Sequences

1. Sequences

2. What is sequence? • A sequence is an ordered collection of objects. • We use sequences to model collections in which order or multiplicity is important. • If we do not wish to impose an order, then we may choose to use bags instead. • A sequence of values of type X is a partial function from natural numbers N to X whose domain is just 1..n, where n is the length of the sequence. • Formally we can define the set of all sequences of values of type X as follows: seq X == { f : N ↛X | (n : N • dom f = 1..n)}

3. Since seq X is a set of functions, the typeofseq X must be ℙ (ℙ(ℤ  X)) • The type of any sequence of values of type X is ℙ(ℤ  X) • The following notation denote a sequence: {1↦u, 2 ↦s, 3 ↦i, 4 ↦n, 5 ↦g, 6 ↦z} ‹u,s,i,n,g,z› ‹› empty sequence

4. If we let s denotes the above sequence, then the following statements are well-formed and true: dom s = 1..6 ran s = {u,s,i,n,g,z} #s = 4 s 4 = n

5. Sequence notation

6. Sequence notation

7. Sequence notation

8. Sequence notation • last s = s #s • front s = {#s} s

9. Filter • If s is a sequence, then s ↾ A is the largest subsequence of s containing only those objects that are elements of A. <a, b, c, d, e, d, c, b, a> ↾ {a, b} = <a, b, b, a>

10. Question

12. Sequences as function

13. Concatenation

15. Bag • A bag is an unordered collection of objects in which multiplicities are important: [[ ]] [[a, b, c, a, b, c]]