Directory UMM :Data Elmu:jurnal:B:Biosystems:Vol54.Issue3.2000:

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Towards a closed description of observation processes

Dieter Gernert

Department of Economics,Technical Uni6ersity of Munich,Arcisstrasse21,D-80333Munich,Germany Received 24 May 1999; received in revised form 26 October 1999; accepted 5 November 1999

Abstract

A closed description of observation processes must necessarily include the observer, too. In order to find a basis for such a unified description, an operator algebra is developed which enables a formal description of at least a significant majority of cognitive processes. It is found that this operator algebra, which is a noncommutative semiring of a type already known in literature, has astonishing similarities with the usual operator algebras in quantum theory. Combined with a method for the formal treatment of perspective notions, the representation scheme proposed here may open a chance for a unified description of a process itself together with the relevant cognitive processes on the observer’s side. By the description of parallel-processing systems on the same basis an operational definition of ‘internal time’ becomes possible. © 2000 Elsevier Science Ireland Ltd. All rights reserved.

Keywords: Observation; Measurement; Unified description; Operator algebra; Quantum theory; Cognitive process; Perspective notion; Semantics

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1. When is a process of observation completely described?

‘‘Any description of the world that someone advocates as being complete… must ‘close the circle’: it must include an account of how we come by that description. In particular, any phys-ical theory that claims such completeness must account for our experience as observers.’’ (But-terfield, 1995) The crucial role of the observer in quantum physics is generally accepted, and there is an abundant literature just on this topic. Prob-lems immediately arise when an ‘inclusion of the observer’ is understood in a strict sense, because this will inevitably imply a study and formal description of processes beyond the usual scope of physics.

‘‘Consciousness exists, but it resists definition.’’ (Flanagan, 1995) This quotation gives a cue to stop (for a moment) searching for a perfect defin-ition of terms like ‘consciousness’, ‘mind’, ‘men-tal’, etc. Rather, an attempt will be made here to supply a basis for a unifying description1_through a ‘change of perspective’, which will not only help circumventing that trouble with definitions, but, hopefully, will also bring about some new insight.

1_{The ‘problem of two languages’ and the necessity of a} ‘unified description’ are particularly emphasized by Lau-rikainen (1988).

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2. Outline of the proposal

After some necessary remarks on the process of observation, a ‘theoretical backbone’, a formalism for the description of cognitive processes and cognitive systems will be developed. This formal-ism is based on five classes of characteristic opera-tors, such that at least a significant majority of all cognitive processes can be represented by suitable combinations thereof (Section 4). It will be found that the algebrasAandA%_{formed by these}

oper-ators have astonishing similarities with the opera-tor algebras used in quantum theory (Section 5). It is proposed here to describe physical pro-cesses as propro-cesses ofinterpretation – e.g. a parti-cle interprets a surrounding field by following a specific curved path derived from that field. Thus the term ‘interpretation’ will become rather com-prehensive, including both physical and cognitive processes, and hence can be a promising candi-date for the basis of a uniform description re-quested here.

3. A new look on the process of observation

3.1. The process of obser6_{ation as a sequence of}

se6_{eral steps}

The term ‘observation’ will be used here in a broad sense including e.g.

any kind of measurement (of a single variable or of a set of variables) in experimental sciences

the scientific analysis of a work of art (e.g. in order to give an interpretation or to classify it as genuine or fake)

observations in the social sciences.

Any process of observation can be understood as an ordered sequence of several consecutive phases. In special cases, one or another of these phases may be less important; it may even remain unnoticed at a first glance, or appear as non-exis-tent. But for the sake of a general and compre-hensive theory (and in order to refute some frequent misunderstandings, see Section 3.2) the following sequence of steps should be kept in mind.

We must make a distinction between the initia -tor of an observation and the obser6_er as such. Both can be identical, but this is not the general case. The observer can be a human individual (like an anthropologist exploring a foreign cul-ture), a team, or a technical device (like an inter-planetary probe); following a proposal due to Matsuno (1985, 1989), we can also include physi-cal probes, like atoms, molecules, rays, waves, etc., entering a physical object under investigation.

Now, a process of observations can be charac-terized by the following phases:

1. The initiator decides what is to be observed and how this shall be done, simultaneously defining the context, e.g. the purpose of the observation, the relative importance of differ-ent features, etc. (definition phase).

2. If necessary, technical preparations are done (preparation phase).

3. The observation as such is performed: there is an internal state change within the observer, which, depending on the context, can be de-scribed as registration, learning, insight, etc. 4. The results — possibly after preprocessing (as

in the case of an interplanetary probe) or after a reformulation as an understandable report2 — are transmitted to the initiator of the observation.

