PROGRAMMING IT IN MIRANDA

This document tells you some ways that you can write programs in Miranda. In the left-hand column are general programming advice, and suggestions about the specifics of writing in Miranda. In the right-hand column you can find examples to illustrate these ideas.

GETTING STARTED

First we need to get a clear idea of the problem. What are the inputs; what are the outputs? What are their types? For example, we may want to find the greatest of three numbers. The inputs are three numbers (num in Miranda), the output is a number.
We have to think about how to use Miranda to represent the types of the problem ; we first look at cases where choices are clear, and revisit types later.
We can then start our definition in Miranda. We have to name the function(s) we are going to write, and give their type(s).

We have to know this information before we start writing our definitions.

In the `greatest of three' example we have the function maxThree of type
num -> num -> num -> num
The type after the last arrow is the type of the result; the others are the types of the arguments (or inputs).
Usually we are not working from scratch; now is the time to review what we know already that might be relevant to the problem:
What functions does the language provide already which might be useful? You can look in the standard environment to find out about these. In the context of the example, max2 is useful as it gives the maximum of two arguments.
Have we solved a simpler problem before? If so, we can perhaps take the definition as a guide, or modify it to work in the new problem. In the book we define maxi to take the maximum of two arguments:
maxi a b = a   , if a>=b
         = b   , otherwise
Have we solved a related problem before? We might already have found the minimum of three numbers; the two problems are close, and we can modify the minimum to maximum.
Can we use a function we have defined already in solving this problem? In our running example, we can in fact use maxi or max2 in defining maxThree:
maxThree a b c = maxi a (maxi b c)
 

DEFINING A FUNCTION

Function definitions in Miranda consist of a number of equations. On the left hand side of each there are patterns, which show to which data each equation applies. Within an equation, there may be multiple right hand sides, representing different cases, and a where clause to hold local definitions. In this section we take the running example of finding the maximum of a list of positive numbers. We can begin by naming it and giving it a type, thus:
maxList :: [num] -> num
An obvious question raised by the specification is what to do about an empty list? Since we have lists of positive numbers, we can signal that a list is empty by returning the result 0 in that case.
We can start by designing the left-hand sides of the equations. Each type has characteristic patterns which are often (but not always) used. In the case of lists we have patterns for an empty and a non-empty list; for natural numbers 0 and (n+1), and so on.
maxList []    = ...
maxList (a:x) = ...
Given the left hand sides we look next at how to work out their right-hand sides. What will help? In the example, we can do the [] case straight away:
maxList []    = 0
For the non-empty list a:x we have to think a bit more…
  • It usually helps to think of examples. These clarify the typical cases, and how the definitions might work.
  • In the example, we might think of
    maxList [4,1,2]
    maxList [2,1,4]
    
    in one case the maximum occurs at the head, in the second it occurs in the tail of the list.
  • Often definitions are recursive: the value at (a:x) is calculated using the value at x, or the value at (n+1) is calculated from the value at n.
  • Here we try to define
    maxList (a:x)
    
    using maxList x

    The problem is that as we saw in the examples above, the result may be maxList x, or it may be a itself, so ...

  • In working out values, we maybe need to divide into cases. These give guards, which follow commas.
  • maxList (a:x) 
      = maxList x    , if maxList x > a
      = a            , otherwise
    
    Can we break down how the value is calculated into a number of smaller calculations?
  • We can use the where clause to make these smaller calculations, for instance.
  • maxList (a:x) = maxL   , if maxL > a
                  = a      , otherwise
                    where
                    maxL = maxList x
    

    MORE COMPLEX DEFINITIONS: BREAKING THE PROBLEM DOWN

    A problem is often solved by breaking it into parts. These parts might be functions which are to be called by other functions, or to be composed together.
  • Function composition is useful in many examples. A task is broken into parts, the inputs being transformed to an intermediate value, then the result is calculated from this value.
  • How many characters in a list of strings?
    charCount :: [string] -> num
    
    First find the length of each string,
    countEach :: [string] -> [num]
    
    then sum the results
    sum :: [num] -> num
    
    giving the definition
    charCount stList
      = sum (countEach stList)
    
    or directly,
    charCount = sum . countEach 
    
  • Built in functions are helpful in suggesting ways of breaking a problem down.
  • We want to count the number of characters in each string in a list, that is apply a function to every member of a list, so
    countEach stList
      = map countString stList
    
    Of course, countString is built in too
      = map (#) stList
    
  • Another way of breaking a problem down is to write the solution using things which then have to be defined themselves in a where clause.
  • If we want to calculate the maximum of three numbers and the number of times that maximum occurs we can write
    maxThreeCount a b c
      = (max,count)
        where
        max   = maxThree a b c
        count = 3  , if a=b=c
              = 2 ...
    
  • The methods suggested here are top down: we work down from the original problem (the top). It can also be useful to work bottom up, writing functions we know we will need in our overall solution.
  • Suppose we are asked to build an index for a document. We will need functions to split the document into lines, words and so on; to order words and entries etc. These can be built and tested separately.
  • Sometimes we have to solve a related problem, in addition to the original one.
  • An example occurs if we are trying to find out whether one string is a substring of another.

    In deciding whether the string st is a substring of (a:x), it will either be a substring of x, or a substring of (a:x) starting at the front: we need a function to decide the latter: frontSubStr.

    subStr st (a:x) 
      =  subStr st x \/
         frontSubStr st (a:x)
    
  • Sometimes we have to generalise a problem, seemingly making it more complicated, in order to get the solution, This happens when trying to write a recursion fails.
  • A good example would be to define [1..n] if it was not already built in. We start by saying
    [1..n] = 1:[2..n]
    
    but where do we go now? We have to define [m..n] instead:
    [m..n] = m:[m+1..n]   , if m<=n
           = []           , otherwise
    

    DESIGNING DATA TYPES

    We need to know the built in types of the language. The base types are num, bool and char. Compound types are tuples (t1,t2,...), lists [t] and function types t1->t2.
    Types can be given names. Type synonyms are given in Miranda thus:
    name == [char]    age == num
    
    The types can be combined to give representations of many more complex objects. A person might be represented by their name and age
    person == (name,age)
    
    In a functional programming language, functions can be thought of as arguments and results of other functions. map takes a function which is to be applied to every element of a list.

    filter takes a property, which is represented as a function taking an element to a bool, as well as the list to be filtered.

    If a type contains different kinds of object, then we might well use an algebraic type to represent it. A simple example is of geometrical shapes on a two-dimensional grid.
  • First we name the type and think of the different kinds of object which it contains.
  • The type will be shape, and will contain circles, lines and triangles.
    shape ::= Circle ... |
              Line ... |
              Triangle ...
    
  • Next we have to think of the components of the different kinds of object. This completes the definition.

    This process works equally well for recursive types like trees.

  • Points on the grid will be represented by point (to be defined). A circle is given by its radius and centre, a line by its end points and a triangle by its three corners:
    shape ::= Circle num point |
              Line point point |
              Triangle point point point
    
    © Simon Thompson, 1996