% CSci 555: Functional Programming \ Functional Programming in Scala \ Functional Data Structures % **H. Conrad Cunningham** % **6 March 2019** --- lang: en --- | Copyright (C) 2016, 2018, 2019, [H. Conrad Cunningham ]() | Professor of [Computer and Information Science ]() | [University of Mississippi ]() | 211 Weir Hall | P.O. Box 1848 | University, MS 38677 | (662) **Note:** This set of notes accompanies my lectures on Chapter 3 of the book *Functional Programming in Scala* \[Chiusano 2015\] (i.e. the Red Book). **Prerequisites**: In this set of notes, I assume the reader is familiar with the programming concepts and Scala features covered in my *Notes on Scala for Java Programmers* \[Cunningham 2019a\], *Recursion Styles, Correctness, and Efficiency (Scala Version)* \[Cunningham 2019b\], and *Type System Concepts* \[Cunningham 2019c\]. **Browser Advisory:** The HTML version of this textbook requires a browser that supports the display of MathML. A good choice as of March 2019 is a recent version of Firefox from Mozilla. \newpage \setcounter{section}{2} # Functional Data Structures ## Introduction To do functional programming, we construct programs from collections of pure functions. Given the same arguments, a *pure function* always returns the same result. The function application is thus referentially transparent. By *referentially transparent* we mean that a name or symbol always denotes the same value in some well-defined context in the program. Such a pure function does not have *side effects*. It does not modify a variable or a data structure in place. It does not set throw an exception or perform input/output. It does nothing that can be seen from outside the function except return its value. Thus the data structures in pure functional programs must be *immutable*, not subject to change as the program executes. (If *mutable* data structures are used, no changes to the structures must be detectable outside the function.) For example, the Scala empty list---written as `Nil`{.scala} or `List()`{.scala}---represents a value as immutable as the numbers `2`{.scala} and `7`{.scala}. Just as evaluating the expression `2 + 7`{.scala} yields a new number `9`{.scala}, the concatenation of list `c`{.scala} and list `d`{.scala} yields a new list (written `c ++ d`{.scala}) with the elements of `c`{.scala} followed by the elements of `d`{.scala}. It does not change the values of the original input lists `c`{.scala} and `d`{.scala}. Perhaps surprisingly, list concatenation does not require both lists to be copied, as we see below. ## A `List` algebraic data type To explore how to build immutable data structures in Scala, we examine a simplified, singly linked list structure implemented as an algebraic data type. This *list data type* is similar to the builtin Scala `List`{.scala} data type. What do we mean by algebraic data type? ### Algebraic data types An *algebraic data type* is a type formed by combining other types, that is, it is a *composite* data type. The data type is created by an algebra of operations of two primary kinds: - a *sum* operation that constructs values to have one variant among several possible variants. These sum types are also called *tagged*, *disjoint union*, or *variant* types. The combining operation is the alternation operator, which denotes the choice of one but not both between two alternatives. - a *product* operation that combines several values (i.e. *fields*) together to construct a single value. These are *tuple* and *record* types. The combining operation is the Cartesian product from set theory. We can combine sums and products recursively into arbitrarily large structures. An *enumerated type* is a sum type in which the constructors take no arguments. Each constructor corresponds to a single value. ### ADT confusion Although sometimes the acronym ADT is used for both, an *algebraic data type* is a different concept from an *abstract data type*. - We specify an algebraic data type with its *syntax* (i.e. structure)---with rules on how to compose and decompose them. - We specify an abstract data type with its *semantics* (i.e. meaning)---with rules about how the operations behave in relation to one another. TODO: Update paragraph below better for Scala course The modules we build with abstract interfaces, contracts, and data abstraction, such as the Rational Arithmetic modules from Chapter 7 of reference \[Cunningham 2018\], are abstract data types. Perhaps to add to the confusion, in functional programming we sometimes use an algebraic data type to help define an abstract data type. ### Using a Scala trait A *list* consists of a sequence of values, all of which have the same type. It is a hierarchical data structure. It is either *empty* or it is a pair consisting of a *head* element and a *tail* that is itself a list of elements. We define `List`{.scala} as an abstract type using a Scala `trait`{.scala}. (We could also use an `abstract class`{.scala} instead of a `trait`{.scala}.) We define the *constructors* for the algebraic data type using the Scala `case class`{.scala} and `case object`{.scala} features. ~~~{.scala} sealed trait List[+A] case object Nil extends List[Nothing] case class Cons[+A](head: A, tail: List[A]) extends List[A] ~~~ Thus `List`{.scala} is a *sum type* with two alternatives: - `Nil`{.scala} constructs the singleton case object that represents the empty list. - `Cons(h,t)`{.scala} constructs a new list from an element `h`{.scala}, called the *head*, and a list `t`{.scala}, called the *tail*. `Cons`{.scala} itself is a *product (tuple) type* with two fields, one of which is itself a `List`{.scala}. The `sealed`{.scala} keyword tells the Scala compiler that all alternative cases (i.e. subtypes) are declared in the current source file. No new cases can be added elsewhere. This enables the compiler to generate safe and efficient code for pattern matching. As we have seen previously, for each `case class`{.scala} and `case object`{.scala}, the Scala compiler generates: - a constructor function (e.g. `Cons`{.scala}) - accessor functions (methods) for each field (e.g. `head`{.scala} and `tail`{.scala} on `Cons`{.scala}) - new definitions for `equals`{.scala}, `hashcode`{.scala}, and `toString`{.scala} In addition, the `case object`{.scala} construct generates a *singleton object*---a new type with exactly one instance. Programs can use the constructors to build instances and use the pattern matching to recognize the structure of instances and decompose them for processing. `List`{.scala} is a polymorphic type. What does polymorphic mean? We examine that in the next subsection. #### Aside on Haskell In Haskell, an algebraic data type similar to the Scala `List[A]`{.scala} is the following: ~~~{.haskell} data List a = Nil | Cons a (List a) deriving (Show, Eq) ~~~ The Haskell `List a`{.haskell} is a type similar to the Scala `List[A]`. However, `Nil`{.scala} and `Cons[A]`{.scala} are subtypes of `List`{.scala} in Scala, but they are not types in Haskell. In Haskell, they are constructor functions that return values of type `List a`{.haskell}. ### Polymorphism *Polymorphism* refers to the property of having "many shapes". In programming languages, we are primarily interested in how *polymorphic* function names (or operator symbols) are associated with implementations of the functions (or operations). Scala is a hybrid, object-functional language. Its type system supports all three types of polymorphism discussed in the notes on [*Type System Concepts* ](<../TypeConcepts/TypeSystemConcepts.html>). - *subtyping* by extending classes and traits - *parametric polymorphism* by using generic type parameters, - *overloading* through both the Java-like mechanisms and Haskell-like "type classes" Scala's *type class pattern* builds on the languages's `implicit`{.scala} classes and conversions. A type class enables a programmer to enrich an existing class with an extended interface and new methods without redefining the class or subclassing it. For example, Scala extends the Java `String`{.scala} class (which is `final`{.scala} and thus cannot be subclassed) with new features from the `RichString`{.scala} wrapper class. The Scala `implicit`{.scala} mechanisms associate the two classes "behind the scene". We defer further discussion of implicits until later in the semester. Note: The type class feature arose from the language Haskell. Similar capabilities are called extension methods in C# and protocols in Clojure and Elixir. The `List`{.scala} data type defined above is polymorphic; it exhibits both subtyping and parametric polymorphism. `Nil`{.scala} and `Cons`{.scala} are subtypes of `List`{.scala}. The generic type parameter `A`{.scala} denotes the type of the elements that occur in the list. For example, `List[Double]`{.scala} denotes a list of double-precision floating point numbers. What does the `+`{.scala} annotation mean in the definition `List[+A]`{.scala}? ### Variance The presence of both subtyping and parametric polymorphism leads to the question of how these features interact---that is, the concept of *variance*. #### Covariance and contravariance Suppose we have a supertype `Fish`{.scala} with a subtype `Bass`{.scala}, which in turn has a subtype `BlackBass`{.scala}. For generic data type `List[A]`{.scala} as defined above, consider `List[Fish]`{.scala} and `List[Bass]`{.scala}. - If `List[Bass]`{.scala} is a subtype of `List[Fish]`{.scala}, preserving the subtyping order, then the relationship is *covariant*. - If `List[Fish]`{.scala} is a subtype of `List[Bass]`{.scala}, reversing the subtyping order, then the relationship is *contravariant*. - If there is no subtype relationship between `List[Fish]`{.scala} and `List[Bass]`{.scala}, the the relationship is *invariant* (sometimes called *nonvariant*). In the Scala definition `List[+A]`{.scala} above, the `+`{.scala} annotation in front of the `A`{.scala} is