Copyright (C) 2016, 2017, H. Conrad Cunningham

Acknowledgements: I adapted and revised much of this work in Summer and Fall 2016 from my previous materials.

In 2017, I continue to develop this chapter.

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.

Advisory: The HTML version of this document may require use of a browser that supports the display of MathML. A good choice as of October 2017 is a recent version of Firefox from Mozilla.

TODO:

2 Basic Haskell Functional Programming

2.1 Chapter Introduction

This chapter introduces the basic features of the Haskell language needed for programming using first-order functions, primitive data types, and tuples. The goal is for the students to be able to use these features to develop small Haskell programs.

Upon successful completion of this chapter, students should be able to:

  1. describe the basic syntax and semantics of first-order Haskell functions

  2. describe the basic Haskell data types

  3. execute Haskell programs from the REPL

  4. develop first-order Haskell functional programs using recursion and the primitive data types and tuples

  5. apply top-down refinement and data abstraction to develop Haskell modules

  6. relate the basic Haskell features to similar features in other languages

2.2 Defining Our First Haskell Functions

Let's look at our first function definition in the Haskell language, a program to implement the factorial function for natural numbers.

The Haskell source file Fact.hs holds the Haskell function definitions for this section. The source file TestFact.hs gives a simple test script.

2.2.1 Factorial function specification

We can give two mathematical definitions of factorial, fact and fact', that are equivalent for all natural number arguments. We can define fact using the product operator as follows:

fact$(n) = \prod_{i=1}^{i=n}\,i$

For example,

fact$(4) = 1 \times 2 \times 3 \times 4$.

By definition

fact$(0) = 1$

which is the identity element of the multiplication operation.

We can also define the factorial function fact' with a recursive definition (or recurrence relation) as follows:

fact'$(n) = 1$, if $n = 0$
fact'$(n) = n \times$ fact'$(n-1)$, if $n \geq 1$

Since the domain of fact' is the set of natural numbers, a set over which induction is defined, we can easily see that this recursive definition is well defined.

In a previous chapter, we proved that fact$(n)$ $=$ fact'$(n)$ by mathematical induction.

The Haskell functions defined in the following subsections must compute fact$(n)$ when applied to argument value $n$.

2.2.2 Factorial function using if-then-else: fact1

One way to translate the recursive definition fact' into Haskell is the following:

    fact1 :: Int -> Int 
    fact1 n = if n == 0 then 
                  1 
              else 
                  n * fact1 (n-1)

In Haskell there is only one way to form more complex expressions from simpler ones: apply a function.

Neither parentheses nor special operator symbols are used to denote function application; it is denoted by simply listing the argument expressions following the function name. For example, a function f applied to argument expressions x and y is written in the following prefix form:

    f x y

However, the usual prefix form for a function application is not a convenient or natural way to write many common expressions. Haskell provides a helpful bit of syntactic sugar, the infix expression. Thus instead of having to write the addition of x and y as

    add x y

we can write it as

    x + y

as we have since elementary school. Here the symbol + represents the addition function.

Function application (i.e., juxtaposition of function names and argument expressions) has higher precedence than other operators. Thus the expression f x + y is the same as (f x) + y.

2.2.3 Factorial function using guards: fact2

An alternative way to differentiate the two cases in the recursive definition is to use a different equation for each case. If the Boolean guard (e.g., n == 0) for an equation evaluates to true, then that equation is used in the evaluation of the function. A guard is written following the | symbol as follows:

    fact2 :: Int -> Int 
    fact2 n 
      | n == 0    = 1 
      | otherwise = n * fact2 (n-1)

Function fact2 is equivalent to the fact1. Haskell evaluates the guards in a top-to-bottom order. The otherwise guard always succeeds; thus it's use above is similar to the trailing else clause on the if-then-else expression used in fact1.

2.2.4 Factorial function using pattern matching: fact3 and fact4

Another equivalent way to differentiate the two cases in the recursive definition is to use pattern matching as follows:

    fact3 :: Int -> Int 
    fact3 0 = 1 
    fact3 n = n * fact3 (n-1)

The parameter pattern 0 in the first leg of the definition only matches arguments with value 0. Since Haskell checks patterns and guards in a top-to-bottom order, the n pattern matches all nonzero values. Thus fact1, fact2, and fact3 are equivalent.

To stop evaluation from going into an “infinite loop” for negative arguments, we can remove the negative integers from the function’s domain. One way to do this is by using guards to narrow the domain to the natural numbers as in the definition of fact4 below:

    fact4 :: Int -> Int 
    fact4 n 
        | n == 0 =  1 
        | n >= 1 =  n * fact4 (n-1)

Function fact4 is undefined for negative arguments. If fact4 is applied to a negative argument, the evaluation of the program encounters an error quickly and returns without going into an infinite loop. It prints an error and halts further evaluation.

We can define our own error message for the negative case using an error call as in fact4' below.

    fact4' :: Int -> Int 
    fact4' n 
      | n == 0    =  1 
      | n >= 1    =  n * fact4' (n-1)
      | otherwise = error "fact4' called with negative argument"

In addition to displaying the custom error message, this also displays a stack trace of the active function calls.

2.2.5 Factorial function using a built-in library function: fact5

The four definitions we have looked at so far use recursive patterns similar to the recurrence relation fact'. Another alternative is to use the library function product and the list-generating expression [1..n] to define a solution that is like the function fact:

    fact5 :: Int -> Int 
    fact5 n = product [1..n]

The list expression [1..n] generates a list of consecutive integers beginning with 1 and ending with n. We study lists in a later chapter.

The library function product computes the product of the elements of a finite list.

If we apply fact5 to a negative argument, the expression [1..n] generates an empty list. Applying product to this empty list yields 0, which is the identity element for multiplication. Defining fact5 to return 0 is consistent with the function fact upon which it is based.

Which of the above definitions for the factorial function is better?

