Int. J. Communications, Network and System Sciences, 2013, 6, 260-276 Published Online May 2013 (
Copyright © 2013 SciRes. IJCNS
All-in-One: Space-Time Body, Function and Metric
A Fundamentally New Approach to Computation
Hermann von Issendorff
Institut für Netzwerkprogrammierung, Hemmoor, Germany
Received February 21, 2013; revised April 21, 2013; accepted May 9, 2013
Every algorithm which can be executed on a computer can at least in principle be realized in hardware, i.e. by a discrete
physical system. The problem is that up to now there is no programming language by which physical systems can con-
structively be described. Such tool, however, is essential for the compact description and automatic production of com-
plex systems. This paper introduces a programming language, called Akton-Algebra, which provides the foundation for
the complete description of discrete physical systems. The approach originates from the finding that every discrete
physical system reduces to a spatiotemporal topological network of nodes, if the functional and metric properties are
deleted. A next finding is that there exists a homeomorphism between the topological network and a sequence of sym-
bols representing a program by which the original nodal network can be reconstructed. Providing Akton-Algebra with
functionality turns it into a flow-controlled general data processing language, which by introducing clock control and
addressing can be further transformed into a classical programming language. Providing Akton-Algebra with metrics,
i.e. the shape and size of the components, turns it into a novel hardware system construction language.
Keywords: Topological space, Dataflow Machine, Many-Sorted Logic, DNA Programming, Layout Simplification
1. Introduction
Living nature demonstrates how to program discrete spa-
tiotemporal systems: It generates chains of amino acids
from genetic code [1]. In a suitable environment the
chain of amino acids then folds to a protein, i.e. a one-
dimensional formation mutates into a three-dimensional
one. The transaction can be reversed by changing the
environment. This shows that there is a bijective and
bicontinuous mapping between the chain of amino acids
and the protein, called homeomorphism [2]. With other
words: The chain of amino acids represents a program
which contains the complete spatial information of the
protein. The spatial structure of the protein arises from
additional weaker binding forces between the amino ac-
ids. By external impact, e.g. by adsorption of another
molecule, the structure of the protein may change into
another metastable state. With other words: A protein has
the ability to store and process information.
Discrete physical systems usually are a composition of
a set of three-dimensional material components or any
abstraction thereof. If the components are active, i.e. if
they produce physical objects or evaluate functions, then
they are temporally directed and are activated in a partial
temporal order. If the components are static, then they
can be assigned a partial assembling order which also
induces a temporal direction.
Abstracting a discrete physical system, for instance a
computer, from its metrics, i.e. from the spatial measures
of its components, the residue is a three-dimensional
directed network of the executable functions which are
realized by the components.
Abstracting a discrete system, for instance again a
computer, from its functionality, the residue is a three-
dimensional directed network of building blocks which
have the size and shape of the system components. Ab-
stracting a discrete system from the metrics and the func-
tionality at the same time, the residue is a directed net-
work of nodes showing their dependencies. There, any
two nodes may be related or not, and each node may be
related to any finite number of preceding and succeeding
If the nodes of the network are provided again with
their original concrete functional and metric properties,
then the original system is regained. The nodal network
is therefore a structural class of all discrete physical sys-
tems. Thus, if there is a formal language for the spatio-
temporal description of a directed relational nodal net-
work, it is a common language for all discrete physical
systems. This even includes every kind of computer, no
matter, if it applies to a von-Neumann architecture or not.
Akton-Algebra, for short AA, provides this capability.
It is important to notice that abstraction from metrics
and functionality does not mean abstraction from space
and time. The latter would reduce a directed discrete
system to a directed graph, i.e. to a mathematical object
of graph theory, which does not have any relation to
space or time. The spatial relations between the nodes,
however, are the very properties on which AA is based
upon. An AA-node, i.e. the metric abstraction of a com-
ponent, does have an arbitrary but non-zero shape and
size, and traversing the node from its front-end to its
back-end takes an arbitrary but non-zero amount of time.
This means that an action of a component does not dis-
appear under metric and functional abstraction but is
only reduced to a rudimentary action of propagation.
This is the reason why the word "Akton" has been chosen
as a general designator of a concrete component and any
of its metrical or functional abstractions.
A spatial description of spatiotemporal structures by
programming has not been considered up to now. Clas-
sical data processing languages are not provided with
spatial semantics. They do not need to because they are
tailored to the sequential execution of the von-Neumann-
computer. At first glance, the graphical calculus pro-
posed by [3] seems to have some similarity to AA. Their
calculus, however, is aimed at quantum informatics and
does not describe classical physical structures. Thus, the
only paper on spatiotemporal structures seems to be an
early one by the author himself [4].
The paper proceeds as follows: In the next section the
fundamental elements of a nodal network are analyzed.
A topological cut needs to be introduced in order to re-
solve crossings in spatial structures, and a second cut to
resolve cycles and crosslinks in planar structures. This
gives rise to a hierarchy of Akton sorts and a hierarchy of
Interface sorts. The language of abstract AA is then syn-
thesized from these two fundamental hierarchies. There
are four sets of production rules, which stepwise generate
a programming language of increasing power. The first
set of production rules introduces an AA language for the
abstract description of planar and antiparallel structures,
the second set extends the AA language to represent
symbolic networks, the third one extends it to describe
digital or analog functional structures, and the fourth one
to even comprise metric structures.
The second part of the paper is devoted to symbolic,
functional and metrical concretizations. The application
of the symbolic language specification already suffices to
describe spatial structures of considerable complexity.
Surprisingly, the language of antiparallel linear structures
generates four fundamental substructures which may be
interpreted as the four genetic instructions of life,
chemically known as the nucleotides guanine, adenine,
thymine, and cytosine.
Introducing functionality into AA requires the exten-
sion of both the Akton and the In terface hierarchies. This
is achieved by introducing digital values as subsorts of
the Interface hierarchy. Although not elaborated in detail,
analog instead of digital values could be introduced as
well. Finally, in order to expand AA into a metric lan-
guage requires the extension of Akton sorts and Interface
sorts. For this purpose, several basic geometrical struc-
tures need to be introduced as for instance multiple links,
multiple forks, multiple joins as well as topological cuts.
This way AA always permits the constructive representa-
tion of every digital or analog electronic circuit on two
layers only. These features considerably simplify the
layout of highly integrated electronic circuits.
The conclusions finally subsume the essential achieve-
ments of AA.
2. Elements and Elementary Structures of
As observed in the introduction, an abstract nodal net-
work is a common structure underlying every discrete
physical system. In general, a nodal network of a discrete
physical system has a three-dimensional structure. A
formal description, on the other hand, is an ordered se-
quence of symbols and thus has a one-dimensional
structure. The aim of this chapter is to show that a
three-dimensional nodal network can be mapped into a
one-dimensional description and vice versa, without los-
ing any structural information.
The class of nodal networks we are dealing with is al-
ways assumed to be directed. If the physical system un-
derlying the network is not directed or does not have an
entry and an exit, these features can be introduced with-
out the loss of generality. Each node of a directed nodal
network has two interfaces, one for the input and one for
the output.