5. The initiator receives and interprets the results. Of particular interest here is the fact that the role of the decision phase (no. 1) is frequently underestimated or ignored (for a possible mathe-matical tool see the Appendix).

3.2. The specific role of perspecti6_{e notions}

Perspecti6_{e notions} _{are terms which — beyond}

the well-known context-dependence of any mean-ing of words — require an explicit statement of the context. A simple example is the term ‘classifi-cation’: e.g. the chemical elements can be classified according to their atomic weights, spe-cific weights, electrochemical or radioactive prop-2_{For the ‘record}_/_{report antagonism’, the observer’s} neces-sity to formulate a report such that it will get a chance of being understood, see Gernert (1998).

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erties, etc. The tasks to classify a single object or to subdivide a given set can be accomplished only after the purpose of the classification or the rele-vant criteria have been disclosed. Of course, that indispensable context is often self-understood among the persons involved, but just this fact brings about an additional complication to be discussed below. The following remarks on per-spective notions, although of interest by them-selves, are a necessary preparation for the main part of this paper.

Four essential perspective notions (or pairs of such notions)

1. meaning and interpretation, 2. complexity,

3. pragmatic information, 4. similarity/dissimilarity

can be represented by the four corners of a (regu-lar) tetrahedron in three-dimensional space, or by a plane drawing as in Fig. 1. The six lines con-necting the corners are to show that each of the four entries is interrelated with each other3

. The meaning of a word, a sentence, a symbol, etc., as well as the process ofinterpretationand its outcome, depend on the situation, the historical and social context, and the purpose of the analy-sis. Without any doubt, thecomplexityof a struc-ture will depend (among other things) on the individual education and training; what seems to be very complex to a newcomer may be simple for an expert. Just as demonstrated above by the example of classification, thesimilarityordissimi -larity of two objects or two structures can by defined only after exposing the purpose or the relevant criteria — within a given set, two objects can be similar or less similar according to their size, shape, appearance, function, etc.

The concept of pragmatic information claims that information can be understood only as an impact upon or an alteration within a receiving system brought about by the arrival of a message:

information begins when the channel has ended. For the argumentation in detail, the history of this concept, and a possible quantification only a reference can be given (Gernert, 1996). What is of interest here is the fact that ‘pragmatic informa-tion’ is a perspective notion, too. The same in-coming message may be considered highly important under one perspective, but irrelevant under a different view. If a message is regarded under the aspects of telecommunication engineer-ing, then we are back to the classical Shannon – Weaver theory (with its possible extensions and ramifications), but the same message can also be judged under aesthetical, pedagogic, or economic aspects, etc.

Exactly the same situation is found with respect to obser6ation. In the general case, the definition phase (Section 3.1) cannot be ignored or circum-vented; the necessary context must be specified and taken into consideration. The term ‘observa-tion’ also includes measurement, which can be regarded as a special case of observation. The situation in which the context is self-evident can be exemplified as follows. If somebody in a labo-ratory is told to ‘measure the temperature’ then the location and the time are clear, as well as

Fig. 1. Four eminent perspective notions and their relation-ships.

3_{The relations between complexity and meaning are studied} in Grassberger (1989) and in Atmanspacher et al. (1992), those between complexity and information in Atmanspacher (1997). For the connections between similarity and pragmatic infor-mation, and between these two concepts and the rest of the diagram see Gernert (1996).

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further circumstances, like the required precision, the available instruments, and the style of docu-mentation. As a consequence, the predominance of situations like that leads to the illusion of a ‘single-step measurement’: it is broadly assumed that only the name of one variable (or observable) must be said and then everything is clear.

By way of contrast, there are situations in which the definition phase, the identification of the context can no longer be skipped and, further-more, will require some labour and a methodical procedure. If, for instance, the similarity or dis-similarity between two structures from a given set is to be quantified, this will require the formula-tion of a suitable graph grammar that generates at least all the structures under consideration (Gern-ert, 1996). As soon as such a graph grammar has been specified, the requested similarity measure is clear. Now, the crucial point is that a graph grammar with the required properties always ex-ists, but it is not uniquely defined. The necessity to select exactly one graph grammar from the multitude of all suitable ones just stands for the compulsory specification of the context, like the goal pursued by that individual measurement of similarity.