Most people in the functional programming community would consider fact4 (or fact4') and fact5 as being better than the others. The choice between them depends upon whether we want to trap the application to negative numbers as an error or to return the value 1.

2.3 Using the Glasgow Haskell Compiler (GHC)

See the Glasgow Haskell Compiler Users Guide for information on the Glasgow Haskell Compiler (GHC) and its use.

GHCi is an environment for using GHC interactively. That is, it is a REPL (Read-Evaluate-Print-Loop) command line interface using Haskell. The "Using GHCi" chapter of the User Guide describes its usage.

Below, we show a GHCi session where we load source code file (module) Fact.hs and apply the factorial functions to various inputs. The instructor ran this in a Terminal session on an iMac running macOS 10.12.6 (Sierra) with ghc 8.2.1 installed.

  1. Start the REPL.

    bash-3.2$ ghci
GHCi, version 8.2.1: http://www.haskell.org/ghc/  :? for help

  2. Load module Fact that holds the factorial function definitions. This assumes the Fact.hs file is in the current directory. The load command can be abbreviated as just :l.

    Prelude> :load Fact
[1 of 1] Compiling Fact             ( Fact.hs, interpreted )
Ok, 1 module loaded.

  3. Inquire about the type of fact1.

    *Fact> :type fact1
fact1 :: Int -> Int

  4. Apply function fact1 to 7, 0, 20, and 21. Note that the factorial of 21 exceeds the Int range.

    *Fact> fact1 7
5040
*Fact> fact1 0
1
*Fact> fact1 20
2432902008176640000
*Fact> fact1 21
-4249290049419214848

  5. Apply functions fact2, fact3, fact4 and fact5 to 7.

    *Fact> fact2 7
5040
*Fact> fact3 7
5040
*Fact> fact4 7
5040
*Fact> fact5 7 
5040

  6. Apply functions fact1, fact2, and fact3 to -1. All go into an infinite recursion, eventually terminating with an error when the runtime stack overflows its allocated space.

    *Fact> fact1 (-1)
*** Exception: stack overflow
*Fact> fact2 (-1)
*** Exception: stack overflow
*Fact> fact3 (-1)
*** Exception: stack overflow

  7. Apply functions fact4 and fact4' to -1. They quickly return with an error.

    *Fact> fact4 (-1)
*** Exception: Fact.hs:(33,1)-(35,29): Non-exhaustive patterns 
    in function fact4
*Fact> fact4' (-1)
*** Exception: fact4' called with negative argument
CallStack (from HasCallStack):
  error, called at Fact.hs:50:17 in main:Fact

  8. Apply function fact5 to -1. It returns a 1 because it is defined for negative integers.

    *Fact> fact5 (-1)
1

  9. Set the +s option to get information about the time and space required and the +t option to get the type of the returned value.

    *Fact> :set +s
*Fact> fact1 20
2432902008176640000
(0.00 secs, 80,672 bytes)
*Fact> :set +t
*Fact> fact1 20
2432902008176640000
it :: Int
(0.00 secs, 80,720 bytes)
*Fact> :unset +s +t
*Fact> fact1 20
2432902008176640000

  10. Exit GHCi.

    :quit
Leaving GHCi.

Suppose we had set the environment variable EDITOR to our favorite text editor in the Terminal window. For example, on a Mac OS system, your instructor might give the following command in shell (or in a startup script such as .bash_profile):

    export EDITOR=Aquamacs

Then the :edit command within GHCi allows us to edit the source code. We can give a filename or default to the last file loaded.

    :edit

Or we could also use a :set command to set the editor within GHCi.

    :set editor Aquamacs
    ...
    :edit

See the Glasgow Haskell Compiler Users Guide for more information about use of GHC and GHCi.

2.4 Surveying the Basic Types

The type system is an important part of Haskell; the compiler or interpreter uses the type information to detect errors in expressions and function definitions. To each expression Haskell assigns a type that describes the kind of value represented by the expression.

Haskell has both built-in types (defined in the language or its standard libraries) and facilities for defining new types. In the following we discuss the primary built-in types. As we have seen, a Haskell type name begins with a capital letter.

In these notes, we sometimes refer to the types Int, Float, Double, Bool, and Char as being primitive because they likely have direct support in the host processor's hardware.

2.4.1 Integers: Int and Integer

The Int data type is usually an integer data type supported directly by the host processor (e.g., 32- or 64-bits on most current processors), but it is guaranteed to have the range of at least a 30-bit, two’s complement integer.

The type Integer is an unbounded precision integer type. Unlike Int, host processors usually do not support this type directly. The Haskell library or runtime system typically supports this type in software.

Haskell supports the usual integer literals (i.e., constants) and operations such as +, -, *, <, etc.

For integer division, Haskell provides functions div and rem such that div m n returns the integral quotient from dividing m by n and rem m n returns the remainder.

2.4.2 Floating point numbers: Float and Double

The Float and Double data types are usually the single and double precision floating point numbers supported directly by the host processor.

Haskell floating point literals must include a decimal point; they may be signed or in scientific notation: 3.14159, 2.0, -2.0, 1.0e4, 5.0e-2, -5.0e-2.

Haskell supports the usual operations on floating point numbers. Division is denoted by / as usual.

2.4.3 Booleans: Bool

The Bool data type is usually supported directly by the host processor as one or more contiguous bits.

The Boolean literals are True and False (note capitals). Haskell supports Boolean operations such as && (and), || (or), and not.

Functions can match against patterns using the Boolean constants. For example, we could define a function myAnd as follows:

    myAnd :: Bool -> Bool -> Bool
    myAnd True  b = b
    myAnd False _ = False

Above the pattern _ is a wildcard that matches any value but does not bind a value that can be used on the right-hand-side of the definition.

2.4.4 Characters: Char

The Char data type is usually supported directly by the host processor by one or more contiguous bytes.

