Note: this is the completed version of lecture Basics.

Haskell Basics

Welcome to Haskell!

If you are reading this text online, you may wish to access it as a Haskell project instead. You can find this module as part of the repository 01-basics on github. We have prepared instructions on using github with CIS 5520 and with installing GHC and VSCode. We strongly encourage you to read this file in the VSCode editor so that you can experiment with it.

This file provides an introduction to the basics of the Haskell language. It explains the components of the language through examples and provides exercises for you to test your understanding.

First, every Haskell file begins with the name of the module (this name must start with a capital letter and be the same as the file name, without the .hs). After that, the next lines specify definitions that are imported from other modules. These import lines must be at the beginning of the file, before any other definitions.

> module Basics where    
> 
> -- library imports must come at the beginning of the module
> import Test.HUnit
>    ( (~?), runTestTT, (~?=), Counts, Test(TestList) )      
> import Prelude hiding (take, sum, const, or, tail)

Observe that Haskell supports two kinds of comments: single line comments start with -- and block comments begin and end with {- and -} respectively.

Understanding a Haskell program is about equality

Our goal in this class is not just for you to be able to write programs. We also want you to be able to reason about them abstractly, so that you understand exactly what they mean.

Haskell is a function programming language, and functional programming means that the semantics of a program can be described mathematically. One principle of mathematics is called Leibniz equality: in any context, we can replace an object with anything equivalent to it. Therefore, in Haskell, we reason about computation by reasoning about equality of sub-expressions.

For example, if we want to know what value an arithmetic expression computes to, we only need to find some number that is equal to it.

3 * (4 + 5)

{ 4+5 is equal to 9 by addition }

== 3 * 9

{ by multiplication }

== 27

That's it!

Furthermore, we can ask VSCode to compute the value of an expression for us with a special form of comment (i.e. a single line comment that starts with '>>>').

Try clicking on "Evaluate..." below. (You will only see "Evaluate..." if your IDE is set up correctly.)

> -- >>> 3 * (4 + 5)
> -- 27

This sort of reasoning isn't so surprising so far. We can do the same thing in almost any other language too. What makes Haskell different is that we will be able to use this equational reasoning in more contexts than just arithmetic.

A Haskell module is a list of definitions

A Haskell module (like this one) is a list of definitions. These definitions allow us to give names to Haskell expressions. In a module, all of the definitions must have unique names. If we try to add another definition of ex, we will get an error.

> ex :: Integer
> ex = 3 * (4 + 5)

We can ask VSCode to calculate with these names, just as we did above.

> -- >>> ex
> -- 27

Haskell definitions do not need to be in order. In a definition for one name, we we can use any other name in the module, even if it appears later in the file.

> ey :: Integer
> ey = ex + ez     -- refers to `ex` above and `ez` below.

> ez :: Integer
> ez = ex + 200

> -- >>> ey
> -- 254

Whenever we give a name to an expression, it is a good idea to also write down its type, as we have done so for ex above. VSCode can often figure out this type for you and will suggest that you add it to your file by clicking on the suggestion. You should do so---as later error messages can be perplexing without these additional hints.

The Integer type is the type of arbitrarily large integers in Haskell.

> bigInteger :: Integer
> bigInteger = 12345678901234567890

This is in contrast to the Int type, for word-sized integers (machine dependent). Numbers are overloaded in Haskell, so the type annotation tells the compiler how to interpret this expression. (And, note the warning issued by the compiler for this out of range number!)

> bigInt :: Int
> bigInt = 12345678901234567890

Compare the value of a extra-large Integer

> -- >>> bigInteger
> -- 12345678901234567890

with an Int

> -- >>> bigInt
> -- -6101065172474983726

Above, we declared the type of an expression separately from giving it a name. However, if we don't want to give a name to an expression, we can still annotate it with its type using ::.

> -- >>> 31 * (42 + 56) :: Integer
> -- 3038

More generally, the type annotation can be attached to any subexpression, not just at the top level.

> -- >>> (31 :: Integer) * (42 + 56)
> -- 3038

It is good style to annotate the type of every declaration in a Haskell program. This helps with error messages, as Haskell operators, like *, and constants like '31', are often overloaded.

Elements of Haskell

So far, we have have seen the following three properties of Haskell:

Haskell code is based on expressions
Expressions evaluate to values
Every expression has a type, which may influence evaluation

You are probably familiar with expressions in other programming languages, where they are often used to compute numeric and boolean values. Haskell also includes these types and operators.

For example, Haskell includes floating point numbers, via the Double type, using the same overloaded syntax.

> -- >>> 31 * (42 + 56) :: Double    -- double precision floating point

Furthermore, you'll also find characters, strings and boolean values.