a *variance annotation*. The `+`{.scala} means that parameter `A`{.scala} is a *covariant* parameter of `List`{.scala}. That is, for all types `X`{.scala} and `Y`{.scala} such that `X`{.scala} is a subtype of `Y`{.scala}, then then `List[X]`{.scala} is a is subtype of `List[Y]`{.scala}. If we leave off the variance annotation, then `List`{.scala} would be *invariant* in the type parameter. Regardless of how types `X`{.scala} and `Y`{.scala} may be related, `List[X]`{.scala} and `List[Y]`{.scala} are unrelated. If we were put a `-`{.scala} annotation in front of `A`{.scala}, then we declare parameter `A`{.scala} to be *contravariant*. That is, for all types `X`{.scala} and `Y`{.scala} such that `X`{.scala} is a subtype of `Y`{.scala}, then then `List[Y]`{.scala} is a is subtype of `List[X]`{.scala}. In the definition of the `List`{.scala} algebraic data type, `Nil`{.scala} extends `List[Nothing]`{.scala}. `Nothing`{.scala} is a subtype of all other types. In conjunction with covariance, the `Nil`{.scala} list can be considered a list of any type. Aside: Note the position of `Nothing`{.scala} in Scala's [unified type hierarchy ](). #### Function subtypes Now consider first-class functions. When is one function a subtype of another? From the notes on [*Type System Concepts* ](<../TypeConcepts/TypeSystemConcepts.html>) \[Cunningham 2019c\], we recall that we should be able to safely *substitute* elements of a subtype `S`{.scala} for elements of type `T`{.scala} because `S`{.scala}'s operations behave the "same" as `T`{.scala}'s operations. That is, the relationship satisfies the *Liskov Substitution Principle* \[Liskov 1987\] \[Wikipedia 2019\]. Using the `Fish`{.scala} type hierarchy above, consider a function of type `Bass => X`{.scala} (for some type `X`{.scala}). It would be unsafe to use a function of type `BlackBass => X`{.scala} in its place. The function would be undefined for any values that are of type `Bass`{.scala} but are not of type `BlackBass`{.scala}. So a function with a input type `BlackBass`{.scala} is not a subtype of a function with input `Bass`{.scala}. However, a function of type `Fish => X`{.scala} would be defined for any value that is of type `Bass`{.scala}. So we need to examine the relationships between the output types to determine what the subtyping relationship is between the functions. Consider a function of type `X => Bass`{.scala}. A function of type `X => BlackBass`{.scala} can be safely used in its place because a `BlackBass`{.scala} is a `Bass`{.scala}, so the function yields a value of the expected type. However, a function of type `X => Fish`{.scala} cannot be safely used in place of the `X => Bass`{.scala} function. It may yield some value that is a `Fish`{.scala} but not a `Bass`{.scala}. Thus we could safely use a `Bass => Bass`{.scala} function in place of a `Bass => Fish`{.scala}, `BlackBass => Bass`{.scala}, or `BlackBass => Fish`{.scala} function. Thus `Bass => Bass`{.scala} is a subtype of the others. However, we could not safely use a `Bass => Bass`{.scala} function in place of a `Bass => BlackBass`{.scala}, `BlackBass => BlackBass`{.scala}, `Fish => Fish`{.scala}, `Fish => Bass`{.scala}, or `Fish => BlackBass`{.scala} function. Thus `Bass => Bass`{.scala} is a not a subtype of the others. Bringing these observations together, a function type `S1 => S2`{.scala} is a subtype of a function type `T1 => T2`{.scala} if and only if: - `T1`{.scala} is a subtype of `S1`{.scala} (i.e. contravariant on the input type) - `S2`{.scala} is a subtype of `T2`{.scala} (i.e. covariant on the output type) This general observation is consistent with the applicable theory. For a Scala function of type `S => T`{.scala}, we thus say the `S`{.scala} is in a *contravariant position* and `T`{.scala} is in a *covariant position*. TODO: May want to discuss multiargument functions. ### Defining functions in companion object The *companion object* for a trait or class is a singleton object with the same name as the trait or class. The companion object for the `List`{.scala} trait is a convenient place to define functions for manipulating the lists. Because `List`{.scala} is a Scala algebraic data type (implemented with case classes), we can use pattern matching in our function definitions. Pattern matching helps enable the *form of the algorithm* to match the *form of the data structure*. Or, in terms that Chiusano and Bjarnason use, it helps in *following types to implementations* \[Chiusano 2015\]. Note: Other writers call this design approach *type-driven development* \[Brady 2017\] or *type-first development* \[Petricek 2012\]. This is considered elegant. It is also pragmatic. The structure of the data often suggests the algorithm needed for a task. In general, lists have two cases that must be handled: the empty list (represented by `Nil`{.scala}) and the nonempty list (represented by `Cons`{.scala}). The first yields a *base leg* of a recursive algorithm; the second yields a *recursive leg*. Breaking a definition for a list-processing function into these two cases is usually a good place to begin. We must ensure the recursion *terminates*---that each successive recursive call gets closer to the base case. ### Function to sum a list Consider a function `sum`{.scala} to add together all the elements in a list of integers. That is, if the list is $v_{1}, v_{2}, v_{3}, \cdots, v_{n}$, then the sum of the list is the value resulting from inserting the addition operator between consecutive elements of the list: > $v_{1} + v_{2} + v_{3} + \cdots + v_{n}$ Because addition is an *associative* operation, the additions can be computed in any order. That is, for any integers $x$, $y$, and $z$: > $(x + y) + z = x + (y + z)$ We can use the form of the data to guide the form of the algorithm---or *follow the type to the implementation* of the function. What is the sum of an empty list? Because there are no numbers to add, then, intuitively, zero seems to be the proper value for the sum. In general, if some binary operation is inserted between the elements of a list, then the result for an empty list is the *identity element* for the operation. Zero is the identity element for addition because, for all integers $x$: > $0 + x = x = x + 0$ Now, how can we compute the sum of a nonempty list? Because a nonempty list has at least one element, we can remove one element and add it to the sum of the rest of the list. Note that the "rest of the list" is a simpler (i.e. shorter) list than the original list. This suggests a recursive definition. The fact that we define lists recursively as a `Cons`{.scala} of a head element with a tail list suggests that we structure the algorithm around the structure of the *beginning* of the list. Bringing together the two cases above, we can define the function `sum`{.scala} in Scala using pattern matching as follows: ~~~{.scala} def sum(ints: List[Int]): Int = ints match { case Nil => 0 case Cons(x,xs) => x + sum(xs) } ~~~ The length of a non-nil argument decreases by one for each successive recursive application. Thus `sum`{.scala} will eventually be applied to a `Nil`{.scala} argument and terminate. For a list consisting of elements 2, 4, 6, and 8, that is, ~~~{.scala} Cons(2,Cons(4,Cons(6,Cons(8,Nil)))) ~~~ function `sum`{.scala} computes: ~~~ 2 + (4 + (6 + (8 + 0))) ~~~ Function `sum`{.scala} is backward linear recursive; its time and space complexity are both O($n$), where $n$ is the length of the input list. We could, of course, redefine this to use a tail-recursive auxiliary function. With *tail call optimization*, the recursion could be converted into a loop. It would still be order O($n$)in time complexity (but with a smaller constant factor) and O(1) space. ### Function to multiply a list Now consider a function `product`{.scala} to multiply together a list of floating point numbers. The product of an empty list is 1 (which is the identity element for multiplication). The product of a nonempty list is the head of the list multiplied by the product of the tail of the list, except that, if a 0 occurs anywhere in the list, the product of the list is 0. We can thus define `product`{.scala} with two bases cases and one recursive case, as follows: ~~~{.scala} def product(ds: List[Double]): Double = ds match { case Nil => 1.0 case Cons(0.0, _) => 0.0 case Cons(x,xs) => x * product(xs) } ~~~ Note: 0 is the *zero element* for the multiplication operation on real numbers. That is, for all real numbers $x$: > $0 * x = x * 0 = 0$ For a list consisting of elements 2.0, 4.0, 6.0, and 8.0, that is, ~~~{.scala} Cons(2.0,Cons(4.0,Cons(6.0,Cons(8.0,Nil)))) ~~~ function `product`{.scala} computes: ~~~{.scala} 2.0 * (4.0 * (6.0 * (8.0 * 1.0))) ~~~ For a list consisting of elements 2.0, 0.0, 6.0, and 8.0, function `product`{.scala} "short circuits" the computation as: ~~~{.scala} 2.0 * 0.0 ~~~ Like `sum`{.scala}, function `product`{.scala} is backward linear recursive; it has a worst-case time complexity of O($n$), where $n$ is the length of the input list. It terminates because the argument of each successive recursive call is one element shorter than the previous call, approaching one of the base cases. ### Function to remove adjacent duplicates Consider the problem of removing adjacent duplicate elements from a list. That is, we want to replace a group of adjacent elements having the same value by a single occurrence of that value. As with the above functions, we let the form of the data guide the form of the algorithm, following the type to the implementation. The notion of adjacency is only meaningful when there are two or more of something. Thus, in approaching this problem, there seem to be three cases to consider: - The argument is a list whose first two elements are duplicates; in which case one of them should be removed from the result. - The argument is a list whose first two elements are not duplicates; in which case both