Haskell uses Unicode for its character data type. Haskell supports character literals enclosed in single quotes—including both the graphic characters (e.g., ’a’, ’0’, and ’Z’) and special codes entered following the escape character backslash \ (e.g., '\n' for newline, '\t' for horizontal tab, and '\\' for backslash itself).

In addition, a backslash character \ followed by a number generates the corresponding Unicode character code. If the first character following the backslash is o, then the number is in octal representation; if followed by x, then in hexadecimal notation; and otherwise in decimal notation.

For example, the exclamation point character can be represented in any of the following ways: ’!’, '\33', '\o41', '\x21'

2.4.5 Functions: t1 -> t2

If t1 and t2 are types then t1 -> t2 is the type of a function that takes an argument of type t1 and returns a result of type t2.

Function and variable names begin with lowercase letters optionally followed by a sequences of characters each of which is a letter, a digit, an apostrophe (') (sometimes pronounced "prime"), or an underscore (_).

Haskell functions are first-class objects. They can be arguments or results of other functions or be components of data structures. Multi-argument functions are curried-–that is, treated as if they take their arguments one at a time.

For example, consider the integer addition operation (+). (Surrounding the binary operator symbol with parentheses refers to the corresponding function.) In mathematics, we normally consider addition as an operation that takes a pair of integers and yields an integer result, which would have the type signature

    (+) :: (Int,Int) -> Int

In Haskell, we give the addition operation the type

    (+) :: Int -> (Int -> Int)

or just

    (+) :: Int -> Int -> Int

since Haskell binds -> from the right.

Thus (+) is a one argument function that takes some Int argument and returns a function of type Int -> Int. Hence, the expression ((+) 5) denotes a function that takes one argument and returns that argument plus 5.

We sometimes speak of this (+) operation as being partially applied (i.e., to one argument instead of two).

This process of replacing a structured argument by a sequence of simpler ones is called currying, named after American logician Haskell B. Curry who first described it.

The Haskell library, called the standard prelude (or just Prelude), contains a wide range of predefined functions including the usual arithmetic, relational, and Boolean operations. Some of these operations are predefined as infix operations.

2.4.6 Tuples: (t1,t2,...,tn)

If t1, t2, $\cdots$, tn are types, where n is finite and n >= 2, then is a type consisting of n-tuples where the various components have the type given for that position.

Each element in a tuple may have different types. The number of elements in a tuple is fixed.

Examples of tuple values with their types include the following:

    ('a',1) :: (Char,Int)
    (0.0,0.0,0.0) :: (Double,Double,Double) 
    (('a',False),(3,4)) :: ((Char, Bool), (Int, Int))

We can also define a type synonym using the type declaration and the use the synonym in further declarations as follows:

    type Complex = (Float,Float)
    makeComplex :: Float -> Float -> Complex
    makeComplex r i = (r,i)`

A type synonym does not define a new type, but it introduces an alias for an existing type. We can use Complex in declarations, but it has the same effect as using (Float,Float) expect that Complex provides better documentation of the intent.

2.4.7 Lists: [t]

The primary built-in data structure in Haskell is the list, a sequence of values. All the elements in a list must have the same type. Thus we declare lists with notation such as [t] to denote a list of zero or more elements of type t.

A list literal is a comma-separated sequence of values enclosed between [ and ]. For example, [] is an empty list and [1,2,3] is a list of the first three positive integers in increasing order.

We will look at programming with lists in a later chapter.

2.4.8 Strings: String

In Haskell, a string is just a list of characters. Thus Haskell defines the data type String as a type synonym :

    type String = [Char]

We examine lists and strings in a later chapter, but, because we use strings in a few examples in this subsection, let's consider them briefly.

A String literal is a sequence of zero or more characters enclosed in double quotes, for example, "Haskell programming".

Strings can contain any graphic character or any special character given as escape code sequence (using backslash). The special escape code \& is used to separate any character sequences that are otherwise ambiguous.

For example, the string literal "Hotty\nToddy!\n" is a string that has two newline characters embedded.

Also the string literal "\12\&3" represents the two-element list ['\12','3'].

Because strings are represented as lists, all of the Prelude functions for manipulating lists also apply to strings. We look at manipulating lists and strings in later chapters of these notes.

2.5 Using Top-Down Stepwise Refinement

As we saw in a previous chapter, there are two processes of abstraction in program design and implementation.

Procedural abstraction:
the separation of the logical properties of an action from the details of how the action is implemented.
Data abstraction:
the separation of the logical properties of data from the details of how the data are represented.

This section focuses on procedural abstraction. Later sections focus on data abstraction.

A useful and intuitive design process for a small program is to begin with a high-level solution and incrementally fill in the details. We call this process top-down stepwise refinement. Here we introduce it with an example.

2.5.1 Developing a square root package

Consider the problem of computing the nonnegative square root of a nonnegative number $x$. Mathematically, we want to find the number $y$ such that

$y \geq 0$ and $y^{2} = x$.

A common algorithm in mathematics for computing the above $y$ is to use Newton's method of successive approximations, which has the following steps for square root:

  1. Guess at the value of $y$.
  2. If the current approximation (guess) is sufficiently close (i.e. good enough), return it and stop; otherwise, continue.
  3. Compute an improved guess by averaging the value of the guess $y$ and $x/y$, then go back to step 2.

To encode this algorithm in Haskell, we work top down to decompose the problem into smaller parts until each part can be solved easily. We begin this top-down stepwise refinement by defining a function with the type signature:

    sqrtIter :: Double -> Double -> Double

We choose type Double (double precision floating point) to approximate the real numbers. Thus we can encode step 2 of the above algorithm in Haskell as follows:

    sqrtIter guess x 
        | goodEnough guess x = guess 
        | otherwise          = sqrtIter (improve guess x) x

We define sqrtIter to take two arguments--the current approximation guess and number x for which we need the square root. We have two cases:

The function improve takes the current guess and x and carries out step 3 of the algorithm, thus averaging guess and x/guess, as follows:

    improve :: Double -> Double -> Double 
    improve guess x = average guess (x/guess)

Here we abstract average into a separate function as follows:

    average :: Double -> Double -> Double
    average x y = (x + y) / 2

The new guess is closer to the square root than the previous guess. Thus the algorithm will terminate assuming a good choice for function goodEnough, which guards the base case of the sqrtIter recursion.

