Function operators

Function operators

The final functional programming technique we will discuss is function operators: functions that take (at least) one function as input and return a function as output. Function operators are similar to functionals, but where functionals abstract away common uses of loops, function operators instead abstract over common uses of anonymous functions. Function operators allow you to add extra functionality to existing functions, or combine multiple existing functions to make new tools.

Here's an example of a simple function operator that makes a function chatty, showing its input and output (albeit in a very naive way). It's useful because it gives a window into functionals, and we can use it to see how lapply() and mclapply() execute code differently.

show_results <- function(f) {
  function(x) {
    res <- f(x)
    cat(format(x), " -> ", format(res, digits = 3), "\n", sep = "")
    res
  }
}
s <- c(0.4, 0.3, 0.2, 0.1))
x2 <- lapply(s, show_results(Sys.sleep))
x2 <- mclapply(s, show_results(Sys.sleep))

Function operators can make it possible to eliminate parameters by encapsulating common variations as function transformations. Like functionals, there's nothing you can't do without them; but they can make your code more readable and expressive. One advantage of using FOs instead of additional arguments is that your functions become more extensible: your users are not limited to functionality that you've thought up - as long as they modify the function in the right way, they can add functionality that you've never dreamed of. This in turn leads to small, simpler functions that are easier to understand and learn.

In the last chapter, we saw that most built-in functionals in R have very few arguments (some have only one!), and we used anonymous functions to modify how they worked. In this chapter, we'll start to build up tools that replace standard anonymous functions with specialised equivalents that allow us to communicate our intent more clearly. For example, in the last chapter we saw how to use Map with some fixed arguments:

Map(function(x, y) f(x, y, zs), xs, ys)

Later in this chapter, we'll learn about partial application, and the partial() function that implements it. Partial application allows us modify our original function directly, leading to the following code that is both more succint and more clear (assuming your vocabulary includes partial()).

Map(partial(f, zs = zs), xs, yz)

In this chapter, we'll explore four classes of function operators (FOs). Function operators can:

• add behaviour, leaving the function otherwise unchanged, like automatically logging when the function is run, ensuring a function is run only once, or delaying the operation of a function.

• change input, like partially evaluating the function, converting a function that takes multiple arguments to a function that takes a list, or automatically vectorising a functional.

• change output, for example, to return a value if the function throws an error, or to negate the result of a logical predictate

• combine functions, for example, combining the results of predicate functions with boolean operators, or composing multiple function calls.

For each class, we'll show you useful function operators, and show you how you can use them as alternative means of describing tasks in R: as combinations of multiple functions instead of combinations of arguments to a single function.

At a higher level, function operators allow you to define specialised languages for solving wide classes of problems. The building blocks are simple functions, which you combine together with function operators to solve more complicated problems. The final section of the chapter shows how you can do this to build a language that specifies what arguments to a function should look like.

Add additional behaviour

The first class of FOs are those that leave the inputs and outputs of a function unchanged, but add some extra behaviour. In this section, we'll see functions that:

• log to disk everytime a function is run
• automatically print how long it took to run
• add a delay to avoid swamping a server
• print to console every n invocations (useful if you want to check on a long running process)
• save time by caching previous function results

To make this concrete, imagine we want to download a long vector of urls with download.file(). That's pretty simple with lapply():

lapply(urls, download.file, quiet = TRUE)

But because it's such a long list we want to print some output so that we know it's working (we'll print a . every ten urls), and we also want to avoid hammering the server, so we add a small delay to the function between each call. That leads to a rather more complicated for loop, since we can no longer use lapply() because we need an external counter:

i <- 1
for(url in urls) {
  i <- i + 1
  if (i %% 10 == 0) cat(".")
  Sys.delay(1)
  download.file(url, quiet = TRUE) 
}

Reading this code is quite hard because we are using low-level functions, and it's not obvious (without some thought), what we're trying to do. In the remainder of this chapter we'll create FO that encapsulate each of the modifications, allowing us to instead do:

lapply(urls, dot_every(10, delay_by(1, download.file)), quiet = TRUE)

Useful behavioural FOs

Implementing the function are straightforward. dot_every is the most complicated because it needs to modify state in the parent environment using <<-.

