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of our topic, whereas algorithm and programemphasizes the logical and math-
ematical side.
2.1.5 Alphabets, Strings and Numbers
Numbers and strings of symbols are fundamental to computer science. Just
as a computation can be thought of as the calculation of numerical values of
a function, it can as well be regarded as the processing of strings of symbols.
The equivalence follows from the fact that any finite set of finite strings of
symbolsdrawnfromafinitealphabetcanbeputinaone-to-onecorrespondence
4This is necessary in order to implement if < condition >then < statement > else
<statement>,andwhile<condition>do<statement> programmingprimitives.
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with a subset of the natural numbers. Any enumeration of the strings in some
lexicographicorderwilldo. Wewillmaketheseconceptssomewhatmoreprecise.
Numbers
Byanumberwemean,unlessotherwisestated,anaturalnumber,i.e. amember
of the infinite set N = 0,1,2,... . This set can be defined inductively start-
{ }
ing from the number 0 and adding, in a step by step fashion, the successors.
Informally,
0 is a natural number,
(cid:26)If n is a natural number, then the succesor n+1 is a natural number.
This is not really a gooddefinition, since in writing n+1 for the successorof n
weareinfactpresupposingthenumberstogetherwithaddition. Butitcaptures
the idea behind the following more formal definition.
The set N of natural numbers are defined by the clauses
0 N
∈ (2.1)
(cid:26)n N S(n) N
∈ β‡’ ∈
where S(n) denotes the successor of n.
Arithmetical operations, like addition and multiplication can be defined on
this basis [14].
By Nd we denote the set of all d-tuples (n ,n ,...,n ) of numbers.
1 2 d
Strings and languages
The concept of a symbol, or a token, will be taken to be intuitively given and
not further analyzed. An alphabet is non-empty a set of symbols, generically
denoted by Ξ£. A string (over an alphabet) results when the symbols taken
from an alphabet are written consecutively. The order in which the symbols
are written within the string matters. There must be no blanks or commas or
other separators between the symbols in a string, instead, if blanks or other
separatorsare needed they should be included among the symbols. A language
is a set of strings. The following notation is useful to have at hand.
The set of all strings over the alphabet Ξ£ is denoted by Ξ£ and it includes
βˆ—
the empty string . An equivalent term for string is word. Ξ£+ denotes the set
E
of strings with the empty string excluded. In some contexts, strings will be
enclosed by ” ”, as is prevalent in computer programming languages. Now, a
language is an arbitrary subset of Ξ£ . This is all we will need from the general
βˆ—
theory of formal languages (cf. [15]).
Bit strings
One particular important kind of strings are the bit strings. They are based
on the alphabet 0,1 . A bit string is any combination of the symbols 0 and 1
{ }
written without any blank separators. By the notation 0,1 k we mean a bit
{ }
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string of length k which will also be more explicitly denoted by ”b b b ”.
1 2 k
Β·Β·Β·
Such a string is naturally interpreted as a k-bit binary number. Then the
corresponding base-10 representation of the binary number can be used as a
shorthand for the string, as in the example ”0101”=5.
2.1.6 Functions
The concept of a function is supposed to be well-known, but we record the
basic definition here for completeness and fixing our notation. A function can
be regarded as a mapping between two sets X and Y, taking an element x in
X and mapping it into a unique element y in Y. This is formally written as
f :X Y. When we want to focus on generic, or particular elements that are
β†’
connected by the mapping, we write y =f(x) or b=f(a). The element b in Y
is called the image (under the mapping f) of the element x in X.
A more formal definition of functions is based on the concept of a relation.
A relation is a subset of all ordered pairs (x,y) where x X and x Y. A
∈ ∈
function f is a special kind of relation for which we require: if (x,y) f and
∈
(x,z) f, then y = z. This condition expresses the uniqueness requirement,
∈
that for each x in X there is a unique y in Y. This is the only restriction on a
function, and consequently, the concept of a function is very general indeed.5
IfthefunctionisdefinedforallelementsinX,itisatotalfunction,otherwise
it is a partial function. The elements of X for which the function is defined is
calledthedomain ofdefinition. TheelementsofY whichareimagesofelements
of X are called the range.
We willalmostexclusivelyconsiderfunctions fromNd to N,from 0,1 k to
{ }
0,1 l, or from Ξ£ to Ξ£ .
βˆ— βˆ—
{ }
Enumerations
A function is said to be one-to-one if the image of two different elements in the
domainaredifferent. IfeveryelementinthesetY isthe imageofanelementin
the set X (i.e. the range of the function is the whole set Y) then the function
is said to be onto.
Furthermore, a total function which is both one-to-one and onto is called a
one-to-onecorrespondence. Suchfunctionsareusefulforcomparingthenumbers
of elements in two sets, as they as the name suggests, sets up a one-to-one