How should we define goodEnough? Given that we are working with the limited precision of computer floating point arithmetic, it is not easy to choose an appropriate test for all situations. Here we simplify this and use a tolerance of 0.001.

We thus postulate the following definition for goodEnough:

    goodEnough :: Double -> Double -> Bool 
    goodEnough guess x = abs (square guess - x) < 0.001

In the above, abs is the built-in absolute value function defined in the standard Prelude library. We define square as the following simple function (but could replace it by just guess * guess):

    square :: Double -> Double 
    square x = x * x

What is a good initial guess? It is sufficient to just use 1. So we can define an overall square root function sqrt' as follows:

    sqrt' :: Double -> Double
    sqrt' x | x >= 0 = sqrtIter 1 x

(A square root function sqrt is defined in the Prelude library, so a different name is needed to avoid the name clash.)

2.5.2 Making the package a Haskell module

We can make this package into a Haskell module by putting the definitions in a file (e.g., named Sqrt.hs) and adding a module header at the beginning as follows:

    module Sqrt
        (sqrt')
    where
      -- give the definitions above for functions sqrt',
      --   sqrtIter, improve, average, and goodEnough,

The header gives the module the name Sqrt and defines the names in parenthesis as being exported to other modules that import this module. The other symbols (e.g., sqrtIter, goodEnough) are local to (i.e., hidden inside) the module.

In the above Haskell code, the symbol "--" denotes the beginning of an end-of-line comment. All text after that symbol is text ignored by the Haskell compiler.

The Haskell module for the Square root case study is in file Sqrt.hs. Limited testing code is in module TestSqrt.hs.

2.5.3 Top-down stepwise refinement

The program design strategy known as top-down stepwise refinement is a relatively intuitive design process that has long been applied in the design of structured programs in imperative procedural languages. It is also useful in the functional setting.

In Haskell, we can apply top-down stepwise refinement as follows.

  1. Start with a high-level solution to the problem consisting of one or more functions. For each function, identify its type signature and functional requirements (i.e., its inputs, outputs, and termination condition).

    Some parts of each function are abstracted as "pseudocode" expressions or as-yet-undefined function calls.

  2. Choose one of the incomplete parts. Consider its type signature and functional requirements. Refine the incomplete part by breaking it into subparts or, if simple, defining it directly in terms of Haskell expressions (including calls to the Prelude or other available library functions).

    When refining an incomplete part, consider the various options according to the relevant design criteria (e.g., time, space, generality, understandability, elegance, etc.)

    The refinement of the function may require a refinement of the data being passed. If so, back up in the refinement process and readdress previous design decisions as needed.

    If it not possible to design an appropriate refinement, back up in the refinement process and readdress previous design decisions.

  3. Continue step 2 until all parts are fully defined in terms of Haskell code and data and the resulting set of functions meets all required criteria.

For as long as possible, we should stay with terminology and notation that is close to the problem being solved. We can do this by choosing appropriate function names and signatures and data types. (In later chapters, we examine Haskell's rich set of builtin and user-defined types.)

For stepwise refinement to work well, we must be willing to back up to earlier design decisions when appropriate. We should keep good documentation of the intermediate design steps.

The stepwise refinement method can work well for small programs, but it may not scale well to large, long-lived, general purpose programs. In particular, stepwise refinement may lead to a module structure in which modules are tightly coupled and not robust with respect to changes in requirements. A combination of techniques may be needed to develop larger software systems.

2.6 Using Data Abstraction

A design technique that can help make a program robust with respect to change in the data is to use data abstraction. As in the previous subsection, let's begin with an example.

2.6.1 Rational number arithmetic

For this example, let's implement a group of Haskell functions to perform rational number arithmetic, assuming that the Haskell library does not contain such a data type.

In mathematics we usually write rational numbers in the form $\frac{\texttt{x}}{\texttt{y}}$ where x and y are integers and y $\neq$ 0.

For now, let's assume we have a special type Rat to represent rational numbers and a constructor function

    makeRat :: Int -> Int -> Rat

to create a rational number instance from its numerator x and denominator y. That is, makeRat x y constructs rational number $\frac{\texttt{x}}{\texttt{y}}$.

Further, let us assume we have selector functions numer and denom with signatures

    numer, denom :: Rat -> Int

that each take a Rat argument and return the numerator and denominator, respectively. That is, they satisfy the equalities:

    numer (makeRat x y) == x
    denom (makeRat x y) == y

We consider how to implement rational numbers in Haskell later, but for now let's look at rational arithmetic using the constructor and selector functions above.

Given the knowledge of rational arithmetic from mathematics, we can define the operations for unary negation, addition, subtraction, multiplication, division, and equality.

    negRat :: Rat -> Rat
    negRat x = makeRat (- numer x) (denom x)

    addRat, subRat, mulRat, divRat :: Rat -> Rat -> Rat 
    addRat x y = makeRat (numer x * denom y + numer y * denom x) 
                         (denom x * denom y)   -- x + y
    subRat x y = makeRat (numer x * denom y - numer y * denom x) 
                         (denom x * denom y)   -- x - y
    mulRat x y = makeRat (numer x * numer y)
                         (denom x * denom y)   -- x * y
    divRat x y = makeRat (numer x * denom y)
                         (denom x * numer y)   -- x / y

    eqRat :: Rat -> Rat -> Bool 
    eqRat x y = (numer x) * (denom y) == (numer y) * (denom x)

Above we give the type signatures for all four functions in the same type declaration by listing them separated by commas.

These functions all use the type Rat, constructor function makeRat, and selector functions numer and denom assumed above. They do not depend upon any specific representation for rational numbers.