> -- >>> 'a' :: Char                 -- characters

> -- >>> "abcd" :: String            -- strings

> -- >>> "cis" ++ "5520"             -- string concatenation

> -- >>> True :: Bool                -- boolean values

> -- >>> 1 <= 3 || False && 3 > 2    -- boolean operators, comparisons

What is a little different about Haskell is that everything is an expression, including conditionals. This means that if can be nested inside other expressions.

> -- >>> (if ex > 28 then 1 else 0) + 2 :: Int

Now the last basic type, shown below, is subtle. It is a special constant, written () and called "unit". The type of this constant is written with the same notation, and also called "unit". You'll need to pay attention to context to know whether we mean the value or the type. The key fact about this basic type is that there is only one value with type (). As a result, we sometimes say that this type is "trivial" --- if we know that a variable has type (), then we know that its value (if it has one) must be equal to ().

> -- >>> () :: ()            -- 'unit' (both value and type have the same syntax)
> -- ()

What is the point of a trivial type? In Haskell, every function is a mathematical function, so must return some value. However, sometimes there isn't anything interesting to return, so the function returns (). This is similar to the "void" type in other languages.

We'll have more to say about unit later.

What is Abstraction?

We're going to talk a lot about abstraction in this course, starting from simple examples and getting dramatically more expressive. But what is abstraction, exactly?

The first answer is: "Pattern Recognition"

31 * (42 + 56)

70 * (12 + 95)

90 * (68 + 12)

What do these expressions have in common? They all follow the same general pattern. We can generalize that pattern to a function by defining an equation.

> pat :: Integer -> Integer -> Integer -> Integer
> pat a b c = a * (b + c)

We call functions by providing them with arguments.

> -- >>> pat 31 42 56
> -- 3038

No parentheses are necessary, unless the argument itself is a compound expression.

> -- >>> pat (30 + 1) 42 56
> -- 3038

The important question is not "What does this function do?" but, instead "What does this function mean?" We can reason about that meaning using what we know about equality.

pat (30 + 1) 42 56

{ function call, replace a b & c in right-hand side of 
equation by (30 + 1) 42 and 56 }

== (30 + 1) * (42 + 56)

{ addition (twice) }

== 31 * 98

{ multiplication }

== 3038

Functions, like pat, are the core abstraction mechanisms in functional programming.

There was something subtle about the example above: we put the expression (30 + 1) in for a in the body of pat. When reasoning in most languages, it is only sound to do this replacement using values (like 31) instead general expressions, like (30 + 1). Haskell, supports more flexible reasoning. We could have also replaced (30 + 1) with 31 in the first step above, before using the equation for the function call, and we would have gotten the same result.

Function types

Like all expressions, functions have types.

The type of a function taking an input of type A and yielding an output of type B is written as

A -> B

For example, the pos function determines whether an Int is strictly greater than zero.

> pos :: Int -> Bool
> pos x = x > 0

The pat function above takes multiple arguments. Therefore, its type has multiple ->s, one for each argument.

The type of a function taking inputs of type A1, A2, and A3 and returning a result of type B is written as

A1 -> A2 -> A3 -> B

Symbolic vs. alphabetic names

Symbolic identifiers (i.e. + and *) are infix by default.

Parentheses around a symbolic name turn it into a regular name.

For example, if we want to define an alphabetic name for the addition function, we can do so.

> plus :: Int -> Int -> Int
> plus = (+)

And we can call operations in parentheses just like "standard" functions, by writing their arguments afterwards.

> p0 :: Int
> p0 = (+) 2 4

Likewise we can use alphabetic name in backquotes as infix.

> p1 :: Int
> p1 = 2 `plus` 2

We can also define new symbolic infix identifiers. For example, the >+ operator below is just like +, but it adds one to the result. When giving its type, we put it in parentheses to refer to it outside of an infix context. However, in its definition, we put the parameters x and y on either side of the operator.

> (>+) :: Int -> Int -> Int
> x >+ y = x + y + 1

> -- >>> 2 >+ 3
> -- 6

Laziness is a virtue

One major difference between Haskell and other programming languages is that Haskell uses a "call-by-need" semantics for evaluation, aka "lazy" evaluation. What this means is that Haskell does not evaluate arguments before calling functions. Instead, expressions are only evaluated when they are needed.

We can observe this behavior in Haskell by seeing what happens when we use error in a subexpresion. The error keyword in Haskell triggers a non-recoverable runtime exception, aborting any computation in progress. It is not a good idea to use error in production code, but it is convenient for incremental development and for learning about how Haskell evaluation works. An error can be used in any context and can be given any type because it does not produce a value. If an error is triggered, then we know that subexpression was evaluated.

For example, addition always needs to evaluate its arguments, so this error will trigger.

> -- >>> 1 + 2 + 3 + error "Here!"
> -- Here!

However, we won't trigger an error that is in dead code, such as in the non-selected part of an if-expression...

> -- >>> if 1 < 3 then 5 else error "Unreachable"

..or that was short-circuited when evaluating a boolean expression.