elements are needed in the result. - The argument is a list with fewer than two elements; in which case the remaining element, if any, is needed in the result. Of course, we must be careful that sequences of more than two duplicates are handled properly. Our algorithm thus can examine the first two elements of the list. If they are equal, then the first is discarded and the process is repeated recursively on the list remaining. If they are not equal, then the first element is retained in the result and the process is repeated on the list remaining. In either case the remaining list is one element shorter than the original list. When the list has fewer than two elements, it is simply returned as the result. In Scala, we can define function `remdups`{.scala} as follows: ~~~{.scala} def remdups[A](ls: List[A]): List[A] = ls match { case Cons(x, Cons(y,ys)) => if (x == y) remdups(Cons(y,ys)) // duplicate else Cons(x,remdups(Cons(y,ys))) // non-duplicate case _ => ls } ~~~ Function `remdups`{.scala} puts the base case last in the pattern match to take advantage of the wildcard match using `_`{.scala}. This needs to match either `Nil`{.scala} and `Cons(_,Nil)`{.scala}. The function also depends upon the ability to compare any two elements of the list for equality. Because `equals`{.scala} is builtin operation on all types in Scala, we can define this function polymorphically Without constraints on the type variable `A`{.scala}. Like the previous functions, `remdups`{.scala} is backward linear recursive; it takes a number of steps that is proportional to the length of the list. This function has a recursive call on both the duplicate and non-duplicate legs. Each of these recursive calls uses a list that is shorter than the previous call, thus moving closer to the base case. ### Variadic function `apply` We can also add a function `apply`{.scala} to the companion object `List`{.scala}. ~~~{.scala} def apply[A](as: A*): List[A] = if (as.isEmpty) Nil else Cons(as.head, apply(as.tail: _*)) ~~~ Scala treats an `apply`{.scala} method in an `object`{.scala} specially. We can invoke the `apply`{.scala} method using a postfix `()`{.scala} operator. Given a singleton object `X`{.scala} with an `apply`{.scala} method, the Scala complier translates the notation `X(p)`{.scala} into the method call `X.apply(p)`{.scala}. In the `List`{.scala} data type, function `apply`{.scala} is a *variadic function*. It accepts zero or more arguments of type `A`{.scala} as denoted by the type annotation `A*`{.scala} in the parameter list. Scala collects these arguments into a `Seq`{.scala} (sequence) data type for processing within the function. The special syntax `_*`{.scala} reverses this and passes a sequence to another function as variadic parameters. Builtin Scala data structures such as lists, queues, and vectors implement `Seq`{.scala}. It provides methods such as the `isEmpty`{.scala}, `head`{.scala}, and `tail`{.scala} methods used in `apply`{.scala}. It is common to define a variadic `apply`{.scala} methods for algebraic data types. This method enables us to create instances of the data type conveniently. For example, `List(1,2,3)`{.scala} creates a three-element list of integers with `1`{.scala} at the head. ## Data sharing Suppose we have the declaration ~~~{.scala} val xs = Cons(1,Cons(2,Cons(3,Nil))) ~~~ or the more concise equivalent using the `apply`{.scala} method: ~~~{.scala} val xs = List(1,2,3) ~~~ As we learned in the data structures course, we can implement this list as a linked list `xs`{.scala} with three cells with the values `1`{.scala}, `2`{.scala}, and `3`{.scala}, as shown in Figure 3-1. ![**Figure 3-1: Data sharing in lists**](fig_03_01.png "Data sharing in lists") Consider the following declarations ~~~{.scala} val ys = Cons(0,xs) val zs = xs.tail ~~~ where - `Cons(0,xs)`{.scala} returns a list that has a new cell containing `0`{.scala} in front of the previous list - `xs.tail`{.scala} returns the list consisting of the last two elements of `xs`{.scala} If the linked list `xs`{.scala} is immutable (i.e. the values and pointers in the three cells cannot be changed), then neither of these operations requires any copying. - The first just constructs a new cell containing `0`{.scala}, links it to the first cell in list `xs`{.scala}, and initializes `ys`{.scala} with a reference to the new cell. - The second just returns a reference to the second cell in list `xs`{.scala} and initializes `zs`{.scala} with this reference. - The original list `xs`{.scala} is still available, unaltered. This is called *data sharing*. It enables the programming language to implement immutable data structures efficiently, without copying in many key cases. Also, such functional data structures are *persistent* because existing references are never changed by operations on the data structure. ### Function to take tail of list Consider a function that takes a `List`{.scala} and returns its tail `List`{.scala}. (This is different from the `tail`{.scala} accessor method on `Cons`{.scala}.) If the `List`{.scala} is a `Cons`{.scala}, then the function can return the `tail`{.scala} element of the cell. What should it do if the list is a `Nil`{.scala}? There are several possibilities: - return `Nil`{.scala} - throw an exception (with perhaps a custom error string) - leave the function undefined in this case (which would result with a standard pattern match exception) Generally speaking, the first choice seems misleading. It seems illogical for an empty list to have a tail. And consider a typical usage of the function. It is normally an error for a program to attempt to get the tail of an empty list. A program can efficiently check whether a list is empty or not. So, in this case, it is probably better to take the second or third approach. We choose to implement `tailList`{.scala} so that it explicitly throws an exception. It can be defined in the companion object for `List`{.scala} as follows: ~~~{.scala} def tailList[A](ls: List[A]): List[A] = ls match { case Nil => sys.error("tail of empty list") case Cons(_,xs) => xs } ~~~ Above, the value of the `head`{.scala} field of the `Cons`{.scala} pattern is irrelevant in the computation on the right-hand side. There is no need to introduce a new variable for that value, so we use the wildcard variable `_`{.scala} to indicate that the value is not needed. Function `tailList`{.scala} is O(1) in time complexity. It does not need to copy the list. It is sufficient for it to just return a reference to the tail of the original immutable list. This return value shares the data with its input argument. ### Function to drop from beginning of list We can generalize `tailList`{.scala} to a function `drop`{.scala} that removes the first `n`{.scala} elements of a list, as follows: ~~~{.scala} def drop[A](ls: List[A], n: Int): List[A] = if (n <= 0) ls else ls match { case Nil => Nil case Cons(_,xs) => drop(xs, n-1) } ~~~ The `drop`{.scala} function terminates when either the list argument is `Nil`{.scala} or the integer argument 0 or negative. The function eventually terminates because each recursive call both shortens the list and decrements the integer. This function takes a different approach to the empty list issue than `tailList`{.scala} does. Although it seems illogical to take the `tailList`{.scala} of an empty list, dropping the first element from an empty list seems subtly different. Given that we often use `drop`{.scala} in cases where the length of the input list is unknown, dropping the first element of an empty list does not necessarily indicate a program error. Suppose `drop`{.scala} throws an exception when called with an empty list. To avoid this situation, the program might need to determine the length of the list argument. This is inefficient, usually requiring a traversal of the entire list to count the elements. ### Function to append lists Consider the definition of an *append* (list concatenation) function. We must define the `append`{.scala} function in terms of the constructors `Nil`{.scala} and `Cons`{.scala}, already defined list functions, and recursive applications of itself. As with previous functions, we follow the type to the implementation---let the form of the data guide the form of the algorithm. The `Cons`{.scala} constructor takes an element as its left operand and a list as its right operand and returns a new list with the left operand as the head and the right operand as the tail. Similarly, append must take a list as its left operand and a list as its right operand and return a new list with the left operand as the initial segment and the right operand as the final segment. Given the definition of `Cons`{.scala}, it seems reasonable that an algorithm for `append`{.scala} must consider the structure of its left operand. Thus we consider the cases for nil and non-nil left operands. - If the left operand is `Nil`{.scala}, then the function can just return the right operand. - If the left operand is a `Cons`{.scala} (that is, non-nil), then the result consists of the left operand’s head followed by the append of the left operand’s tail to the right operand. In following the type to the implementation, we use the form of the left operand in a pattern match. We define `append`{.scala} as follows: ~~~{.scala} def append[A](ls: List[A], rs: List[A]): List[A] = ls match { case Nil => rs case Cons(x,xs) => Cons(x, append(xs, rs)) } ~~~ For the recursive application of `append`{.scala}, the length of the left operand decreases by one. Hence the left operand of an `append`{.scala} application eventually becomes `Nil`{.scala}, allowing the evaluation to terminate. The number of steps needed to evaluate `append(as,bs)`{.scala} is proportional to the length of `as`{.scala}, the left operand. That is, it is O($n$), where $n$ is the length of list `as`. Moreover, `append(as,bs)`{.scala} only needs to copy the list `as`{.scala}. The list `bs`{.scala} is