The above six functions work on rational numbers as a data abstraction defined by the type Rat, constructor function makeRat, and selector functions numer and denom.

The goal of a data abstraction is to separate the logical properties of data from the details of how the data are represented.

2.6.2 Rational number data representation

Now, how can we represent rational numbers?

For this package, we define a type synonym Rat to denote this type:

    type Rat = (Int, Int)

For example, (1,7), (-1,-7), (3,21), and (168,1176) all represent $\frac{\texttt{1}}{\texttt{7}}$.

As with any value that can be expressed in many different ways, it is useful to define a single canonical (or normal) form for representing values in the rational number type Rat.

It is convenient for us to choose a rational number representation (x,y) that satisfies the following property, which we call an invariant:

y > 0, x and y are relatively prime, and zero is denoted uniquely by (0,1).

By relatively prime, we mean that the two integers have no common divisors except 1.

By invariant, we mean that the logical assertion always holds for every rational number created by makeRat and manipulated only by the operations in the RationalCore and Rational modules.

This representation has the advantage that the magnitudes of the numerator x and denominator y are kept small, thus reducing problems with overflow arising during arithmetic operations.

We thus provide a function for constructing rational numbers in this canonical form. We define constructor makeRat as follows.

    makeRat :: Int -> Int -> Rat
    makeRat x 0 = error ( "Cannot construct a rational number "
                           ++ showRat (x,0) ++ "\n" ) 
    makeRat 0 _ = (0,1) 
    makeRat x y = (x' `div` d, y' `div` d) 
        where x' = (signum' y) * x 
              y' = abs' y 
              d  = gcd' x' y'

Above we use features of Haskell we have not used in the previous examples:

The function signum' (similar to the more general function signum in the Prelude) takes an integer and returns the integer -1, 0, or 1 when the number is negative, zero, or positive, respectively.

    signum' :: Int -> Int 
    signum' n | n == 0 =  0 
              | n > 0  =  1 
              | n < 0  = -1

The function abs' (similar to the more general function abs in the Prelude) takes an integer and returns its absolute value.

    abs' :: Int -> Int
    abs' n | n >= 0 =  n
           | n <  0 = -n

The function gcd' (similar to the more general function gcd in the Prelude) takes two integers and returns their greatest common divisor.

    gcd' :: Int -> Int -> Int 
    gcd' x y = gcd'' (abs' x) (abs' y) 
             where gcd'' x 0 = x 
                   gcd'' x y = gcd'' y (x `rem` y)

Prelude operation rem returns the remainder from dividing its first operand by its second.

Given makeRat defined as above, we can define numer and denom as follows:

    numer, denom :: Rat -> Int
    numer (x,_) = x
    denom (_,y) = y

Finally, to allow rational numbers to be displayed in the normal fractional representation, we include function showRat in the package. We use function show, found in the Prelude, here to convert an integer to the usual string format and use the list operator ++ to concatenate the two strings into one.

    showRat :: Rat -> String
    showRat x = show (numer x) ++ "/" ++ show (denom x)

Unlike Rat, makeRat, numer, and denom, function showRat (as implemented) does not use knowledge of the data representation, but it is used by makeRat. We could optimize it slightly by allowing it to access the structure of the tuple directly.

2.6.3 Modularization

There are three groups of functions in this package:

  1. the six public rational arithmetic functions negRat, addRat, subRat, mulRat, divRat, and eqRat

  2. the public type Rat, public constructor function makeRat, public selector functions numer and denom, and string conversion function showRat

  3. the private utility functions called only by the second group, but just reimplementations of Prelude functions anyway

As we have seen, Rat, makeRat, numer, denom, and showRat are the interface to the data abstraction that hides the information about the representation of the data. We can encapsulate this group of functions in a Haskell module as follows. This source code must also be in a file named RationalCore.hs.

    module RationalCore
        (Rat, makeRat, numer, denom, showRat)
    where
        -- Rat, makeRat, numer, denom, showRat definitions

We can encapsulate the utility functions in a separate module, which would enable them to be used by several other modules.

However, given that the only use of the utility functions is within the data representation module, we choose not to separate them at this time. We leave them in the data abstraction module. Of course, we could also eliminate them and use the corresponding Prelude functions directly.

Similarly, negRat, addRat, subRat, mulRat, divRat, and eqRat use the core data abstraction and, in turn, extend the interface to include rational number arithmetic operations. We can encapsulate these in another Haskell module that imports the module giving the data representation. This module must be in a file named Rational.hs.

    module Rational 
      ( Rat, makeRat, numer, denom, showRat, -- from RatioalCore
        negRat, addRat, subRat, mulRat, divRat, eqRat )
    where
      import RationalCore
      -- negRat, addRat, subRat, mulRat, divRat, eqRat definitions

Other modules that use the rational number package can import module Rational.

This modular approach to program design and implementation offers the potential of scalability and robustness with respect to change.

The key to this information-hiding approach to design is to identify the aspects of a software system that are most likely to change from one version to another and make each a design secret of some module.

The secret of the RationalCore module is the rational number data representation used. The secret of the Rational module itself is the methods used for rational number arithmetic.

2.6.4 Alternative rational number data representation

In the rational number data representation above, constructor makeRat creates pairs in which the two integers are relatively prime and the sign is on the numerator. Selector functions numer and denom just return these stored values.

An alternative representation is to reverse this approach, as shown in the following module (in file RationalDeferGCD.hs.)

    module RationalDeferGCD
      (Rat, makeRat, numer, denom, showRat)
    where

    type Rat = (Int,Int)

    makeRat :: Int -> Int -> Rat
    makeRat x 0 = error ( "Cannot construct a rational number "
                           ++ showRat (x,0) ++ "\n" ) 
    makeRat 0 y = (0,1) 
    makeRat x y = (x,y)

    numer :: Rat -> Int
    numer (x,y) = x' `div` d
        where x' = (signum' y) * x 
              y' = abs' y 
              d  = gcd' x' y'

    denom :: Rat -> Int
    denom (x,y) = y' `div` d
        where x' = (signum' y) * x 
              y' = abs' y 
              d  = gcd' x' y'

    showRat :: Rat -> String
    showRat x = show (numer x) ++ "/" ++ show (denom x)

This approach defers the calculation of the greatest common divisor until a selector is called.

