Specification of Reactive Hardware/Software Systems - Electronic ...

7.4 Mathematical Preliminaries 237
an image in W is called the domain of f . It is denoted by Dom(f ). The collection of all
elements in W that are image of some element in V is called the range of f . The range of
f is denoted by Rng(f ).
Undefinedness
Not all elements of V have necessarily an image in W under f . If an element v has no
image, i.e. if ¡ ¡ v Dom(f ), we often write f (v) undef, denoting that f (v) is ’undefined’.
v¥ v¡ ¡ ¡
Variant Notation
Let f be a partial function from V to W, let V and w W. then f £
© v¦ w equals
function f , except in domain element v, there it is w. f £
© v¦ w is defined by
f £
w © v¦ (v¡ )
¡
¡ £
¢
f (v¡ ) if v¡ ¡
v and v¡ ¡ Dom(f )
w if v¡ v
undef if v¡ ¡
v and v¡ ¡
Dom(f ) ¡
The notation is called the variant notation for functions. In the context of defining a
language semantics, it appears to be a very convenient notation.
Restriction
Another convenient notation for functions is the restriction operator ¤ . Let f be a partial
function from V to W and let Z ¥ V be a subset of V. Then f ¤ Z denotes the partial
function from V to W that is the same as f except that its domain is restricted to Z. Let
v ¡ V. Then f ¤ Z is defined by
£ § f Z
undef if ¡ ¡ v Dom ¢
£ § f Z
f ¤ Z(v) ¦¥ f (v) if v ¡ Dom ¢
£ w1© ¥¡ ¡ ¡ ¥ ©
Complete Functions
A special kind of partial function is a complete function. If f is a complete function from
V to W, then each element in V has an image in W. This can also be expressed as
Dom(f ) V. To denote that f is a complete function from V to W we will use the
notation f : V W. We will often write v1 wn vn to denote a function f for
which f ¢
§ vi£ § wi (1 i n) and f ¢
v£ v otherwise.
7.4.5 Inductive Definitions and Proofs by Induction
Inductive Definitions
The main mathematical tool that we will use is induction. Many relations, functions
and sets will be defined in an inductive way. There exist many notations for inductive
definitions. In principle, however, they all boil down to a single notion. Basically, an
inductive definition consists of a number of inference rules. An inference rule is of the
form
premises
conclusion
if condition