14 April 2016 (minor edit 4 February 2018)
Copyright (C) 2016, 2018, H. Conrad Cunningham
Acknowledgements: This is a set of notes written to accompany my lectures on Chapter 4 of the book Functional Programming in Scala by Paul Chiusano and Runar Bjarnason (Manning, 2015). I constructed the notes around the ideas and Scala examples from that chapter. I encourage all students to study that chapter.
I also patterned some of the discussion of for-comprehensions on Chapter 10 of the document Scala by Example by Martin Odersky, on Chapter 23 of the book Programming in Scala, Second Edition, by Marin Odersky, Lex Spoon, and Bill Venners (Artima Press, 2010), on the relevant parts of the Scala language specification, and on a relevant FAQ in the Scala documentation.
Prerequisite: This discussion assumes the reader is familiar with the programming concepts and Scala features covered in my Notes on Scala for Java Programmers, Recursion Concepts and Terminology, and Functional Data Structures.
Advisory: The HTML version of this document uses MathML in a few locations. For best results, use a browser that supports the display of MathML. A good choice as of April 2016 seems to be a recent version of Firefox from Mozilla.
TODO: Change to use code blocks, restructure to match current style
A benefit of Java or Scala exception handling (i.e., using try-catch blocks) is that it consolidates error handling to well-defined places in the code.
However, code that throws exceptions typically exhibits two problems for functional programming.
It is not referentially transparent. It has side effects. Its meaning is dependent upon the context in which it is executed.
It is not type safe. Exceptions may cause effects that are of a different type than the return value of the function.
The key idea from Chapter 4 is to use ordinary data values to represent failures and exceptions in programs. This preserves referential transparency and type safety while also preserving the benefit of exception-handling mechanisms, that is, the consolidation of error-handling logic.
To do this, we introduce the Option and Either algebraic data types. These are standard types in the Scala library, but for pedagogical purposes Chapter 4 introduces its own definition that is similar as do the standard one.
This set of notes also introduces Scala features that we have not previously discussed extensively: type variance, call-by-name parameter passing, and for-comprehensions.
Option Algebraic Data TypeWe define an algebraic data type Option using sealed trait Option with case class Some to wrap a value and case object None to denote an empty value. We specify the operations on the data type using the method-chaining style.
import scala.{Option => _, Either => _, _} // hide standard
sealed trait Option[+A] {
def map[B](f: A => B): Option[B]
def getOrElse[B >: A](default: => B): B
def flatMap[B](f: A => Option[B]): Option[B]
def orElse[B >: A](ob: => Option[B]): Option[B]
def filter(f: A => Boolean): Option[A]
}
case class Some[+A](get: A) extends Option[A]
case object None extends Option[Nothing]The import statement above hides the standard definitions of Option and Either from the Scala standard library.
Before we move on to the definition of these methods, let’s consider two issues raised in the signatures of the getOrElse and orElse methods: one related to the generic parameters and the other to the default parameters.
As we have discussed previously, the +A annotation in Option[+A] declares that parameter A is covariant. That is, if S is a subtype of T, then Option[S] is a subtype of Option[T]. Also remember that Nothing is a subtype of all other types.
For example, suppose type Beagle is a subtype of type Dog, which, in turn, is a subtype of type Mammal. Then, because of the covariant definition, Option[Beagle] is a subtype of Option[Dog], which is a subtype of Option[Mammal]. This is intuitively what we expect.
However, in getOrElse and orElse, we use type constraint B >: A. This means that B must be equal to or a supertype of A. We also define value parameter of these functions to have type Option[B] instead of Option[A].
Why must we have this constraint?
See the chapter notes for the Functional Programming in Scala book (https://github.com/fpinscala/fpinscala/wiki) for more detail on this complicated issue. We sketch the argument below.
In some sense, the +A annotation declares that, in all contexts, it is safe to cast this type A to some supertype of A. The Scala compiler does not allow us to use this annotation unless we can cast all members of a type safely.
Suppose we declare orElse (incorrectly) as follows:
trait Option[+A] {
def orElse(o: Option[A]): Option[A]
...
}
We have a problem because this declaration of orElse only allows us to cast A to a subtype of A.
Why?
As with any function, orElse can only be passed a subtype of its declared argument type. That is, a function of type Dog => R can be passed an object of subtype like Beagle or of type Dog itself, but it cannot be passed a supertype object of Dog such as Mammal.
But orElse has a value parameter type of Option[A]. Because of the covariance of A (declared in the trait), this parameter only allows subtypes of A–not supertypes as required by the covariance.
So, we have a contradiction.
For the incorrect signature of orElse, the Scala compiler generates an error message such as “Covariant type A occurs in contravariant position.” We can get around this error by using a more complicated signature that does not mention A in any of the function arguments, such as:
def orElse[B >: A](o: Option[B]): Option[B]Now consider the second new feature appearing in the signatures of getOrElse and orElse–the => B and => Option[B] types for the parameters.