A characteristic feature of perspective notions is the necessity of atwo-step proceeding, such that a definition phase precedes. A mathematical treat-ment is possible, but this is distinguished from the customary style: a measure will no more be sup-plied by a single formula, nor by several formulas, but by a proceeding in which, in addition, a mathematical structure must (and can) be set up that accounts just for the peculiarity of the per-spective notion. The above-mentioned illusion of a single-step measurement, the expectation that problem data simply can be inserted into a couple of formulas, turns out to be a widespread tacit assumption4

. It seems plausible that just this tacit

assumption is one of the causes why the concept of pragmatic information is accepted in such a reluctant and hesitating manner.

4. A systematic description of cognitive processes

4.1. O6_er6_iew

The formalism for the description of cognitive processes which is to be developed here may be of interest for cognitive science, too, but this is not the primary goal. Rather, the formalism is in-tended as an intermediate step; the central issue is the analogy with operator algebras in quantum theory (Section 5). We presuppose a cognitive system of any kind, that is a system capable of performing processes which can be interpreted as learning, concluding, forgetting, etc.; this system may be a human individual, an animal, or a technical device. Formally, the system is charac-terized by

1. states, which can vary in time and can be represented by vectors from a finite-dimen-sional real vector space Rn

(with a fixed posi-tive integer n), and

2. transitions, which lead from one state to an-other and which are represented byoperators. In this sense, an operator stands for an elemen-tary state change within the underlying system, e.g. for a single act of learning. Five classes of operators are proposed as follows:

1. Learning: L={L1, L2,...} C={C1,C2,...} 2. Conclusion:

3. Valuation: V={V1,V2,...} R={R1,R2,...} 4. Re6_ision_:

5. Tentati6_e_: _T₌_{_T₁_,_T₂_,...}

Some well-known types of cognitive processes, like forgetting, concept formation, or problem-solving, will not be recognized in this list, but it will be shown later (Section 4.3) that some charac-teristic cognitive processes can be represented by suitable combinations of these elementary opera-tors. At least a great majority of all cognitive processes can be represented in this way.

4_{Descartes planned to present a universal method for the} solution of problems, which can be roughly outlined as fol-lows: ‘‘First, reduce any kind of problem to a mathematical problem. Second, reduce any kind of mathematical problem to a problem of algebra. Third, reduce any problem of algebra to the solution of a single equation.’’ (Polya, 1962) In quoting this, Polya immediately attaches his reservations concerning the validity and reach of this general rule.

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4.2. The fundamental operators and their properties in detail

4.2.1. Operators in class L:learning

Any cognitive system has a system environ-ment. The operators in L(learning) describe pro-cesses by which a system accepts information from outside. The reception of some new information can lead to a change of the system behaviour or to a modification of its internal structure such that its repertoire for future behaviour is extended.

If the underlying systemSis fixed, we can write Li instead of Li(S); and LiLk(S), which means that S first undergoes the operation Lkand then

Li, can be abridged as LiLk. In the general case this composition of operators is not commutative: LiLk"LkLi. As an illustration two different tasks of learning are contrasted. If ‘serious’ material is to be learned, that is material with an internal structure, then the temporal order of its presenta-tion can be relevant, whereas in extreme cases of rote-learning the temporal order of input opera-tions may be irrelevant. This crucial feature of noncommutati6_ity _{will be discussed later (Section}

6.1).

4.2.2. Operations in class C: conclusion

Operations in class C, written as Ci,Ck,…, de-note conclusions derived from entries already ex-isting within the system. We writeCiCk(short for

CiCk(S)) for the fact that the conclusion Ck is performed first (with the knowledge contributed by Ck being stored in the system) and Ci is performed afterwards.

It would be irrelevant from a logical viewpoint which of two possible conclusions is achieved first. Here, however, only ‘realistic’ systems are consid-ered, such that labour, time, or energy consumed play a role, and hence commutativity can no longer be maintained. The new findings obtained by the conclusion operation Ck can simplify the subsequent operation Cisignificantly, whereas no such reduction of labour may occur if both opera-tors are applied in the inverse order. Therefore in the general case CiCk"CkCi. Which of the many possible operators in C will really be activated may be triggered by a process of valuation as described in the next section.

4.2.3. Operations in class V:6_aluation

An important class of operations occurring within cognitive systems can be united under the term ‘valuation’. The class V includes the opera-tors Vi,Vk,…

The object of a single process of valuation can be

a single item of knowledge already stored in the system

the present state of the system (when it is checked, e.g. whether a solution to a certain problem already has been found)

a recent state change (caused e.g. by some new incoming information)

a series of recent state changes (e.g. the result of a series of conclusion operations).

