Records/tuples/structures/object_states are shown using the notation of Ada. Here is a record with fields a,b,... and values xa,xb, respectively:
(a=>xa, b=>xb, ...).The set of all objects with fields a,b,... of type A,B, .. (respectively) is shown like this:
$ Net{ a:A, b:B, ... }.Usually however it pays to give a name to the abstract structure
RECORD::=Net{ a:A, b:B, ... }.This allows the structure itself to be referred to, extendded, used, inherritted in the definitions of other structures. The dollar sign is put infront of the name to indicate the set of object that fit the named structure:
$ $RECORD
Example
Class or Set:
phones::=$ Net{name:Strings, number:Strings}.
Object in phones:
(name=>"John Smith", number=>"714-123-5678") in phones.
Formally such a set of structures defines (and is defined by) a collection of functions: one for each component. If X=$ Net{a:A,b:B,c:C,... } then
a in X -> A,
b in X -> B,
c in X -> C,and if x:X then
a(x)=x.a=`the a of x`,
b(x)=x.b=`the b of x`,
c(x)=x.c=`the c of x`.
Example - Phone numbers
(name=>"John Smith", number=>"714-123-5678").name ="John Smith"
(name=>"John Smith", number=>"714-123-5678").number ="714-123-5678"
or
name(name=>"John Smith", number=>"714-123-5678") ="John Smith"
number(name=>"John Smith",number=>"714-123-5678") ="714-123-5678"
UML
In UML terms the field names (a,b,c,...) are role names in the
class X which would be linked to classes representing A,B,C, ...
The link normally has cardinality (0..*) at the X end.
The link must have cardinallity 1 at the other end.
The reverse role is not (usually) named in MATHS. There is a slight
abuse of notation in that the name of the role is also
used as the name of the relationship.
At the conceptual level it is best to not indicate whether the links are aggregations, generalisations, etc. These physical properties are not shown in the simpler MATHS model.
The symbols
(M)|-are short for "Because of the definition of M the following statemnt is true".
This is the theoretical justification for
It is convenient to give a name to a collection of
declarations, predicates and definitions. There are two forms -
short and long:
Short form
S::=Net{ a:A, b:B, c:C, ..., P, Q, ..., x:=e,... }
S::=following
.Net
a::A,
b::B,
c::C,
...
|-P,
|-Q,
...,
x::=e,
...
.Close.Net
Comments and theorems can also be included in the long form.
(reasons)|- Theorem.Anything that is not recognised as a declaraction, definition, assumption(axiom), formula, or theorem is a comment.
Either-Or
There is another way to make structures from components - the discriminated
union.
U::=\/{ a:A, b:B, c:C,.... z:Z}means that for any x:A, (a=>x) in U, for any x in B, (b=>x) in U, and so on. Further, if x:U then
type(x) = the Tag t such that (t=>x.t)=x.Also if S is one of the sets A,B,C,....Z and s:S then
s.U = ( t=>s) in U for the 'tag' t.
Inheritance
There is a simple notation when we want to say that one set
is a subset of another set, but with certain extra properties:
Extended_set::= Super_set with{ Extra_component and properties}
Given a structure S=$ N, where
N=Net{a:A, b:B, c:C, ..., P, Q, R, x:=e,...}it is not always necessary to specify all the components to determine an unique object. For example if N=Net{ a,b:Real, a=2*b+1} then there is exactly one object in $ N which has a=3, namely the one with b=1. So
(a=>3, b=>1) = the n:$ $N( n.a=3 ).A shorthand notation can be used:
the N(a=3) = (a=>3, b=>1).In general
the N(W) = the $(N and W) = the N with(W).whenever there is exactly one object in $(N and W).
Constrained Structures are Subsets of Type
MATHS lets you include constraints in a structure definition. For example the mathematical structure of a point in a plane as something that has both Cartesian(x,y) and Polar(arg,abs) coordinates, as long as they both refer to the same point:
|-used above include an unproved assumption into the meaning of POINT.
This means that
the POINT(x=>1,y=>1) = the POINT(abs=>1, arg=45.degrees).
Thus at the conceptual, logical, and mathematical level we can document properties that must be true - without describing any means for this to be done.
This and some other more advanced techniques makes it possible to document the properties of real life entities.
. . . . . . . . . ( end of section Records and Labeled Tuples) <<Contents | Index>>