The representation of a directed abstract nodal net-
work can be done at different levels of detail. As de-
picted by the hierarchy in Figure 1, new levels of sorts
with more and more properties can be added. At the top
level, the network is represented by the general sort Ak-
ton. At the second level, called the fundamental level,
there are four sorts of nodes called Head, Body, Ta il and
CS (Closed System). The relations between the sorts are
specified by means of interfaces. At the fundamental
level, there are only two primitive sorts of interfaces, the
non-empty one (symbolized by ε) and the empty one
(symbolized by ε). Sort Head has no preceding nodes, i.e.
an empty input but at least one succeeding node, i.e. a
non-empty output. Sort Tail has no succeeding nodes but
at least one preceding node, i.e. a non-empty input and
an empty output. Sort Body has at least one preceding
Copyright © 2013 SciRes. IJCNS
node and at least one succeeding node, i.e. a non-empty
input and a non-empty output.
Figure 1. (a) The hierarchy of abstract Akton sorts. These
three levels are common to all discrete systems. The third
level comprises the set of basic abstract sorts. (b) The
hierarchy of Interface sorts is depicted at the right. At the
third level, the non-empty Interface sort is quantified by a
basic element called Pin.
Finally, sort CS has no preceding and no succeeding
nodes, i.e. input and output are empty.
The third hierarchical level, called abstract structural
level, consists of basic abstract sorts. Their non-empty
inputs and outputs are specified by a quantified basic
Interface element called Pin. There are three basic sub-
sorts of Body called Fork, Join and Link. Fork has one
Pin in the input and two Pins in the outpu t, Join has two
Pins in the input and one Pin in the output, and Link has
one Pin in the input and one in the output. The basic
subsorts of Head are called Up and Set each of them
having a single Pin in the output. The basic subsorts of
Tail are called Down and Off, each having a single Pin in
the input.
The subsorts of Head and Tail differ by their seman-
tics. Up and Down as well as Set and Off are necessary
for mapping the three dimensions of nodal networks to
the single dimension of a string of symbols, i.e. the pro-
gram code. This mapping needs to be explained more
Because of the abstraction from metrics the structure
of a nodal network is a topological one. A topological
structure preserves the adjacency of the nodes, if it is
mapped into another shape by a function called homeo-
morphism. Even cuts are admissible as long as the corre-
lation between the cutting ends is guaranteed. Homeo-
morphism is bijective and bicontinuous. This means that
an original topological structure may arbitrarily be dis-
torted but can always be regained by reversing the ho-
In order to describe the topological properties of the
nodal network, a topological frame of reference is
needed. Since there is no metrics, the frame of reference
can only be relational. Such frame of reference can be
defined by referring to a human observer who physically
differentiates between three independent pairs of inverse
spatial relations, i.e. left-right, above-below and front-
In addition, a privileged direction in the frame of refer-
ence needs to be selected in order to orient the directed
nodal system, i.e. on which side to place the elements of
sort Head and on which side the elements of sort Tail.
Following the reading standard of the Western Hemi-
sphere, the direction from left to right is chosen. Since
every node represents an action and action takes time,
the orientation also introduces a direction of time. Thus
left will also be interpreted as earlier and right as later.
1) The mapping of the nodal network from a three-
dimensional representation to the one-dimensional de-
scription of AA requires several steps: The nodal network
has to be oriented so that all system Entries are at the left
side and all system Exits are at the right side.
2) The nodal network has to be projected to an ori-
ented plane of observation spanned between the left-right
axis and the above-below axis and positioned between
the nodal network and the observer. Usually, this project-
tion will give rise to crossings of nodal connections (see
Figure 2). Since the crossings are spatial residues they
must be removed. This is achieved by cutting the rear
connection and replacing the cutting ends by a pair of
Down and Up, Down being a subsort of Tail and Up be-
ing a subsort of Head as mentioned before.
3) The resulting planar network may still contain
non-orientable structures like cycles or crosslinks (see
Figure 3). This problem can be solved in a way similar
to the crossing problem by cutting the structures and in-
serting a pair Off and Set, representing a cut in the plane.
Off is a sub sort of Tail and Set is a subsort of Head as
mentioned before.
Figure 2. Planarization of a spatial structure, i.e. the re-
moval of a crossing, is achieved by cutting the rear con-
nection and inserting a pair of Aktons instead called Down
and Up.
Figure 3. Orientation of cycles (a) and crosslinks (b) from
left to right by cutting and inserting a pair of Aktons instead
called Off and Set.
opyright © 2013 SciRes. IJCNS
4) The separate utilization of spatial and planar cuts
does not suffice to linearize every spatial nodal network.
There are nodal structures where both cuts are to be ap-
plied crosswise. These structures can be characterized by
two antisense strands which are interconnected by sev-
eral crosslinks. A simplified structure of this kind is de-
picted in Figure 4 showing two antisense strands which
are connected by two crosslinks. The directions of these
crosslinks are not specified on purpose. An orientation of
them can be accomplished by either left- or right-twisting
the contrarily oriented strand thus generating different
crossings of the crosslinks.
Figure 4. Simplified structure of two antisense strands
connected by two undirected crosslinks.
The planarization of the crossings is achieved by ap-
plying a spatial cut to the rear crosslink as shown by the
dotted red line, and a planar cut to the upper one as
shown by the dashed blue line. Finally assigning direc-
tions to the crosslinks amounts to four different twin-cuts
for the left-twisted as well as the right-twisted structure.
The twin-cuts of the left-twisted structures are depicted
in detail in Figure 5.
A nodal network can now formally be represented by a
string of symbols, i.e. in linear form. To this end, two
adjacent independent subnetworks x and y are related by
an infix symbol "/", called Juxta, where x/y means x lies
above y. Likewise, two adjacent dependent subnetworks
x and y are related by an infix symbol ">", called Next,
where x>y means x precedes y in space and time. In or-
der to reduce the amount of parentheses Juxta is assumed
to bind stronger than Next.
3. Definition of Abstract AA
In the previous chapter we discovered a way how to turn
a three-dimensional network of nodes into a linear ori-
ented network of a nodal string, i.e. a string of abstract
Aktons. A nodal string can be read and processed like a
program, which under abstraction from functionality and
metrics means to reconstruct the original spatial topo-
logical structure of the nodal network. Thus we already
know that there exists a formal language for the descrip-
tion of nodal networks. In particular, we discovered the
set of basic structural nodes of AA. However, what we
cannot do up to now is to synthesize a nodal system, be-
cause we do not know the rules by which nodal subnet-
works are to be related. In formal language terms, we
need to know the grammar of AA. That is what we will
do in this chapter.
Figure 5. Detailed view of four left-twisted twin-cut
structures (a), (b), (c), (d) which can be derived from the
undirected twin-cut structure of figure 4.
AA is a many-sorted term-algebra, throughout defined
by first order logic. It is built up from a hierarchy of Ak-
ton sorts and a hierarchy of Akton Interfaces. The hier-
archies can be formally described by a general function
abstract which maps each lower level set of subsorts to
its immediate upper sort. The grammar of abstract AA is
systematically derived descending the two hierarchies of
sorts as shown in Figure 1. There, the first hierarchy is
headed by sort Akton, the other one by sort Interface.
These general notions will step by step be specified by
adding more properties while descending the hierarchical
levels further down.
Aktons and the relations Next and Ju xta are destined to
describe directed nodal networks. Adjacent directed Ak-
ton terms may either be dependent or independent. De-
pendency is expressed by the relation Next and inde-
pendency by the relation Juxta. Next and Juxta produce a
free semigroup FA+ of fundamental Akton ter m s.
