A Calculus of Number Based on Spatial Forms - University of ...

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The third rule creates new elements <strong>of</strong> B by collecting previous elements. These
collections are unordered so permutations are irrelevant. This lack <strong>of</strong> order means
that the form 2 is considered identical to 2: Collection introduces the following
elements into B:
; 2; 22; (); 2(); ()(); ; 2; 22; 222:
Between the three rules, deep and wide expressions are introduced into B, including
the forms: ([][]); (([[]][])):
4.2.2 Equivalence Axioms
Elements in the two-boundary calculus are related by two equivalence axioms: involution
and distribution. The involution axiom denes a symmetric relationship
between the two boundaries.
Axiom 1 (Involution) Instance and abstract are functional inverses:
([A]) =A : =[(A)]:
:
Involution allows the removal or introduction <strong>of</strong> instance-abstract pairs, including
void equivalents when A is void, (2)=[]= . Involution does not apply to a pair
with something lying between the boundaries; it can neither introduce or remove a
conguration such as([]); where lies outside <strong>of</strong> the abstract and more than
one instance lies within it. Acceptable examples <strong>of</strong> involution are shown in Figure
4.1.
The distribution axiom denes an asymmetric relationship between the two boundaries.
Axiom 2 (Distribution) An instance around abstract distributes over the contents
<strong>of</strong> the abstract:
(A[BC]) =(A[B])(A[C]):
:
Distribution manipulates the modier form, (A[:::]); composed <strong>of</strong> instance, the
template variable A, and abstract. It threads the modier form over the contents
<strong>of</strong> its inner abstract boundary. Distribution states that the form can modify these
contents collectively or separately. Unlike involution, distribution matches a pattern
lying between the two boundaries. Examples <strong>of</strong> distribution are shown in Figure 4.2.
These two axioms dene the transformational basis <strong>of</strong> two-boundary expressions.