Scala’s primary parameter-passing mode is call by value as in Java and C. That is, the program evaluates the caller’s argument and the resulting value is bound to the corresponding formal parameter within the called function.
If the argument’s value is of a primitive type, then the value itself is passed. If the value is an object, then the address of (i.e., reference to) the object is passed.
A call-by-value parameter is called strict because the called function always requires that parameter’s value before it can execute. The corresponding argument must be evaluated eagerly before transferring control to the called function.
Scala also has call-by-name parameter passing. Consider the default: => B feature in the declaration
def getOrElse[B >: A](default: => B): BThe type notation => B means the calling program leaves the argument of type B unevaluated. That is, the calling program wraps the argument expression in an parameterless function and passes the function to the called method. This automatically generated parameterless function is sometimes called a thunk.
Every reference to the corresponding parameter causes the thunk to be evaluated. If the method does not access the corresponding parameter during some execution, then the parameter is never evaluated.
As with all higher-order arguments in Scala, a thunk is passed as a closure. In addition to the function, the closure captures any free variables occurring in the expression–that is, the variables defined in the caller’s environment but not within the expression itself.
Note: The closure actually captures the variable itself, not its value. So if the free variable is either a reference to a var or to a mutable data structure, then changes in the value are seen inside the called function. But in this course, we normally use immutable data structures and val declarations.
A call-by-name parameter is called nonstrict because the called function does not always require that parameter’s value for its execution. The corresponding argument can thus be evaluated lazily, that is, evaluated only if and when its value is needed.
We look at more implications of strict and nonstrict functions in Chapter 5.
Now, let’s define the methods in the Option data type.
Option methodsThe Option data type is similar to a list that is either empty or has one element.
The map method applies a function to its implicit Option argument. If the implicit argument is a Some, the method applies the function to the wrapped value and returns the resulting Some. If it is a None, the method just returns None. We can implement map using pattern matching directly as follows:
def map[B](f: A => B): Option[B] = this match {
case None => None
case Some(a) => Some(f(a))
}Similarly, we can use pattern matching directly to implement the getOrElse function. If the implicit argument is of type Some, this function returns the value it wraps. If the implicit argument is of type None, this function returns the value denoted by the default argument. By passing default by name, the argument is only evaluated when its value is needed.
def getOrElse[B >: A](default: => B): B = this match {
case None => default // evaluate the thunk
case Some(a) => a
}Function flatMap applies its argument function f, which might fail, to its implicit Option argument when this value is not None. We can define flatMap in terms of map and getOrElse as shown below. (Reminder: If we apply method names as operators in an infix manner, they associate to the left.)
def flatMap[B](f: A => Option[B]): Option[B] =
map(f) getOrElse NoneWe can also define flatMap using pattern matching directly.
def flatMap_1[B](f: A => Option[B]): Option[B] = this match {
case None => None
case Some(a) => f(a)
}Function orElse returns the implicit Option argument if is not None; otherwise, it returns the explicit Option argument. We can define orElse in terms of map and getOrElse or by directly using pattern matching.
def orElse[B >: A](ob: => Option[B]): Option[B] =
this map (Some(_)) getOrElse ob
def orElse_1[B>:A](ob: => Option[B]): Option[B] =
this match {
case None => ob
case _ => this
}The filter function converts its implicit argument from Some to None if it does not satisfy the boolean function p. We can define filter by using pattern matching directly or by using flatMap.
def filter(p: A => Boolean): Option[A] =
this match {
case Some(a) if p(a) => this
case _ => None
}
def filter_1(p: A => Boolean): Option[A] =
flatMap(a => if (p(a)) Some(a) else None)
Option for statistical mean and varianceConsider a function to calculate and return the mean (i.e., average value) of a list of numbers. This function must sum the list of numbers and divide by the number of elements. It might have a signature such as:
def mean(xs: List[Double]): DoubleBut what should be returned for empty lists?
We can modify the signature to use Option and define the function as follows:
def mean(xs: Seq[Double]): Option[Double] =
if (xs.isEmpty)
None
else
Some(xs.sum / xs.length)The return type now allows the possibility that the mean may be undefined. We thus extend a partial function to a total function in a meaningful way.
Above we also generalize the List type to its supertype Seq from the Scala library. Type Seq denotes a family of sequential data types that includes the List type, array-like collections, etc. This type defines the methods isEmpty, sum, and length.
If the mean of a sequence is , then the (statistical) variance of the sequence is the mean of the sequence formed by the terms for each (perserving the order).