The Akton term variables defined here and further
down by means of tables are represented by using the
proper names of the sets of term sorts, i.e. if Z+ desig-
nates a set of terms sorts then the term variable is called
Z. The representation of the production rules by tables is
nothing but an inverse BNF.
The next level below the general sort Akton comprises
the sorts Head, Body, Tail, CS. Their properties have
been informally defined in the previous chapter. The four
Copyright © 2013 SciRes. IJCNS
sorts establish a set FA of fundamental Akton sorts .
Definition A.1 (Fundamental Akton Sorts)
The set FA of fundamental akton sorts is defined as
FA := {Head, Body, Tail, CS}, where
abstract: FA
abstract(x):= Akton, if x
{Head, Body, Tail, CS}
An Interface at the second level of the specification
hierarchy (see Figure 1) is either empty or not empty,
represented by ε or ε. The two elements establish a set
FI of fundamental in terface sorts .
Definition A.2 (Fundamental Interface Sorts)
The set FI of fundamental interface sorts is defined as
FI := {
ε ,ε}, where
abstract: FI
abstract(x):= Interface, if x
ε, ε}
Every Akton term x has two Interfaces, an input and an
output, formally described by in(x) and out(x). Depend-
ing on the level of concretization, the Interfaces may be
refined by introducing more properties. This gives rise to
a hierarchy of Interface sorts, similar to the hierarchy of
Akton sorts. At the fundamental level, these functions
have the following definition.
Definition A.3 (Fundamental Term Level Functions
in, out)
The fundamental term level functions in, out: FA+
are inductively defined as:
ε, if zHeadCS
ε, if z BodyTail
in(x)/in(y),if z = (x/y)
in(x), if z = (x>y)
ε, if zTailCS
ε, if z HeadBody
out(x)/out(y),if z = (x/y)
out(y), if z = (x>y)
The in- and out-interfaces of Juxta-related Akton terms
are packed on top of each other, in the same way as the
Akton terms themselves. This means that the Jux-
ta-symbol is used for relating Akton terms as well as In-
terface elements. This way the order of the Interface el-
ements matches the linear order of adjacent independent
nodes, i.e., the order serves as a local physical addressing
Definition A.4 (Fundamental Interface Terms)
The set of fundamental interface terms is defined as:
FI*: (x/y)
FI*, if
ε ε
ε ε
The third level of the hierarchy of Akton sorts introduces
structural properties. Consequently, it is called the struc-
tural level. Concerning Aktons of sort Body, there are
three subsorts representing the structures fork, join and
link. They are called Fork, Join and Link accordingly.
As discussed in the previous chapter, the mapping of
spatial structures to planar structures requires a cut, rep-
resented by the pair of Akton sorts Down and Up. Like-
wise, the complete sequencing of planar structures re-
quires another pair of Akton sorts Off and Set. Thus,
Head comprises the subsorts Up and Set and Tail the
subsorts Down and Off.
Definition A.5 (Structural Akton Sorts)
The set A of structural Akton sorts is defined as
A := {Up, Set}
{Down, Off}
{Fork, Join, Link}
{CS}, where
abstract: A
abstract(x):= Head, if x
{ Up, Set},
abstract(x):= Tail, if x
{ Down, Off},
abstract(x):= Body, if x
{Fork, Join,Link},
abstract(x):= CS, if x=CS
At the third level of the specification hierarchy (see
Figure 1) an Interface consists of a set of ordered ele-
ments of sort Pin or is empty. This set of structural in-
terface sorts is designated by I.
Definition A.6 (Structural Interface Sorts)
The set I of structural interface sorts is defined as
I := {Pin, ε}, where
abstract: I
ε, if x=Pin
abstract(x):=ε, if x=ε
Definition A.7 (Structural Term Level Functions in,
The term level functions in, out: A+
I* are inductively
defined as:
ε, if zHeadCS
Pin, if zTailLink
Pin, if z = Fork
Pin/Pin, if z = Join
in(x)/in(y), if z = (x/y)
in(x), if z = (x>y)
ε, if zTailCS
Pin, if zHeadLink
Pin/Pin, if z = Fork
Pin, if z = Join
out(x)/out(y), if z = (x/y)
out(y), if z = (x>y)
Definition A.8 (Structural Interface Terms)
The set of structural interface terms is defined as the
smallest set satisfying:
I*: (x/y)
opyright © 2013 SciRes. IJCNS
The set of Akton terms A+ is a subset of FA+. A first
restriction of A+ is that two dependent terms can only be
Next-related if their adjacent Interfaces are identical. The
other restriction is that every relation of terms must con-
form to a real topological structure.
A crossing for instance is characterized by the cut
terms D, D
D+, and U, U
U+, with a separate term B,
B+ in between. This gives rise to four special term
sorts, designated by DB+, BD+, UB+, BU+ (see Figure 2).
There are four of them because every crossing owns a
chirality that can be either left- or right-handed. The
planar structures of cycles and crosslinks on the other
hand are characterized by the cut terms O, O
O+, and S,
S+, which are not separated by another term (see Fig-
ure 3). A pair of cut terms can thus only be positioned
either at the left or at the right side of a B-term, which
results in four special term sorts OB+, BO+, SB+, BS+.
Four more special term sorts OBO+, OBS+, SBO+, SBS+
need to be introduced because independent cut terms
may occur at both sides of a B-term.
The set of production rules P consists of four subsets
Pi, i
{1,2,3,4} . P1 lists the fundamental rules, P2 the
planarizing rules, P3 the linearizing rules and P4 the
twin-cut rules.
Definition A.9 (Terms of the Akton Language)
The set of Akton terms A+ is inductively defined as the
smallest set satisfying:
A+ := B+
B+, Head
H+, Tail
T+, CS
, Up
, Down
, Set
, Off
A+: (x>y)
A+, if out(x)=in(y) and (x>y)
A+: (x/y)
A+, if (x/y)
Pi:= P1
Definition A.10 (Fundamental Production Rules of
the Akton Language)
The set of fundamental production rules P1 is defined as:
(x/y) H B T CS
(x>y) B T
Definition A.11 (Planarizing Production Rules of the
Akton Language)
The set of planarizing production rules P2 is defined as:
(x>y)DB BU UB
The complete definition of crossings (see Figure 2) re-
quires the additional definition of the implicit
in-out-relations between U- and D-terms. The following
two definitions regard the mirrored structures alterna-
tively crossing a B-term either from above or below.
Definition A.12 (Implicit Cut-Relations in Crossings)
The implicit cut-relations in crossings are defined as:
out(y):= in(v), if (v/w>x/y)
B+ and y
U+ and v
out(x):= in(w), if (v/w>x/y)
B+ and x
U+ and w
Similarly, the complete definition of cycles and cross-
links requires the additional definition of the implicit
in-out relations between O- and S-terms. Recall that the
feedback of a cycle is always located either above or
below a B-term (see Figure 3a). A crosslink, on the oth-
er hand, is always located between two B-terms con-
necting an upper B-term with a lower B-term or vice
versa (see Figure 3b).
