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Chapter No. 9

Sequences

Introduction

• It is sometimes necessary to record the order in which objects are arranged, e.g., - Data may be indexed by an ordered collection of keys - Messages may be stored in order of arrival - Tasks may be performed in order of importance.
• Now, we introduce a notion of a sequence which is an ordered collection of objects

Introduction

• Sequence: A sequence is an ordered collection of objects.
• Empty Sequence : If there are no objects in the collection, the sequence is the empty sequence, and is written  .
• Otherwise, the sequence is written as a list of objects between angle brackets, e.g.  a, b, c  denotes the sequence containing objects a, b, and c, in that order.

Sequence Notation

• A useful way of composing sequences is concatenation
• In concatenation two sequences are combined in such a way that the elements of one follow elements of the other, order is maintained.
• If s and t are sequences, we write s  t to denote the concatenation of s and t, e.g.,  a, b, c    d, e  =  a, b, c, d, e 

Example: Destination Board at Railway Station

Time From To 10:15 Oxford London Paddington 10:38 London Paddington Edinburgh 10:40 Great Malvern London Paddington 11:15 Manchester Poole 11:20 Oxford Reading 11:40 London Paddington Manchester

Modeling using Sequence

Model of Above Destination Board

The destination board displays a list of trains, arranged in order of departure. Model it as a sequence of pairs, each recording a time and a destination

trains =  (10 :15, london), (10:38, edinburgh), (10:40, london), (11:15, birmingham), (11:20, reading), (11:40, london) 

Head and Tail Operators

head  a, b, c, d, e  = a tail  a, b, c, d, e  =  b, c, d, e 

it is to be noted that

s =  head s   tail s

Neither of these operators is defined upon the empty sequence.

Example 9.3: Head and Tail Operators

Sally wants to take the first train to London. From the list of trains on the destination board, she knows that this is the 10:15, head (trains  { t : Time  (t, london)}) = (10 :15, london) If we assume that she will not reach in time, then the list of available trains is given by tail trains =  (10:38, edinburgh), (10:40, london), (11:15, birmingham), (11:20, reading), (11:40, london) 

A Model of Sequences

• If X is a set, then the set of all finite sequences of objects from X is defined by the following abbreviation: seq X == {s :   X |  n :   dom s = 1..n}
• This definition makes explicit an assumption about sequences, that every element of a given sequence must share the same type.
• The expression  1, (1, 2)  does not make any sense to us

Formalizing Concatenation Operator

• If a function takes more than one argument, then it may be defined as an infix symbol,
• We use underscores to indicate its position

[ X ] _  _ : seq X  seq X  seq X

 s, t : seq X 

( s  t) = # s + # t

 i :1.. # s  (s  t) i = s i  j : 1.. # t  (s  t ) (# s + j )= t j

Formalizing Tail Operator

On the other hand, tail operator requires a translation. Tail operator is also partial

[ X ]

_ tail _ : seq X  seq X

 s : seq X | s    

tail s = # s -

 i: 1.. #s -1  (tail s) i = s (i+ 1)

Non-empty and Injective Sequences

• we give a name to the set of all non-empty sequences over X as well, seq1 X == {s : seq X | s   }
• “head” and „tail‟ are total when defined upon this
• Another special set of sequences is set of all injective sequences i.e. sequences in which no element appears more than once. We write iseq X to denote set of all injective sequences over set X, iseq X == {s : seq X | s    X}
• Such sequences are used to represent ordered collections of distinct objects.

Bags

• A sequence stores information about the multiplicity and ordering of its elements.
• For example, in sequence a, b, c, a, b, c we can see that there are exactly two occurrences of a, and that these occupy the first and fourth positions in the sequence.
• Sometimes this is more information than we need, and hence we use bag

Bags

• Suppose that only the number of occurrences of a is important.
• If this is the case, then the sequence above contains more detail than is necessary
• It is not a fully abstract representation.
• The set {a, b, c}, on the other hand, is not an adequate representation
• It records that a is present, but does not record how many times it occurs.