Functional Programming Principles in Scala
这一周的主要内容是函数。函数是scala语言最重要的概念，既可以当作函数的参数，也可以作为返回值。函数还可以拥有多个参数列表。
因此，这次作业就是要把函数作为参数和返回值，实现set数据类型的一些方法。而set本身却是一个函数，这一点非常奇妙。另外测试程序也要自己实现。
编程过程中，发现有些函数实现的方法非常复杂，不知道是否有简化的方法。
作业要求如下：
In this assignment, you will work with a functional representation of sets based on the mathematical notion of characteristic functions. The goal is to gain practice with higher-order functions.
Download the funsets.zip handout archive file and extract it somewhere on your machine. Write your solutions by completing the stubs in the FunSets.scala file.
Write your own tests! For this assignment, we don’t give you tests but instead the FunSetSuite.scala file contains hints on how to write your own tests for the assignment.
Representation
We will work with sets of integers.
As an example to motivate our representation, how would you represent the set of all negative integers? You cannot list them all… one way would be so say: if you give me an integer, I can tell you whether it’s in the set or not: for 3, I say ‘no’; for -1, I say yes.
Mathematically, we call the function which takes an integer as argument and which returns a boolean indicating whether the given integer belongs to a set, the characteristic function of the set. For example, we can characterize the set of negative integers by the characteristic function (x: Int) => x < 0.
Therefore, we choose to represent a set by its characterisitc function and define a type alias for this representation:
type Set = Int => Boolean
Using this representation, we define a function that tests for the presence of a value in a set:
def contains(s: Set, elem: Int): Boolean = s(elem)
2.1 Basic Functions on Sets
Let’s start by implementing basic functions on sets.
Define a function which creates a singleton set from one integer value: the set represents the set of the one given element. Its signature is as follows:
def singletonSet(elem: Int): Set
Now that we have a way to create singleton sets, we want to define a function that allow us to build bigger sets from smaller ones.
Define the functions union, intersect, and diff, which takes two sets, and return, respectively, their union, intersection and differences. diff(s, t) returns a set which contains all the elements of the set s that are not in the set t. These functions have the following signatures:
def union(s: Set, t: Set): Set
def intersect(s: Set, t: Set): Set
def diff(s: Set, t: Set): Set
Define the function filter which selects only the elements of a set that are accepted by a given predicate p. The filtered elements are returned as a new set. The signature of filter is as follows:
def filter(s: Set, p: Int => Boolean): Set
2.2 Queries and Transformations on Sets
In this part, we are interested in functions used to make requests on elements of a set. The first function tests whether a given predicate is true for all elements of the set. This forall function has the following signature:
def forall(s: Set, p: Int => Boolean): Boolean
Note that there is no direct way to find which elements are in a set. contains only allows to know whether a given element is included. Thus, if we wish to do something to all elements of a set, then we have to iterate over all integers, testing each time whether it is included in the set, and if so, to do something with it. Here, we consider that an integer x has the property -1000 <= x <= 1000 in order to limit the search space.
Implement forall using linear recursion. For this, use a helper function nested in forall. Its structure is as follows (replace the ???):
def forall(s: Set, p: Int => Boolean): Boolean = {
def iter(a: Int): Boolean = {
if (???) ???
else if (???) ???
else iter(???)
}
iter(???)
}
Using forall, implement a function exists which tests whether a set contains at least one element for which the given predicate is true. Note that the functions forall and exists behave like the universal and existential quantifiers of first-order logic.
def exists(s: Set, p: Int => Boolean): Boolean
Finally, write a function map which transforms a given set into another one by applying to each of its elements the given function. map has the following signature:
def map(s: Set, f: Int => Int): Set
Extra Hints
Be attentive in the video lectures on how to write anonymous functions in Scala.
Sets are represented as functions. Think about what it means for an element to belong to a set, in terms of function evaluation. For example, how do you represent a set that contains all numbers between 1 and 100?
Most of the solutions for this assignment can be written as one-liners. If you have more, you probably need to rethink your solution. In other words, this assignment needs more thinking (whiteboard, pen and paper) than coding ;-).
If you are having some trouble with terminology, have a look at the glossary.
实现的程序如下：
package funsets
import common._
/**
* 2. Purely Functional Sets.
*/
object FunSets {
/**
* We represent a set by its characteristic function, i.e.
* its `contains` predicate.
*/
type Set = Int => Boolean
/**
* Indicates whether a set contains a given element.
*/
def contains(s: Set, elem: Int): Boolean = s(elem)
/**
* Returns the set of the one given element.
*/
def singletonSet(elem: Int): Set = x=>x==elem
/**
* Returns the union of the two given sets,
* the sets of all elements that are in either `s` or `t`.
*/
def union(s: Set, t: Set): Set = x=>s(x) || t(x)
/**
* Returns the intersection of the two given sets,
* the set of all elements that are both in `s` and `t`.
*/
def intersect(s: Set, t: Set): Set = x=>s(x) && t(x)
/**
* Returns the difference of the two given sets,
* the set of all elements of `s` that are not in `t`.
*/
def diff(s: Set, t: Set): Set = x=>s(x) && (!t(x))
/**
* Returns the subset of `s` for which `p` holds.
*/
def filter(s: Set, p: Int => Boolean): Set = x=> s(x) && p(x)
/**
* The bounds for `forall` and `exists` are +/- 1000.
*/
val bound = 1000
/**
* Returns whether all bounded integers within `s` satisfy `p`.
*/
def forall(s: Set, p: Int => Boolean): Boolean = {
def iter(a: Int): Boolean = {
if (a== -bound-1) true
else if (s(a) && !p(a)) false
else iter(a-1)
}
iter(bound)
}
/**
* Returns whether there exists a bounded integer within `s`
* that satisfies `p`.
*/
def exists(s: Set, p: Int => Boolean): Boolean = ! forall(s, x => !p(x))
/**
* Returns a set transformed by applying `f` to each element of `s`.
*/
def map(s: Set, f: Int => Int): Set = x=>exists(s,elem=>f(elem)==x)
/**
* Displays the contents of a set
*/
def toString(s: Set): String = {
val xs = for (i <- -bound to bound if contains(s, i)) yield i
xs.mkString("{", ",", "}")
}
/**
* Prints the contents of a set on the console.
*/
def printSet(s: Set) {
println(toString(s))
}
}
测试部分代码片段如下：
test("union contains all elements") {
new TestSets {
val s = union(s1, s2)
assert(contains(s, 1), "Union 1")
assert(contains(s, 2), "Union 2")
assert(!contains(s, 3), "Union 3")
}
}
test("union intersect diff"){
def a(x:Int)= x<20
def b(x:Int)= x>10
assert(contains(union(a,b), 55), "Union")
assert(contains(union(a,b), 5), "Union")
assert(contains(intersect(a,b), 15), "intersect")
assert(!contains(diff(a,b), 15), "diff")
assert(contains(diff(a,b), 5), "diff")
}
test("forall exists map"){
def a(x:Int)=x<10
assert(forall(a,x=>x*x>=0),"forall")
assert(!forall(a,x=>x*x<60),"forall")
assert(exists(a,x=>x*10==50),"exists")
assert(!exists(a,x=>x*10==120),"exists")
assert(contains(map(a,x=>x*x),64),"contains")
assert(!contains(map(a,x=>x*x),640),"contains")
}