Module 09 Lab

Install Haskell if you do not have it on your computer. We recommend you install the Haskell platform. If for some reason you want to get started on the lab without installing Haskell, you can use Repl.it (click “new repl” and choose Haskell) for the first part of the lab. You can also just log into stu if you wish.
Open a terminal window and go to a directory where you can make some files. Start up the interactive Haskell interpreter by typing ghci (which stands for Glasgow Haskell Compiler Interpreter—the compiler is called ghc and they just added the i for the interpreter).
Do some arithmetic in the normal way. ghci is a very nice calculator. You will have to use `div` and `mod` (as infix operators, including the back-quotes) to do integer division. Also, you must put parentheses around negative numbers (like (-3)). Note that you can use the arrow keys to move around and see expressions that you typed before. The variable it always holds the value of the last expression evaluated. Values can be arbitrarily large. For example, type 2^200.
Named functions in Haskell are all prefix operators and do not allow parentheses around arguments, nor commas separating arguments. Hence where in mathematics we might write f(x,y,z), in Haskell it would be f x y z. For example, div 8 3 and mod 8 3 are the “usual” ways to call these functions. Haskell allows functions with two arguments to be called in an infix form if back-quotes are placed around the function name, as above. You can also put parentheses around a function application (as in (f x y z)) to make it more readable or to enforce precedence.
Haskell has a large library of predefined functions in the Prelude module, which is loaded automatically whenever the Haskell compiler or interpreter runs. The Prelude contains declarations for dozens of types and operations sufficient for many applications. Try out some functions like even, odd, sqrt, sin, and gcd (note that the latter requires two parameters!).
Haskell is a functional language. This means that it has no variables in the way we are used to from imperative language. This is not obvious in ghci because ghci appears to have variables and allows you to associate different values with the same name using the let expression: let <id> = <value>. However, this is a feature of the interpreter and will not work in a Haskell program: in Haskell, the form <id> = <value> associates an identifier with a value for the duration of the program, and let <id> = <value> associates an identifier with a value only in a limited context. These expressions are more like constant definitions than variable assignments.
As a functional language, Haskell does everything with functions: it has no control structures like those we find in imperative languages. Try the following: if (even 5) then "even" else "odd" Although this may look like a statement, it is an expression. The expression after the if is evaluated, and if it is true, the value of the entire expression is the value of the expression after then, otherwise it is the value of the expression after else. These sorts of conditional and case expressions take the place of conditional and switch statements, and recursion stands in for loops in functional languages. We will see a lot more of this later.
A very important type in all functional languages is the list. In Haskell, a list can be formed by enclosing values of the same type in square brackets, separated by commas: [2, 3, 5, 7, 11], for example. The infix operator ++ is the list concatenation operator. Try it.
In Haskell, strings are lists of characters, so "abc" is the same as ['a', 'b', 'c']. Concatenate "hello" and "world" with a space between.
There are other ways to make lists: one way is to use the cons operator : (colon) which concatenates a value on the front of a list. For example 5 : [6, 7] is [5, 6, 7]. In fact, you can create an entire list just using cons, values, and the empty list (actually, lists are internally built in exactly this way). Write the list [5, 6, 7] using only 5, 6, 7, cons, and the empty list (cons is right-associative).
Lists can be indexed using the indexing operator !! and compared using the standard comparison operators. The Prelude also contains a whole bunch of list operations, including head, tail, length, reverse, sum, and take (Note that the latter has two parameters: an integer n and a list l. It returns a new list with the first n elements from l). Try all of these out on some lists.
Another way to make lists is to use a range: [1..20] is the list with all the numbers between 1 and 20. Write an expression that produces a list with all the numbers between 25 and 35.
Yet another way to make lists is to filter elements out of other lists using the built-in filter function. This function is unlike the others we’ve seen so far because it takes a function as a parameter! In fact, it requires both a function and a list, and it returns a new list with only the elements from the original list that produce true when passed to the given function. Try evaluating the following expression: filter odd [1..10]