Definition A.13 (Linearizing Production Rules of the
Akton Language)
The set of linearizing production rules P3 is defined as:
Copyright © 2013 SciRes. IJCNS
Definition A.14 (Implicit Cut-Relations in Cycles)
The implicit cut-relations in cycles are defined as:
out(v):=in(x), if (v/w>x/y)
B+ and v
S+ and x
O or
out(w):=in(y), if (v/w>x/y)
B+ and w
S+ and y
Definition A.15 (Implicit Cut-Relations in Crosslinks)
The implicit cut-relations in crosslinks are defined as:
A+: out(v):=in(y), if (u>x/y)/(v/w>z)
and v
S+ and y
O+ or
B+and v
S+and y
The final structure to be defined by production rules is
the twin-cut structures as outlined in section 2.4 of the
previous chapter. As mentioned the twin-cut structures
may be either left- or right-twisted. Changing the direc-
tion of the spatial as well as the planar cut gives rise to
four different structures each. The left-twisted structures
are depicted in Figure 5.
Definition A.16 (Twin-Cut Production Rules of the
Akton Language)
The set of the twin-cut production rules P4 is defined as:
(x>y) BO BS DB UB
Definition A.17 (Implicit Twin-Cut Relations in
Left-twisted Antiparallel Structures)
The implicit cut-relations in left-twisted antiparallel
structures are defined as:
+: out(t):=in(y) and out(u):=in(x), if
B+ and t
U+ and y
D+ and x
and u
+: out(x):=in(u) and out(y):=in(t), if
B+ and y
U+ and t
D+ and u
and x
+: out(y):=in(t) and out(u):=in(x), if
B+ and y
U+ and t
D+ and x
and u
A+: out(t):=in(y) and out(x):=in(u), if
B+ and t
U+ and y
D+ and u
and x
Definition A.18 (Implicit Twin-Cut Relations in
Right-twisted Antiparallel Structures)
The implicit cut-relations in right-twisted antiparallel
structures are defined as:
+: out(t):=in(y) and out(u):=in(x), if
B+ and u
U+ and x
D+ and y
and t
+: out(x):=in(u) and out(y):=in(t), if
B+ and x
U+ and u
D+ and t
and y
+: out(y):=in(t) and out(u):=in(x), if
B+ and x
U+ and u
D+ and y
and t
+: out(t):=in(y) and out(x):=in(u), if
B+ and u
U+ and x
D+ and t
and y
Single Forks, Joins and Links have a planar structure,
and multiple Links are also planar. The structures of mul-
tiple Forks and multiple Joins, however, are always spa-
tial, i.e. projecting them on a plane can only be achieved
by means of Down/Up-cuts. A multiple Fork structure
duplicates a structure of multiple Links into two identi-
cally ordered structures of multiple Links, and a multiple
Join term does the reverse. In order to formally describe
these structures, we need to introduce the separation
functions pre and suc, which split a Next-relation into a
preceding and a succeeding part.
Definition A.19 (Separation Functions pre, suc)
The functions pre, suc: A+
A+ are defined as:
pre(x>y): = x,
suc(x>y): = y
Definition A.20 (Multiple Link)
The set of link terms L+ is inductively defined as the
smallest set satisfying:
L+: Link/x
Multiple Forks as well as multiple Joins can be real-
opyright © 2013 SciRes. IJCNS
ized by either left-hand or right-hand twisting the other-
wise identical structures. Accordingly, there are two sets
of multiple Forks and of multiple Joins. They will be
distinguished by the subscripts l and r indicating whether
the twist is left- or right-handed.
Definition A.21 (Multiple Fork)
The sets of multiple fork terms are recursively defined as:
+ Fr
+ B+,
(Fork>Down/Link) > (Link/Up) Fl
x Fl
+:(pre(x)/(Fork>Down/Link) > (Link/suc(x)/Up))
(Fork > Link/Down) > (Up/Link) Fr
+:(pre(x)/(Fork > Link/Down) > (Up/suc(x)/Link))
Definition A.22 (Multiple Join)
The sets of multiple join terms are recursively defined as:
+ Jr
+ B+,
(Down/Link) > (Link/Up > Join) Jl
+:(Down/pre(x)/Link > suc(x)/(Link/Up>Join))
(Link/Down) > (Up/Link > Join) Jr
+:(Link/pre(x)/Down > suc(x)/(Up/Link>Join))
Modularity, i.e. the capability of combining several
modules into a single one, is an indispensable require-
ment for the design of complex systems. In AA modular-
ity is easily incorporated because all modules are Aktons,
and every Akton term, how big it ever may be, can be
concealed into a single Akton. Concealing means hiding
the structure of an Akton term into an Akton while pre-
serving the visibility of the input and the output. The new
Akton is added to set A, and of course needs to be pro-
vided with a distinct name. In contrast, regarding con-
ventional digital programming languages, modularization
and information hiding can be quite a problem [5].
Definition A.23 (Function conceal)
The function conceal: A+
A is defined as:
conceal(x) := y, if not xA
In order to densify the Akton expressions later on we
introduce a count function for sequences of identical
Next-terms and a count function for regular Ju xta -terms.
Next-counts are represented by symbol ‘*’(called star)
and Juxta-counts by symbol ‘^’(called roof). Syntacti-
cally, both bind stronger than Next and Juxta.
Definition A.24 (Counting Functions *, ^ of Next- and
The counting function *: N
+ is inductively
defined as:
(n+1)*x = n*x>x, 1*x = x, nN, xA+
The counting function ^: A+
+ is inductively de-
fined as: x^(n+1)= x^n/x, x^1 = x, nN, xA+
We also introduce the counting function ^ for sequences
of regular Interface terms.
Definition A.25 (Counting Function ^ of Inter-
The counting function ^: I*
I* is inductively defined
x^(n+1)= x^n/x, x^1 = x, nN, xI*
4. Dependency Preserving Term
The structure of a given nodal network can be modified
in different ways without affecting the dependencies
between the terms. Formally the modifications are
achieved by term replacement according to the rules of
Tab. 1. The rules say that the left term may be replaced
by the right term provided that the constraint at the right
side holds. The ""-symbol says that the terms are mu-
tually replaceable.
Table 1: Dependency preserving term replacement rules
a. Link-Rules: x(y>x), if in(y)=in(x)
x(x>y), if out(y)=out(x)
b. Expansion-Rules: x(y/x), if yCS+
x(x/y), if yCS+
c. Associativity-Rules:((x>y)>z )(x>(y>z)), if true
((x/y)/z)(x/(y/z)), if true
d. Distributivity: ((w>x)/(y>z ))(w/y>x/z),if true
e. Connectivity-Rules:((w>x)/(y>z ))(w>x/y>z), if
out(x)=ε and in(y)=ε
((w>x)/(y>z ))(y>w/z>x), if
in(w)=ε and out(z)=ε
The link-rules a. add a Link term y to a term x or delete
it. The term y may either precede or succeed term x as
stated by the two rules. The constraints are that the out-
put of the left term must fit the input of the right term.
Usually term y will just consist of a strip of Links. The
expansion-rules b. place or remove a dead term y, i.e. a
neutral place, above or below a term x. Term x is of sort
A+, term y is of sort CS+. Both expansion-rules play an
important role in the layout process. The associativ-
ity-rules c. modify the structure of Next- and
Juxta-related te r ms . The first of the distributivity-rules d.
states that distributivity of Juxta over Next always holds
while the other rule states that distributivity of Next over
Juxta is restricted. The connectivity-rules e. splice two
independent Juxta-terms into a single term or vice versa.