The invariant for this rational number representation requires that, for (x,y),

y $\neq$ 0 and zero is represented uniquely by (0,1).

Furthermore, function numer and denom satisfy the equalities

    numer (makeRat x y) == x'
    denom (makeRat x y) == y'

where y' > 0, x' and y' are relatively prime, and $\frac{\texttt{x}}{\texttt{y}} = \frac{\texttt{x'}}{\texttt{y'}}$.

Question:

What are the advantages and disadvantages of the two data representations?

Like module RationalCore, the design secret for module RationalDeferGCD is the rational number data representation.

Regardless of which approach is used, the definitions of the arithmetic and comparison functions do not change. Thus the Rational module can import data representation module RationalCore or RationalDeferGCD.

The figure below shows the dependencies among the modules we have examined in the rational arithmetic case study.

Module Dependency Diagram for Rational Arithmetic Case Study

Module Dependency Diagram for Rational Arithmetic Case Study

We can consider the RationalCore and RationalDeferGCD modules as two concrete instances (Haskell modules) of a more abstract module we call "RationalRep" in the diagram.

The abstract module "Rational" relies on the abstract module "RationalRep" for an implemenation of rational numbers. In the Haskell code above, there are really two versions of the Haskell module Rational that differ only in whether they import RationalCore or RationalDeferGCD.

We could also replace alias Rat by a user-defined type to get another alternative definition of "RationalRep", as long as the interface functions do not have to work with types other than Int.

2.7 Modular Design and Programming

Now let's step back from the rational arithmetic case study and consider the general issues of data abstraction and modular design and programming.

Software engineering pioneer David Parnas defines a module as "a work assignment given to a programmer or group of programmers" [Parnas 1978]. This is a software engineering view of module.

In a programming language like Haskell, a module is also a program unit defined with a construct or convention. This is a programming language view of a module.

Ideally, a language's module features should support the software engineering module methods.

2.7.1 Information-hiding modules

According to Parnas, the goals of modular design are to [Parnas 1972]:

  1. enable programmers to understand the system by focusing on one module at a time (i.e., comprehensibility).

  2. shorten development time by minimizing required communication among groups (i.e., independent development).

  3. make the software system flexible by limiting the `number of modules affected by significant changes (i.e., changeability).

Parnas advocates the use of a principle called information hiding to guide decomposition of a system into appropriate modules (i.e. work assignments). He points out that the connections among the modules should have as few information requirements as possible [Parnas 1972].

In the Parnas approach, an information-hiding module:

Aspects likely to change independently of each other become secrets of separate modules. Aspects unlikely to change can become interactions (connections) among modules.

This approach supports the goal of changeability (goal 2). When care is taken to design the modules as clean abstractions with well-defined and documented interfaces, the approach also supports the goals of independent development (goal 1) and comprehensibility (goal 3).

Information hiding has been absorbed into the dogma of contemporary object-oriented programming. However, information hiding is often oversimplified as merely hiding the data and their representations [Weiss 2001].

The secret of a well-designed module may be much more than that. It may include such knowledge as a specific functional requirement stated in the requirements document, the processing algorithm used, the nature of external devices accessed, or even the presence or absence of other modules or programs in the system [Parnas 1972, 1979, 1985]. These are important aspects that may change as the system evolves.

2.7.2 Interfaces

It is important for information-hiding modules to have well-defined and stable interfaces.

According to Britton et al, an interface is a "set of assumptions ... each programmer needs to make about the other program ... to demonstrate the correctness of his own program" [Britton 1981].

An interface includes the type signature of each function (i.e., name, arguments, and return value) and the constraints on the environment and argument values (e.g., the invariants).

An abstract interface is an interface that does not change when one module implementation is substituted for another [Britton 1981; Parnas 1978]. It concentrates on module's essential aspects and obscures incidental aspects that vary among implementations.

Information-hiding modules and abstract interfaces enable us to design and build software systems with multiple versions. The information-hiding approach seeks to identify aspects of a software design that might change from one version to another and to hide them within independent modules behind well-defined abstract interfaces.

We can reuse the software design across several similar systems. We can reuse an existing module implementation when appropriate. When we need a new implementation, we can create one by following the specification of the module's abstract interface.

2.7.3 Haskell information-hiding modules

As we have seen, in Haskell the module construct can be used to encapsulate an information-hiding module. The features exported form part of the interface to the module. One module can import another module, specifying its dependence on the interface of the other module.

We define each Haskell module in a separate file. The Haskell compiler can compile a module independently of others except that the modules it depends on must be previously compiled. The Haskell build and package management tools cabal-install and stack support Haskell modules as their primary units.

The interface of a Haskell module consists of the names and type signatures of its exported types and functions plus the constraints on the functions and the expected properties of the "objects" manipulated.

In the Rational Arithmetic case study, we defined two information-hiding modules:

  1. "RationalRep", whose secret is how to represent the rational number data and whose interface consists of the data type Rat, operations (functions) makeRat, numer, denom, and showRat, and the constraints on these types and functions

  2. "Rational", whose secret is how to implement the rational number arithmetic and whose interface consists of operations (functions) negRat, addRat, subRat, mulRat, divRat, and eqRat, the other module's interface, and the constraints on these types and functions

We developed two distinct Haskell modules, RationalCore and RationalDeferGCD, to implement the "RationalRep" information-hiding module. We developed one distinct Haskell module, Rational, to implement the "Rational" information-hiding module. Haskell module Rational can be paired (i.e., by chaning the import statement) either of the other two variants of "RationalRep".