What happens when you switch the odd function to the even function? Write an expression that produces a list containing all of the even numbers less than 50. Being able to pass a function to another function is part of what makes Haskell a “functional” programming language. We will see more implications of this later.
A list must contain values of the same type, but it can have an arbitrary number of them. A tuple must contain a fixed number of values, but their types need not be the same. Tuples are specified using parentheses with their components separated by commas. For example, (1, True), ("a", 34, []), and ([0]) are tuples. There are several built-in functions for manipulating tuples, such as fst and snd to get the first and second values of a 2-element tuple, respectively.
All values in Haskell have types, and Haskell is VERY picky about types. But you don’t have to declare types if you don’t want to because Haskell has a powerful type inferencing mechanism built into it: whenever you create a value (including function definitions) Haskell will figure out what all the types of all the expressions have to be, and complain about type errors. Try adding a number and a string and see what ghci says. But you should always declare the types of your functions anyway. This is because if what you think the type of your function should be does not correspond with what Haskell has computed it actually is, then you know you have a problem. So Haskell can help you debug your programs before you execute them (which is a big advantage of statically-typed languages). The bottom line is that you need to understand how Haskell specifies types.
The :type or :t commands in ghci show the type of a value. For example, :type "abc" is [Char] because a string is a list of characters. Find the types of True, [True], and (True,"abc"). Make sure you understand these types.
Query the type of the empty list []. Notice that there is an a between the brackets. This is a type variable, not unlike a generic type in Java. There are no rules imposed on a, which means that the empty list is flexibly typed.
Query the type of the not function. Notice that it looks like the mathematical notation for maps (functions), in this case a function that maps a Bool to Bool. Now find the types of the even and the length functions. These functions map what type to what type?
Find the type of the take function. What’s up with that? Shouldn’t it be Int, [a] -> [a]? You can think of all the parameters as being separated by arrows as well as the return value, so this is a function that takes an Int and a list and returns a list. Try filter and map and see what you get. What does this mean?
Things are a bit more complicated with numbers. There are several numeric types in Haskell, including Int, Integer, Float, and Double. But many values (like 5) can be used in contexts requiring any of these types. So Haskell has type classes, which are like interfaces in other languages, as they specify groups of operations. The most general numeric type class is Num. So if you find the type of 5, Haskell says it is Num a => a, which means, if a is a type that implements the Num type class, then 5 can have that type. Find the types of sqrt, div, and (/) to see a few more type classes (you have to put operator symbols in parentheses to indicate that you are talking about the symbol, not using it).
We can define our own functions, of course. ghci does not like things to be more than one line long, and we will soon be ready to start making functions that are more than one line long. So open another window, go to the directory where ghci is open, and start editing a file called lab.hs. Put your name at the top in a comment. Comments begin with -- and extend to the end of the line.
We define functions similarly to how we define constants: write the function name followed by the parameter name(s), an equal sign, and the definition. For example, the following definition is for a function that computes the sum of the first n natural numbers using the equation n*(n+1)/2:
```
sumOfFirstN :: Integral a => a -> a
sumOfFirstN n = n * (n+1) `div` 2
```
Type this into your lab.hs file, then in ghci type :load lab. This loads the file into ghci (you can reload by typing this line again, or entering just :reload). Now you can call this function on various values; sumOfFirstN 100 should be 5050. Notice that we included a type declaration for this function (it takes an Integral value and returns one). These are optional in Haskell but it is considered good programming practice to include them.
Write a function divBy15 that takes a number and returns true if that number is divisible by 15. (The equality operator is == in Haskell.) Reload your file in ghci and test your function. Write an expression that produces all the numbers between 1 and 100 that are divisible by 15 (Hint: use filter!).

