the parallel execution of operators Section 4.4. These composite operators are always defined.
The class A will be handled first, since the opera- tors in T are distinguished by a special feature
Section 5.3.
It was shown for each of the classes L, C, V, R
separately that generally two operators are non- commutati6e with respect to o if both belong to
the same class. The same holds if both operators are taken from different classes in A. Altogether
we would have to check six pairs of operators from different classes: L
i
oC
k
, L
i
oV
k
, …,V
i
oR
k
. In order to avoid boring legwork we only consider
L
i
oC
k
, L
i
oV
k
, and L
i
oR
k
. The three operators C
k
, V
k
, and R
k
share the property that their execution can be simplified by some new information L
i
coming in before they are started, whereas the belated arrival of the same information need not
offer that advantage. Hence, in the general case L
i
oC
k
, L
i
oV
k
, and L
i
oR
k
are noncommutative; similar arguments hold for the rest.
Both laws of associativity hold: XoYoZ = XoYoZ
since both expressions mean that Z, Y, X act upon the system in that temporal order; and
X + Y + Z = X + Y + Z since both sides of this equation mean that X, Y,
and Z act in parallel. Next, both laws of distribu- tivity hold:
X + YoZ = XoZ + YoZ, ZoX + Y = ZoX + ZoY.
The reason for this lies in the definition of ‘ + ’. The mechanism can be seen from an example:
Let X
+
Y oZ mean that a process of learning,
Z, is performed first, and afterwards the conclu- sions X and Y are performed in parallel; then
XoZ
+
YoZ means that the two operations
learn Z, conclude X and
learn Z, conclude Y are
done in parallel. This is merely a difference in style, but the outcomes are equal. Similar argu-
ments hold if X, Y, Z are taken from other classes in A.
As a result, we have an algebra A =A, +, o
such that
A,
+
is a commutative semigroup,
A, o
is a noncommutative semigroup,
A,
+
, o is a semiring.
By the definition of ‘ + ’ we have X
+
X = X for every X, and the ‘addition’ is idempotent. There-
fore A is a semiring in which the addition is commutative and idempotent, or an ACI-semiring
for short. Properties of semirings were studied, e.g. by Steinfeld 1959, Kuich and Salomaa
1986, and Morak 1997; survey is given by Hebisch and Weinert 1996. The particular struc-
ture of ACI-semirings, with its relations to the theories
of semigroups
and semilattices,
is analysed in detail by Haftendorn 1975.
5
.
2
. Similarities between A and operator algebras
in quantum theory Any attempt of a bridge-building between
quantum theory and one or another discipline or problem field will require strong precaution. In
the context of a possible extension of quantum theory Atmanspacher 1997 comments: ‘‘Quan-
tum mechanics in its contemporary understand- ing, though very advanced in many respects, is
still a theory of the material world and nothing else’’. Gro¨ssing 1996 warns against ‘‘magical
mystifications of quantum theory’’, and Omnes 1990 focuses a specific kind of misunderstanding
when he is suspicious at ‘‘the view according to which quantum mechanics is a part of informa-
tion theory’’.
Any observation begins and ends in the observ- er’s head Section 3.1. For heuristic purposes
only not motivated by metaphysical assumptions an intermediate step is proposed here: Just try and
test to what extent physical processes can be described as processes of interpretation — and to
what extent such tentative descriptions will offer a surplus of new insight. On the basis of such a
style of description we can state that,
an unmoved billiard ball hit by another ball ‘interprets’ this event by starting in a certain
direction
a particle in a field ‘interprets’ this field by following a specific path,
water masses ‘interpret’ a dry valley by follow- ing the curves defined by that valley whereas
they would ignore a prohibitory sign.
