Modeling Data with Functional Programming

Description

Lambda.r is a language extension that supports a functional programming style in R. As an alternative to the object-oriented systems, lambda.r offers a functional syntax for defining types and functions. Functions can be defined with multiple distinct function clauses similar to how multipart mathematical functions are defined. There is also support for pattern matching and guard expressions to finely control function dispatching, all the while still supporting standard features of R. Lambda.r also introduces its own type system with intuitive type constructors are and type constraints that can optionally be added to function definitions. Attributes are also given the attention they deserve with a clean and convenient syntax that reduces type clutter.

Details

Data analysis relies so much on mathematical operations, transformations, and computations that a functional approach is better suited for these types of applications. The reason is that object models rarely make sense in data analysis since so many transformations are applied to data sets. Trying to define classes and attach methods to them results in a futile enterprise rife with arbitrary choices and hierarchies. Functional programming avoids this unnecessary quandry by making objects and functions first class and preserving them as two distinct entities.

R provides many functional programming concepts mostly inherited from Scheme. Concepts like first class functions and lazy evaluation are key components to a functional language, yet R lacks some of the more advanced features of modern functional programming languages. Lambda.r introduces a syntax for writing applications using a declarative notation that facilitates reasoning about your program in addition to making programs modular and easier to maintain.

Function Definition

Functions are defined using the %as% symbol in place of <-. Simple functions can be defined as simply as

f(x) %as% x 

and can be called like any other function.

f(1) 

Functions that have a more complicated body require braces.

f(x) %as% { 2 * x }

g(x, y) %as% {
  z <- x + y
  sqrt(z)
}

Multipart functions and guards

Many functions are defined in multiple parts. For example absolute value is typically defined in two parts: one covering negative numbers and one covering everything else. Using guard expressions and the %when% keyword, these parts can be easily captured.

abs(x) %when% { x < 0 } %as% -x
abs(x) %as% x

Any number of guard expressions can be in a guard block, such that all guard expressions must evaluate to true.

abs(x) %when% {
  is.numeric(x)
  length(x) == 1
  x < 0
} %as% -x

abs(x) %when% {
  is.numeric(x)
  length(x) == 1
} %as% x

If a guard is not satisfied, then the next clause is tried. If no function clauses are satisfied, then an error is thrown.

Pattern matching

Simple scalar values can be specified in a function definition in place of a variable name. These scalar values become patterns that must be matched exactly in order for the function clause to execute. This syntactic technique is known as pattern matching.

Recursive functions can be defined simply using pattern matching. For example the famed Fibonacci sequence can be defined recursively.

fib(0) %as% 1
fib(1) %as% 1
fib(n) %as% { fib(n-1) + fib(n-2) }

This is also useful for conditionally executing a function. The reason you would do this is that it becomes easy to symbolically transform the code, making it easier to reason about.

pad(x, length, TRUE) %as% c(rep(NA,length), x)
pad(x, length, FALSE) %as% x

It is also possible to match on NULL and NA.

sizeof(NULL) %as% 0
sizeof(x) %as% length(x)

Types

A type is a custom data structure with meaning. Formally a type is defined by its type constructor, which codifies how to create objects of the given type. The lambda.r type system is fully compatible with the built-in S3 system. Types in lambda.r must start with a capital letter.

Type constructors

A type constructor is responsible for creating objects of a given type. This is simply a function that has the name of the type. So to create a type Point create its type constructor.

Point(x,y) %as% list(x=x,y=y) 

Note that any built-in data structure can be used as a base type. Lambda.r simply extends the base type with additional type information.

Types are then created by calling their type constructor.

p <- Point(3,4)

To check whether an object is of a given type, use the %isa% operator.

p %isa% Point

Type constraints

Once a type is defined, it can be used to limit execution of a function. R is a dynamically typed language, but with type constraints it is possible to add static typing to certain functions. S4 does the same thing, albeit in a more complicated manner.

