# Encapsulation theory: the minimised, uniformly violational radial branch.

# Edmund Kirwan^{*}

# www.EdmundKirwan.com

# Abstract

This paper investigates how the hidden elements of a single radial branch should be distributed so as to minimise the branch's potential coupling.

# Keywords

Encapsulation theory, radial encapsulation, branch, potential coupling.

# 1. Introduction

In the absolute encapsulation context when an encapsulated set is uniformly-distributed in violational elements, then its potential coupling is minimised when it is also uniformly-distributed in hidden elements (see proposition 1.11 in [1]). In the radial encapsulation context, however, this is not the case. This paper demonstrates this phenomenon using an example branch and derives the equation showing the hidden element distribution that achieves a minimum potential coupling.

This paper considers sets of radial information-hiding only.

# 2. Selected branch potential couplings

Consider the branch shown in figure 1, showing a branch of three disjoint primary sets, where the information-hiding and the information-hiding violation of the sets are shown within their representative symbols.

*Figure
1: A branch of three disjoint primary sets.*

Thus,
taking the root set, *K*_{1}, for example, we see
that *K*_{1} has 10 violational elements,*=10*,
and 5 hidden elements,*=5*;
as we are concerned here with uniform violational distributions, we
will discuss the specific violation of each set, and use the variable
that we normally use for specific violations, hence *d=10*,
though the meaning of both is, of course, the same. We note that
violational and hidden elements of the branch are distributed
uniformly throughout the sets.

Let us calculate the potential coupling of this branch.

The
potential coupling of *K*_{3} is the sum of its
internal and external potential couplings. The internal potential
coupling of *K*_{3} is the total number of
contained elements multiplied by this number minus one, i.e.,
*15x14=210*. The external potential coupling of *K*_{3}
is the total number of contained elements multiplied by the total
number of violational elements it can see "below" it, i.e.,
*15x(10+10)=300*. So the potential coupling of *K*_{3}
is *210+300=510*.

As
*K*_{2} has the same number of contained elements
then it will have the same internal potential coupling as *K*_{3},
i.e., *210*. *K*_{2} can only see *10*
violational elements "below" it, so its external potential
coupling is *15x10=150*. So the potential coupling of *K*_{2}
is *210+150=360*.

*K*_{1}
has the same internal potential coupling as the other two, i.e., *210*,
and *K*_{1} has, like all root sets, no external
potential coupling (there are no sets "below" it for it to
see), so its potential coupling is the same as its internal potential
coupling, *210*.

The
branch's potential coupling is then: 5*10*+*360+210=1080*.

We
know that, in the absolute encapsulation context, potential coupling
is minimised (ignoring A.M.C.s) when both the violational and hidden
element are distributed uniformly over all sets. Let us test to see
whether this also holds in the radial encapsulation context: let us
move a hidden element from *K*_{2} to *K*_{1},
see figure 2.

*Figure
2: A branch of three disjoint primary sets.*

Now let us calculate the potential coupling of this modified branch.

The
potential coupling of *K*_{3} is unchanged.

The
potential coupling of *K*_{2} has now fallen from
*360* to *14x13+14x10=322*.

The
potential coupling of *K*_{1} has now risen from
*210 *to* 16x15 = 240*.

The
branch's total potential coupling has now fallen from *1080* to
*510+322+240=1072*.

Thus by moving a hidden element, thereby rendering the branch non-uniformly distributed in hidden elements, we have reduced the branch's potential coupling. We note that the branch is still uniformly distributed in violational elements. This then raises the obvious question: given a branch that is uniformly distributed in violational elements, what distribution of hidden elements will minimise its potential coupling?

This question is of practical relevance as we suspect that a branch's minimum potential coupling will be achieved when each set in the branch has the minimum number of violational elements, and that minimum is one. In putting one element in each set, however, we are automatically uniformly distributing the violational elements, hence the configuration of a branch at its minimum potential coupling is one in which it is uniformly distributed in violational elements.

It
can be shown that, to attain the minimum potential coupling in this
case, the number of hidden elements in the *i*^{th}
disjoint primary set is given by the following equation (see
proposition 9.12):

With this equation we can calculate the number of hidden elements in each of the three sets examined above to minimise the potential coupling.