5. Abstract Structural Models
The properties of abstract AA are exemplified by four
symbolic nodal structures. Each of the descriptions is
accompanied by an AA-program by which every struc-
Copyright © 2013 SciRes. IJCNS
ture can completely be reconstructed.
5.1. Tetrahedron
The first example (see Figure 6) deals with a
tetrahedron showing in detail the successive steps
of mapping from the spatial structure to the linear
program. In a first step a rear edge of the
tetrahedron is cut, as marked by a thin red line, and
the free ends are marked by an Up/Down-pair.
Figure 6. The mapping of a tetrahedron from the spatial
structure to a linear AA-program needs three steps. It is
first planarized by an Up/Down cut along the red line, then
fully oriented by a Set/Off cut along the blue line, and
finally turned into an AA- program.
This makes it possible to spread the tetrahedron on a
plane, and to orient the planar structure from left to right
according to the direction introduced by the
Up/Down-plane, and to orient the planar structure from
left to right according to the direction introduced by the
Up/Down-pair. The planar structure is then cut again, as
marked by a long thin blue line. While the cuts of both
outer edges are healed by inserting Links the cut of the
crosslink is marked by a Set/Off-pair, all shown in blue.
This provides the crosslink with a unique direction (see
Figure 6). (A reversely ordered Off/Set-pair would of
course reverse the direction of the crosslink.) The result-
ing structure is represented by a linear program.
5.2. Helix and Sheet
The next two examples have been selected in order to
indicate the relation between AA and the as yet unknown
protein programming language, which programs the spa-
tial structure of proteins by chains of amino acids [1].
Since the vocabulary of AA is just derived from a few
general principles it can be conjectured that the amino
acids are also describable by AA-expressions. The struc-
ture (a) of Figure 7 represents a model of two loops of a
right-handed α-helix. Since a α-helix is a spatial structure,
it takes Up/Down-pairs to planarize it and Set/Off-pairs
to fully orient it from left to right. The structure (b) of
Figure 7 represents a model of a ß-pleated sheet. Since
this structure is planar, it only takes S et/Off-pairs in order
to stretch it into a programming code.
Figure 7. Model and AA-program of a right-handed α-helix
(a), and model and AA-program of a β-sheet (b.)
5.3. Modeling DNA
A first particular application of twin-cuts concerns the
modeling of the genetic code.
Figure 8. Nucleotide models of the four elements of DNA-
code. Akton term (A) is pair-wise complementary to Akton
term (T), and Akton term (G) is pair-wise complementary to
Akton term (C).
opyright © 2013 SciRes. IJCNS
Copyright © 2013 SciRes. IJCNS
Figure 9. Model of a double stranded DNA, partially split into two single strands modeling RNA. Figure (a)
shows the planar AA-model, Figure (b) the linear AA-expression.
As well-known, the basic information on the construc-
tion of all organisms are expressed by a 4-symbol pro-
gramming language, where the symbols represent the
nucleobases called Adenine (A), Guanine (G), Cytosine
(C) and Thymine (T) [6]. Here we will use the same ab-
breviations for the nucleotides, whereupon each nucleo-
base is extended by a piece of backbone. An important
feature of the nucleotides is that they are pair-wise com-
plementary, i.e. A matches with T, and G matches with C.
The twin-cuts of AA offer exactly the same property and
thus are perfectly suited to model DNA. Since AA dis-
tinguishes between above and below, there are two rep-
resentations for each of the four nucleotides. This is
visualized by Figure 8, where the four twin-cuts of Fig-
ure 5 are now described by akton terms according to the
production rules P4.
Assuming that Adenine (A) is represented by the Ak-
ton term BUBO then Thymine (T) is represented by
SBDB. Likewise, assuming that Guanine (G) is repre-
sented by BDBO then Cytosine (C) is represented by
SBUB. Figure 8 models the four elements of DNA-code.
The elements are pairwise complementary to each other.
Exchanging U and D (the red balls) as well as S and O
(the blue balls) turns term (A) into term (T), and term (G)
into term (C). This is exactly the matching property of
Adenine/Thymine-pair and the Guanine/Cytosine-pair.
The four elements of the DNA-code are perfectly
suited to describe a double stranded helix as well as
RNA. Figure 9a shows an example and Figure 9b the
A second particular feature of the twin-cuts is that their
left- or right-twist is suited to model the chirality of a
double helix. This statement is substantiated by the fol-
lowing arguments: Since topological nodes have a finite
albeit unknown volume and twin-cuts are crossings of
two pairs of nodes, twin-cuts do have a natural skew. As
shown in Figure 10, a left-twisted twin-cut causes a
right-twisted skew between the strands and vice versa. It
is this skew that causes two strands which are intercon-
nected by twin-cuts to turn into a double helix.
Figure 10. Simplified sketch of a twin-cut visualizing its
Another example of a twin-cut structure is the circuit
diagram of an SR-Flipflop (see Figure 12) which will be
treated in section 6.2.
6. Concretizing AA: Symbolic Systems
The language of AA, as defined up to this point, de-
scribes the topological structure of abstract discrete spa-
tial systems. It can now step by step be concretized to-
wards special discrete systems by introducing new Ak-
tons as subsorts of the abstract Akton sorts. The new
subsorts inherit the properties of their hierarchical an-
cestors and may be provided with additional properties.
However, none of the additional properties may ever
conflict with the inherited properties. There are three
main ways how to concretize abstract AA. Most trivially,
it could be done by introducing new subsorts and desig-
nating them only symbolically, i.e. without adding new
properties. Typical examples are wiring diagrams of an-
alog or digital circuitry.
The new akton sorts can be provided with functionality
by extending the Interface hierarchie giving rise to func-
tional systems. Finally, the Akton and the Interface
hierarchie can be provided with a metric giving rise to
concrete spatial systems. Each of the three modes will be
studied in the sequel.
6.1. Digital Circuit Description
As well-known, every binary function can be realized by
a single sort Nand or by a single sort Nor. Usually how-
ever, several sorts are applied. Here, we introduce the
subsorts And, Or, No t and Wire, where Not denotes the
inversion function and Wire the 1-function. And and Or
are subsorts of Join, and Not and Wire are subsorts of
Link. The extension of the Akton hierarchy by concrete
subsorts is depicted in Figure 11(a). The set of digital
subsorts is designated by dA.
Figure 11. Extending so rt Join of the akton hierarchy (a) by
the subsorts And and Or, and sort Link by the subsorts Wire
and Not, keeping the Interface hierarchy (b) unchanged,
turns AA into a digital circuit description language.
Definition D.1 (Digital Akton Sorts)
The set dA of digital akton sorts is defined as:
dA:={Up, Set} {Fork} {And, Or} {Wire, Not}
{Down, Off} {CS}, where
abstract: dA
abstract(x):= Join, if x{And, Or}
abstract(x):= Link, if x{Wire, Not}
abstract(x):= x, if x{Up, Set, Fork, Down, Off, CS}
Definition D.2 (Digital Akton Terms)
The set of digital akton terms dA+ is inductively defined
as the smallest set satisfying:
dA dA+
x,y dA+: (x>y) dA+
x,y dA+: (x/y) dA+
The properties of the digital circuit description lan-
guage defined on the set of digital akton terms dA+ are
subsequently demonstrated by three digital circuits and
their description by an AA-program.