> -- >>> True || error "Unreachable"

In contrast, you can see that if the first argument were False instead, it does not short circuit and does trigger the error.

> -- >>> False || error "Ooops!"

In most languages, if and || are defined via special constructs because they include sub-expressions that are not always evaluated. However, in Haskell, these constructs are less special. For example, you can define your own short-circuiting version of the or operator. Suppose you would like this operator to have a textual name instead of (infix) symbols:

> or :: Bool -> Bool -> Bool 
> or a b = if a then True else b

Through laziness, this definition short circuits, just like the Prelude version of ||.

> -- >>> or True (error "Unreachable")

More generally, because Haskell is lazy, the language enables more abstraction. Functions and operators that we define can have nontrivial control behavior.

Laziness is also the reason that we can reason about Haskell programs just by thinking about equalities. For example, there is a function in the Prelude with the following definition:

> const :: a -> b -> b
> const x y = y

In most languages if you see a subexpression like const (f 3) 4, you have to know whether the expression f 3 produces a normal value first before you can know that the result is 4. However, in Haskell, you can use substitution: the pattern above says that with any call to const, the result is the value of the second argument.

Thus:

> -- >>> const (error "Here!") 4

Laziness is unique to Haskell, and we'll see more examples of it throughout the semester. Sometimes we use the word "strictness" to contrast with laziness. If an argument will always be evaluated before the function is called, we call this argument "strict." In most languages all functions are strict.

However, in Haskell, functions can be either lazy or strict, and lazy is the default. For example, both arguments in an addition operation are strict, because we need to know what the numbers are to sum them together. However, only the first argument in || is strict because it does not evaluate the second argument when the value of the first one is True.

Making Haskell DO something

Programs often interact with the world:

Read files
Display graphics
Broadcast packets
Run test cases and print success or failure

They don't just compute values.

How does this fit with values & equalities above?

Note, we've gotten far without doing any I/O. That's fairly standard in Haskell. Working with VSCode means that we can see the answers directly, we don't need an action to print them out. However, a standalone executable needs to do something, so we demonstrate that next.

The GHC System

We'll start with a few examples just using the interactive toplevel for the Haskell language. Although Haskell is a compiled language, the interactive toplevel, "ghci" is available for experimentation. You can access this toplevel using any command prompt (i.e. Terminal), as long as you have GHC installed. The examples below also assume that you have the "stack" tool available and that you have started the command prompt in the same directory that contains this source code. Instructions for installing "stack" and other tools are available.

First use the terminal to start ghci and instruct it to load the Basics module.

sweirich@sixteen 01-basics % stack ghci Basics.hs
Using configuration for cis5520-basics:lib to load /Users/sweirich/552/cis5520-23fa/lectures/01-basics/Basics.lhs
cis5520-basics> configure (lib)
Configuring cis5520-basics-0.1.0.0...
cis5520-basics> initial-build-steps (lib)
Configuring GHCi with the following packages: cis5520-basics
GHCi, version 9.2.5: https://www.haskell.org/ghc/  :? for help
[1 of 1] Compiling Basics           ( /Users/sweirich/5520/cis5520-23fa/lectures/01-basics/Basics.lhs, interpreted )
Ok, one module loaded.
Loaded GHCi configuration from /private/var/folders/p3/pkythxvx6rq9q054797y4bb80000gn/T/haskell-stack-ghci/48e82592/ghci-script
*Basics>

Now, with the prompt from the module, you can ask for the values, types and information of expressions.

*Basics> ex
27
*Basics> :t ex
ex :: Integer
*Basics> :i ex
ex :: Integer     -- Defined at Basics.hs:5:1

Obligatory Hello World

"IO actions" are a new sort of sort of value that describe an effect on the world.

IO a  --  Type of an action that returns an `a`  (a can be anything!)

Actions that do something but return nothing have the type IO ().

putStr :: String -> IO ()

So putStr takes in a string and returns an action that writes the string to stdout.

Because putStr doesn't evaluate to a value that we can print, we can't play with it using the IDE. Even if you hit Evaluate... you will see nothing because the IDE hides the printing action and only displays the value.

> -- >>> putStr "Say what?"

Instead, to observe the printing action, we need to use GHCi. Let's give it name first:

> hw :: IO ()
> hw = putStr "Hello World!\n"

and then we can give it a try.

*Basics> hw
Hello World!
*Basics>

Just 'do' it

How can we do many actions? By composing small actions.

The do syntax allows us to create a compound action that sequences one action after another. The definition of many below is a compound action that outputs the three strings in order. (Try it out in ghci!)

> many :: IO ()
> many = do putStr "Hello"     -- each line in the sequence
>           putStr " World!"   -- must be an IO action
>           putStr "\n"        -- don't forget the newline

Note: white-space is significant here. The do notation sequences actions, but each action in the sequence must start at the same character offset: all of the putStrs must be lined up.