shared between the second operand and the result. If we did a similar function to append two (mutable) arrays, we would need to copy both input arrays to create the output array. Thus, in this case, a linked list is more efficient than arrays! The append operation has a number of useful mathematical (algebraic) properties, for example, associativity and an identity element. Associativity---For any finite lists `xs`{.scala}, `ys`{.scala}, and `zs`{.scala}: ~~~{.scala} append(xs,append(ys,zs)) = append(append(xs,ys),zs) ~~~ Identity---For any finite list `xs`{.scala}: ~~~{.scala} append(Nil,xs) = append(xs,Nil) = xs ~~~ Scala's builtin `List`{.scala} type uses the infix operator `++`{.scala} for the "append" operation. For this operator, associativity can be stated conveniently with the equation: ~~~{.scala} xs ++ (ys ++ zs) = (xs ++ ys) ++ zs ~~~ Mathematically, the `List`{.scala} data type and the binary operation `append`{.scala} form a kind of abstract algebra called a *monoid*. Function `append`{.scala} is closed (i.e. it takes two lists and gives a list back), is associative, and has an identity element. Aside: For more information on operations and algebraic structures, see Chapter 80, ["Review of Relevant Mathematics" ](), in the Haskell-based book [*Exploring Languages with Interpreters and Functional Programming* ]() \[Cunningham 2018\]. For discussion of how to prove properties like those above, see Chapter 25, ["Proving Haskell Laws" ](), in the same book. ## Other list functions ### Tail recursive function reverse Consider the problem of reversing the order of the elements in a list. Again we can use the structure of the data to guide the algorithm development. If the argument is a nil list, then the function returns a nil list. If the argument is a non-nil list, then the function can append the head element at the back of the reversed tail. ~~~{.scala} def rev[A](ls: List[A]): List[A] = ls match { case Nil => Nil case Cons(x,xs) => append(rev(xs),List(x)) } ~~~ Given that evaluation of `append`{.scala} terminates, the evaluation of `rev`{.scala} also terminates because all recursive applications decrease the length of the argument by one. How efficient is this function? The evaluation of `rev`{.scala} takes O($n^{2}$) steps, where $n$ is the length of the argument. There are O($n$) applications of `rev`{.scala} . For each application of `rev`{.scala} there are O($n$) applications of `append`{.scala} . The initial list and its reverse do not share data. Function `rev`{.scala} has a number of useful properties, for example a distribution and an inverse properties. Distribution---For any finite lists `xs`{.scala} and `ys`{.scala}: ~~~{.scala} rev(append(xs,ys)) = append(rev(ys), rev(xs)) ~~~ Inverse---For any finite list `xs`{.scala}: ~~~{.scala} rev(rev(xs)) = xs ~~~ Can we define a function to reverse a list using a "more efficient" tail recursive solution? As we have seen, a common technique for converting a backward linear recursive definition like `rev`{.scala} into a *tail recursive* definition is to use an *accumulating parameter* to build up the desired result incrementally. A possible definition for a tail recursive auxiliary function is: ~~~{.scala} def revAux[A](ls: List[A], as: List[A]): List[A] = ls match { case Nil => as case Cons(x,xs) => revAux(xs,Cons(x,as)) } ~~~ In this definition parameter `as`{.scala} is the accumulating parameter. The head of the first argument becomes the new head of the accumulating parameter for the tail recursive call. The tail of the first argument becomes the new first argument for the tail recursive call. We know that `revAux`{.scala} terminates because, for each recursive application, the length of the first argument decreases toward the base case of `Nil`{.scala}. We note that `rev(xs)`{.scala} is equivalent to `revAux(xs,Nil)`{.scala} . To define a single-argument replacement for `rev`{.scala} , we can embed the definition of `revAux’`{.scala} as an *auxiliary* function within the definition of a new function `reverse`{.scala} . ~~~{.scala} def reverse[A](ls: List[A]): List[A] = { def revAux[A](rs: List[A], as: List[A]): List[A] = rs match { case Nil => as case Cons(x,xs) => revAux(xs,Cons(x,as)) } revAux(ls,Nil) } ~~~ Function `reverse(xs)`{.scala} returns the value from `revAux(xs,Nil)`{.scala}. How efficient is this function? The evaluation of `reverse`{.scala} takes O($n$) steps, where $n$ is the length of the argument. There is one application of `revAux`{.scala} for each element; `revAux`{.scala} requires a single O(1) `Cons`{.scala} operation in the accumulating parameter. Where did the increase in efficiency come from? Each application of `rev`{.scala} applies `append`{.scala}, a linear time (i.e. O($n$)) function. In `revAux`, we replaced the applications of `append`{.scala} by applications of `Cons`{.scala}, a constant time (i.e. O(1)) function. In addition, a compiler or interpreter that does tail call optimization can translate this tail recursive call into a loop on the host machine. ### Higher-order function dropWhile Consider a function `dropWhile`{.scala} that removes elements from the front of a `List`{.scala} while its predicate argument (a Boolean function) holds. ~~~{.scala} def dropWhile [A](ls: List[A], f: A => Boolean): List[A] = ls match { case Cons(x,xs) if f(x) => dropWhile(xs, f) case _ => ls } ~~~ This higher-order function terminates when either the list is empty or the head of the list makes the predicate false. For each successive recursive call, the list argument is one element shorter than the previous call, so the function eventually terminates. If evaluation of function argument `p`{.scala} is O(1), then function `dropWhile`{.scala} has worst-case time complexity O($n$), where $n$ is the length of its first operand. The result list shares data with the input list. ### Curried function dropWhile We often pass *anonymous functions* to higher-order utility functions like `dropwhile`{.scala}, which has the signature: ~~~{.scala} def dropWhile[A](ls: List[A], f: A => Boolean): List[A] ~~~ When we call `dropWhile`{.scala} with an anonymous function for `f`{.scala}, we must specify the type of its argument, as follows: ~~~{.scala} val xs: List[Int] = List(1,2,3,4,5) val ex1 = dropWhile(xs, (x: Int) => x < 4) ~~~ Even though it is clear from the first argument that higher order argument `f`{.scala} must take an integer as its argument, the Scala *type inference* mechanism cannot detect this. However, if we rewrite `dropWhile`{.scala} in the following form, type inference can work as we want: ~~~{.scala} def dropWhile2[A](ls: List[A])(f: A => Boolean): List[A] = ls match { case Cons(x,xs) if f(x) => dropWhile2(xs)(f) case _ => ls } ~~~ Function `dropWhile2`{.scala} is written in *curried* form above. In this form, a function that takes two arguments can be represented as a function that takes the first argument and returns a function, which itself takes the second argument. If we apply `dropWhile2`{.scala} to just the first argument, we get a function. We call this a *partial application* of `dropWhile2`{.scala}. More generally, a function that takes multiple arguments can be represented by a function that takes its arguments in groups of one or more from left to right. If the function is partially applied to the first group, it returns a function that takes the remaining groups, and so forth. Currying and partial application are directly useful in a number of ways in our programs. Here currying is indirectly useful by assisting type inference. If a function is defined with multiple groups of arguments, the type information flows from one group to another, left to right. In `dropWhile2`{.scala}, the first argument group binds type variable `A`{.scala} to `Int`{.scala}. Then this binding can be used in the second argument group. ## Generalizing to Higher Order Functions ### Fold Right Consider the `sum`{.scala} and `product`{.scala} functions we defined above, ignoring the short-cut handling of the zero element in `product`{.scala}. ~~~{.scala} def sum(ints: List[Int]): Int = ints match { case Nil => 0 case Cons(x,xs) => x + sum(xs) } def product(ds: List[Double]): Double = ds match { case Nil => 1.0 case Cons(x,xs) => x * product(xs) } ~~~ What do `sum`{.scala} and `product`{.scala} have in common? What differs? Both exhibit the same *pattern of computation*. - Both take a list as input. But the element type differs. Function `sum`{.scala} takes a list of `Int`{.scala} values and `product`{.scala} takes a list of `Double`{.scala} values. - Both insert a binary operator between all the consecutive elements of the list in order to reduce the list to a single value. But the binary operation differs. Function `sum`{.scala} applies integer addition and `product`{.scala} applies double-precision floating-point multiplication. - Both group the operations from the right to the left. - Both functions return some value for an empty list. The function extends nonempty input lists to implicitly include this value as the "rightmost" value of the input list. But the actual value differs. Function `sum`{.scala} returns integer 0, the (right) identity element for addition. Function `product`{.scala} returns 1.0, the (right) identity element for multiplication. In general, this value could be something other than the (right) identity element. - Both return a value of the same element type as the input list. But the input type differs, as we noted above. Both functions insert operations of type `(A,A) => A`{.scala} between elements a list of type `List[A]`{.scala}, for some generic type `A`{.scala}. But these are special cases of more general operations of type `(A,B) => B`{.scala}. In this case, the value returned must be of type `B`{.scala} in the case of both empty and nonempty lists. Whenever we recognize a pattern like this, we can systematically *generalize the function* definition as follows: 1. Do a *scope-commonality-variability* (SCV) analysis on the set of related functions. That is, identify what is to be included and what not (i.e. the *scope*), the parts of functions that are the same (the *commonalities* or *frozen spots*), and the parts that differ (the *variabilities* or *hot spots*). 