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## 0.6 Data Structures in Axiom

This chapter is an overview of some of the data structures provided by Axiom.

### 0.6.1 Lists

The Axiom List type constructor is used to create homogenous lists of finite size. The notation for lists and the names of the functions that operate over them are similar to those found in functional languages such as ML.

Lists can be created by placing a comma separated list of values inside square brackets or if a list with just one element is desired then the function list is available:

[4]

[4]

Type: List PositiveInteger
list(4)

[4]

Type: List PositiveInteger
[1,2,3,5,7,11]

[1, 2, 3, 5, 7, 11]

Type: List PositiveInteger

The function append takes two lists as arguments and returns the list consisting of the second argument appended to the first. A single element can be added to the front of a list using cons:

append([1,2,3,5],[7,11])

[1, 2, 3, 5, 7, 11]

Type: List PositiveInteger
cons(23,[65,42,19])

[23, 65, 42, 19]

Type: List PositiveInteger

Lists are accessed sequentially so if Axiom is asked for the value of the twentieth element in the list it will move from the start of the list over nineteen elements before it reaches the desired element. Each element of a list is stored as a node consisting of the value of the element and a pointer to the rest of the list. As a result the two main operations on a list are called first and rest. Both of these functions take a second optional argument which specifies the length of the first part of the list:

first([1,5,6,2,3])

1

Type: PositiveInteger
first([1,5,6,2,3],2)

[1, 5]

Type: List PositiveInteger
rest([1,5,6,2,3])

[5, 6, 2, 3]

Type: List PositiveInteger
rest([1,5,6,2,3],2)

[6, 2, 3]

Type: List PositiveInteger

Other functions are empty? which tests to see if a list contains no elements, member? which tests to see if the first argument is a member of the second, reverse which reverses the order of the list, sort which sorts a list, and removeDuplicates which removes any duplicates. The length of a list can be obtained using the `` #'' operator.

empty?([7,2,-1,2])

false

Type: Boolean
member?(-1,[7,2,-1,2])

true

Type: Boolean
reverse([7,2,-1,2])

[2,-1, 2, 7]

Type: List Integer
sort([7,2,-1,2])

[-1, 2, 2, 7]

Type: List Integer
removeDuplicates([1,5,3,5,1,1,2])

[1, 5, 3, 2]

Type: List PositiveInteger
#[7,2,-1,2]

4

Type: PositiveInteger

Lists in Axiom are mutable and so their contents (the elements and the links) can be modified in place. Functions that operate over lists in this way have names ending in the symbol ``!''. For example, concat! takes two lists as arguments and appends the second argument to the first (except when the first argument is an empty list) and setrest! changes the link emanating from the first argument to point to the second argument:

u := [9,2,4,7]

[9, 2, 4, 7]

Type: List PositiveInteger
concat!(u,[1,5,42]); u

[9, 2, 4, 7, 1, 5, 42]

Type: List PositiveInteger
endOfu := rest(u,4)

[1, 5, 42]

Type: List PositiveInteger
partOfu := rest(u,2)

[4, 7, 1, 5, 42]

Type: List PositiveInteger
setrest!(endOfu,partOfu); u

[9, 2, 4, 7, 1‾]

Type: List PositiveInteger

From this it can be seen that the lists returned by first and rest are pointers to the original list and not a copy. Thus great care must be taken when dealing with lists in Axiom.

Although the nth element of the list l can be obtained by applying the first function to n-1 applications of rest to l, Axiom provides a more useful access method in the form of the ``.'' operator:

u.3

4

Type: PositiveInteger
u.5

1

Type: PositiveInteger
u.6

4

Type: PositiveInteger
first rest rest u -- Same as u.3

4

Type: PositiveInteger
u.first

9

Type: PositiveInteger
u(3)

4

Type: PositiveInteger

The operation u.i is referred to as indexing into u or elting into u. The latter term comes from the elt function which is used to extract elements (the first element of the list is at index 1).

elt(u,4)

7

Type: PositiveInteger

If a list has no cycles then any attempt to access an element beyond the end of the list will generate an error. However, in the example above there was a cycle starting at the third element so the access to the sixth element wrapped around to give the third element. Since lists are mutable it is possible to modify elements directly:

u.3 := 42; u

[9, 2, 42, 7, 1‾]

Type: List PositiveInteger

Other list operations are:

L := [9,3,4,7]; #L

4

Type: PositiveInteger
last(L)

7

Type: PositiveInteger
L.last

7

Type: PositiveInteger
L.( #L - 1)

4

Type: PositiveInteger

Note that using the `` #'' operator on a list with cycles causes Axiom to enter an infinite loop.