Sometimes people put the do on a line by itself and then start the list of actions on the next line. This saves column width in larger developments.

> many' :: IO ()
> many' = do
>   putStr "Hello"
>   putStr " World!"
>   putStr "\n"

Example: Input Action

Actions can also return a value.

getLine :: IO String

This action reads and returns a line from stdin. We can name the result as part of a do sequence, with this notation, called "bind".

x <- action

Here x is a variable that can be used to refer to the result of the action in later code. For example, below, n is new String variable that is initialized by reading from stdin.

> query :: IO ()
> query = do putStr "What is your name? "
>            n <- getLine
>            putStrLn ("Welcome to CIS 5520 " ++ n)

Any IO t action can be used with bind to create a variable of type t. However, when working with type IO () the result is a special value, written () and called "unit". This special value indicates that the IO action did not have anything interesting to return---similar to a method that returns "void" in other programming languages. What is different in Haskell is that we do have a value of type (), and we can name it with bind, as shown in query' below. However, we probably won't do anything with this value. (Note that this example also demonstrates the Haskell convention to name variables that are unused with names that start with an underscore.)

> query' :: IO ()
> query' = do _m <- putStr "What is your name? "
>             n <- getLine
>             putStrLn ("Welcome to CIS 5520 " ++ n)

Note that you cannot name the last action in a sequence. Names are there so that you can use their results later. If you want to return the value instead, the last action should be a return.

> query2 :: IO String   -- compare this type to `query` above.
> query2 = do putStr "What is your name? "
>             n <- getLine
>             return n

Furthermore, there is no need to name a value if it is just going to be returned right away. This version is also equivalent.

> query2' :: IO String
> query2' = do putStr "What is your name? "
>              getLine

Example: Testing Actions

The hunit library contains definitions for constructing unit tests for your programs. You must import this library at the top of your module (with import Test.HUnit) before you can access these definitions. This library defines aTest` type for test cases.

> t1 :: Test
> t1 = (1 + 2 :: Int) ~?= 3
> -- check that the result of the computation matches the expected value `3`

Haskell is lazy by default, so these definitions create tests, but don't actually run them yet. We'll do that below.
The (~?=) operator is overloaded. You can create tests that compare expressions at many different types. When the expressions themselves are also overloaded (such as those with numbers), we run into ambiguity---what type of expressions should this test actually use? We resolve that ambiguity with an internal typing annotation (3 :: Int).

To run the test case, we use the function runTestTT.

       runTestTT :: Test -> IO Counts

> numTest :: IO Counts
> numTest = runTestTT t1

This expression is an action that runs the test case(s) and returns a data structure (of type Counts) recording how many tests cases were run and how many failed or produced errors. A failing test case is caused by a false assertion, for example True ~?= False. An erroroneous testcase is one that produced a run-time error, such as division by zero.

Although we can evaluate numTest in the IDE, we get more information about failing and erroroneous tests using ghci.

      *Basics> runTestTT (True ~?= False)
      ### Failure:                              
      <interactive>:16
      expected: False
      but got: True
      Cases: 1  Tried: 1  Errors: 0  Failures: 1
      Counts {cases = 1, tried = 1, errors = 0, failures = 1}

      *Basics> runTestTT ( 1 `div` 0 ~?= (1 :: Int))
      ### Error:                                
      divide by zero
      Cases: 1  Tried: 1  Errors: 1  Failures: 0
      Counts {cases = 1, tried = 1, errors = 1, failures = 0}

Structured Data

Tuples

An ordered sequence of values is called a tuple and its type is written as (where A1 ... An are the types of the values in the sequence).

    (A1, ..., An)

For example,

> tup1 :: (Char, Int)
> tup1 = ('a', 5)                         
> tup2 :: (Char, Double, Int)
> tup2 = ('a', 5.2, 7)                    
> tup3 :: ((Int, Double), Bool)
> tup3 = ((7, 5.2), True)

There can be any number of elements in a tuple, but the structure must match the type.

Pattern Matching extracts values from tuples.

A function that takes a tuple as an argument looks like it has multiple arguments, but in reality, it has just one. We use a pattern to name the three components of the tuple for use in the function.

> tpat :: (Int, Int, Int) -> Int
> tpat (a, b, c) = a * (b + c)

We can put anything in a tuple

We can have tuples of tuples. These three values below have three different types. Look closely! (We use the word 'pair' to describe a tuple with two values.)

> tup4 :: ((Int,Int),Int)
> tup4 = ((1,2),3)              -- a pair of a pair and a number

> tup5 :: (Int,(Int,Int))
> tup5 = (1,(2,3))              -- a pair of a number and a pair

> tup6 :: (Int, Int, Int)
> tup6 = (1, 2, 3)              -- a three-tuple, or triple

Note that the pattern that names the variables must match the structure of the type exactly.

