the figure on the right. The benefits of having multi-methods at our disposal are apparent: the code is simpler and, notably, adding new behaviours and classes to the system does not need modifications of existing code. For instance, we can introduce a Piranha, which eats unhealthy sharks instead of swimming away from them, by defining the requisite class and methods, without any modification whatsoever to the already defined ones.
On the downside, we have still to deal with the complications associated with internal state. Enter the magic world of prototype-based systems.
The ultimate dynamic
If you like dynamic languages, chances are you'll find prototype-based system an almost perfect development environment. Prototype-based languages emerged as an evolution of Smalltalk with the invention of Self by David Ungar and Randall B. Smith during the late eighties. The key idea behind Self is noticing that, most of the time, class definitions needlessly coerce and complicate your object model.
A class definition becomes a contract to be satisfied by any instance, and it is all too easy to miss future or particular needs of your objects (class-based inheritance is just a partial solution to this problem, as shown, for instance, by the so-called fragile base class problem). But, if you look around you, objects change in internal behaviour and data content continously, and our attempts at distilling their Platonic nature are often in vain.
In prototype-based programming, instead of providing a plan for constructing objects, you simply clone existing instances and modify their behaviour by directly changing the new instance's slots (which provide uniform access to methods and state). New clones contain a pointer to their parent, from which they inherit non-modified slots: there is no way to access state other than via messages sent to instances, which simplifies tackling with state.
Class-based languages oblige you to keep two relationships in mind to characterize object instances: the "is-a" relationship of the object with its class, and the "kind-of" relationship of that class with its parent. In self, inheritance (or behaviour delegation) is the only one needed. As you can see, Self is all about making working with objects as simple as possible. No wonder Ungar and Smith's seminal paper was titled Self: The Power of Simplicity. Needless to say, a must read.
The figure above shows how our running example would look in selfish pseudo-code. As promised, state is no longer surfacing in our method implementation's logic. Unfortunately, we have lost the benefits of multi-methods in the process. But fear not, for, as we will see, you can eat your cake and have it too. Instead of pseudo-code, you can use Self itself, provided you are the happy owner of a Mac or a Sun workstation. Or you can spend 20 fun minutes seeing the Self video, which features the graphical environment accompanying the system. Like Smalltalk, Self provides you with a computing environment where objects are created, by cloning, and interact with you. The system is as organic and incremental as one can possibly get.
The best of both worlds
Prototyping and multiple dispatch are, at first sight, at odds. After all, method dispatching based on arguments' type needs, well, a type for each argument, doesn't it? As it happens, Lee Salzman and Brian Rice have envisioned a way of combining the power of both paradigms into Slate. In fact, proving how this is possible is the crux of Lee's article. In addition, Slate aims at providing a complete development environment in the vein of Smalltalk or Self. Too good to be true? In future installments of this blog category, we'll see how and why it's true, but, if you cannot wait, just run-not-walk to Slate's site. You'll have a great time.
Back in the old days i was a macho C++ programmer, one of those sneering at Java or any other language but C, willing to manage my memory and pointers and mystified by the complexity of the template syntax (it was difficult and cumbersome, ergo it had to be good). Everyone has a past.
Things began to change when i decided to add Guile extensibility to GNU MDK. I was using the project as an excuse to learn everything one has to learn to write free software, from parsers with flex and bison, to documentation generators like texinfo and doxygen or localisation via gettext. Next thing was scriptability, and those days Scheme was still the way to extend your GNU programs (last time i checked, GNOME was full of XML, a windows-like registry and, oh my, C#… no Scheme (or good taste) to be seen).
So, when i first encountered Scheme i was high on static type checking, object oriented programming in its narrow C++ flavour, and all that jazz. I didn't understand immediately what was so cool about having an interpreter, and defining functions without the compiler checking their signature at every single call made me feel uneasy. I was told that i still had strong type checking in Lisp, but that it is deferred to run time, instead of at the apparently safer compile phase. I didn't get it. Thanks god, SICP was so fun that i kept on learning, and i kept wondering for a while what was so great about interpreters and dynamic typing.
Problem was, i was writing C programs in Scheme. In a compiled language (a la C) and, to some degree, in any statically typed one, your code is dead. You write pages and pages of inert code. You compile it. Still dead. Only when you launch that binary does it come to life, only that it lives elsewhere, beyond your reach. Admittedly, i'm exaggerating: you can reach it in a convoluted way via a debugger. But still. A debugger is an awkward beast, and it will only work with the whole lot: all your program compiled, linked and whatnot.
Enter a dynamic language. Enter its REPL. When you have a, say, Lisp interpreter at your disposal you don't write your code first and load it later (that's what i was doing at first). You enter your code piecewise, function by function, variable by variable at that innocent looking prompt. You develop incrementally, and, at every single moment, your objects and functions are alive: you can access them, inspect them, modify them. Your code becomes an organic creature, plastic. Its almost not programming, but experimenting.
Maybe you're raising a skeptical eyebrow. Maybe you have one of those modern visual-something debugger that lets you modify your compiled code on the fly and continue running your code using the new definitions and you think that's what i'm talking about… Well, no, sorry, that's only part of what i'm talking about. To begin with, you continue executing your program. I can do whatever i want. But that's not all. We are talking about a dynamically typed language. That means that me and my little REPL have much more leeway to modify the living code, and thus much more margin to grow up and evolve the code.
At the end of the day, dynamically typed languages give me freedom. Programming is a creative process and greatly benefits from that freedom. At first, abandoning the safety net provided by static typing was a little bit scary, but as i grew up as a programmer i felt more and more confident, and gradually the initially uneasy feeling morphed into joy. The joy of REPL.
Richard P. Gabriel has made a far better job in beautifully conveying what i'm trying to express in his excellent introduction to David Lamkins' book Successful Lisp, entitled The Art of Lisp and Writing. Unfortunately, i haven't found it online – you can read the first few pages in amazon.com's "look inside this book" section for this book. And also in his essay Do Programmers Need Seat Belts?. Paul Graham has famously argued in favour of bottom-up development in many of his essays, and specially in Programming Bottom-Up:
It's worth emphasizing that bottom-up design doesn't mean just writing the same program in a different order. When you work bottom-up, you usually end up with a different program. Instead of a single, monolithic program, you will get a larger language with more abstract operators, and a smaller program written in it. Instead of a lintel, you'll get an arch.
Finally, please note that i'm well aware that the static vs. dynamic typing debate is still open, and that decent type systems like those in Haskell and ML have, arguably, much to offer in the way to solid software engineering. Type theory has also a powerful and beautiful mathematical foundation. The above is just my gut feeling and current position on these issues, and i don't pretend to have backed it with solid technical argumentation. Nor was it my goal. I'm more interested here in programming as a creative activity than as engineering.