Suppose we want to define a distance function for Point. Since it is only meaningful for Points we do not want to execute it for other types. This is achieved by using a type constraint, which declares the function argument types as well as the type of the return value. Type constraints are defined by declaring the function signature followed by type arguments.

distance(a,b) %::% Point : Point : numeric
distance(a,b) %as% { sqrt((b$x - a$x)^2 + (b$y - a$y)^2) }

With this type constraint distance will only be called if both arguments are of type Point. After the function is applied, a further requirement is that the return value must be of type numeric. Otherwise lambda.r will throw an error. Note that it is perfectly legal to mix and match lambda.r types with S3 types in type constraints.

Type variables

Declaring types explicitly gives a lot of control, but it also limits the natural polymorphic properties of R functions. Sometimes all that is needed is to define the relationship between arguments. These relationships can be captured by a type variable, which is simply any single lower case letter in a type constraint.

In the distance example, suppose we do not want to restrict the function to just Points, but whatever type is used must be consistent for both arguments. In this case a type variable is sufficient.

distance(a,b) %::% z : z : numeric
distance(a,b) %as% { sqrt((b$x - a$x)^2 + (b$y - a$y)^2) }

The letter z was used to avoid confusion with the names of the arguments, although it would have been just as valid to use a.

Type constraints and type variables can be applied to any lambda.r function, including type constructors.

The ellipsis type

The ellipsis can be inserted in a type constraint. This has interesting properties as the ellipsis represents a set of arguments. To specify that input values should be captured by the ellipsis, use ... within the type constraint. For example, suppose you want a function that multiplies the sum of a set of numbers. The ellipsis type tells lambda.r to bind the types associated with the ellipsis type.

sumprod(x, ..., na.rm=TRUE) %::% numeric : ... : logical : numeric
sumprod(x, ..., na.rm=TRUE) %as% { x * sum(..., na.rm=na.rm) }

> sumprod(4, 1,2,3,4)
[1] 40

Alternatively, suppose you want all the values bound to the ellipsis to be of a certain type. Then you can append “'...“' to a concrete type.

sumprod(x, ..., na.rm=TRUE) %::% numeric : numeric... : logical : numeric
sumprod(x, ..., na.rm=TRUE) %as% { x * sum(..., na.rm=na.rm) }

> sumprod(4, 1,2,3,4)
[1] 40
> sumprod(4, 1,2,3,4,'a')
Error in UseFunction(sumprod, "sumprod", ...) :
  No valid function for 'sumprod(4,1,2,3,4,a)' 

If you want to preserve polymorphism but still constrain values bound to the ellipsis to a single type, you can use a type variable. Note that the same rules for type variables apply. Hence a type variable represents a type that is not specified elsewhere.

sumprod(x, ..., na.rm=TRUE) %::% a : a... : logical : a
sumprod(x, ..., na.rm=TRUE) %as% { x * sum(..., na.rm=na.rm) }

> sumprod(4, 1,2,3,4)
[1] 40
> sumprod(4, 1,2,3,4,'a')
Error in UseFunction(sumprod, "sumprod", ...) :
  No valid function for 'sumprod(4,1,2,3,4,a)' 

The don't-care type

Sometimes it is useful to ignore a specific type in a constraint. Since we are not inferring all types in a program, this is an acceptable action. Using the “'.“' within a type constraint tells lambda.r to not check the type for the given argument.

For example in f(x, y) %::% . : numeric : numeric, the type of x will not be checked.

Attributes

The attribute system in R is a vital, yet often overlooked feature. This orthogonal data structure is essentially a list attached to any object. The benefit of using attributes is that it reduces the need for types since it is often simpler to reuse existing data structures rather than create new types.

Suppose there are two kinds of Points: those defined as Cartesian coordinates and those as Polar coordinates. Rather than create a type hierarchy, you can attach an attribute to the object. This keeps the data clean and separate from meta-data that only exists to describe the data.