Unfortunately, Haskell 2010 has a relatively weak module system that does not support multiple implementations as well as we might like. There is no way to declare that multiple Haskell modules have the same interface other than copying the common code into each module and documenting the interface carefully. We must also have multiple versions of Rational that differ only in which other module is imported.

Together the Glasgow Haskell Compiler (GHC) release 8.2 (July 2017) and the Cabal-Install package manager release 2.0 (August 2017) support a new extension, the Backpack mixin package system. This new system remedies the above shortcoming. In this new approach, we would define the abstract module "RationalRep" as a signature file and require that RationalCore and RationalDeferGCD conform to it.

Further discussion of this new module system is beyond the scope of this chapter.

2.7.4 Invariants

As we saw in the Rational Arithmetic case study, a module that provides a data abstraction must ensure that the objects it creates and manipulates maintain their integrity--always have a valid structure and state. An invariant for the data abstraction can help us design and implement such objects.

Invariant:

A logical assertion that must always true for every "object" created by the public constructors and manipulated only by the public operations of the data abstraction.

Often, we separate an invariant into two parts.

Interface invariant:

An invariant stated in terms of the public features and abstract properties of the "object".

Implementation (representation) invariant:

A detailed invariant giving the required relationships among the internal features of the implementation of an "object"

An interface invariant is a key aspect of the abstract interface of a module. It is useful to the users of the module, as well to the developers.

In the Rational Arithmetic case study, the interface invariant for the "RationalRep" abstract module is the following.

For all integers x and nonzero integers y,

        numer (makeRat x y) == x' 
        denom (makeRat x y) == y'

where y' > 0, x' and y' are relatively prime, $\frac{\texttt{x}}{\texttt{y}} = \frac{\texttt{x'}}{\texttt{y'}}$ and if x' = 0, then y' = 1.

An implementation invariant guides the developers in the design and implementation of the internal details of a module. It relates the internal details to the interface invariant.

We can state an implementation invariant for the RationalCore module as follows.

For all integers x and nonzero integers y,

        makeRat x y == (x',y')

where y' > 0, x' and y' are relatively prime, $\frac{\texttt{x}}{\texttt{y}} = \frac{\texttt{x'}}{\texttt{y'}}$ and if x' = 0, then y' = 1.

The implementation invariant implies the interface invariant. Although makeRat does quite a bit of work, numer and denom are simple.

We can state an implementation invariant for the RationalDeferGCD module as follows.

For all integers x and nonzero integers y,

        makeRat x y == (x,y)

In this module implementation, makeRat is trivial, thus numer and denom must do most of the work to establish the interface invariant.

We will return to the invariant concepts in later chapters.

2.7.5 Design criteria for interfaces

What makes a good interface for an information-hiding module?

In designing an interface for a module, we should also consider the following criteria. Of course, some of these criteria conflict with one another; a designer must carefully balance the criteria to achieve a good interface design.

Note: These are general principles; they are not limited to Haskell or functional programming. In object-oriented languages, these criteria apply to class interfaces.

We must trade off conflicts among the criteria. For example, we must balance:

We must also balance these design criteria against efficiency and functionality.

2.8 Exercises

TODO: Add more exercises for the techniques and features introduced in this section. Make sure what is here still make sense.

For each of the following exercises, develop and test a Haskell function or set of functions.

  1. Develop a Haskell function sumsqbig that takes three numbers as arguments and returns the sum of the squares of the two larger numbers. That is sumsqbig 2.0 1.0 3.0 yields 13.

  2. Develop a Haskell function xor that takes two Booleans and returns the "exclusive-or" of the two values. An exclusive-or operation returns True when exactly one of its arguments is True and returns False otherwise.

  3. Develop a Haskell Boolean function div23n5 such that div23n5 n returns True if and only if n is divisible by 2 or divisible by 3 but not divisible by 5. That is, div23n5 6 yields True and div23n5 30 yields False.

  4. Develop a Haskell function notDiv such that notDiv n d returns True if and only if integer n is not divisible by d. That is, notDiv 10 5 yields False and notDiv 11 5 yields True.

  5. Develop a Haskell function mult that takes two natural numbers (i.e., nonnegative integers) and returns their product. The function must not use the multiplication (*) or division (div) operators. Hint: Multiplication can be done by repeated addition.

  6. Develop a Haskell function addTax that takes two Double values such that addTax c p returns c with a sales tax of p percent added. For example, addTax 2.0 9.0 returns 2.18.

    Also develop a function subTax that is the inverse of addTax. That is, subTax (addTax c p) p yields c.

  7. The time of day can be represented by a tuple (hours,minutes,m) where m indicates either "AM" or "PM". Develop a Boolean Haskell function comesBefore that takes two time-of-day tuples and determines whether the first is an earlier time than the second.

  8. Develop a Haskell function

    minf :: (Int -> Int) -> Int

    such that minf g returns the smallest integer m such that 0 <= m <= 10000000 and g m == 0 (if such an integer exists).

  9. Develop a Haskell module (or modules) for line segments on the two-dimensional coordinate plane using the rectangular coordinate system.

    We can represent a line segment with two points--the starting point and the ending point. Develop the following Haskell functions:

    We normally represent the plane with a rectangular coordinate system. That is, we use two axes--an x axis and a y axis--intersecting at a right angle. We call the intersection point the origin and label it with 0 on both axes. We normally draw the x axis horizontally and label it with increasing numbers to the right and decreasing numbers to the left. We also draw the y axis vertically with increasing numbers upward and decreasing numbers downward. Any point in the plane is uniquely identified by its x-coordinate and y-coordinate.

    Define a data representation for points in the rectangular coordinate system and develop the following Haskell functions:

    Now, using the various constructors and selectors, also develop the Haskell functions for line segments:

    Note that newSeg, startPt, endPt, midPt, and showSeg can be implemented independently from how the points are represented.

