[Index] || [Contents] || [Source] || [Definitions] || [Search] || [Notation] || [Copyright] || [Comment] Fri Jan 16 12:32:03 PST 2004

- Records and Labeled Tuples
- : Example
- : Example - Phone numbers
- : UML
- : Example - pairs
- : Example - LISP functions
- : Example - triples
- : Triples and Pairs
- : Sets of Tuples
- : Example - a set of tuples
- : Short form
- : Long Form
- : Example - CIRCLE
- : Either-Or
- : Inheritance
- : Specifying Unique Objects
- : Constrained Structures are Subsets of Type
- Index of defined terms etc.

- A><B::=$ Net{1st:A, 2nd:B}, (a,b) ::=(1st=>a, 2nd=>b), So we can deduce the following formulae must also hold:
- (a,b).1st=a,
- (a,b).2nd=b.
- 1st in A><B->A,
- 2nd in A><B->B.
## Example - LISP functions

- CAR::=1st,
- CDR::=2nd,
- CONS::=fun[a,b](a,b).
## Example - triples

- A><B><C::=$ Net{1st:A, 2nd:B, 3rd:C }.
(a,b,c) ::A><B><C=(1st=>a, 2nd=>b, 3rd=>c).
## Triples and Pairs

Notice that A><B><C is equivalent to A><(B><C) in the sense that we can construct a one-one function from any (a,b,c) in A><B><C into A><(B><C). The function*M*that does this is defined by: - For all a:A, b:b, c:C, M(a,b,c)::=
*CONS*( a ,*CONS*( b, c) ). - (M) |-
*CAR*(*M*(x))=x.1st and*CAR*(*CDR*(M(x)))=x.2nd and*CAR*(*CDR*(*CDR*(*M*(x))))=x.3rd.The symbols

(M)|-

are short for "Because of the definition of M the following statemnt is true".This is the theoretical justification for

- LISP's notation for lists
- Computer Science's obssession with binary trees.

## Sets of Tuples

Let $*X*be a type of tuple with variables x,y,z,... then the predicates in the variables x,y,z,... define susbsets of the set of all possible tuples. If the predicate is P(x,y,z,...) then - $ Net{X, P(x,y,z) }
can be written for the set of tuples satisfying P.
## Example - a set of tuples

- circle1::= $ Net{x,y:real, r:Real&Positive, x^2+y^2<=r^2}.
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,... }

## Long Form

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.## Example - CIRCLE

- LONG_CIRCULAR::=following

Net- x::Real.
- y::Real.
- r::Real & Positive.
- |- (circle_axiom): x^2+y^2 <= r^2.
- (above) |- (circle_theorem): if r=0 then x=y=0.

(End of Net)

- CIRCULAR::=Net{x,y:real, r:Real&Positive, x^2+y^2<=r^2},
- circle::=$
*CIRCULAR*. So this definition of*circle*has the same effect as the definition of*circle1*above. We can even prove this be substituting equal eaxpressions: - (circle, circle1) |- circle1 = circle.
## 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 thentype(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 thens.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}

## Specifying Unique Objects

Given a structure S=$

*N*, whereN=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 generalthe 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:

- POINT::=following

Net

(End of Net)

The symbol|-

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.

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

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`.

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.

*. . . . . . . . . ( end of section Records and Labeled Tuples) *<<Contents | Index>>

- abs (Declaration)
- (Function Declaration)
- (Function definition)
- A><B (Definition)
- A><B><C (Definition)
- arg (Declaration)
- CAR (Definition)
- CDR (Definition)
- circle1 (Definition)
- circle_axiom (Axiom)
- circle (Definition)
- circle_theorem (Theorem)
- CIRCULAR (Definition)
- CONS (Definition)
- LONG_CIRCULAR (Definition)
- M (Function definition)
- POINT (Definition)
- r (Declaration)
- x (Declaration)
- y (Declaration)