> pat4 :: ((Int,Int),Int) -> Int
> pat4 ((a, b), c) = a * (b + c)

> -- >>> pat4 tup4

> pat5 :: (Int, (Int, Int)) -> Int
> pat5 (a, (b, c)) = a * (b + c)

> -- >>> pat5 tup5

> pat6 :: (Int,Int,Int) -> Int
> pat6 (a, b, c) = a * (b + c)

> -- >>> pat6 tup6

We can stick anything in tuples, even IO actions.

> act2 :: (IO (), IO ())
> act2 = (putStr "Hello", putStr "Hello")

This doesn't actually run both actions, it just creates a pair holding two IO computations. Haskell doesn't evaluate the components of a tuple when it is constructed. It waits until you actually need it.

Compare the difference between these definitions in ghci. Try to predict what they will do.

> runAct2 :: IO ()
> runAct2 = do
>    let (x, y) = act2     -- pattern match in `do` sequences using `let`
>    x                     -- run the first action
>    y                     -- then run the second

> runAct2' :: IO ()
> runAct2' = do
>    let (x, y) = act2     -- pattern match
>    y                     -- run the second action
>    x                     -- then run the first

> runAct2'' :: IO ()
> runAct2'' = do
>    let (x, y) = act2     -- pattern match
>    x                     -- run the first action
>    x                     -- then run it again!

Optional values

The type of "optional" or "partial" values is written

    Maybe A

This type describes either a value of type A, or else nothing.

> m1 :: Maybe Int
> m1 = Just 2       -- the 'Just' tag tells the compiler that we have a value

> m2 :: Maybe Int
> m2 = Nothing      -- the 'Nothing' tag means there is no value

The Nothing value of the Maybe type often plays the role that null does in other languages. But note that Nothing is not a value of type Int or String or any other type. It is a value of type Maybe Int. This means that we can't accidentally use Nothing where we need an Int. This is a good thing, because it means that we can't be surprised by null --- the types tell us exactly when a value could be Nothing.

'Maybe' is useful for partial functions

A common use of the Maybe type is for functions that don't always have a value to return. For example, suppose we know the location of some, but not all, classes in the CIS department. For the ones that we have no information about, we can return Nothing.

> location :: String -> Maybe String
> location "cis5010" = Just "Wu & Chen"
> location "cis5020" = Just "Heilmeier"
> location "cis5200" = Just "Wu & Chen"
> location "cis5520" = Just "3401 Walnut, 401B"
> location _         = Nothing    -- wildcard pattern, matches anything

Extracting values from 'Maybe's

Pattern matching extracts values from Maybes; we need a pattern for each case.

> pat'' :: Maybe Int -> Int
> pat'' (Just x) = x
> pat'' Nothing  = 2

Patterns can be nested, too.

> jn :: Maybe (Maybe a) -> Maybe a
> jn (Just (Just x)) = Just x
> jn (Just Nothing) = Nothing
> jn Nothing = Nothing

Quiz: See if you can come up with a slightly simpler way to write jn using two patterns instead of three. The undefined expression is one that produces a run-time error if it is ever evaluated.

> jn' :: Maybe (Maybe a) -> Maybe a
> 
> jn' (Just x) = x
> jn' Nothing = Nothing

Lists

The type of a list of values, each of type A is written

[A]

A list is a sequence of values of the same type. There is no limit to the number of values that can be stored in a list. We notate lists as a sequence of comma-separated values inside square brackets.

> l1 :: [Double]
> l1 = [1.0,2.0,3.0,4.0]

> l2 :: [Int]
> l2 = [1,2,3]

Lists can contain structured data...

> l3 :: [(Int,Bool)]
> l3 = [ (1,True), (2, False) ]

...and can be nested:

> l4 :: [[Int]]
> l4 = [ [1,2,3], [4,5], [] ]

List elements must have the same type.

> -- l5 :: [Int]
> -- l5 = [ 1 , True ]  -- doesn't type check

(Observe the type error that results when you uncomment the definition above.)

The empty list is written [] and pronounced "nil".

> l6 :: [a]
> l6 = []

Note: String is just another name for a list of characters ([Char]).

> l7 :: String
> l7 = ['h','e','l','l','o',' ','5','5','2','0','!']

What is the value of l7?

> -- >>> l7
> -- "hello 5520!"

"Cons"tructing Lists

The infix operator : constructs a new list, by adding a new element to the front of an existing list. (Note, the existing list is not modified.) We call this operator cons.

(The a in the type of cons below means that this function works for lists containing any type of element. In other words, we say that this function is polymorphic. More on this later.)

> cons :: a -> [a] -> [a]
> cons = (:)

> c1 :: [Bool]
> c1 = True : [False, False]

> c2 :: [Int]
> c2 = 1 : []

Try evaluating c1 and c2.