Using the mean function defined above, we can compute the variance of a sequence of numbers as follows:
def variance(xs: Seq[Double]): Option[Double] =
mean(xs) flatMap
(m => mean(xs.map(x => math.pow(x - m, 2))))Option in the labelled digraphIn the doubly labelled Digraph case study, we defined the following method to return the label on a vertex:
def getVertexLabel(ov: A): B
In the case study’s DigraphList implementation, we defined this function as follows. The function terminates with an error when the the argument vertex is not present in the digraph.
def getVertexLabel(ov: A): B =
(vs dropWhile (w => w._1 != ov)) match {
case Nil => sys.error("Vertex " + ov + " not in digraph")
case ((_,l)::_) => l
} We can avoid the error termination in this function by changing the method signature to return an Option[B] instead of B.
def getVertexLabelOption(ov: A): Option[B] =
(vs dropWhile (w => w._1 != ov)) match {
case Nil => None
case ((_,l)::_) => Some(l)
} Code that uses this new Digraph method can call the various Option methods (or directly use pattern matching) to process the result appropriately.
For example, if the vertex label is a string, it may be appropriate in some scenarios to just use a null string for the label of a nonexistent vertex. Let g be a Digraph and be v be a possible vertex, then
(g getVertexLabelOption v) getOrElse ""would either return the label string for v if it exists or the null string if v does not exist.
Similarly, the code
(g getVertexLabelOption v) getOrElse
(sys.error("undefined vertex " + v))would still terminate with an error. However, this design enables the user of the Digraph library to decide under what circumstances and at what point in the code to terminate.
Idiom: A common pattern for computing with the Option type is to use map, flatMap, and filter to transform values generated by a function like getVertexLabelOption and then to use getOrElse for error handling at the end.
It seems that that deciding to use of Option could cause lots of changes to ripple through our code, much like the introduction of extensive exception-handling would. However, we can avoid that somewhat by using a technique called lifting.
For example, the Option type’s map function enables us to transform values of type Option[A] into values of type Option[B] using a function of type A => B.
Alternatively, we could consider map as transforming a function of type A => B into a function of type Option[A] => Option[B]. That is, we lift an ordinary function into a function on Option.
We can formalize this alternative view with the following function:
def lift[A,B](f: A => B): Option[A] => Option[B] = _.map(f)Thus any existing function can be transformed to work within the context of a single Option value. For example, we can lift the square root function from type Double to work with Option[Double] as follows:
def sqrtO: Option[Double] => Option[Double] = lift(math.sqrt _)We can now use sqrtO such as sqrtO(Some(4)). This evaluates to Some(2).
Chapter 4 of Functional Programming in Scala gives several examples where Option types can be used effectively in realistic scenarios.
Another useful function on Option data types is the function map2 that combines two Option values by lifting a binary function. If either of the arguments are None, then the result should also be None. We can define map2 as follows:
def map2[A,B,C](a: Option[A], b: Option[B])(f: (A, B) => C): Option[C] =
a flatMap (aa =>
b map (bb =>
f(aa, bb)))
This function applies a series of map and flatMap calls.
Because lifting is so common in functional programming, Scala provides a syntactic construct called a for-comprehension to facilitate its use. This construct is really syntactic sugar for a series of applications of map, flatMap, and withFilter. (Method withFilter works like filter except it filters on demand, without creating a new collection as a result.)
Here is the same code expressed as a for-comprehension:
def map2fc[A,B,C](a: Option[A], b: Option[B])(f: (A, B) => C):
Option[C] =
for {
aa <- a
bb <- b
} yield f(aa, bb)Of course, for-comprehensions are more general than just their use with Option. They can be used for the lists, arrays, iterators, ranges, streams, and other types in the Scala standard library that support the map, flatMap, and withFilter (or filter) operations. m Consider a list persons of person objects with name and age fields. We can collect the names of all persons who are age 21 and above as follows:
for (p <- persons if p.age >= 21) yield p.nameThis is equivalent to the following List expression
filter (p => p.age >= 21) map (p => p.name)
In general, a for-comprehension
evaluates expression e for each binding generated by the enumerator sequence enums and collects the results. An enumerator sequence begins with a generator and may be followed by additional generators, value definitions, and guards.
A generator p <- e produces a sequence of zero or more bindings from expression e by matching each value against pattern p.
A value definition p = e binds the names in pattern p to the result of evaluating the expression e.
A guard if e restricts the bindings to those that satisfy the boolean expression e.
We can translate (or desugar) for-comprehension (more or less) as follows:
We replace every generator p <- e, where p is a pattern and e is an expression, by
Here we use withFilter to filter out those items that do not match the pattern p.
While all comprehension have not been eliminated, repeat the following:
Translate for-comprehension
to the expression
Translate for-comprehension
where ... is a (possibly empty) sequence of generators, definitions, or guards, to the expression
Translate a generator p <- e followed by a guard if g to a single generator
where x,...,xn are the free variables of p.