Interpretation can have two facets: we can
‘read something out’, and understand a text, a situation, etc. in a specific sense, or
perform an ‘active’ or even ‘expressive’ inter- pretation, when e.g. orchestral music is pro-
duced, given only the score. Viewing at nondeterministic physical processes,
too, we find that both facets have their counter- parts in physical processes
10
. ‘Interpretation’ will become a comprehensive term, including both
physical and cognitive processes, and can serve as a common basis for studying relationships with a
minimum of prior assumptions.
Because of their special properties the operators
in T will be considered later Section 5.3. The operator algebra A as defined in Section 5.1
shows some analogies with quantum theory, but also different features. Both in the description of
cognitive processes proposed here and in quantum theory a system is characterized by states and
transitions between these states; in both cases the transitions are represented by operators, and these
operators form operator algebras. For the sake of brevity we simply let Q denote some operator
algebra from quantum theory, such that we have to compare Q and A.
Now Q forms a commutative semigroup for +
and a noncommutative semigroup for the juxta- position of operators, and hence a semiring. Both
A and Q are
additi6ely commutati6e semirings.
Operators from an important subclass of Q, the projection operators, are idempotent P
2
= P.
Similarly, operators in A are idempotent with respect to o: we have XoX = X for all X A. The
semigroups A, + and A, o, and as well their counterparts in Q, each have an identity or neu-
tral element; therefore these four semigroups are monoids. Another important subclass of Q, the
selectors, may suggest analogies with operators in V Section 4.2.3. Disparities lie in the fact that in
Fig. 2. Illustration for a non-idempotent operator, as ex- plained in the text.
A there is no underlying structure like Hilbert space; in A the difference of two operators is not
yet defined because of its particular difficulties the latter point will be handled separately in
Section 5.4.
5
.
3
. Special properties of A Since A is defined by A = A T, we now
have to study the operators in T under algebraic aspects. Part of these operators, which describe
the generation of a new pattern from an existing one, are idempotent T
i 2
= T
i
, whereas the other are not. This can be demonstrated by two differ-
ent transformations of simple graphs, that is finite undirected graphs without multiple edges. For a
first example, assume that in a graph G
1
two vertices 6 and 6 are not connected by an edge,
and that T
1
just inserts an edge between 6 and 6. Then a second application of T
1
will bring about no change, since an edge between 6 and 6 already
exists and multiple edges are excluded. As an example for the contrary, consider the graph G
2
which has exactly one vertex of degree 1 one edge leaving that vertex, and an operator T
2
which introduces a new vertex that is connected with the
unique vertex of degree 1. Then T
2
can be applied repeatedly, always generating a graph different
from the earlier ones, and T
2 m + 1
T
2 m
for all m ] 1 Fig. 2.
Apart from the fact that part of the operators
in T are not idempotent, A has the same alge- braic properties as A. Particularly A is an addi-
tively commutative semiring. A united with the subset of all idempotent operators in T forms an
ACI-semiring see Section 5.1.
5
.
4
. Time-windows and in6erses of operators As remarked before Section 5.2, expressions
like ‘ − X’ and ‘X − Y’ are not yet defined for operators X, Y A. A definition for these ex-
10
Interpretation does not imply arbitrariness. Apparently this misunderstanding is fostered by the disastrous tradition
which wants to split the scientific disciplines into ‘Naturwis- senschaften’ natural sciences and ‘Geisteswissenschaften’ let-
ters, arts, humanities, cultural sciences. Models from classical mechanics, where identical settings lead to identical outcomes,
are also comprehended here: in this case the interpretations for identical settings simply will be the same.
pressions, which is to be compatible with the ‘usual understanding’ at least as far as possible,
requires special carefulness. Since X
+
Y denotes that the two operators act in parallel, we may be
tempted to introduce a neutral element ¥ such
that X +
¥=¥+X=X for all X A.
This means that in a system with parallel pro- cesses one component can undergo the transfor-
mation X, whereas in another component no change occurs, such that the outcome only de-
pends on X.