Note that any operation on a list L that returns a list L L′ will, in general, be such that any changes to L L′ will have the side-effect of altering L. For example:

m := rest(L,2)

[4, 7]

Type: List PositiveInteger
m.1 := 20; L

[9, 3, 20, 7]

Type: List PositiveInteger
n := L

[9, 3, 20, 7]

Type: List PositiveInteger
n.2 := 99; L

[9, 99, 20, 7]

Type: List PositiveInteger
n

[9, 99, 20, 7]

Type: List PositiveInteger

Thus the only save way of copying lists is to copy each element from one to another and not use the assignment operator:

p := [i for i in n] -- Same as `p := copy(n)'

[9, 99, 20, 7]

Type: List PositiveInteger
p.2 := 5; p

[9, 5, 20, 7]

Type: List PositiveInteger
n

[9, 99, 20, 7]

Type: List PositiveInteger

In the previous example a new way of constructing lists was given. This is a powerful method which gives the reader more information about the contents of the list than before and which is extremely flexible. The example

[i for i in 1..10]

[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Type: List PositiveInteger

should be read as

``Using the expression i, generate each element of the list by iterating the symbol i over the range of integers [1,10]''

To generate the list of the squares of the first ten elements we just use:

[i**2 for i in 1..10]

[1, 4, 9, 16, 25, 36, 49, 64, 81, 100]

Type: List PositiveInteger

For more complex lists we can apply a condition to the elements that are to be placed into the list to obtain a list of even numbers between 0 and 11:

[i for i in 1..10 | even?(i)]

[2, 4, 6, 8, 10]

Type: List PositiveInteger

This example should be read as:

``Using the expression i, generate each element of the list by iterating the symbol i over the range of integers [1,10] such that i is even''

The following achieves the same result:

[i for i in 2..10 by 2]

[2, 4, 6, 8, 10]

Type: List PositiveInteger

### 0.6.2 Segmented Lists

A segmented list is one in which some of the elements are ranges of values. The expand function converts lists of this type into ordinary lists:

[1..10]

[1..10]

Type: List Segment PositiveInteger
[1..3,5,6,8..10]

[1..3, 5..5, 6..6, 8..10]

Type: List Segment PositiveInteger
expand(%)

[1, 2, 3, 5, 6, 8, 9, 10]

Type: List Integer

If the upper bound of a segment is omitted then a different type of segmented list is obtained and expanding it will produce a stream (which will be considered in the next section):

[1..]

[1..]

Type: List UniversalSegment PositiveInteger
expand(%)

[1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…]

Type: Stream Integer

### 0.6.3 Streams

Streams are infinite lists which have the ability to calculate the next element should it be required. For example, a stream of positive integers and a list of prime numbers can be generated by:

[i for i in 1..]

[1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…]

Type: Stream PositiveInteger
[i for i in 1.. | prime?(i)]

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29,…]

Type: Stream PositiveInteger

In each case the first few elements of the stream are calculated for display purposes but the rest of the stream remains unevaluated. The value of items in a stream are only calculated when they are needed which gives rise to their alternative name of ``lazy lists''.

Another method of creating streams is to use the generate(f,a) function. This applies its first argument repeatedly onto its second to produce the stream [a, f(a),f(f(a)),f(f(f(a)))…]. Given that the function nextPrime returns the lowest prime number greater than its argument we can generate a stream of primes as follows:

generate(nextPrime,2)$Stream Integer

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29,…]

Type: Stream Integer

As a longer example a stream of Fibonacci numbers will be computed. The Fibonacci numbers start at 1 and each following number is the addition of the two numbers that precede it so the Fibonacci sequence is: 1, 1, 2, 3, 5, 8,….

Since the generation of any Fibonacci number only relies on knowing the previous two numbers we can look at the series through a window of two elements. To create the series the window is placed at the start over the values [1, 1] and their sum obtained. The window is now shifted to the right by one position and the sum placed into the empty slot of the window; the process is then repeated. To implement this we require a function that takes a list of two elements (the current view of the window), adds them, and outputs the new window. The result is the function [a, b]→b, a+b]:

win : List Integer -> List Integer
Type: Void
win(x) == [x.2, x.1 + x.2]
Type: Void
win([1,1])

[1, 2]

Type: List Integer
win(%)

[2, 3]

Type: List Integer

Thus it can be seen that repeatedly applying win to the results of the previous invocation each element of the series is obtained. Clearly win is an ideal function to construct streams using the generate function:

fibs := [generate(win,[1,1])]

[[1, 1],[1, 2],[2, 3],[3, 5],[5, 8],[8, 13],[13, 21],[21, 34],[34, 55],[55, 89],…]

Type: Stream List Integer

This isn't quite what is wanted -- we need to extract the first element of each list and place that in our series:

fibs := [i.1 for i in [generate(win,[1,1])] ]

[1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…]

Type: Stream Integer

Obtaining the 200th Fibonacci number is trivial:

fibs.200

280571172992510140037611932413038677189525

Type: PositiveInteger

One other function of interest is complete which expands a finite stream derived from an infinite one (and thus was still stored as an infinite stream) to form a finite stream.

### 0.6.4 Arrays, Vectors, Strings, and Bits

The simplest array data structure is the one-dimensional array which can be obtained by applying the oneDimensionalArray function to a list:

oneDimensionalArray([7,2,5,4,1,9])

[7, 2, 5, 4, 1, 9]

Type: OneDimensionalArray PositiveInteger

One-dimensional arrays are homogenous (all elements must have the same type) and mutable (elements can be changed) like lists but unlike lists they are constant in size and have uniform access times (it is just as quick to read the last element of a one-dimensional array as it is to read the first; this is not true for lists).