Translate a generator p <- e followed by a value definition p1 = e1 to the following generator of pairs of values, where x and x1 are fresh names:
The Scala notation x@p means that name x is bound to the value of the expression p.
Note: Above we do not consider the imperative for-loops. These can also be translated as above except that the imperative method forEach is also needed.
As an example, we can translate (desugar) the for-comprehension
into expression:
As a second example, we can also translate the for-comprehension
into expression:
If no withFilter method is available, we can instead use:
As a third example, we can translate for-comprehension
into expression
The Scala language has no typing rules for the for-comprehensions themselves. The Scala compiler first translates for-comprehensions into calls on the various method and then checks the types. It does not require that methods map, flatMap, and withFilter have particular type signatures. However, a particular setup for some collection type C with elements of type A is the following:
trait C[A] {
def map[B](f: A => B): C[B]
def flatMap[B](f: A => C[B]): C[B]
def withFilter(p: A => Boolean): C[A]
}We can define our own data types to support for-comprehension by providing one or more of the required operations above.
If the data type defines just map, Scala allows for-comprehensions consisting of a single generator.
If the data type defines both flatMap and map, Scala allows for-comprehensions consisting of several generators.
If the data type defines withFilter, Scala allows for-comprehensions with guards. (If withFilter is not defined but filter is, Scala will currently use filter instead. However, this gives a deprecation warning, so this fallback feature may be eliminated in a future release of Scala.)
We added for-comprehensions to our own Option type earlier. We do the same for the Either type in the next section.
Note: A for-comprehension is, in general, convenient syntactic sugar for expressing compositions of monadic operators. If time allows, we will discuss monads later in the semester.
Either Algebraic Data TypeWe can use data type Option to encode that a failure or exception has occurred. However, it does not give any information about what went wrong.
We can encode this additional information using the algebraic data type Either.
import scala.{Option => _, Either => _, _} // hide builtin
sealed trait Either[+E,+A] {
def map[B](f: A => B): Either[E,B]
def flatMap[EE >: E, B](f: A => Either[EE,B]): Either[EE,B]
def orElse[EE >: E, AA >: A](b: => Either[EE,AA]): Either[EE,AA]
def map2[EE >: E, B, C](b: Either[EE, B])(f: (A,B) => C): Either[EE, C]
}
case class Left[+E](get: E) extends Either[E,Nothing]
case class Right[+A](get: A) extends Either[Nothing,A]By convention, we use the constructor Right to denote success and constructor Left to denote failure.
We can implement map, flatMap, and orElse directly using pattern matching on the Either type.
def map[B](f: A => B): Either[E, B] =
this match {
case Left(e) => Left(e)
case Right(a) => Right(f(a))
}
def flatMap[EE >: E, B](f: A => Either[EE, B]): Either[EE, B] =
this match {
case Left(e) => Left(e)
case Right(a) => f(a)
}
def orElse[EE >: E, AA >: A](b: => Either[EE, AA]): Either[EE, AA] =
this match {
case Left(_) => b
case Right(a) => Right(a)
}The availability of flatMap and map enable us to use for-comprehension generators with Either. We can thus implement map2 using a for-comprehension, as follows:
def map2[EE >: E, B, C](b: Either[EE, B])(f: (A, B) => C): Either[EE, C] =
for { a <- this; b1 <- b } yield f(a,b1)Let’s again use mean as an example and use a String to describe the failure in Left.
def mean(xs: IndexedSeq[Double]): Either[String, Double] =
if (xs.isEmpty)
Left("mean of empty list!")
else
Right(xs.sum / xs.length)We can use the Left value to encode more information, such as the location of the error in the program. For example, we might catch and return the value of an Exception generated as we do in the safeDiv function below.
def safeDiv(x: Int, y: Int): Either[Exception, Int] =
try Right(x / y)
catch { case e: Exception => Left(e) }We can abstract the computational pattern in the safeDiv function as function Try defined below:
def Try[A](a: => A): Either[Exception, A] =
try Right(a) // evaluate thunk
catch { case e: Exception => Left(e) }Chapter 4 of Functional Programming in Scala describes other functions for type Either.
Both Option and Either appear in the standard library.
The standard library Option type is similar to the one developed here, but the library version is missing some of the extended functions described in Chapter 4.
The standard library Either type is similar but more complicated, using projections on the left and right. It is also missing some of the extended functions from Chapter 4.
Study the Scala API documentation for more information on these data types.
The big idea in this chapter is to use ordinary values to represent exceptions and use higher-order functions for handling and propagating errors. As examples, we considered the algebraic data types Option and Either and functions such as map, flatMap, filter, and orElse to process their values.
We will use this general technique of using values to represent effects in the subsequent studies in this course.
We introduced the idea of nonstrict functions. We examine the implications and use of these more in Chapter 5.