> -- >>> c1
> -- [True, False, False]
> -- >>> c2
> -- [1]

And check out the type of c3.

> c3 = [] : []

Strings

The String type in Haskell is syntactic sugar for a list of characters. Haskell doesn't distinguish between the types [Char] and String. For example, here are two different ways to write the same list:

> s1 :: [Char]
> s1 = "abc"
> 
> s2 :: String
> s2 = ['a', 'b', 'c']

Try evaluating s1 and s2.

> -- >>> s1
> -- "abc"
> -- >>> s2
> -- "abc"

Syntactic Sugar for lists

Haskell views the notation

[x1, x2, .. , xn]

as short for

x1 : x2 : .. : xn : []

This means that we can think of lists as a sequence of cons'ed elements, ending with nil. For example,

[1,2,3,4]  and  1 : 2 : 3 : 4 : []

are the same list.

> -- >>> [1,2,3,4] == 1:2:3:4:[]
> -- True

Function practice: List Generation

Example: Write a function that, given an argument x and a number n, returns a list containing n copies of x.

Step 1: Write test cases for the function.

We're using HUnit, a library for defining unit tests in Haskell and using the ~?= operator to construct unit Tests by comparing the computed result (first argument) with an expected result (second argument).

> testClone1, testClone2, testClone3, testClone4 :: Test
> testClone1 = clone 'a' 4 ~?= ['a','a','a','a']
> testClone2 = clone 'a' 0 ~?= []
> testClone3 = clone (1.1 :: Double) 3 ~?= [1.1, 1.1, 1.1]
> testClone4 = clone 'a' (-1) ~?= []

Step 2: Declare the type of the function.

This function replicates any type of value, so the type of the first argument is polymorphic.

> clone :: a -> Int -> [a]

Step 3: Implement the function.

We implement this function by recursion on the integer argument.

> clone x n = if n<=0 then [] else x : clone x (n-1)

Step 4: Run the tests.

The HUnit function runTestTT actually runs a given unit test and prints its result to the standard output stream. (That is why its result type is IO Counts. The IO in the type means that this computation does IO.)

> cl1, cl2, cl3, cl4 :: IO Counts
> cl1 = runTestTT testClone1
> cl2 = runTestTT testClone2
> cl3 = runTestTT testClone3
> cl4 = runTestTT testClone4

> cls :: IO Counts
> cls = runTestTT (TestList [ testClone1, testClone2, testClone3, testClone4 ])

You can run the tests by evaluating the definition cls at the ghci prompt.

ghci> cls
Cases: 4  Tried: 4  Errors: 0  Failures: 0
Counts {cases = 4, tried = 4, errors = 0, failures = 0}

Function practice

Quiz: Define a function called range that, given two integers i and j, returns a list containing all of the numbers at least as big as i but no bigger than j, in order.

Step 1: Write test cases. We can define multiple test cases at once using a list.

> testRange :: Test
> testRange = TestList [ range 3  6  ~?= [3,4,5,6],
>                        range 42 42 ~?= [42],
>                        range 10 5  ~?= [] ]

Step 2: Declare the type of the function.

> range :: Int -> Int -> [Int]

Step 3: Define the function. This part is for you to do for your quiz.

> 
> range i j = if i > j then [] else i : range (i+1) j

Step 4: Run the tests.

> runRTests :: IO Counts
> runRTests = runTestTT testRange

Pattern matching with lists

The examples so far have constructed various lists. Of course, sometimes we would like to write functions that use lists. We can use a list by pattern matching...

> isHi :: String -> Bool
> isHi ['H','i'] = True
> isHi _ = False

If we have a list of characters, we can use string constants as patterns too.

> isGreeting :: String -> Bool
> isGreeting "Hi" = True
> isGreeting "Hello" = True
> isGreeting "Bonjour" = True
> isGreeting "Guten Tag" = True
> isGreeting _ = False

We can also work with lists more abstractly, for example determining if we have a list of length one...

> isSingleton :: [a] -> Bool
> isSingleton [_] = True
> isSingleton _   = False

...or of length greater than two. Complete this one yourself using a pattern to identify lists that have three or more elements.

> isLong :: [a] -> Bool
> 
> isLong (_:_:_:_) = True
> isLong _         = False

> testIsLong :: Test
> testIsLong = TestList [ not (isLong [])   ~? "nil",    -- can convert booleans to tests by naming them via `~?`
>                         not (isLong "a")  ~? "one",
>                         not (isLong "ab") ~? "two",
>                         isLong "abc"      ~? "three" ]

Finally, all of the patterns we have shown you so far have been part of definitions. We can define a function, like isSingleton or isGreeting above, by cases, using multiple lines. This style is common in Haskell, but if you are coming from OCaml, you might be more familiar with match, or a separate expression form of pattern matching. Such a form is also available in Haskell, using the case and of keywords.

For example, we can rewrite isGreeting using case instead.