Now we can approach a definition of − X by the property
X + − X = ¥
1 and introduce the short notation X − Y instead of
X
+
− Y
. Eq. 1 may mean that in two compo- nents of the system simultaneously two operations
occur which exactly compensate each other, such that the system remains unchanged.
But a closer look will reveal that there are three problems hidden behind Eq. 1:
1. Keeping in mind that we are studying cogni- tive processes, is it really possible that two
simultaneous operations exactly compensate each other — or will their remain some
‘residue’ or side effect?
2. Given an operator X, is − X uniquely defined? 3. What is the exact meaning of ‘simultaneously’?
If L
i
, L
k
L and L
i
+
L
k
= ¥ or L
k
= − L
i
, this may mean that two acts of learning are
performed simultaneously by two independent components, and a natural assumption is that the
information obtained by L
k
is the negate of the information received by L
i
. Thus, neither L
i
nor L
k
will be accepted as a new information, and in this sense the system remains unchanged. But,
depending on the global organization of the sys- tem, it may be registered that two contradicting
messages came in and that at least one of the two underlying sources is not reliable
11
. If the definition of − L
i
is to be unique then −
L
i
must be defined as the negate of L
i
. All other possibilities which also would leave the system
unchanged must be excluded here. For example, let X be the process of accepting the inequality
x B
1
, and Y the same with the incompatible statement x \
2
, then it is possible with use of some background knowledge to recognize that
contradiction, but just the same would be true if the second statement would read x \
3
. In a similar manner the meaning of − X can be
identified if X C or X V. If C
i
and − C
i
are two processes of conclusion performed by differ-
ent components with opposite results, then no direct or immediate action may be taken, whereas
a discrepancy between the underlying local data may be noticed. The same reason may lie behind
the simultaneous occurrence of V
i
and − V
i
, two acts of valuation of the same object with oppo-
site outcomes; − V
i
means that the valuation V
i
is not ascribed to the object under consideration. On
the other hand, it is impossible to define − R
i
or −
T
i
correspondingly
12
. As an intermediate result we can state that
operators X L, C, or V have inverses in the sense of Eq. 1; it depends on the overall organization
of the system whether the simultaneous execution of X and − X leaves the system unchanged in a
strict sense or whether possibly some additional information about inconsistencies between exter-
nal sources or between internal data or processes is obtained.
It is still open how the term
‘
simultaneous
’
is to be understood in the present context. Any concept
of a continuously flowing time will be inadequate here. There must be some internal regulation
which defines discrete and finite intervals of time with the following property demonstrated by the
example of two acts of learning: If and only if two operations L
i
and − L
i
are performed within
12
It follows from noncommutativity that an inverse of X
with respect to o cannot be consistently defined for X L, C, V. An inversion of
X R, T can be possible under special
conditions, and it can make sense in a system with parallel processing, where different components can take advantage
from unequal representation schemes or different tentative steps.
11
The evaluation of cues concerning the reliability of het- erogeneous sources is a central issue in ‘governmental intelli-
gence’ Watson et al., 1990. For a formal treatment within the theory of doxastic states see Adamatzky 1999.
such a time-slice, the system will recognize the contradiction between the two underlying messages
coming from outside and act correspondingly as described above. In a similar manner, a contradic-
tion between C
i
and − C
i
or between V
i
and − V
i
will be detected if and only if the two operations are executed within one such time-slice.
To sum up, two operations are called simulta- neous if they are both performed within the same
internally defined discrete time interval time-slice. The ‘length’ or ‘duration’ of such a time-slice
measured on the basis of our common understand- ing of time or of a traditional ‘reference time’ may
depend on one or another parameter:
It may be different for the three classes L, C, and V.
It may vary with technical side conditions, like the lengths of incoming sequences of signals or
the complexity of internal derivation processes.
It may change within the lifetime of the system.
There may be different ‘local time scales’ in disjoint parts of the system.
6. Related topics and possible applications
