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

236 Introduction to the Semantics of POOSL
n-tuples v1¥¡ ¡ ¡ ¥ vn with vi ¡ Vi for i 1¥¡ ¡ ¡ ¥ n. If w is an n-tuple, then the i-th element
of w is denoted by w(i), so w w(1)¥¡ ¡ ¡ ¥ w(n) .
Lists
An n-tuple is a list of fixed length. Sometimes it is convenient to denote lists of arbitrary
length. For set V and natural number n we will let Vn denote the n-fold cartesian product
of V. Note that V1 denotes the collection of all 1-tuples of the form ¡ v with v V. For
n 0 we define V0 to consist of the empty list only. The empty list is denoted by
, so
V0 £
¦ £
¨
. To denote the collection of all lists with entries from V we use the notation
V . More formally, V Vn ¡ ¡ ¡ ¡ ¦ n . The concatenation of lists v V and w W is
again a list and is denoted by v w. V W denotes the set £
¥ ¡ ¡ ¦ ¡ v w v V w W . To
indicate the length of a list v, i.e. the amount of entries of v, we write ¡ v ¡ .
7.4.3 Binary Relations
¥
¡ ¡
Binary Relations
A binary relation R on two sets V and W is a subset of V W, so R V W. If two
elements v V and w W are related under R, we can write v¥ w ¡ R. Often, however,
we will use an infix notation and write v R w instead of v¥ w ¡ R. In case infix notations
are used, relations are often denoted by symbols such as ¡ , ¢
£ ¤
relation on V and V is simply called a binary relation on V.
, , and . A binary
Relation Composition
Binary relations can be composed into a new relation. Let R be a relation on V and W
and let S be a relation on W and Z. Then the relation composition R ¥ S is a relation on V
and Z. For v ¡ V and z ¡ Z, v¥ z ¡ R ¥ S if and only if there exists a w ¡ W such that
v¥ w ¡ R and w¥ z ¡ S. Instead of R ¥ S we will often simply write RS.
Reflective Transitive Closures
It is often convenient to apply a relation zero or more times. Let R be a binary relation
on V and let n be a natural number. We define R n to be the n-fold composition of relation
R. For n 1, R 1 R. For n 0, R 0 is the identity relation Id on V. It is defined as follows.
v¥ ¡ For each w V, v¥ £
w ¡
¨
Id if and only if v w. To denote that R can be composed an
arbitrary amount of times, we use the notation R . More formally, R Rn ¦ ¡ ¡ n .
R is called the reflexive transitive closure of relation R.
7.4.4 Functions
Partial Functions
A partial function f from V to W is a binary relation on sets V and W such that for each
element v ¡ V there exists at most one element w ¡ W with v¥ w ¡ f . Instead of v¥ w ¡ f
we will in general write f (v) w. The object f (v) is called the image of v under f .
Domains and Ranges
To denote that f is a partial function from V to W we write f : V ¦£ W. V is called the
definition domain and W the co-domain of f . The collection of all elements in V that have