6.2. SR-Flipflop
The SR-Flipflop shown in Figure 12 serves to emphasize
the important property of AA to analytically describe
feedback circuits. With this property AA overcomes the
severe restriction of register-transfer-level programming
languages which only describe the logical expressions
between two storage cycles [7].
Figure 12. Symbolic Circuit diagram and AA-program of
an SR-Flipflop.
6.3. Half- and Full-Adder
The examples of a half- and a full-adder shown in Fig-
ure 13 serve to demonstrate how more complex systems
can be built up from low level systems either by desig-
nating Akton terms by abbreviations or by concealing
them into Aktons. Systems of arbitrary complexity can
thus be treated just as every simple system.
Figure 13. Symbolic Circuit diagrams and AA-programs of
a half-adder HAd (a), and a full-adder FAd (b). The
structure and the Akton term on top of Figure 13 show the
circuit diagram of the half-adder HAd (a), and those at the
bottom the circuit diagram of the full adder FAd (b). Both
Akton terms are of sort B+. F2 designates a two-Fork
structure doubling 2 inputs into 4 outputs.
The structure and the Akton term on top of Figure 13a
show the circuit diagram and the AA-program of a
half-adder HAd. Figure 13b at the bottom depicts the
circuit diagram and AA-program of a full adder FAd.
Both Akton terms are of sort B+. F2 designates a
two-Fork structure doubling two inputs into four outputs.
7. Concretizing AA: Functional Systems
In the previous section we concretized AA symbolically,
i.e. only by extending the Akton hierarchy by additional
Akton sorts, however without providing them with any
extra properties. This step alone was sufficient to create a
opyright © 2013 SciRes. IJCNS
programming language for symbolic system description
and design.
We now proceed to provide the newly introduced Ak-
ton sorts with functional properties. This enhances AA to
a general data processing language. The enhancement is
achieved by introducing data values as subsorts of the
interface sort Pin and by defining functions between the
input and the output of the Aktons. This way several
kinds of functionality can be implemented, e.g. digital or
analog functions or even both together. Moreover, be-
cause AA can be equipped with all the elementary func-
tions of analog or digital circuitry and because these
functions can arbitrarily be composed to more complex
functions, a plethora of low or high level programming
languages can be created this way.
It should also be noted that abstract AA does not im-
pose any restrictions on the system behaviour. Since ab-
stract AA does not make use of the notion of states, the
execution of an AA-program behaves flow-controlled or,
if digital data processing is introduced, as data driven [8].
However, the data driven behaviour can easily be turned
into a state driven one by starting the evaluation of a
succeeding akton only after the output of the preceding
aktons is fully defined. A clock driven behaviour, i.e. the
behaviour of most computers, can then be achieved by
supplying each akton term with a storage function and by
fitting the Akton evaluation time into the clock cycle.
7.1. Digital Data Processing
In this section, we concentrate on data driven digital data
processing. To this end, we extend sor t Join of the Ak-
ton-hierarchy not just by the subsorts And and Or, and
sort Link by the subsorts Wire and Not (see figure 14a).
Figure 14: Extending sort Join by the subsorts And and Or,
and sort Link by the subsorts Wire and Not, turns AA into a
digital circuit description language (a). Further extending
sort Pin of the Interface hierarchie (b) by the subsorts 0,1,#
and defining the functions in and out by them, as done by
definition D.5, generates a digital data processing language.
In addition, we now expand sort Pin of the inter-
face-hierarchy (Figure 14b) by the digital values (0, 1,
#), where value # indicates the state of the Pin before
processing. Each of the functions contained in an Akton
can only be evaluated, if its individual input data are
available. Moreover, since the evaluation of the functions
takes a different finite amount of time, the output of an
Akton is always delayed and if there are several output
Pins, they are never exactly synchronized.
Definition D.3 (Digital Interface Sorts)
The set of digital interface sorts dI is defined as:
dI := {0, 1, #} {ε}, where
abstract: dI
abstract(x):= Pin, if x{0, 1, #}
abstract(x):= ε, if x=ε
Definition D.4 (Digital Interface Terms)
The set of digital interface terms dI* is inductively de-
fined as the smallest set satisfying:
dI dI*,
x,y dI*: (x/y)dI*
Definition D.5 (Digital Functions in, out)
The digital functions in, out: dA+
dI* are defined as:
out(And) :=1, if in(And)=1/1
out(And) :=0, if (in(And)=0/x or in(And)=x/0)
out(And) :=#, if (in(And)=#/x or in(And)=x/#)
out(Or):=0, if in(Or)=0/0
out(Or):=1, if (in(Or)=1/x or in(Or)=x/1)
out(Or):=#, if (in(Or)=#/x or in(Or)=x/#)
out(Not):=1, if in(Not)=0
out(Not):=0, if in(Not)=1
out(Not):=#, if in(Not)=#
out(Wire):= x, if in(Wire)=x
out(Fork):=x/x,if in(Fork)=x
out(Head):=x, if in(Head)=ε
out(Tail):=ε, if in(Tail)=x
Establishing the transmission of data between
Next-related Akton terms extends AA from a description
language of static systems to a description language of
dynamic systems. In particular, this renders it possible to
describe positive or negative feedback, i.e. either the
storage of data or the repetition of functions. The fol-
lowing definition can therefore be regarded as a basic
rule of computation.
Definition D.6 (Digital Data Transmission)
The digital data transmission is defined as:
x,y dA+: in(y):=out(x), if (x>y)
7.2. Systems Behaviour
The behaviour of linear AA-programs like those of the
half-adder or the full-adder is immediately understand-
able just by providing them with data and then tracing
their execution step-by-step. However, a cyclic program
is not that simple. For this reason we inspect the behav-
iour of a feedback cycle as depicted in Figure 15. Part (a)
shows the structure of the feedback cycle and its program,
part (b) the behaviour. Initially, every Akton is in the
undefined state #. Supplying state 0 to the input, i.e.
Copyright © 2013 SciRes. IJCNS
out(Set1):=0, results in a steady state, where all Aktons
are defined. Recall that according to Def. A.14 Off
transfers its input to the output of Set. If subsequently
state 1 is supplied to the inp ut , i.e. out(S et1):=1, the
feedback cycle starts oscillating.
Figure 15: Structural and behavioural representation
of a feedback circuit. Part (a) depicts the planar and
linear representation of the circuit. Part (b) describes
the states of the circuit components. In order to
enter into a continuous loop the circuit first needs to
be initialized by out(Set1):=0 followed by out(Set1):=1.
8. Concretizing AA: Metric Systems
Recall that a correct AA-expression completely describes
the topological structure of a physical system no matter,
whether the structure is spatial, planar or linear, and no
matter, whether the system representation is abstract or
concrete. In order to fully reconstruct the metric of a
given physical system from its abstract topological
structure, we only have to reintroduce the original com-
ponents and to eliminate the spatial cuts. This is the nov-
el benefit of AA which has not been available up to now.
However, our actual objective is more ambitious: We are
aiming at a powerful tool to construct new systems and
to adapt their concrete structures to arbitrary technical
requirements. This makes it necessary to introduce a
frame of reference by which the metric of the compo-
nents, i.e. their shape and size, can consistently be de-
A simple frame of reference can be introduced by as-
suming all basic Aktons having the same quadratic re-
spectively cubical size.