The result of an act of valuation can be

a predicate, like ‘true’/‘false’, ‘relevant’/ ‘irrele-vant’, etc.

a mathematical object, like a number, a vector, a matrix, a function, a network, or a system of relations

the identification of an item of information which fulfills given requirements.

There is a variety of reasons why valuation operators are necessary, and different purposes are pursued by them:

In ‘realistic’ systems a distinction must be made between relevant and irrelevant parts of incom-ing information.

In a similar way it must be decided whether the result of a series of conclusion operations is to be stored or not.

Among the items of knowledge already present in the system those must be identified and selected which are likely to fit to a given task.

It must be decided whether a certain strategy makes sense (or by which different one it should be replaced).

It must be recognized whether an operation called ‘revision’ (Section 4.2.4) or ‘tentative’ (Section 4.2.5) becomes necessary, or at least useful, and which will be the proper side condi-tions for such an operation.

Valuation means that an object is confronted with a predefined standard. In simple cases a procedure takes certain features of the given ob-ject as its input and supplies, e.g. an index, a

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score, or a Boolean value (like ‘acceptable’/‘not acceptable’). In the general case, however, the outcome of a valuation process is not necessarily a single value. Rather, it can take on the shape of a vector, a matrix, a function, etc. which represent the discrepancies between the ideal standard and the real situation with respect to several criteria (deviation profile). The result of a valuation pro-cess may also point forward to actions to be taken.

A necessary tool for many valuation processes is the measurement of the similarity or dissimilar-ity between two complex structures (see Section 3.2). For example, incoming information must be compared with the available information, in order to avoid redundant entries, but also with the information requirements of the system, in order to exclude irrelevant information. The search for some information that is likely to fit to a certain task can be an issue of similarity measurement, too.

Just as the operators in L and C, also the operators inVare noncommutative in the general case: ViVk"VkVi. If e.g. Vi has identified and selected a set of objects with required properties, then the subsequent operation Vk can focus on just these objects — the overall effort or effi-ciency may depend on the order of both operators.

4.2.4. Operators in class R: re6ision

Every ‘realistic’, and hence finite system has a limited ability to accept, to store, and to process information. Therefore such a system is forced to economize on these limited capacities. If new knowledge is permanently accepted and accumu-lated, if numerous results of conclusions are con-sidered worth storing, then a revision of the underlying representation scheme will become compulsory from time to time in order to main-tain an efficient usage of capacities.

The following examples show two different situ-ations, but also two techniques for a transition to a new representation scheme:

1. If there is a simple (finite) graph with a rela-tively small number of edges, then it is reason-able to represent it by the list of its pairs of connected vertices. But if, step by step, new

edges are inserted, then there will be a critical point beyond which it will be more advanta-geous to store the graph as its adjacency matrix.

2. A series of measurement data can be stored as a long list of pairs (xi,yi), but also by a short string (or code) representing an approximating formula like y=ax ory=a log x.

In the first example the transitions from one representation to the other and back are re-versible. By way of contrast, the second example stands for the frequent situation that a change of representation implies a loss of information: the original version can no more be reconstructed from the ‘condensed’ form (in category theory the term ‘forgetful functors’ is used).

A theoretical framework is supplied by a con-cept named belief re6ision, knowledge re6ision, or theory change, which is now pursued in an inter-disciplinary effort in philosophy, logic, and com-puter science (Rott, 1996). Comcom-puter scientists mainly study the problem of how to revise a body of knowledge if updating must be performed un-der capacity restrictions (Fuhrmann and Mor-reau, 1991; Wrobel, 1994) and the problem of consistency maintenance in knowledge-based sys-tems. Logic and philosophy, however, focus on the necessary modifications of an ensemble of propositions provoked by new, frequently incom-patible information5_.

The transition from a bulk of original data to a formula (the second example above) can be re-garded as a primitive type of theory formation. Under a unifying view we find a quasi-continuous transition from

a merely ‘technical’ revision, which is enforced by capacity restrictions (the first example above) and permits conversions in both direc-tions without any loss of information, at one end of a scale, to

a fundamental revision of an established theory — a paradigm change — at the opposite end of that scale.

We can use the terms ‘weak re6_ision’ and ‘strong re6ision’ for these two extreme cases, provided 5_{For a recent multidisciplinary overview of belief revision} see Gabbay and Smets (1998).

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that the quasi-continuous transition between them will be kept in mind.