> isGreeting2 :: String -> Bool
> isGreeting2 s = 
>   case s of 
>     "Hi" -> True
>     "Hello" -> True
>     "Bonjour" -> True
>     "Guten Tag" -> True
>     _ -> False

Note that all of the patterns in a case expression must start in the same column of your source file. This expression is layout sensitive, and if things don't line up, you will get a compilation error. Case expressions are particularly good for nested patterns, where you might want to match again inside of a branch. Here's a silly definition of isGreeting that demonstrates a nested pattern. Note how the layout determines where the patterns for the inner case end and the ones for the outer case resume.

> isGreeting3 :: String -> Bool
> isGreeting3 s = 
>   case s of 
>     ('H':r) -> case r of 
>                  "i" -> True
>                  "ello" -> True
>                  _ -> False
>     "Bonjour" -> True
>     "Guten Tag" -> True
>     _ -> False

Function practice: List Recursion

**Example*: Define a function called listSum that, given a list of Ints returns their sum.

Step 1: Write test cases.

> sumTests :: Test
> sumTests = TestList [ sum [1,2,3] ~?= 6,
>                           sum [] ~?= 0 ]

Step 2: Declare the type of the function.

> sum :: [Int] -> Int

Step 3: Define the function. (Use pattern matching to define the function by case analysis.)

> sum []       = 0
> sum (x : xs) = x + sum xs

Step 4: Run the tests.

> runSumTests :: IO Counts
> runSumTests = runTestTT sumTests

Note that listAdd follows a general pattern of working with lists called list recursion. We can define lists as follows.

A list is either

[], the empty list, or
x : xs, an element x cons'ed onto another list xs.

This is a recursive definition, as we are defining the notion of lists in terms of itself. Recursive functions that work with lists will follow the pattern of this definition:

f :: [a] -> ...
f [] =  ...       -- case for the empty list
f (x : xs) = ...  -- case for a nonempty list, will use `f xs`
                  -- recursively somehow.

Function practice: List access

Define a function, called take, that, given a number n and a list, returns the first n items in the list, or the whole list if there are fewer than n items.

Step 1: Write test cases.

> takeTests :: Test
> takeTests =
>  TestList [ take 1 [1,2,3] ~?= [1],
>             take 5 [1,2,3] ~?= [1,2,3] ]

Step 2: Declare the type of the function. This function is polymorphic and works with any element type.

> take :: Int -> [a] -> [a]

Step 3: Define the function.

> take 0 xs = []
> take n [] = []
> take n (x:xs) = x : take (n - 1) xs

Step 4: Run the tests.

> runTakeTests :: IO Counts
> runTakeTests = runTestTT takeTests

Function practice: List transformation

Define a function, called listIncr, that, given a list of ints, returns a new list where each number has been incremented.

Step 1: Write test cases.

> listIncrTests :: Test
> listIncrTests =
>  TestList [ listIncr [1,2,3] ~?= [2,3,4],
>             listIncr [42]    ~?= [43] ]

Step 2: Declare the type of the function.

> listIncr :: [Int] -> [Int]

Step 3: Define the function.

> 
> listIncr []       = []
> listIncr (x : xs) = x + 1 : listIncr xs

Step 4: Run the tests.

> runLITests :: IO Counts
> runLITests = runTestTT listIncrTests

Function practice: Double List transformation

Define a function, called listAdd, that, given two lists of numbers, adds them together pointwise. Any extra numbers are ignored.

Step 1: Write test cases.

> listAddTests :: Test
> listAddTests =
>  TestList [ listAdd [1,2,3] [2,4,5] ~?= [3,6,8],
>             listAdd [42] []   ~?= [] ]

Step 2: Declare the type of the function.

> listAdd :: [Int] -> [Int] -> [Int]

Step 3: Define the function.

> 
> listAdd []  _     = []
> listAdd _ []      = []
> listAdd (x : xs) (y:ys) = x + y : listAdd xs ys

Step 4: Run the tests.

> runLAddTests :: IO Counts
> runLAddTests = runTestTT listAddTests

> -- >>> runLAddTests
> -- Counts {cases = 2, tried = 2, errors = 0, failures = 0}

Function practice: "Infinite" lists

The : operator is lazy in Haskell. When we create a list we don't need to know the value of all of the elements.

> -- >>> take 2 (1 : 2 : 3 : error "unreachable")
> -- [1,2]

Because : is lazy, we can define lists in terms of themselves. This list has as many ones as you want it to contain.

> ones :: [Int]
> ones = 1 : ones

Do NOT try to print this list.

> -- >>> ones

But, we can work with any finite prefix of the list without trouble.

> -- >>> take 17 ones
> -- [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

And, we can use this technique to define many different series of numbers. For example, we can define a simple incrementing series.