• Delay a function by delay seconds before executing:

  delay_by <- function(delay, f) {
    function(...) {
      Sys.sleep(delay)
      f(...)
    }
  }

• Print a dot to the console every n invocations of the function:

  dot_every <- function(n, f) {
    i <- 1
    function(...) {
      if (i %% n == 0) cat(".")
      i <<- i + 1
      f(...)
    }
  }

• Log a time stamp and message to a file everytime a function is run:

  log_to <- function(path, message, f) {
    stopifnot(file.exists(path))
  
    function(...) {
      cat(Sys.time(), ": ", message, sep = "", file = path, 
        append = TRUE)
      f(...)
    }
  }

• Print how long it took a function to execute:

  time_it <- function(f) {
    function(...) {
      start <- proc.time()
      res <- f(...)
      end <- proc.time()
  
      print(end - start)
      out
    }
  }

• Ensure that if the first input is NULL the output is NULL (the name is inspired by Haskell's maybe monad which fills a similar role in Haskell, making it possible for functions to work with a default empty value).

  maybe <- function(f) {
    function(x, ...) {
      if (is.null(x)) return(NULL)
      f(x, ...)
    }
  }

Notice that I've made the function the last argument to each FO, this make it reads a little better when we compose multiple function operators. If the function was the first argument, then instead of:

download <- dot_every(10, delay_by(1, download.file))

we'd have

download <- dot_every(delay_by(download.file, 1), 10)

which I think is a little harder to follow because the argument to dot_every() is far away from the function call. That's sometimes called the Dagwood sandwhich problem: you have too much filling (too many long arguments) between your slices of bread (parentheses).

Memoisation

Another thing you might worry about when downloading multiple file is downloading the same file multiple times: that's a waste of time. You could work around it by calling unique on the list of input urls, or manually managing a data structure that mapped the url to the result. An alternative approach is to use memoisation: a way of modifying a function to automatically cache its results.

library(memoise)
slow_function <- function(x) {
  Sys.sleep(1)
  10
}
system.time(slow_function())
system.time(slow_function())
fast_function <- memoise(slow_function)
system.time(fast_function())
system.time(fast_function())

Memoisation is an example of a classic tradeoff in computer science: we are trading space for speed. A memoised function uses a lot more memory (because it stores all of the previous inputs and outputs), but is much much faster.

A slightly more realistic use case is implementing the Fibonacci series (a topic we'll come back to software systems). The Fibonacci series is defined recursively: the first two values are 1 and 1, then f(n) = f(n - 1) + f(n - 2). A naive version implemented in R is very slow because (e.g.) fib(10) computes fib(9) and fib(8), and fib(9) computes fib(8) and fib(7), and so on, so that the value for each location gets computed many many times. Memoising fib() makes the implementation much faster because each value only needs to be computed once.

fib <- function(n) {
  if (n < 2) return(1)
  fib(n - 2) + fib(n - 1)
}
system.time(fib(23))
system.time(fib(24))

fib2 <- memoise(function(n) {
  if (n < 2) return(1)
  fib2(n - 2) + fib2(n - 1)
})
system.time(fib2(23))
system.time(fib2(24))

It doesn't make sense to memoise all functions. The example below shows that a memoised random number generator is no longer random:

runifm <- memoise(runif)
runif(10)
runif(10)

Once we understand memoise(), it's straightforward to apply it to our modified download.file():

download <- dot_every(10, memoise(delay_by(1, download.file)))

Exercises

• What does the following function do? What would be a good name for it?

  f <- function(g) {
    g <- match.fun(g)
    result <- NULL
    function(...) {
      if (is.null(result)) {
        result <<- g(...)
      }
      result
    }
  }
  runif2 <- f(runif)
  runif2(10)

• Modify delay_by so that instead of delaying by a fixed amount of time, it ensures that a certain amount of time has elapsed since the function was last called. That is, if you called f(); Sys.sleep(2); f() there shouldn't be an extra delay.

• There are three places we could have added a memoise call: why did we choose the one we did?

  download <- memoise(dot_every(10, delay_by(1, download.file)))
  download <- dot_every(10, memoise(delay_by(1, download.file)))
  download <- dot_every(10, delay_by(1, memoise(download.file)))

Input modification

• modify an existing function by changing the default arguments (pryr::curry)
• convert a function that works with a data frame to a function that works with a matrix (plyr::colwise)
• convert a function of multiple parameters to a function of a single list parameter (plyr::splat)
• vectorise a scalar function (base::Vectorise)

splat <- function (f) {
  f <- match.fun(f)
  function(args) {
    do.call(f, args)
  }
}

Partial function evaluation

A common task is making a variant of a function that has certain arguments "filled in" already. Instead of doing:

x <- function(a) y(a, b = 1)
x <- partial_eval(y, b = 1)

compact <- function(x) Filter(Negate(is.null), x)
compact <- curry(Filter, Negate(is.null))

One way to implement curry is as follows:

curry <- function(FUN, ...) { 
  .orig <- list(...)
  function(...) {
    do.call(FUN, c(.orig, list(...)))
  }
}

But implementing it like this prevents arguments from being lazily evaluated, so pryr::curry() has a more complicated implementation that works by creating the same anonymous function that you'd created by hand, using techniques from the computing on the language chapter.