Since these arrays are mutable all the warnings that apply to lists apply to arrays. That is, it is possible to modify an element in a copy of an array and change the original:

x := oneDimensionalArray([7,2,5,4,1,9])

[7, 2, 5, 4, 1, 9]

Type: OneDimensionalArray PositiveInteger
y := x

[7, 2, 5, 4, 1, 9]

Type: OneDimensionalArray PositiveInteger
y.3 := 20 ; x

[7, 2, 20, 4, 1, 9]

Type: OneDimensionalArray PositiveInteger

Note that because these arrays are of fixed size the concat! function cannot be applied to them without generating an error. If arrays of this type are required use the FlexibleArray constructor.

One-dimensional arrays can be created using new which specifies the size of the array and the initial value for each of the elements. Other operations that can be applied to one-dimensional arrays are map! which applies a mapping onto each element, swap! which swaps two elements and copyInto!(a,b,c) which copies the array b onto a starting at position c.

a : ARRAY1 PositiveInteger := new(10,3)

[3, 3, 3, 3, 3, 3, 3, 3, 3, 3]

Type: OneDimensionalArray PositiveInteger

(note that ARRAY1 is an abbreviation for the type OneDimensionalArray.) Other types based on one-dimensional arrays are Vector, String, and Bits.

map!(i +-> i+1,a); a

[4, 4, 4, 4, 4, 4, 4, 4, 4, 4]

Type: OneDimensionalArray PositiveInteger
b := oneDimensionalArray([2,3,4,5,6])

[2, 3, 4, 5, 6]

Type: OneDimensionalArray PositiveInteger
swap!(b,2,3); b

[2, 4, 3, 5, 6]

Type: OneDimensionalArray PositiveInteger
copyInto!(a,b,3)

[4, 4, 2, 4, 3, 5, 6, 4, 4, 4]

Type: OneDimensionalArray PositiveInteger
a

[4, 4, 2, 4, 3, 5, 6, 4, 4, 4]

Type: OneDimensionalArray PositiveInteger
vector([1/2,1/3,1/14])

[1 2, 1 3, 1 14]

Type: Vector Fraction Integer
"Hello, World"

"Hello,World"

Type: String
bits(8,true)

"11111111"

Type: Bits

A vector is similar to a one-dimensional array except that if its components belong to a ring then arithmetic operations are provided.

### 0.6.5 Flexible Arrays

Flexible arrays are designed to provide the efficiency of one-dimensional arrays while retaining the flexibility of lists. They are implemented by allocating a fixed block of storage for the array. If the array needs to be expanded then a larger block of storage is allocated and the contents of the old block are copied into the new one.

There are several operations that can be applied to this type, most of which modify the array in place. As a result these functions all have names ending in ``!''. The physicalLength returns the actual length of the array as stored in memory while the physicalLength! allows this value to be changed by the user.

f : FARRAY INT := new(6,1)

[1, 1, 1, 1, 1, 1]

Type: FlexibleArray Integer
f.1:=4; f.2:=3 ; f.3:=8 ; f.5:=2 ; f

[4, 3, 8, 1, 2, 1]

Type: FlexibleArray Integer
insert!(42,f,3); f

[4, 3, 42, 8, 1, 2, 1]

Type: FlexibleArray Integer
insert!(28,f,8); f

[4, 3, 42, 8, 1, 2, 1, 28]

Type: FlexibleArray Integer
removeDuplicates!(f)

[4, 3, 42, 8, 1, 2, 28]

Type: FlexibleArray Integer
delete!(f,5)

[4, 3, 42, 8, 2, 28]

Type: FlexibleArray Integer
g:=f(3..5)

[42, 8, 2]

Type: FlexibleArray Integer
g.2:=7; f

[4, 3, 42, 8, 2, 28]

Type: FlexibleArray Integer
insert!(g,f,1)

[42, 7, 2, 4, 3, 42, 8, 2, 28]

Type: FlexibleArray Integer
physicalLength(f)

10

Type: PositiveInteger
physicalLength!(f,20)

[42, 7, 2, 4, 3, 42, 8, 2, 28]

Type: FlexibleArray Integer
merge!(sort!(f),sort!(g))

[2, 2, 2, 3, 4, 7, 7, 8, 28, 42, 42, 42]

Type: FlexibleArray Integer
shrinkable(false)$FlexibleArray(Integer)

true

Type: Boolean

There are several things to point out concerning these examples. First, although flexible arrays are mutable, making copies of these arrays creates separate entities. This can be seen by the fact that the modification of element g.2 above did not alter f. Second, the merge! function can take an extra argument before the two arrays are merged. The argument is a comparison function and defaults to ``<='' if omitted. Lastly, shrinkable tells the system whether or not to let flexible arrays contract when elements are deleted from them. An explicit package reference must be given as in the example above.

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