The dimensionality of the frame of reference can be
expressed by the directions a wayfarer would have to
take to follow the directions of the AA -structure. Three
directions are needed for a planar frame, i.e. straight, left
and right, and two more for a spatial frame, i.e. up and
down. Since there is no difference between a planar and a
spatial frame of reference except for the number of di-
rections, we confine ourselves to a planar one.
8.1. Rectangular Metric Systems
The planar directions are introduced into AA by means of
structured subsorts of the sorts Fork, Join and Link.
There are three basic aktons to each subsort,
lr,Fls,Fsr}, {Jlr,Jls,Jsr}, {Ls,Ll,Lr}, where F, J, L are ab-
breviations of Fork, Join and Link, and the subscripts s, l,
r are abbreviations of straight, left and right. (In a
3-dimensional reference frame the number of basic Ak-
ton would rise to 10 for subsorts of Fork and Join and to
5 for subsorts of Link.) The extended hierarchy of Akton
sorts is shown in Figure 16 (a). A uniform metric is
achieved by extending the interface of sort ε by a subsort
Gap representing an empty place with the same width as
sort Pin, as shown in Figure 16 (b).
igure 16. Extending the sorts Fork, Join and Link by
An arbitrary metric Akton term built up from quadratic
s, L, Lr}
if x{Flr, Fls, Fsr},
tructural interface is
tric Interface Sorts)
d as:
face Terms)
ductively de-
structured metric subsorts (a) and introducing Gap as an
identically sized companion of Pin (b) turns AA into a
metric spatial programming language.
sic components will generally not have a quadratic
shape. However, an Akton term may always be given a
rectangular shape by adding or deleting basic Akton
terms being subsorts of CS. Without loss of generality it
can therefore be assumed that every metric Akton term
can always be given a rectangular shape.
Definition M.1 (Metric Akton Sorts)
The set of metric akton sorts mA is defined
mA := {Head} {Flr, Fls, Fsr} {Jlr, Jls, Jsr} {L l
{Tail}, where
abstract: mA
abstract(x):= Fork,
abstract(x):= Join, if x{Jlr, Jls, Jsr},
abstract(x):= Link, if x{Ls, Ll, Lr}
As defined in A.7, an abstract s
ther empty or contains Pins. With the metric refine-
ment an interface of an Akton term may now contain
Pins, Gaps or both.
Definition M.2 (Me
The set of metric interface sorts mI is define
mI := {Pin} { Gap }, where
abstract: mI
abstract(x):= x, if
abstract(x):= ε, if x=Gap
Definition M.3 (Metric Inter
The set of metric interface terms mI+ is in
fined as the smallest set satisfying:
opyright © 2013 SciRes. IJCNS
+: (x/y)
introducing a metric each Akton
hile i=Gap/j,
t the metric interfaces of the four
5 (Functions si)
}, sidei mI+,
Pin,Gap}, if z=Head
{Gap,Pin,Pin,Pin}, if z=Jlr
Next we defins the out-
defined as:
Definiti ngularton Terms)
(x>y) mA+,
of rigid rectangular Aktons
M.8 (Tilt Functions tl (anticlockwise), tr
s tl, tr: mA
mAare defined as:
output directions to the Body Aktons
On the other hand, by
rm now attains an individual size. Even the adjacent
sides of two Next-related Akton terms x>y may differ in
size, because of different numbers of Gaps above and
below their proper interfaces. These Gaps can formally
be removed by introducing two functions, called atrim
and btrim, where atrim eliminates all Gaps above a first
Pin and btrim all Gaps below a last Pin. The function
composition of atrim and btrim, i.e. the function trim,
produces an interface beginning and ending with a Pin.
This Interface can be interpreted as a plug. Plugging the
Interfaces of two Next-related terms x>y then means to
shift both terms into their correct relative Juxta-position.
Definition M.4 (Functions atrim, btrim, trim)
The functions atrim, btrim, trim: mI+
mI+ are
ductively defined as:
atrim(i) := atrim(j) w
btrim(i) := btrim(j) while i=j/Gap
trim := atrimbtrim.
In order to extrac
des of a rectangular Akton term we introduce the func-
tions si. The sides are clockwise enumerated, s0 denoting
the left-hand side. We first define the sides of the basic
Definition M.
The functions si: mA+
i{0,1,2,3} are defined as:
{Pin,Gap,Gap,Gap}, if z=Tail
{Gap,Gap,Gap,Gap}, if z=CS
si {Pin,Pin,Gap,Pin}, if z=Flr
{Pin,Pin,Pin,Gap}, if z=Fls
{Pin,Gap,Pin,Pin}, if z=Fsr
{Pin,Pin,Pin,Gap}, if z=Jls
{Pin,Gap,Pin,Pin}, if z=Jsr
si {Pin,Gap,Pin,Gap}, if z=Ls
{Pin,Pin,Gap,Gap}, if z=Ll
{Pin,Gap,Gap,Pin}, if z=Lr
e the structural relationbetween
put and the input of each metric subsort.
Definition M.6 (Metric Functions in, ou
The metric functions in, out: mA+
mI+ are
out(Ls) ):= s2, if in(Ls)= s0
out(Ll):= s1, if in(Ll)= s0
out(Lr):= s3, if in(Lr)= s0
out(Fls):= s1/s if in(Fls)=s 0
out(Fsr):= s2/s3, if in(Fsr )= s0
out(Flr):= s1/s3, if in(Flr)= s0
out(Jsr):= s2, if in(Jls )= s1/s
out(Jls):= s2, if in(Jsr )= s0/s3
out(Jlr):= s2, if in(Jlr )= s1/s3
out(Head):= s
in(Tail):= s0
on M.7 (Recta Ak
The set of rectangular akton terms mA+ is i
defined as the smallest set satisfying:
mA mA+
x,y mA+:
if trim(out(x)) = trim(in(y))
In order to fold a network
to a planar area, we need a means to turn the Akton
terms into another direction. This can be achieved by
introducing two tilt functions tl and tr, where tl turns an
Akton term orthogonally to the left and tr orthogonally to
the right.
The tilt function++
t(x>y) = t(x)>t(y),
t(x/y) = t(x)/t(y), t
tl(tr(x)) = tr(tl(x)) = x
tl(tl(x)) = tr(tr(x))
Assigning input/
tends them to three subsorts each. The basic sorts of
structured metric Aktons are depicted in Figure 17.
igure 17. Basic sorts of structured metric Aktons. There
bstract multiple Links, multiple Forks and multiple
are three subsorts of Fork, Join and Link each, which differ
in the direction they proceed. Their paths turn either to the
left, to the right or proceed straight. In addition, the basic
metric aktons of D (Down), U (Up) and CS are shown. The
metric aktons Set and Off are skipped.
ins have already been defined by Defs. A.21, A.22 and
A.23. These structures are important in the layout proc-
ess of digital systems.
Copyright © 2013 SciRes. IJCNS
According to the rectangular metric being assumed
here, there are three structures of metric multiple Links,
which can be generated making use of the basic metric
subsorts Ls, Ll, Lr. While a straight multiple Link can just
be generated by Juxta-relating Links to columns and then
Next-relating the columns to strips of any finite length,
tilted multiple structures need to be realized by Jux-
ta-related structures of single chains of ascending and
descending length which together form a square. The
centerpiece of a left-tilted chain is an element of sort Ll
(as shown later on in Figure 19(a)) and the centerpiece
of a right-tilted chain is an element of sort Lr.