Point(r,theta, 'polar') %as% {
  o <- list(r=r,theta=theta)
  o@system <- 'polar'
  o
}

Point(x,y, 'cartesian') %as% {
  o <- list(x=x,y=y)
  o@system <- 'cartesian'
  o
}

Then the distance function can be defined according to the coordinate system.

distance(a,b) %::% z : z : numeric
distance(a,b) %when% {
  a@system == 'cartesian'
  b@system == 'cartesian'
} %as% {
  sqrt((b$x - a$x)^2 + (b$y - a$y)^2)
}

distance(a,b) %when% {
  a@system == 'polar'
  b@system == 'polar'
} %as% {
  sqrt(a$r^2 + b$r^2 - 2 * a$r * b$r * cos(a$theta - b$theta))
}

Note that the type constraint applies to both function clauses.

Debugging

As much as we would like, our code is not perfect. To help troubleshoot any problems that exist, lambda.r provides hooks into the standard debugging system. Use debug.lr as a drop-in replacement for debug and undebug.lr for undebug. In addition to being aware of multipart functions, lambda.r's debugging system keeps track of what is being debugged, so you can quickly determine which functions are being debugged. To see which functions are currently marked for debugging, call which.debug. Note that if you use debug.lr for all debugging then lambda.r will keep track of all debugging in your R session. Here is a short example demonstrating this.

> f(x) %as% x
> debug.lr(f)
> debug.lr(mean)
>
> which.debug()
[1] "f"    "mean"

Note

Stable releases are uploaded to CRAN about once a year. The most recent package is always available on github [2] and can be installed via 'rpackage' in 'crant' [3].

rpackage https://github.com/zatonovo/lambda.r/archive/master.zip

Author(s)

Brian Lee Yung Rowe

Maintainer: Brian Lee Yung Rowe <r@zatonovo.com>

References

[1] Blog posts on lambda.r: http://cartesianfaith.com/category/r/lambda-r/

[2] Lambda.r source code, https://github.com/muxspace/lambda.r

[3] Crant, https://github.com/muxspace/crant

See Also

%as%, describe, debug.lr, %isa%

Examples

is.wholenumber <-
  function(x, tol = .Machine$double.eps^0.5)  abs(x - round(x)) < tol

## Use built in types for type checking
fib(n) %::% numeric : numeric
fib(0) %as% 1
fib(1) %as% 1
fib(n) %when% {
  is.wholenumber(n)
} %as% {
  fib(n-1) + fib(n-2)
}

fib(5)


## Using custom types
Integer(x) %when% { is.wholenumber(x) } %as% x

fib.a(n) %::% Integer : Integer
fib.a(0) %as% Integer(1)
fib.a(1) %as% Integer(1)
fib.a(n) %as% { Integer(fib.a(n-1) + fib.a(n-2)) }

fib.a(Integer(5))


## Newton-Raphson optimization
converged <- function(x1, x0, tolerance=1e-6) abs(x1 - x0) < tolerance
minimize <- function(x0, algo, max.steps=100)
{
  step <- 0
  old.x <- x0
  while (step < max.steps)
  {
    new.x <- iterate(old.x, algo)
    if (converged(new.x, old.x)) break
    old.x <- new.x
  }
  new.x
}

iterate(x, algo) %::% numeric : NewtonRaphson : numeric
iterate(x, algo) %as% { x - algo$f1(x) / algo$f2(x) }

iterate(x, algo) %::% numeric : GradientDescent : numeric
iterate(x, algo) %as% { x - algo$step * algo$f1(x) }

NewtonRaphson(f1, f2) %as% list(f1=f1, f2=f2)
GradientDescent(f1, step=0.01) %as% list(f1=f1, step=step)


fx <- function(x) x^2 - 4
f1 <- function(x) 2*x
f2 <- function(x) 2

algo <- NewtonRaphson(f1,f2)
minimize(3, algo)

algo <- GradientDescent(f1, step=0.1)
minimize(3, algo)

Results