Suppose we want to write a function to compute the points awarded for places in a swim meet as follows:

Rank	Points
1st	10
2nd	8
3rd	6
4th	5
5th	4
6th	3
7th	2
8th	1

We can write this function using pattern matching for the values of the parameter. Type the following into your lab file.

points :: Integral a => a -> a
points 1 = 10
points 2 = 8
points 3 = 6
points 4 = 5
points 5 = 4
points 6 = 3
points 7 = 2
points 8 = 1
points _ = 0

When faced with a call of the points function, Haskell will search through these alternatives from the first to the last and use the clause that matches. The _ matches any value, so this clause of the definition will be used if all others fail. Save your lab file, reload it in ghci and try this out.

We can simplify the definition of the points function using “guards”, which allow us to use expressions to decide which definitional clause to use. Edit your lab file so the definition of points looks like the following.
```
points n
  | (9 < n) || (n < 1) = 0
  | (n <= 3)           = 12 - 2*n
  | otherwise          = 9 - n
```
The guards follow the bar character and must be boolean expressions. These are evaluated in order and the first one that is true determines which clause is used. The keyword otherwise is just short for True, which of course always succeeds, so the otherwise clause should appear last as a catch-all guard. Save and reload your file and try this out.
Write a function called nextOdd that takes an integer x and returns the next odd number after x. (Hint: If x is even, the next odd number is x+1.)
The last thing we need to define functions is recursive definitions. We can often split our base and recursive cases up with pattern matching. For example, type the following definition for the factorial function.
```
factorial :: Integral a => a -> a
factorial 0 = 1
factorial n = n * factorial (n-1)
```
Notice how we use a literal for the base case and a catch-all pattern for the recursive case. Try factorial 200.
For lists, the base case is the empty list [] and the recursive case is a pattern like [x:xs] where x represents the first element in the list and xs is the rest of the list. Here’s an example implementation of a sumList function that takes a list and returns the sum of all elements in the list:
```
sumList :: Num p => [p] -> p
sumList [] = 0
sumList (x:xs) = x + (sumList xs)
```
Add this definition to your file and test it on a few lists. Note the recursive call to sumList that is part of its own definition. This is a common pattern in functional languages like Haskell.

Use this kind of pattern to Write your own version of head, called myHead, that returns the first value in a list (don’t worry about the empty list) using the pattern (x:xs). What is the type of the myHead function (you can check it again the type of the built-in head function)?
Lets practice writing more functions that work with lists. Write myLength, your own version of the list length function, whose type is [a] -> Int. It should calculate the length of whatever list you pass it. (Hint: What is the length of an empty list? What is the length of any list that you know has at least one element?)
Write myConcat that concatenates two lists like ++. Its type is [a] -> [a] -> [a]. This one is a little tricky! Try to think about “tearing apart” the first list one element at a time, gradually building up a new list that ultimately contains the second list as the tail after all of the elements from the first list. (If you get stuck on this one, keep going and come back to it later!)
Write fib n that returns the nth Fibonacci number, where fib 0 is 1 and fib 1 is 1.
Perhaps you wrote the fib function in the obvious way, which is perfectly correct but incredibly inefficient. Try fib 30 and see what happens. This version is slow because it recomputes the same values over and over again. In an imperative language, we would use a loop and two counters to compute this function efficiently. How do we do an equivalent thing with recursion? We still use accumulators, but we must carry them along in the recursion, and the only way to do this (since we don’t have variables) is as parameters. Consider the following definition.
```
ffib :: Integral a => a -> a
ffib n = fibAccum 1 1 n
fibAccum e0 e1 k
| (k == 0) = e0
| (k == 1) = e1
| otherwise = fibAccum e1 (e0+e1) (k-1)
```
Function ffib (for “fast Fibonacci”) is defined in terms of a helper function fibAccum that has two accumulators and a counter. The accumulators are used just as in the imperative version to keep track of successive values in the sequence, and the counter is used to control the recursion. Put this definition in your lab file, save it and load it, and try ffib 30. Not try ffib 300; there should be no problem—but don’t try fib 300 or you will sit there waiting for years.
Using accumulators and counters like this is an important technique in functional languages. Write indexOf with type Eq a => a -> [a] -> Int that returns the first index of a value in a list, or -1 if it is not in the list. For example, indexOf 'e' "fast times" is 8. ( The Eq a => in the type signature means that a is any type whose values can be compared for equality.)
You can exit ghci by typing :q and return.
For a different take on this material, you can read either the Haskell Basics section and the very first sub-section of the Elementary Haskell section of the Haskell Wikibook or the first four chapters of Learn You a Haskell for Great Good!.

As a final (optional) challenge, you should implement a function called myFilter that performs the same operation as the built-in filter function.

This lab was originally written by Dr. Chris Fox.

JMU CS 430 Spring 2021

Programming Languages

Module 09 Lab