Alternative to providing ... to user supplied functions.

Map(function(x, y) f(x, y, zs), xs, ys)
Map(Curry(f, zs = zs), xs, ys)

Vectorise

Vectorize takes a non-vectorised function and vectorises with respect to the arguments given in the vectorise.args parameter. This doesn't give you any magical performance improvements, but it is useful if you want a quick and dirty way of making a vectorised function.

An mildly useful extension of sample would be to vectorize it with respect to size: this would allow you to generate multiple samples in one call.

sample2 <- Vectorize(sample, "size", SIMPLIFY = FALSE)
sample2(1:10, rep(5, 4))
sample2(1:10, 2:5)

In this example we have used SIMPLIFY = FALSE to ensure that our newly vectorised function always returns a list. This is usually a good idea. Vectorize does not work with primitive functions.

Output modifications

• negate the result of a predicate function (base::Negate)
• return a default value if the function throws an error (fail_with)
• convert a function that prints output to a function that returns output
• return the time it took to run the function instead of the result

Negate <- function(f) {
  f <- match.fun(f)
  function(...) !f(...)
}

capture_it <- function(f) {
  function(...) {
    capture.output(f(...))
  }
}
str_out <- capture_it(str)
str(1:10)
str_out(1:10)

failwith <- function(default = NULL, f, quiet = FALSE) {
  f <- match.fun(f)
  function(...) {
    out <- default
    try(f(...), silent = quiet)
    out
  }
}
log("a")
failwith(NA, log)("a")
failwith(NA, log, quiet = TRUE)("a")

Or taken one of the examples from the functional programming chapter:

timers <- lapply(compute_mean, time_it)
lapply(timers, call_fun, x)

Negate takes a function that returns a logical vector, and returns the negation of that function. This can be a useful shortcut when the function you have returns the opposite of what you need.

Negate <- function(f) {
  f <- match.fun(f)
  function(...) !f(...)
}

(Negate(is.null))(NULL)

One function I find handy based on this is compact: it removes all non-null elements from a list:

compact <- function(x) Filter(Negate(is.null), x)

Exercises

• The evaluate package makes it easy to capture all the outputs (results, text, messages, warnings, errors and plots) from an expression.

Combine multiple functions

• combine two functions together (pryr::compose)
• combine the results of two vectorised functions into a matrix (plyr::each)
• combining logical predicates with boolean operators (the topic of the following section)

This type of programming is called point-free (sometimes derogatorily known as pointless) because it you don't explicitly refer to variables (which are called points in some areas of computer science.) Another way of looking at it is that because we're using only functions and not parameters we use verbs and not nouns, so code in this style tends to focus on what's being done, not what it's being done to.

Common patterns

Most function operators follow a similar pattern:

funop <- function(f, otherargs) {
  f <- match.fun(f)
  function(...) {
    # do something
    res <- f(...)
    # do something else
    res
  }
}

Anonymous functions vs. computing on the language

The disadvantage of this technique is that when you print the function you won't get informative arguments. One way around this is to write a function that replaces ... with the concrete arguments from a specified function by computing on the language.

undot <- function(closure, f) {
  # Can't find out arguments to primitive function, so give up.
  if (is.primitive(f)) return(closure)

  body(closure) <- replace_dots(body(closure), formals(f))
  formals(closure) <- formals(f)

  closure
}

replace_dots <- function(expr, replacement) {
  if (!is.recursive(x)) return(x)
  if (!is.call(x)) {
    stop("Unknown language class: ", paste(class(x), collapse = "/"),
      call. = FALSE)
  }

  pieces <- lapply(y, modify_lang, replacement = replacement)
  as.call(pieces)
}

match.fun

It's often useful to be able to pass in either the name of a function, or a function. match.fun(). Also useful because it forces the evaluation of the argument: this is good because it raises an error right away (not later when the function is called), and makes it possible to use with lapply.

Caveat: http://stackoverflow.com/questions/14183766

Also need the opposite: to get the name of the function. There are two basic cases: the user has supplied the name of the function, or they've supplied the function itself. We cover this in more detail on computing in the language. But unfortunately it's difficult to

fname <- function(call) {
  f <- eval(call, parent.frame())
  if (is.character(f)) {
    fname <- f
    f <- match.fun(f)
  } else if (is.function(f)) {
    fname <- if (is.symbol(call)) as.character(call) else "<anonymous>"
  }
  list(f, fname)
}
f <- function(f) {
  fname(substitute(f))
}
f("mean")
f(mean)
f(function(x) mean(x))

Function composition

"%.%" <- compose <- function(f, g) {
  f <- match.fun(f)
  g <- match.fun(g)
  function(...) f(g(...))
}
compose(sqrt, "+")(1, 8)
(sqrt %.% `+`)(1, 8)

Then we could implement Negate as

Negate <- curry(compose, `!`)

Exercises

• What does the following function do? What would be a good name for it?

  g <- function(f1, f2) {
    function(...) f1(...) || f2(...)
  } 
  Filter(g(is.character, is.factor), mtcars)

  Can you extend the function to take any number of functions as input? You'll probably need a loop.