2. Leave the commonalities in the generalized function's body. 3. Move the variabilities into the generalized function’s header---its type signature and parameter list. - If the part moved to the generalized function’s parameter list is an expression, then make that part a function with a parameter for each local variable accessed. - If a data type potentially differs from a specific type used in the set of related functions, then add a type parameter to the generalized function. - If the same data value or type appears in multiple roles, then consider adding distinct type or value parameters for each role. 4. Consider other approaches if the generalized function’s type signature and parameter list become too complex. For example, we can introduce new data or procedural abstractions for parts of the generalized function. These may be in the same "module" (i.e. object, class, etc.) as the generalized function, in an appropriately defined separate "module" that is imported, etc. A separate module may better accomplish the desired parameterization of the function. A similar approach can be used to generalize a whole class. Following the above guidelines, we can express the common pattern from `sum`{.scala} and `product`{.scala} as a new (broadly useful) polymorphic, higher-order function `foldRight`{.scala}, which we define as follows: ~~~{.scala} def foldRight[A,B](ls: List[A], z: B)(f: (A, B) => B): B = ls match { case Nil => z case Cons(x,xs) => f(x, foldRight(xs, z)(f)) } ~~~ This function: - passes in the binary operation `f`{.scala} that combines the list elements - passes in the element `z`{.scala} to be returned for empty lists (often the right identity element for the operation, but this is not required) - uses two type parameters `A`{.scala} and `B`{.scala}---one for the type of elements in the list and one for the type of the result The `foldRight`{.scala} function "folds" the list elements (of type `A`{.scala}) into a value (of type `B`{.scala}) by "inserting" operation `f`{.scala} between the elements, with value `z`{.scala} "appended" as the rightmost element. For example, `foldRight(List(1,2,3),z)(f)`{.scala} expands to `f(1,f(2,f(3,z)))`{.scala}. Function `foldRight`{.scala} is not tail recursive, so it needs a new stack frame for each element of the input list. If its list argument is long or the folding function itself is expensive, then the function can terminate with a *stack overflow* error. We can specialize `foldRight`{.scala} to have the same functionality as `sum`{.scala} and `product`{.scala}. ~~~{.scala} def sum2(ns: List[Int]) = foldRight(ns, 0)((x,y) => x + y) def product2(ns: List[Double]) = foldRight(ns, 1.0)(_ * _) ~~~ The expression `(_ * _)`{.scala} in `product2`{.scala} is a concise notation for the anonymous function `(x,y) => x * y`{.scala}. The two underscores denote two distinct anonymous variables. This concise notation can be used in a context where Scala's type inference mechanism can determine the types of the anonymous variables. ]\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ We can construct a recursive function to compute the length of a polymorphic list. However, we can also express this computation using `foldRight`{.scala}, as follows: ~~~{.scala} def length[A](ls: List[A]): Int = foldRight(ls, 0)((_,acc) => acc + 1) ~~~ We use the `z`{.scala} parameter to accumulate the count, starting it at 0. Higher order argument `f`{.scala} is a function that takes an element of the list as its left argument and the previous accumulator as its right argument and returns it incremented by 1. In this application, `z`{.scala} is not the identity element for `f`{.scala} by a convenient beginning value for the counter. We can construct an "append" function that uses `foldRight`{.scala} as follows: ~~~{.scala} def append2[A](ls: List[A], rs: List[A]): List[A] = foldRight(ls, rs)(Cons(_,_)) ~~~ Here the the list that `foldRight`{.scala} operates on the first argument of the append. The `z`{.scala} parameter is the entire second argument and the combining function is just `Cons`. So the effect is to replace the `Nil`{.scala} at the end of the first list by the entire second list. We can construct a recursive function that takes a list of lists and returns a "flat" list that has the same elements in the same order. We can also express this `concat`{.scala} function in terms of `foldRight`{.scala}, as follows: ~~~{.scala} def concat[A](ls: List[List[A]]): List[A] = foldRight(ls, Nil: List[A])(append) ~~~ Function `append`{.scala} takes time proportional to the length of its first list argument. This argument does not grow larger because of right associativity of `foldRight`{.scala}. Thus `concat`{.scala} takes time proportional to the total length of all the lists. Above, we "pass" the `append`{.scala} function without writing an explicit anonymous function definition (i.e. *function literal*) such as `(xs,ys) => append(xs,ys)`{.scala} or `append(_,_)`{.scala}. In `concat`{.scala}, for which Scala can infer the types of `append`{.scala}'s arguments, the compiler can generate the needed function literal. In other cases, we would need to use *partial application* notation such as ~~~{.scala} append _ ~~~ or an explicit function literal such as ~~~{.scala} (xs: List[A], ys: List[A]) => append(xs,ys) ~~~ to enable the compiler to infer the types. Above we defined function `foldRight`{.scala} as a backward recursive function that processes the elements of a list one by one. However, as we have seen, it is often more useful to think of `foldRight`{.scala} as a powerful list operator that reduces the element of the list into a single value. We can combine `foldRight`{.scala} with other operators to conveniently construct list processing programs. ### Fold Left We designed function `foldRight`{.scala} above as a backward linear recursive function with the signature: ~~~{.scala} foldRight[A,B](as: List[A], z: B)(f: (A, B) => B): B ~~~ As noted: ~~~{.scala} foldRight(List(1,2,3),z)(f) == f(1,f(2,f(3,z))) ~~~ Consider a function `foldLeft`{.scala} such that: ~~~{.scala} foldLeft(List(1,2,3),z)(f) == f(f(f(z,1),2),3) ~~~ This function folds from the left. It offers us the opportunity to use parameter `z`{.scala} as an accumulating parameter in a tail recursive implementation, as follows: ~~~{.scala} @annotation.tailrec def foldLeft[A,B](ls: List[A], z: B)(f: (B, A) => B): B = ls match { case Nil => z case Cons(x,xs) => foldLeft(xs, f(z,x))(f) } ~~~ In the first line above, we *annotate* function `foldLeft`{.scala} as tail recursive using `@annotation.tailrec`{.scala}. If the function is not tail recursive, the compiler gives an error, rather than silently generating code that does not use tail call optimization (i.e. does not convert the recursion to a loop). We can implement list sum, product, and length functions with `foldLeft`{.scala}, similar to what we did with `foldRight`{.scala}. ~~~{.scala} def sum3(ns: List[Int]) = foldLeft(ns, 0)(_ + _) def product3(ns: List[Double]) = foldLeft(ns, 1.0)(_ * _) ~~~ Given that addition and multiplication of numbers are associative and have identity elements, `sum3`{.scala} and `product3`{.scala} use the same values for parameters `z`{.scala} and `f`{.scala} as `foldRight`{.scala}. Function `length2`{.scala} that uses `foldLeft`{.scala} is like `length`{.scala} except that the arguments of function `f`{.scala} are reversed. ~~~{.scala} def length2[A](ls: List[A]): Int = foldLeft(ls, 0)((acc,_) => acc + 1) ~~~ We can also implement list reversal using `foldLeft`{.scala} as follows: ~~~{.scala} def reverse2[A](ls: List[A]): List[A] = foldLeft(ls, List[A]())((acc,x) => Cons(x,acc)) ~~~ This gives a solution similar to the tail recursive `reverse`{.scala} function above. The `z`{.scala} value is initially an empty list; the folding function `f`{.scala} uses `Cons`{.scala} to "attach" each element of the list to front of the accumulator, incrementally building the list in reverse order. Because `foldLeft`{.scala} is tail recursive and `foldRight`{.scala} is not, `foldLeft`{.scala} is usually safer and more efficient to use in than `foldRight`{.scala}. (If the list argument is lazily evaluated or the function argument `f`{.scala} is nonstrict in at least one of its arguments, then there are other factors to consider. We will discuss what we mean by "lazily evaluated" and "nonstrict" later in the course.) To avoid the stack overflow situation with `foldRight`{.scala}, we can first apply `reverse`{.scala} to the list argument and then apply `foldLeft`{.scala} as follows: ~~~{.scala} def foldRight2[A,B](ls: List[A], z: B)(f: (A,B) => B): B = foldLeft(reverse(ls), z)((b,a) => f(a,b)) ~~~ The combining function in the call to `foldLeft`{.scala} is the same as the one passed to `foldRight2`{.scala} except that its arguments are reversed. ### Map Consider the following two functions, noting their type signatures and patterns of recursion. The first, `squareAll`{.scala}, takes a list of integers and returns the corresponding list of squares of the integers. ~~~{.scala} def squareAll(ns: List[Int]): List[Int] = ns match { case Nil => Nil case Cons(x, xs) => Cons(x*x, squareAll(xs)) } ~~~ The second, `lengthAll`{.scala}, takes a list of lists and returns the corresponding list of the lengths of the element lists ~~~{.scala} def lengthAll[A](lss: List[List[A]]): List[Int] = lss match { case Nil => Nil case Cons(xs, xss) => Cons(length(xs),lengthAll(xss)) } ~~~ Although these functions take different kinds of data (a list of integers versus a list of polymorphically typed lists) and apply different