Definition M.9 (Metric Multiple Links)
The sets of metric multiple Links are recursively defined
x(1):= Ll
N: x(i):=x(i)/((i-1)*Ls>Ll>tl((i-1)*Ls)), if 1i
x(1):= Lr
N: x(i):= x(i)/((i-1)*Ls>Lr>tr((i-1)*Ls)), if 1i
Metric multiple Fork as well as Join structures have
some special features. Firstly, a metric multiple Fork
cannot be constructed from a basic element Flr, because
the two outputs of such a structure are ordered reversely.
The same applies for the two inputs of a metric multiple
Join if it constructed by a basic element Jlr.. Secondly, as
already stated by Defs. A.21 and A.22, regular abstract
multiple Forks and Joins are built up by crossings and
therefore are either left- or right-handed. This chirality is
preserved, if a metric is introduced. However, coming
along with the metrisation is a path orientation which in
case the chirality is left-handed is either oriented straight
or left-tilted, and in case it is right-handed is either ori-
ented straight or right-tilted. This gives rise to two mul-
tiple Fork constructs and two multiple Join constructs as
represented in Figure 18.
Figure 18 further demonstrates that the constructs can be
concealed to individual Aktons (see A.23), and since
each side of the Aktons contains at most one Pin and a
via they can be reduced to unit square size. This way the
concealed structures get the appearance of a combination
of a Link and a via. The concealed and minimized struc-
tures are designated by Fl,l and Fr,r, resp. Jl,l and Jr,r,
where the first index defines the chirality of the via and
the second the orientation of the Link.
Fl,l:=conceal(tl(D/L s)/(F ls>Ll)), Fl,l
Fr,r:=conceal((Fsr>Lr)/tr( L s/D)), Fr,r
Jl,l:=concea l((tr(U/L s)/(tr(Ll)>J ls)), Jl,l
Jr,r:=conceal((tl(Lr)> Jsr )/tl(L s/U)), Jr,r
Figure 18. A concealed structure can be scaled down into a
unit sqare rectangle if no Pin and no via points in the same
Definition M.10 (Metric Multiple Forks)
The sets of metric multiple Forks are recursively defined
x(1):= Fl,l
x(i):=x(i)/((i-1)*Ls>Fl,l>tl((i-1)*Ls))>tl(Ls^i/CS)/U^i , if
x(1):= Fr,r
+>tr((i-1)*Ls))>tr(CS/Ls^i)/U^i ,
if 1i
Definitions M.11 (Metric Multiple Joins)
The sets of metric multiple Joins are recursively defined
x(1):= Jl,l
x(i):=x(i)/((i-1)*Ls>tl(Jl,l)>tr (( i- 1)* Ls))> tr(Ls^i/CS)/U^i ,
if 1i
x(1):= Jr,r
N:x(i):=x(i)/((i- 1)*Ls>tr(Jr,r
+)>tl (( i- 1)* Ls))> tl(CS/Ls
opyright © 2013 SciRes. IJCNS
^i)/U^i , if 1i
8.2. Layout of Metric Triple Links and Triple
Metric Links and metric Forks are important structures
for the planar layout of electronic systems, because they
allow treating a bundle of parallel connections like a
single one. Metric Links are particularly indispensable
for folding a straight Akton structure into a desired planar
layout. Each time the structure is to be expanded a metric
multiple Link of subsort mLs has to be inserted, and each
time it is to be tilted to the left or to the right either
subsort mLl or mLr has to be inserted. Figure 19(a)
shows the structure and the program of a left-tilted strip
of three chains of Links. Figure 19 (b) shows the
structure and the program of a multiple Fork of three
strips. Sort Fl,l represents a unit square concealment of
the term (tl(D/Ls)/(Fls>Ll)).
Figure 19. Two layout examples: (a) Structure and program
of a strip of three chains turning left, and (b) structure and
program of a strip of three Forks forking straight and left.
Mind that in the second Akton term of program (b) each Up
relates exactly to one Down concealed in Fl,l according to
Def. A. 21 and that also the straight Links Ls are metrically
fixed, thus giving rise to its rectangular structure.
9. Conclusions
Every discrete physical system can be converted into a
spatiotemporal network of nodes by abstracting from
functions and metrics, as demonstrated in this paper.
Apparently, this kind of formal description of spatio-
temporal structures has not been considered up to now.
Any two nodes of an abstract network are either de-
pendent or independent. Converting a three-dimensional
network into a linear one requires two homeomorphic
cuts, a first one that planarizes the network and a second
one that linearizes it into a sequence of symbols. Total
execution of the topological program reconstructs the
original spatiotemporal network. An abstract network
represents an algebra with a set of fundamental sorts.
These sets can be refined and concretized by subsorts,
providing them with a wealth of features.
Treated this way, the spatiotemporal approach offers
several surprising properties:
1) The linearization of spatiotemporal networks by
means of homeomorphic twin-cuts directly generates the
four letter code of DNA, which even naturally explains
the skew of the double helix.
2) Extending sort Join of the Akton hierarchy by Boo-
lean subsorts like And, Or, and sort Link by subsorts like
Wire and Not turns the algebra into a digital circuit de-
scription language. Thus, even storage elements like
Flipflops can be represented, i.e. the proper obstacle of
the register transfer level calamity.
3) Reintroducing the functionality of the physical sys-
tem by means of ternary interface subsorts turns the ab-
stract programming language into a flow-controlled gen-
eral data processing language. The functions may either
be analog or digital. The flow-control can be restricted
by partial synchronization or by total clock-control. The
latter squeezes digital data processing into the
word-at-time scheme of the von-Neumann-architecture,
as most digital programming languages of today.
4) Reintroducing the metrics of the physical system,
i.e. the shape and size of the components, turns abstract
Akton-Algebra into a novel hardware system construc-
tion language.
5) An important property regards the layout of elec-
tronic circuitry. Every electronic circuit can always be
realized by two layers only.
6) The metric algebra of AA can be extended to a third
dimension by introducing two more tilt functions (see
Def. M.8) for the vertical direction. It is then possible to
describe devices of any shape or size by employing the
dependency rules of table 1.
7) The original von-Neumann-architecture can simply
be restored by pulling all stored information from the
AA-network and collecting it in a separate main memory,
and by placing a set of accumulation registers in be-
Copyright © 2013 SciRes. IJCNS
Copyright © 2013 SciRes. IJCNS
8) Aside from the considerable technical improve-
ments of the AA-network-architecture the enormous
augmentation in processing speed should be mentioned.
10. Acknowledgements
There are several colleagues who notably supported me
during my long work towards Akton-Algebra. One of the
first was Rudolf Albrecht, Innsbruck, who showed great
interest in my approach when we first met in 1993 in
Lessach and with whom I have had many very fruitful
discussions since then. A next one to be mentioned
gratefully is Walter Dosch, Lübeck, for several lucid
comments. Moreover, I am most grateful to Bernd
Braßel, Kiel, with whom I was in an ongoing email dis-
cussion for a couple of years. Finally, I am much obliged
to Manfred Broy for some valuable hints.
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