*K*_{3}
should contain the following number of hidden elements:

*K*_{2}
should contain the following number of hidden elements:

*K*_{1}
should contain the following number of hidden elements:

The branch thus configured is shown in figure 3.

*Figure
3: A branch non-uniformly distributed in hidden elements with
minimised potential coupling.*

Calculating the potential coupling of this minimised branch we see the following.

The
potential coupling of *K*_{3} is *10x9+10x20=290*.

The
potential coupling of *K*_{2} is *15x14+15x10=360*.

The
potential coupling of *K*_{1} is* 20x19 = 380*.

The
branch's total potential coupling is *290+360+380=1030*, down
from the original, uniformly distributed figure of *1080*.

# 3. Observations

The
reduction of potential coupling achievable using the equation above
to guide hidden element distribution is a function of the specific
violation, *d*, the number of violational elements per disjoint
primary set. In a well-encapsulated set, *d* is small, therefore
the reduction of potential coupling achievable using this
distribution will not be great in well-encapsulated sets of sets.

It is noteworthy, however, that this hidden element distribution favours packing the hidden elements towards the lower end of the branch. In software development, for example, this argues against, "Top heavy," branches and recommends that leaf subsystems and leaf packages should not contain an above average number of program units.

# 4. Conclusion

When a branch is uniformly distributed in violational elements, its minimum potential coupling is not achieved by also uniformly distributing its hidden elements. Instead, an equation describes the hidden element distribution to achieve this minimum potential coupling and this equation suggests placing more hidden elements towards the lower part of the branch than the higher.

# 5. Appendix A

## 5.1 Propositions

### Proposition 9.1

Given
a branch *B* in encapsulated set *G *of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the internal
potential coupling of the *i*^{th} primary set,,
is given by:

*Proof:*

By definition [D1.2] in [1],
given a primary set *Q*_{i} the internal
potential couplingis
given by:

(i)

By definitions [D1.2], [D1.6]
and [D1.7] of [1], the elements in *K*_{i} are
those in the intersection of *K*_{i} with *H*
and *V*, or:

(ii)

By definition [D1.1] in [1], *H*
and *V* are disjoint so taking the cardinality of (ii) gives:

(iii)

By definition, each disjoint
primary set has an information-hiding violation of *d*, so
substituting into (iii) gives:

(iv)

Substituting (iv) into (i) gives:

=

*QED*

### Proposition 9.2

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the external
potential coupling of the *i*^{th} primary set,,
is given by:

*Proof:*

By proposition 8.11 in [3],
given a primary set *Q*_{i} of a single branch
the external potential coupling isgiven
by:

(i)

By definitions [D1.2], [D1.6]
and [D1.7] of [1], the elements in *K*_{i} are
those in the intersection of *K*_{i} with *H*
and *V*, or:

(ii)

By definition [D1.1] in [1], *H*
and *V* are disjoint so taking the cardinality of (ii) gives:

(iii)

By definition, each disjoint
primary set has an information-hiding violation of *d*, so
substituting into (iii) gives:

(iv)

Substituting (iv) into (i) gives:

=

*QED*

### Proposition 9.3

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the potential
coupling of the *i*^{th} primary set,,
is given by:

*Proof:*

By proposition 9.1, the internal
potential coupling of the *i*^{th} primary set,,
is given by:

(i)

By proposition 9.2, the external
potential coupling of the *i*^{th} primary set,,
is given by:

(ii)

By proposition 8.7 in [3], the
potential coupling of *Q*_{i} is the sum of the
external potential coupling sets of *Q*_{i} and
the internal potential coupling of *Q*_{i}, or:

(iii)

Substituting (i) and (ii) into (iii) gives:

*QED*

### Proposition 9.4

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the potential
coupling of *B*,,
is given by:

*Proof:*

By definition I SHOULD PROVE THIS ????????????????????????xxx, the potential coupling of a branch is the sum of the potential couplings of all its sets, or:

(i)

By proposition 9.3, the
potential coupling of the *i*^{th} primary set,,
is given by:

(ii)

Substituting (ii) into (i) gives:

=

*QED*

### Proposition 9.5

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the potential
coupling of *B*,,
is given by:

*Proof:*

By proposition 9.4, the
potential coupling of *B*,,
is given by:

(i)

By definition:

(ii)

Applying (ii) to (i) gives:

=

*QED*

### Proposition 9.6

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the number of
elements in *B* is
given by:

*Proof:*

By definition, the number of elements in a branch is the sum of the number of elements in all its disjoint primary sets, or:

(i)

By definitions [D1.2], [D1.6]
and [D1.7] of [1], the elements in *K*_{i} are
those in the intersection of *K*_{i} with *H*
and *V*, or:

(ii)

By definition [D1.1] in [1], *H*
and *V* are disjoint so taking the cardinality of (ii) gives:

(iii)

By definition, each disjoint
primary set has an information-hiding violation of *d*, so
substituting into (iii) gives:

(iv)

Substituting (iv) into (i) gives:

=

=

*QED*

### Proposition 9.7

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the number of
information-hidden elements in the first primary set,,
is given by:

*Proof:*

By proposition 9.6 the number of
elements in *B* is
given by:

=

*QED*

### Proposition 9.8

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the square of the
number of information-hidden elements in the first primary set,,
is given by:

*Proof:*

By proposition 9.7 the number of information-hidden elements in the first primary set,, is given by:

(i)

Squaring both sides of (i) gives:

*QED*

### Proposition 9.9

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the potential
coupling of *B*,,
is given by:

*Proof:*

By proposition 9.5 the potential
coupling of *B*,,
is given by:

(i)

By proposition 9.7 the number of information-hidden elements in the first primary set,, is given by:

(ii)

By proposition 9.8 the square of the number of information-hidden elements in the first primary set,, is given by:

(iii)

Substituting (ii) and (iii) into (i) gives:

=

=

*QED*

### Proposition 9.10

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the number of
information-hidden elements in the *i*^{th}
primary set,,
where,
which minimises the potential coupling of *B* is given by:

*Proof:*

By proposition 9.9, the
potential coupling of *B*,,
is given by:

(i)

To find the value ofwhich minimises this potential coupling we must differentiate (i) w.r.t.and set to zero:

(ii)

By proposition 9.7, the number of information-hidden elements in the first primary set,, is given by:

(iii)

Substituting (iii) into (ii) gives:

=

*QED*

### Proposition 9.11

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the number of
information-hidden elements in the first primary set,which
minimises the potential coupling of *B* is given by:

*Proof:*

By proposition 9.7, the number of information-hidden elements in the first primary set,, is given by:

(i)

By proposition 9.10, the number
of information-hidden elements in the *i*^{th}
primary set,,
where,
which minimises the potential coupling of *B* is given by:

(ii)

Substituting (ii) into (i) gives:

=

=

=

=

=

=

=

*QED*

### Proposition 9.12

Given
a branch *B* in encapsulated set *G* of *n* elements
and of *r* disjoint primary sets, with each disjoint primary set
having an information-hiding violation of *d*, the number of
information-hidden elements in the *i*^{th}
primary set,which
minimises the potential coupling of *B* is given by:

*Proof:*

By proposition 9.10, the number
of information-hidden elements in the *i*^{th}
primary set,,
where,
which minimises the potential coupling of *B* is given by:

(i)

By proposition 9.11, the number
of information-hidden elements in the first primary set,which
minimises the potential coupling of *B* is given by:

(ii)

Substituting (ii) into (i) gives:

=

=

*QED*

# 6. References

[1] - "Encapsulation theory fundamentals," Ed Kirwan, www.EdmundKirwan.com/pub/paper1.pdf

[2] - "Encapsulation theory: the minimised, uniformly violational radial branch," Ed Kirwan, www.EdmundKirwan.com/pub/paper9.pdf

[3] - "Encapsulation theory: radial encapsulation," Ed Kirwan, www.EdmundKirwan.com/pub/paper8.pdf

*©
Edmund Kirwan 2009. Revision
1.0, December 20^{th}
2009. (Revision 1.0: December 20^{th},
2009.) All entities may republish, but not for profit, all or
part of this material provided reference is made to the author and
title of this paper. The latest revision of this paper is available
at [2].