operations (squaring versus list length), they exhibit the same pattern of computation. That is, both take a list and apply some function to each element to generate a resulting list of the same size as the original. As with the fold functions, the combination of polymorphic typing and higher-order functions allows us to abstract this pattern of computation into a higher-order function. We can abstract the pattern of computation common to `squareAll`{.scala} and `lengthAll`{.scala} as the (broadly useful) function `map`{.scala}, defined as follows: ~~~{.scala} def map[A,B](ls: List[A])(f: A => B): List[B] = ls match { case Nil => Nil case Cons(x,xs) => Cons(f(x),map(xs)(f)) } ~~~ Function `map`{.scala} takes a list of type `A`{.scala} elements, applies function `f`{.scala} of type `A => B`{.scala} to each element, and returns a list of the resulting type `B`{.scala} elements. Thus we can redefine `squareAll`{.scala} and `lengthAll`{.scala} using `map`{.scala} as follows: ~~~{.scala} def squareAll2(ns: List[Int]): List[Int] = map(ns)(x => x*x) def lengthAll2[A](lss: List[List[A]]): List[Int] = map(lss)(length) ~~~ We can implement `map`{.scala} itself using `foldRight`{.scala} as follows: ~~~{.scala} def map1[A,B](ls: List[A])(f: A => B): List[B] = foldRight(ls, Nil: List[B])((x,xs) => Cons(f(x),xs)) ~~~ The folding function `(x,xs) => Cons(f(x),xs)`{.scala} applies the mapping function `f`{.scala} to the next element of the list (moving right to left) and attaches the result on the front of the processed tail. As implemented above, function `map`{.scala} is backward recursive; it thus requires a stack frame for each element of its list argument. For long lists, the recursion can cause a stack overflow error. Function `map1`{.scala} uses `foldRight`{.scala}, which has similar characteristics. So we need to use these functions with care. However, we can use the reversal technique illustrated in `foldRight2`{.scala} if necessary. We could also optimize function `map`{.scala} using *local mutation*. That is, we can use a mutable data structure within the `map`{.scala} function but not allow this structure to be accessed outside of `map`{.scala}. The following function takes that approach, using a `ListBuffer`{.scala}: ~~~{.scala} def map2[A,B](ls: List[A])(f: A => B): List[B] = { val buf = new collection.mutable.ListBuffer[B] @annotation.tailrec def go(ls: List[A]): Unit = ls match { case Nil => () case Cons(x,xs) => buf += f(x); go(xs) } go(ls) List(buf.toList: _*) } ~~~ A `ListBuffer`{.scala} is a mutable list data structure from the Scala library. The operation `+=`{.scala} appends a single element to the end of the buffer in constant time. The method `toList`{.scala} converts the `ListBuffer`{.scala} to a Scala immutable list, which is similar to the data structure we are developing in this module. ### Filter Consider the following two functions. The first, `getEven`{.scala}, takes a list of integers and returns the list of those integers that are even (i.e. are multiples of 2). The function preserves the relative order of the elements in the list. ~~~{.scala} def getEven(ns: List[Int]): List[Int] = ns match { case Nil => Nil case Cons(x,xs) => if (x % 2 == 0) // divisible evenly by 2 Cons(x,getEven(xs)) else getEven(xs) } ~~~ The second, `doublePos`{.scala}, takes a list of integers and returns the list of doubles of the positive integers from the input list; it preserves the order of the elements. ~~~{.scala} def doublePos(ns: List[Int]): List[Int] = ns match { case Nil => Nil case Cons(x,xs) => if (0 < x) Cons(2*x, doublePos(xs)) else doublePos(xs) } ~~~ We can abstract the pattern of computation common to `getEven`{.scala} and `doublePos`{.scala} as the (broadly useful) function `filter`{.scala}, defined as follows: ~~~{.scala} def filter[A](ls: List[A])(p: A => Boolean): List[A] = ls match { case Nil => Nil case Cons(x,xs) => val fs = filter(xs)(p) if (p(x)) Cons(x,fs) else fs } ~~~ Function `filter`{.scala} takes a predicate `p`{.scala} of type `A => Boolean`{.scala} a list of type `List[A]`{.scala} and returns a list containing those elements that satisfy `p`{.scala}, in the same order as the input list. Therefore, we can redefine `getEven`{.scala} and `doublePos`{.scala} as follows: ~~~{.scala} def getEven2(ns: List[Int]): List[Int] = filter(ns)(x => x % 2 == 0) def doublePos2(ns: List[Int]): List[Int] = map(filter(ns)(x => 0 < x))(y => 2 * y) ~~~ Function `doublePos2`{.scala} exhibits both the `filter`{.scala} and the `map`{.scala} patterns of computation. The higher-order functions `map`{.scala} and `filter`{.scala} allowed us to restate the definitions of `getEven`{.scala} and `doublePos`{.scala} in a succinct form. We can implement `filter`{.scala} in terms of `foldRight`{.scala} as follows: ~~~{.scala} def filter1[A](ls: List[A])(p: A => Boolean): List[A] = foldRight(ls, Nil:List[A])((x,xs) => if (p(x)) Cons(x,xs) else xs) ~~~ Above, the folding function ~~~{.scala} (x,xs) => if (p(x)) Cons(x,xs) else xs ~~~ applies the filter predicate `p`{.scala} to the next element of the list (moving right to left). If the predicate evaluates to true, the folding function attaches that element on the front of the processed tail; otherwise, it omits the element from the result. ### Flat Map The higher-order function `map`{.scala} applies its function argument `f`{.scala} to every element of a list and returns the list of results. If the function argument `f`{.scala} returns a list, then the result is a list of lists. Often we wish to flatten this into a single list, that is, apply a function like `concat`{.scala} defined in a previous section. This computation is sufficiently common that we give it the name `flatMap`{.scala}. We can define it in terms of `map`{.scala} and `concat`{.scala} as ~~~{.scala} def flatMap[A,B](ls: List[A])(f: A => List[B]): List[B] = concat(map(ls)(f)) ~~~ or by combining `map`{.scala} and `concat`{.scala} into one `foldRight`{.scala} as: ~~~{.scala} def flatMap1[A,B](ls: List[A])(f: A => List[B]): List[B] = foldRight(ls, Nil: List[B])( (x: A, ys: List[B]) => append(f(x),ys)) ~~~ Above, the function argument to `foldRight`{.scala} applies the `flatMap`{.scala} function argument `f`{.scala} to each element of the list argument and then appends the resulting list in front of the result from processing the elements to the right. We can also define `filter`{.scala} in terms of `flatMap`{.scala} as follows: ~~~{.scala} def filter2[A](ls: List[A])(p: A => Boolean): List[A] = flatMap(ls)(x => if (p(x)) List(x) else Nil) ~~~ The function argument to `flatMap`{.scala} generates a one-element list if the filter predicate `p`{.scala} is true and an empty list if it is false. ## Classic algorithms on lists ### Insertion sort and bounded generics Consider a function to sort the elements of a list into ascending order. A simple algorithm to do this is *insertion sort*. To sort a non-empty list with head `x`{.scala} and tail `xs`{.scala}, sort the tail `xs`{.scala} and insert the element `x`{.scala} at the right position in the result. To sort an empty list, just return it. If we restrict the function to integer lists, we get the following Scala functions: ~~~{.scala} def isort(ls: List[Int]): List[Int] = ls match { case Nil => Nil case Cons(x,xs) => insert(x,isort(xs)) } def insert(x: Int, xs: List[Int]): List[Int] = xs match { case Nil => List(x) case Cons(y,ys) => if (x <= y) Cons(x,xs) else Cons(y,insert(x,ys)) } ~~~ Insertion sort has a (worst and average case) time complexity of O($n^{2}$) where $n$ is the length of the input list. (Function `isort`{.scala} requires $n$ consecutive recursive calls; each call uses function `insert`{.scala} which itself requires on the order of $n$ recursive calls.) Now suppose we want to generalize the sorting function and make it polymorphic. We cannot just add a type parameter `A`{.scala} and substitute it for `Int`{.scala} everywhere. Although all Scala data types support equality and inequality comparison, not all types can be compared on a *total ordering* (`<`{.scala}, `<=`{.scala}, `>`{.scala}, and `>=`{.scala} as well). Fortunately, the Scala library provides a trait `Ordered`{.scala}. Any class that provides the other comparisons can extend this trait; the standard types in the library do so. This trait adds the comparison operators as methods so that they can be called in infix form. ~~~{.scala} trait Ordered[A] { def compare(that: A): Int def < (that: A): Boolean = (this compare that) < 0 def > (that: A): Boolean = (this compare that) > 0 def <=(that: A): Boolean = (this compare that) <= 0 def >=(that: A): Boolean = (this compare that) >= 0 def compareTo(that: a) = compare(that) } ~~~ We thus need to restrict the polymorphism on `A`{.scala} to be a subtype of `Ordered[A]`{.scala} by putting an *upper bound* on the type as follows: ~~~{.scala} def isort[A <: Ordered[A]](ls: List[A]): List[A] ~~~ Note: In addition to upper bounds, we can use a *lower bound*. A constraint `A :> T`{.scala} requires type `A`{.scala} to be a supertype of type `T`{.scala}. We can also specify both an upper and a lower bound on a type such as `T1 <: A <: T2`{.scala}, By using the upper bound constraint, we can sort data from any type that extends `Ordered`{.scala}. However, the primitive types inherited from Java do not extend `Ordered`{.scala}. Fortunately, the Scala library defines implicit conversions between the Java primitive types and Scala's enriched wrapper types. (This is the "type class" mechanism we discussed earlier.) We can use a weaker *view bound* constraint, denoted by `<%`{.scala} instead of `<:`{.scala}. This `A`{.scala} to be any type that is a subtype of or convertible to `Ordered[A]`{.scala}. ~~~{.scala} def isort1[A <% Ordered[A]](ls: List[A]): List[A] = ls match { case Nil => Nil case Cons(x,xs) => insert1(x,isort1(xs)) } def insert1[A <% Ordered[A]](x: A, xs: List[A]): List[A] = xs match { case Nil => List(x) case Cons(y,ys) => if (x <= y) Cons(x,xs) else Cons(y,insert1(x,ys)) } ~~~ We could define `insert`{.scala} inside `isort`{.scala} and avoid the separate type parameterization. But `insert`{.scala} is separately useful, so it is reasonable to leave it external. An alternative to use of the bound would be to pass in the needed comparison predicate, as follows: ~~~{.scala} def isort2[A](ls: List[A])(leq: (A,A) => Boolean): List[A] = ls match { case Nil => Nil case Cons(x,xs) => insert2(x,isort2(xs)(leq))(leq) } def insert2[A](x:A, xs:List[A])(leq:(A,A)=>Boolean):List[A] = xs match { case Nil => List(x) case Cons(y,ys) => if (leq(x,y)) Cons(x,xs) else Cons(y,insert2(x,ys)(leq)) } ~~~ Above we expressed both functions in curried form. By putting the comparison function last, we enabled the compiler to infer the argument types for the function. If we placed the function in the first argument group, the user of the function would have to supply the types. However, putting the comparison function first might allow a more useful partial application of the `isort`{.scala} to a comparison function. ### Merge sort The insertion sort given in the previous section has an average case time complexity of O($n^{2}$) where $n$ is the length of the input list. We now consider a more efficient function to sort the elements of a list: *merge sort*. Merge sort works as follows: - If the list has fewer than two elements, then it is already sorted. - If the list has two or more elements, then we split it into two sublists, each with about half the elements, and sort each recursively. - We merge the two ascending sublists into an ascending list. For a general implementation, we specify the type of list elements and the function to be used for the comparison of elements, giving the following implementation: ~~~{.scala} def msort[A](less: (A, A) => Boolean)(ls: List[A]): List [A] = { def merge(as: List[A], bs: List[A]): List[A] = (as,bs) match { case (Nil,_) => bs case (_,Nil) => as case (Cons(x,xs),Cons(y,ys)) => if (less(x,y)) Cons(x,merge(xs,bs)) else Cons(y,merge(as,ys)) } val n = length(ls)/2 if (n == 0) ls else merge(msort(less)(take(ls,n)), msort(less)(drop(ls,n))) } ~~~ The `merge`{.scala} forms a tuple of the two lists and does pattern matching against that tuple. This allowed the pattern match to be expressed more symmetrically. The above function uses a function we have not yet defined. ~~~{.scala} def take[A](ls: List[A], n: Int): List[A] ~~~ returns the first `n`{.scala} elements of the list; it is the dual of `drop`{.scala}. By nesting the definition of `merge`{.scala}, we enabled it to directly access the the parameters of `msort`{.scala}. In particular, we did not need to pass the comparison function to `merge`{.scala}. The average case time complexity of `msort`{.scala} is O($n\; \log(n)$), where $n$ is the length of the input list. - Each call level requires splitting of the list in half and merging of the two sorted lists. This takes time proportional to the length of the list argument. - Each call of `msort`{.scala} for lists longer than one results in two recursive calls of `msort`{.scala}. - But each successive call of `msort`{.scala} halves the number of elements in its input, so there are O($\log(n)$) recursive calls. So the total cost is O($n\; \log(n)$). The cost is independent of distribution of elements in the original list. We can apply `msort`{.scala} as follows: ~~~{.scala} msort((x: Int, y: Int) => x < y)(List(5, 7, 1, 3)) ~~~ We defined `msort`{.scala} in curried form with the comparison function first (unlike what we did with `isort1`{.scala}). This enables us to conveniently specialize `msort`{.scala} with a specific comparison function. For example, ~~~{.scala} val intSort = msort((x: Int, y: Int) => x < y) _ val descendSort = msort((x: Int, y: Int) => x > y) _ ~~~ However, we do have to give explicit type annotations for the parameters of the comparison function. ## Lists in the Scala standard library In this discussion (and in Chapter 3 of *Functional Programming in Scala* \[Chuisano 2015\]), we developed several functions for a simple `List`{.scala} module. Our module is related to the builtin Scala `List`{.scala} module (from `scala.collection.immutable`{.scala}), but it differs in several ways. Our `List`{.scala} module is standalone module; the Scala `List`{.scala} inherits from an abstract class with several traits mixed in. These classes and traits structure the interfaces shared among several data structures in the Scala library. Many of the functions work for different data structures. For example, in Scala release 2.12.8 `List`{.scala} is defined as follows: ~~~{.scala} sealed abstract class List[+A] extends AbstractSeq[A] with LinearSeq[A] with Product with GenericTraversableTemplate[A, List] with LinearSeqOptimized[A, List[A]] with scala.Serializable ~~~ Our `List`{.scala} module consists of functions in which all arguments must be given explicitly; the Scala `List`{.scala} consists of methods on the `List`{.scala} class. Scala enables methods with one implicit argument (i.e. `this`) and one explicit argument to be called as infix operators with different associativities. It allows symbols such as `<`{.scala} to be used for method names. Scala's approach to functional programming uses *method chaining* in its object system to support composition of pure functions. Each method returns an immutable object that becomes the receiver of the subsequent method call in the same statement. Extensive use of method chaining in an object-oriented program with mutable objects---sometimes called a *train wreck*---can make programs difficult to understand. However, disciplined use of method chaining helps make the functional and object-oriented aspects of Scala work together. (In different ways, method chaining is also useful in development of fluent library interfaces for domain-specific languages.) Our `Cons(x,xs)`{.scala} is written as `x :: xs`{.scala} using the standard Scala library. The `::`{.scala} is a method that has one implicit argument (the tail list) and one explicit argument (the head element). Any Scala method name that ends with a `:`{.scala} is right associative. Thus method `x :: xs`{.scala} represents the method call `xs.::(x)`{.scala}, which in turn calls the data constructor. We can write `x :: y :: z :: zs`{.scala} without parentheses to mean `x :: (y :: (z :: zs))`{.scala}. We can also use multiple `::`{.scala} constructors in cases for pattern matching. For example, where we wrote the pattern ~~~{.scala} case Cons(x, Cons(y,ys)) ~~~ in the `remdups`{.scala} function, we can write the pattern: ~~~{.scala} case x :: y :: ys ~~~ Our `append`{.scala} function is normally written with the infix operator `++`{.scala} in the Scala library. (But there are several variations for special circumstances.) Several of our functions with a single list parameter may appear as parameterless methods with the same name in the Scala library. These include `sum`{.scala}, `product`{.scala}, `reverse`{.scala}, and `length`{.scala}. There is also a `head`{.scala} function to retrieve the head element of a nonempty list. Our `concat`{.scala} function is parameterless method `flatten`{.scala} in the Scala library. Our functions with two parameters, a list and a modifier, are one-parameter methods with the same name in the Scala library, and, hence, usable as infix operators. These include `drop`{.scala}, `dropWhile`{.scala}, `map`{.scala}, `filter`{.scala}, and `flatMap`{.scala}. There are also analogous functions `take`{.scala} and `takeWhile`{.scala}. Our functions `foldRight`{.scala} and `foldLeft`{.scala}, which have three parameters, are methods in the Scala library with two curried parameters. The list argument becomes implicit; the other arguments are in the same order. The Scala library contains several folding and reducing functions with related functionality. Other than `head`{.scala}, `take`{.scala}, `takeWhile`{.scala}, and the appending and folding methods mentioned above, the Scala List library has other useful methods such as `forall`{.scala}, `exists`{.scala}, `scanLeft`{.scala}, `scanRight`{.scala}, `zip`{.scala}, and `zipWith`{.scala}. Check out the Scala API documentation on the Scala website. ## Source Code for Chapter - Chapter 3 source: [`List2.scala` ]() - Chapter 3 tests: [`TestList2.scala` ]() ## Exercise Set A TODO: Edit and reorder these exercises. This order corresponds to Assignment \#2 in Spring 2019. In the following exercises, extend the [`List2.scala`]() algebraic data type implementation developed in these notes to add the following functions. In the descriptions below, type `List`{.scala} refers to the trait defined in that package, not the standard Scala list. 1. Write a Scala function `orList`{.scala} that takes a `List`{.scala} of `Boolean`{.scala} values and returns the logical `or` of the values (i.e. true if any are true, otherwise false). #. Write a Scala function `andList`{.scala} that takes a `List`{.scala} of `Boolean`{.scala} values and returns the logical `and` of the values (i.e. true if all are true, otherwise false). #. Write a Scala function `maxList`{.scala} that takes a nonempty `List`{.scala} of values and returns its maximum value. Hint: First solve this with `Int`{.scala}, then generalize to a generic type. Study the subsection on insertion sort in this set of notes. #. Write a Scala `remdups1`{.scala} that is like `remdups`{.scala} except that it is implemented using either `foldRight`{.scala} or `foldLeft`{.scala}. #. Write a Scala function `total`{.scala} that takes a nonnegative integer `n`{.scala} and a function `f`{.scala} of an appropriate type and returns `f(0) + f(1) + ... f(n)`{.scala}. #. Write a Scala function `flip`{.scala} that takes a function of polymorphic type `(A,B) => C`{.scala} and returns a function of type `(B,A) => C`{.scala} such that, for all `x`{.scala} and `y`{.scala}: ~~~{.scala} f(x,y) == flip(f)(y,x) ~~~ #. Write the following Scala functions using tail recursive definitions: a. `sumT`{.scala} with same functionality as `sum`{.scala} #. `productT`{.scala} with the same functionality as `product`{.scala} #. Write a Scala function `mean`{.haskell} that takes a nonempty `List`{.scala} of `Double`{.scala} values and returns its mean (i.e. average) value. #. Write a Scala function `adjPairs`{.scala} that takes a `List`{.scala} of pairs (i.e. two-tuples) and returns the list of all pairs of adjacent elements. For example, `adjPairs(List(2,1,11,4))`{.scala} returns `List((2,1),(1,11),(11,4))`{.scala}. #. Write a Scala function `splitAt`{.scala} that takes a `List`{.scala} of values and an integer `n`{.scala} and returns a pair (i.e. two tuple) of `List`{.scala}s, where the first component consists of the first `n`{.scala} elements of the input list (in order) and the second component consists of the remaining elements (in order). #. Number base conversion. a. Write a Scala function `natToBin`{.scala} that takes a natural number and returns its binary representation as a `List`{.scala} of `0`{.scala}’s and `1`{.scala}'s with the most significant digit at the head. For example, `natToBin(23)`{.scala} returns `List(1,0,1,1,1)`{.scala}. In computer science, we usually consider 0 as natural number along with the positive integers. #. Generalize `natToBin`{.scala} to Scala function `natToBase`{.scala} that takes base `b`{.scala} (`b >= 2`{.scala}) as its first paramenter and the natural number as its second. The function should return the base `b`{.scala} representation of the natural number as a list of integer digits with the most significant digit at the head. For example, `natToBase(5,42)`{.scala} returns `List(1,3,2)`{.scala}. #. Write Scala function `baseToNat`{.scala} that is the inverse of the `natToBase`{.scala} function. For any base `b`{.scala} (`b >= 2`{.scala}) and natural number `n`{.scala}: ~~~{.scala} baseToNat(b,natToBase(b,n)) == n ~~~ #. For each of the following specifications, write a Scala function that has the given arguments and result. Use the higher functions from the `List`{.scala} algebraic data type from these notes, such as `map`{.scala}, `filter`{.scala}, `foldRight`{.scala}, and `foldLeft`{.scala}, as appropriate. a. Function `numof`{.scala} takes a value and a list and returns the number of occurrences of the value in the list. #. Function `ellen`{.scala} takes a list of lists and returns a list of the lengths of the corresponding lists. #. Function `ssp`{.scala} takes a list of integers and returns the sum of the squares of the positive elements of the list. #. Write a Scala function `scalarProd`{.scala} with type ~~~{.scala} (List[Int],List[Int]):: Int ~~~ to compute the scalar product of two lists of integers (e.g. representing vectors). The *scalar product* is the sum of the products of the elements in corresponding positions in the lists. That is, the scalar product of two lists `xs`{.scala} and `ys`{.scala}, of length `n`, is: > $\sum\limits_{i=0}^{i=n}xs_i * ys_{i}$ For example, `scalarprod(List(1,2,3),List(3,3,3))`{.scala} yields `18`{.scala}. #. Write a Scala function `mapper`{.scala} that takes a list of functions and a list of values and returns the list of results of applying each function in the first list to the corresponding value in the second list. #. Write a Scala function `removeFirst`{.scala} that takes a predicate (i.e. Boolean function) and a list of values and returns the list with the first element that satisfies the predicate removed. #. Define a Scala function `removeLast`{.scala} that takes a predicate (i.e. Boolean function) and a list of values and returns the list with the last element that satisfies the predicate removed. How could you define it using `removeFirst`{.scala}? ## General Tree Algebraic Data Type A *general tree* is a hierarchical data structure in which each node of the tree has zero or more subtrees. We can define this as a Scala algebraic data type as follows: ~~~{.scala} sealed trait GTree[+A] case class Leaf[+A](value: A) extends GTree[A] case class Gnode[+A](gnode: List[GTree[A]]) extends GTree[A] ~~~ An object of class `Leaf(x)`{.scala} represents a *leaf* of the tree holding some value `x`{.scala} of generic type `A`{.scala}. A leaf does not have any subtrees. It has height 1. An object of type `Gnode`{.scala} represents an *internal* (i.e. non-leaf) node of the tree. It consists of a nonempty list of subtrees, ordered left-to-right. A `Gnode`{.scala} has a height that is one more than the maximum height of its subtrees. ## Exercise Set B In the following exercises, write the Scala functions to operate on the `GTree`{.scala}s. You may use functions from the extended `List`{.scala} module as needed. 1. Write Scala function `numLeavesGT`{.scala} that takes a `GTree`{.scala} and returns the count of its leaves. #. Write Scala function `heightGT`{.scala} that takes a `GTree`{.scala} and returns its height (i.e. the number of levels). #. Write Scala function `sumGT`{.scala} that takes a `GTree`{.scala} of integers and returns the sum of the values. #. Write Scala function `findGT`{.scala} that takes a `GTree`{.scala} and a value and returns `true`{.scala} if and only if the element appears in some leaf in the tree. #. Write Scala function `mapGT`{.scala} that takes a `GTree`{.scala} and a function and returns a `GTree`{.scala} with the structure but with the function applied to all the values in the tree. #. Write Scala function `flattenGT`{.scala} that takes a `GTree`{.scala} and returns a `List`{.scala} with the values from the tree leaves ordered according to a left-to-right traversal of the tree. ## Acknowledgements In Spring 2016, I wrote this set of notes to accompany my lectures on Chapter 3 of the book *Functional Programming in Scala* \[Chiusano 2015\] (i.e. the Red Book). I constructed the notes around the ideas, general structure, and Scala examples from that chapter and its associated materials. I also adapted some text and examples from my *Notes on Functional Programming with Haskell* \[Cunningham 2014\]. I expanded the discussion of algebraic data types, polymorphism, and variance. For this expansion, I examined other materials including the Wikipedia articles on Algebraic Data Type, Abstract Data Type, Polymorphism, Ad Hoc Polymorphism, Parametric Polymorphism, Subtyping, Liskov Substitution Principle, Function Overloading, and Covariance and Contravariance \[Wikipedia 2019\]. I also examined the discussion of variance in the textbook *Programming Scala* \[Wampler 2014\]. I adapted the sorting algorithms from *Scala by Example* \[Odersky 2014\]. In 2018 and 2019, I updated the format to be more compatible with evolving document structures. In Spring 2019, I also moved the discussion of the kinds of polymorphism to the new notes on *Type System Concepts*, expanded the discussion of Variance, and added two exercise sets. Several items from Exercise Set A were adapted from the list processing chapters of ELIFP \[Cunningham 2018\]. I maintain these notes as text in Pandoc's dialect of Markdown using embedded LaTeX markup for the mathematical formulas and then translate the notes to HTML, PDF, and other forms as needed. ## References \[Brady 2017\]: : Edwin Bradley. *Type-Driven Development with Idris*, Manning, 2017. \[Chiusano 2015\]: : Paul Chiusano and Runar Bjarnason. *Functional Programming in Scala*, Manning, 2015. This book is sometimes called the Red Book. The chapter notes for this book are available on GitHub at . \[Cunningham 2014\]: : H. Conrad Cunningham. [*Notes on Functional Programming with Haskell* ](), 1993-2014. \[Cunningham 2018\]: : H. Conrad Cunningham. *Exploring Languages with Interpreters and Functional Programming*, 2018. Available at . \[Cunningham 2019a\]: : H. Conrad Cunningham. [*Notes on Scala for Java Programmers* ](<../ScalaForJava/ScalaForJava.html>), 2019. \[Cunningham 2019b\]: : H. Conrad Cunningham. [*Recursion Styles, Correctness, and Efficiency (Scala Version)* ](<../RecursionStyles/RecursionStylesScala.html>), 2019. \[Cunningham 2019c\]: : H. Conrad Cunningham. [*Type System Concepts* ](<../TypeConcepts/TypeSystemConcepts.html>), 2019. \[Odersky 2014\]: : Martin Odersky. *Scala by Example*, EPFL, 2014. \[Odersky 2016\]: : Martin Odersky, Lex Spoon, and Bill Venners. *Programming in Scala*, 3rd Edition, Artima Inc, 2016; 1st Edition, 2007; 2nd Edition, 2010. \[Wampler 2014\]: : Dean Wampler and Alex Payne. *Programming Scala*, Second Edition, O'Reilly, 2014. \[Liskov 1987\]: : Barbara Liskov. Keynote Address---Data Abstraction and Hierarchy, In the Addendum to the *Proceedings on Object-Oriented Programming Systems, Languages, and Applications (OOPSLA '87)*, Leigh Power and Zvi Weiss, Editors, ACM, 1987. \[[local ](<../../localcopy/Liskov\_Data\_Abstraction\_and\_Hierarchy\_1987.pdf>)\] \[Petricek 2012\]: : Tomas Petricek. *Why Type-first Development Matters*, blog entry, August 2012, Accessed 4 March 2019 at . \[Wikipedia 2019\]: : *Wikipedia*. Articles on Algebraic Data Type, Abstract Data Type, Polymorphism (computer science), Ad Hoc Polymorphism, Parametric Polymorphism, Subtyping, Liskov Substitution Principle, Function Overloading, and Covariance and Contravariance (computer science); last accessed 15 February 2019. ## Terms and Concepts TODO: Update Function, pure function, referential transparency, side effects, mutable, immutable, list data type (head, tail, empty), algebraic data type (composite, sum, product, enumerated), abstract data type, ADT, syntax, semantics, `trait`{.scala}, `sealed trait`{.scala}, `case class`{.scala}, `case object`{.scala}, singleton object, polymorphism, subtyping, parametric polymorphism, generics, overloading, type classes, variance (covariant, contravariant, invariant/nonvariant), following types to implementations (type-driven or type-first development).