> allNums :: [Int]
> allNums = 1 : listIncr allNums

> -- >>> take 17 allNums
> -- [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]

And we can define the Fibonacci series. (The tail function below is also defined in the Haskell standard library and returns the list without its first element. Note: in general you should avoid this function when working with finite lists as it could trigger an error. In some projects, we have configured hlint to warn you about this function.)

> tail :: [Int] -> [Int]
> tail (_ : xs) = xs
> tail [] = error "tail of empty list"

> fibs :: [Int]
> fibs = 1 : 1 : listAdd fibs (tail fibs)

> -- >>> take 17 fibs
> -- [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597]

How do you reason about definitions such as fibs above? The technique that we've seen above, replacing equal subterms by equal subterms works!

fibs = -- definition above 
      1 : 1 : listAdd fibs (tail fibs)
       -- name a subexpression
     == 1 : 1 : l1 
        where l1 = listAdd fibs (tail fibs)
                    -- unfold definition of fibs (twice) to simplify this subexpression
                 == listAdd (1 : 1 : listAdd fibs (tail fibs)) (tail (1 : 1 : listAdd fibs (tail fibs)))
                    -- replace `tail` with its definition
                 == listAdd (1 : 1 : listAdd fibs (tail fibs)) (1 : listAdd fibs (tail fibs))
                    -- replace `listAdd` with its definition, in the case of nonempty lists
                 == 1 + 1 : listAdd (1 : listAdd fibs (tail fibs)) (listAdd fibs (tail fibs))
                    -- add numbers, replace subterms equivalent to l1
                 == 2 : listAdd (1 : l1) l1
       -- replace l1 with simpler version
     == 1 : 1 : 2 : listAdd (1 : l1) l1
       -- name a new subexpression
     == 1 : 1 : 2 : l2
        where l2 = listAdd (1 : l1) l1
                    -- unfold definition of l1 
                 == listAdd (1 : l1) (2 : listAdd (1 : l1) l1)
                    -- replace `listAdd` with its definition
                 == 1 + 2 : listAdd l1 (listAdd (1 : l1) l1)
                    -- add numbers, replace subterm equivalent to l2
                 == 3 : listAdd l1 l2
       -- replace l2 with simpler version
     == 1 : 1 : 2 : 3 : listAdd l1 l2
       -- name a subexpression, etc
     == 1 : 1 : 2 : 3 : l3
        where l3 = listAdd l1 l2
                 == listAdd (2 : listAdd (1 : l1) l1) (3 : listAdd l1 l2)
                 == 2 + 3 : listAdd (listAdd (1 : l1) l1) (listAdd l1 l2)
                 == 5 : l2 + l3
     == 1 : 1 : 2 : 3 : 5 : l2 + l3

In the equations above, we are only just replacing subterms by equal subterms. However, if we do this carefully, we can see what the values of the list will be, if we ever need them for some computation.

More generally, because of laziness, our code is more modular. We have separated the generation of the sequence of numbers from deciding just how many of those numbers that we need. We don't compute all of the numbers when we define this series --- we only compute them as we need them. This is a powerful idea which will re-occur throughout the semester.

Here is a derivation about allNums. The important facts to note are that listIncr (x : xs) = x + 1 : listIncr xs and allNums = 1 : listIncr allNums

We start with the latter:

allNums == 1 : listIncr allNums
           -- replace definition of allNums
        == 1 : listIncr (1 : listIncr allNums)
           -- unfold definition of listIncr, x=1 and xs=listIncr allNums
        == 1 : 1 + 1 : listIncr (listIncr allNums)
           -- simplify addition
        == 1 : 2 : listIncr (listIncr allNums)
           -- replace definition of allNums again
        == 1 : 2 : listIncr (listIncr (1 : listIncr allNums))
           -- unfold definition of listIncr again (and simplify)
        == 1 : 2 : listIncr (2 : listIncr (listIncr allNums))
           -- unfold definition of listIncr again (and simplify)
        == 1 : 2 : 3 : listIncr (listIncr (listIncr allNums))

For Penn students: There is a quiz associated with this module on Canvas. Please complete this quiz before the next class.

CIS 5520: Advanced Programming

Fall 2023

Haskell Basics

Welcome to Haskell!

Understanding a Haskell program is about equality

A Haskell module is a list of definitions

Elements of Haskell

What is Abstraction?

Function types

Symbolic vs. alphabetic names

Laziness is a virtue

Making Haskell DO something

The GHC System

Obligatory Hello World

Just 'do' it

Example: Input Action

Example: Testing Actions

Structured Data

Tuples

We can put anything in a tuple

Optional values

'Maybe' is useful for partial functions

Extracting values from 'Maybe's

Lists

"Cons"tructing Lists

Strings

Syntactic Sugar for lists

Function practice: List Generation

Function practice

Pattern matching with lists

Function practice: List Recursion

Function practice: List access

Function practice: List transformation

Function practice: Double List transformation

Function practice: "Infinite" lists