  10. Develop a Haskell module (or modules) for line segments that represents points using the polar coordinate system instead of the rectangular coordinate system used in the previous exercise.

    A polar coordinate system represents a point in the plane by its radial coordinate r (i.e., the distance from the pole) and its angular coordinate t (i.e., the angle from the polar axis in the reference direction). We sometimes call r the magnitude and t the angle.

    By convention, we align the rectangular and polar coordinate systems by making the origin the pole, the positive portion of the x axis the polar axis, and let the first quadrant (where both x and y are positive) be the smallest positive angles in the reference direction. That is, with a traditional drawing of the coordinate systems, we measure and the radial coordinate r as the distance from the origin measure the angular coordinate t counterclockwise from the positive x axis.

    Using knowledge of trigonometry, we can convert among rectangular coordinates (x,y) and polar coordinates (r,t) using the equations:

        x = r * cos(t)
    y = r * sin(t)
    r = sqrt(x^2 + y^2)
    t = arctan2(y,x)

    Define a data representation for points in the polar coordinate system and develop the following Haskell functions:

    Functions newSeg, startPt, endPt, midPt, and showSeg should work as in the previous exercise.

  11. Modify the solutions to the previous two line-segment module exercises to enable the line segment functions to be in one module that works properly if composed with either of the two data representation modules. (The solutions may have already done this.)

  12. Modify the solution to the previous line-segment exercise to use the Backpack module system.

  13. Modify the modules in the previous exercise to enable the line segment module to work with both data representations in the same program.

  14. Modify the solution to the Rational Arithmetic case study to use the Backpack module system.

2.9 References

[Abelson-Sussman 1996]
Harold Abelson and Gerald Jay Sussman. Structure and Interpretation of Computer Programs (SICP), Second Edition, MIT Press, 1996.
[Bird-Wadler 1998]
Richard Bird and Philip Wadler. Introduction to Functional Programming, Second Edition, Addison Wesley, 1998. [First Edition, 1988]
[Britton 1981]
K. H. Britton, R. A. Parker, and D. L. Parnas. "A Procedure for Designing Abstract Interfaces for Device Interface Modules," In Proceedings of the 5th International Conference on Software Engineering, pp. 195-204, March 1981.
[Chiusano-Bjarnason 2015]]
Paul Chiusano and Runar Bjarnason, Functional Programming in Scala, Manning, 2015.
[Cunningham 2014]
H. Conrad Cunningham. Notes on Functional Programming with Haskell, 1994-2014.
[Dale-Walker 1996]
Nell Dale and Henry M. Walker. Abstract Data Types: Specifications, Implementations, and Applications, D. C. Heath, 1996. (Especially chapter 1 on "Abstract Specification Techniques")
[Horstmann 1995]
Cay S. Horstmann. Mastering Object-Oriented Design in C++, Wiley, 1995. (Especially chapters 3-6 on "Implementing Classes", "Interfaces", "Object-Oriented Design", and "Invariants")
[Meyer 1997]
Bertrand Meyer. Object-Oriented Program Construction, Second Edition, Prentice Hall, 1997. (Especially chapter 6 on "Abstract Data Types" and chapter 11 on "Design by Contract")
[Mossenbock 1995]
Hanspeter Mossenbock. Object-Oriented Programming in Oberon-2, Springer-Verlag, 1995. (Especially chapter 3 on "Data Abstraction")
[Parnas 1972]
D. L. Parnas. "On the Criteria to Be Used in Decomposing Systems into Modules," Communications of the ACM, Vol. 15, No. 12, pp.1053-1058, 1972.
[Parnas 1976]
D. L. Parnas. "On the Design and Development of Program Families," IEEE Transactions on Software Engineering, Vol. SE-2, No. 1, pp. 1-9, March 1976.
[Parnas 1978]
D. L. Parnas. "Some Software Engineering Principles," Infotech State of the Art Report on Structured Analysis and Design, Infotech International, 10 pages, 1978. Reprinted in Software Fundamentals: Collected Papers by David L. Parnas, D. M. Hoffman and D. M. Weiss, editors, Addison-Wesley, 2001.
[Parnas 1979]
D. L. Parnas. "Designing Software for Ease of Extension and Contraction," IEEE Transactions on Software Engineering, Vol. SE-5, No. 1, pp. 128-138, March 1979.
[Parnas 1985]
D. L. Parnas, P. C. Clements, and D. M. Weiss. "The Modular Structure of Complex Systems," IEEE Transactions on Software Engineering, Vol. SE-11, No. 3, pp. 259-266, March 1985.
[Thompson 2011]
Simon Thompon. Haskell: The Craft of Programming, Third Edition, Pearson, 2011.
[Weiss 2001]
D. M. Weiss. "Introduction: On the Criteria to Be Used in Decomposing Systems into Modules," In Software Fundamentals: Collected Papers by David L. Parnas, D. M. Hoffman and D. M. Weiss, editors, Addison-Wesley, 2001.

2.10 Terms and Concepts

Factorials, function definition and application, recursion, function domains, error, if, guards, basic types (Int, Integer, Bool, Char, functions, tuples, lists, String), type aliases, library (Prelude) functions, REPL, ghci commands and use, procedural abstraction, top-down stepwise refinement, abstract code, termination condition for recursion, Newton's method, Haskell module, module exports and imports, rational number arithmetic, data abstraction, properties of data, data representation, invariant, interface invariant, implementation or representation invariant, canonical or normal forms, information hiding, module secret, encapsulation, interface, abstract interface, design criteria for interfaces, software reuse, Haskell where local definition, type inference, use of Haskell modules to implement information-hiding modules rational number arithmetic, data abstraction, abstract properties of data, data representation, invariant, interface invariant, implementation or representation invariant, canonical or normal forms, information hiding, module secret, encapsulation, interface, abstract interface, design criteria for interfaces, software reuse, Haskell where local definition, type inference, use of Haskell modules to implement information-hiding modules