• Write a function and that takes two function as input and returns a single function as an output that ands together the results of the two functions. Write a function or that combines the results with or. Add a not function and you now have a complete set of boolean operators for predicate functions.

Case study: checking function inputs and boolean algebra

We will explore function operators in the context of avoiding a common R programming problem: supplying the wrong type of input to a function. We want to develop a flexible way of specifying what a function needs, using a minimum amount of typing. To do that we'll define some simple building blocks and tools to combine them. Finally, we'll see how we can use S3 methods for operators (like +, |, etc.) to make the description even less invasive.

The goal is to be able to succinctly express conditions about function inputs to make functions safer without imposing additional constraints. Of course it's possible to do that already using stopifnot():

f <- function(x, y) {
  stopifnot(length(x) == 1 && is.character(x))
  stopifnot(is.null(y) || 
    (is.data.frame(y) && ncol(y) > 0 && nrow(y) > 0))
}

What we want to be able to express the same idea more evocatively.

f <- function(x, y) {
  assert(x, and(eq(length, 1), is.character))
  assert(y, or(is.null, 
    and(is.data.frame, and(gt(nrow, 0), gt(ncol, 0)))))
}
f <- function(x, y) {
  assert(x, length %==% 1 %&% is.character)
  assert(y, is.null %|% 
    (is.data.frame %&% (nrow %>% 0) %&% (ncol %>% 0)))
}
f <- function(x, y) {
  assert(x, (length) == 1 && (is.character))
  assert(y, (is.null) || ((is.data.frame) & !empty))
}

is.string <- (length) == 0 && (is.character)
f <- function(x, y) {
  assert(x, (is.string))
  assert(y, (is.null) || ((is.data.frame) & !(empty)))
}

We'll start by implementation the assert() function. It should take two arguments, an object and a function.

assert <- function(x, predicate) {
  if (predicate(x)) return()

  x_str <- deparse(match.call()$x)
  p_str <- strwrap(deparse(match.call()$predicate), exdent = 2)
  stop(x_str, " does not satisfy condition:\n", p_str, call. = FALSE)
}
x <- 1:10
assert(x, is.numeric)
assert(x, is.character)

and <- function(f1, f2) {
  function(...) {
    f1(...) && f2(...)
  }
}
or <- function(f1, f2) {
  function(...) {
    f1(...) || f2(...)
  }
}
not <- function(f1) {
  function(...) {
    !f1(...)
  }
}

has_length <- function(n) {
  function(x) length(x) == n
}
or(and(is.character, has_length(4)), is.null)

It would be cool if we could rewrite to be:

(is.character & has_length(4)) | is.null

but due to limitations of S3 it's not possible. The closest we could get is:

"%|%" <- function(e1, e2) function(...) e1(...) || e2(...)
"%&%" <- function(e1, e2) function(...) e1(...) && e2(...)

(is.character %&% has_length(4)) %|% is.null

Another approach would be do something like:

Function <- function(x) structure(x, class = "function")
Ops.function <- function(e1, e2) {
  f <- function(y) {
    if (is.function(e1)) e1 <- e1(y)
    if (is.function(e2)) e2 <- e2(y)
    match.fun(.Generic)(e1, e2)
  }
  Function(f)
}
length <- Function(length)
length > 5
length * length + 3 > 5

is.character <- Function(is.character)
is.numeric <- Function(is.numeric)
is.null <- Function(is.null)

is.null | (is.character & length > 5)

If you wanted to make the syntax less invasive (so you didn't have to manually cast functions to Functions) you could maybe override the parenthesis:

"(" <- function(x) if (is.function(x)) Function(x) else x 
(is.null) | ((is.character) & (length) > 5)

If we wanted to eliminate the use of () we could extract all variables from the expression, look at the variables that are functions and then wrap them automatically, put them in a new environment and then call in that environment.

Exercises

• Something with Negate

• Extend and, or and not to deal with any number of input functions. Can you keep them lazy?

• Implement a corresponding xor function. Why can't you give it the most natural name? What might you call it instead? Should you rename and, or and not to match your new naming scheme?

• Once you have read the S3 chapter, replace and, or and not with appropriate methods of &, | and !. Does xor work?