Not in all cases the physical deletion of entries represented in the ‘old’ style will be compulsory. In special cases it can make sense to store an entry both in the old and in the shorter new representa-tion, such that both versions can be used alterna-tively (Section 4.3).

Here the operators in R, written as Ri, Rk,…, denote a ‘revision’, that is a transition to a differ-ent represdiffer-entation scheme. Apart from special cases6

these operators, again, are noncommuta-tive: RiRk"RkRi — each of both operators can entail its specific loss of information, and the situation found by the operator acting later can have been significantly altered by the operator which had acted first.

4.2.5. Operators in class T:tentati6e

In many methods of heuristic problem-solving (heuristic programming, genetic algorithms, etc.) sequences of patterns or configurations are gener-ated in a tentati6_e _{manner. It is the purpose of}

these attempts to eventually find a configuration fitting given requirements, or to proceed from a preliminary solution to a better one (even if an optimum cannot always be guaranteed). Such pat-terns can be generated — apart from random processes — by sets of recursive rules, as for instance rewriting rules, graph grammars, shape grammars7_{, or the transition rules typical of} ge-netic algorithms.

An example may be a possible heuristic ap-proach to the travelling salesman problem. Given a list of cities together with the distances between each pair of them, an optimal route is to be found that visits each city exactly once and (as assumed here for the sake of simplicity) finally leads back to the starting-point. A tentative configuration may be a closed loop which, however, does not yet include all cities and hence must be expanded

by a stepwise inclusion of further cities, or a closed loop through all cities which is not yet optimal, but can be improved by a series of local exchange operations.

Formally, the class T (tentative) consists of all operators T1,T2,…; each operator stands for a process by which exactly one new pattern is gener-ated. If two production rules of a graph grammar (or a related system) are applied one after an-other, the result strongly depends on the order of execution, and in the general case two operators in Tare noncommutative:TiTk " TkTi.

4.3. The question of completeness

As mentioned before, there are well-known types of cognitive processes that cannot be found in the list of ‘elementary operators’ (Section 4.2). Of course, a proof of ‘completeness’ — the possi-bility to describe every cognitive process by a combination of operators of the five kinds pro-posed here — is excluded, but there is at least some plausibility that a significant part of the field is covered8_.

For cognitive processes of some essential types it can be shown that such a representation is possible. Regularly conclusions (C) are necessary, and new information from outside (L)can inter-vene; hence operators inCor inLare not always explicitly addressed.

An ubiquitous type, that should not be forgot-ten, is forgetting. Some information stored in the system can be deleted as a conseqence of a new valuation of its relevance. This can be a by-product of a revision, and the deletion of a single item may be considered a boundary case of revi-sion. The operators in V and R are sufficient to represent forgetting.

Three central types of cognitive processes — classification, concept formation, and pattern

8_{No metaphysical assumptions are underlying here.} Particu-larly, it is not assumed that human beings can be completely described as ‘information-processing systems’ or something like that; nor is it intended to join the debate on limitations of computers. If some class of cognitive processes would be identified that cannot be represented by combinations of the operators proposed here, this would be a positive result, too. 6_{Two operators} _Ri _and _Rk _{may be ‘independent’ if they}

modify disjoint sets of entries.

7_{A survey of graph grammars and some applications can be} found in Gernert (1997), for shape grammars see the mono-graph by Gips (1975).

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recognition — will be treated jointly. They are mutually connected, and they have in common that perspective notions and similarity measures (Section 3.2) play a dominant role. All objects that are assigned to the same class within a given classification scheme are connected by their ‘intra-class similarity’, and the same holds for all objects subsumed under the same concept. If no classifi-cation scheme is previously defined, as in a method termed ‘cluster analysis’, then solely the similarity measure (and the given overall number of classes) will steer the assignment of the objects to equal or different classes. To each of those classes a newly created term can be assigned afterwards; here again we find a hint to the affinity between classification and concept formation.

Pattern recognition can be understood as the identification of those structures which have a sufficient similarity with one or another element from a predefined set of ‘standard patterns’. For example, character recognition means that a sin-gle scribble is identified (if possible) with the best fitting letter from a given alphabet.

Concept formation can essentially take on one of the following shapes:

1. As already mentioned, after a process of clsification a characterizing notion can be as-signed to each of the classes.

2. It can be recognized that part of the objects in a certain class have a characteristic feature in common and hence can be subsumed under a new notion. (For example, some materials share a ‘medium conductivity’ and thus are named ‘semiconductors’.)

3. If a variable remains constant in spite of varia-tions of other variables or of the overall sys-tem state, then that special variable may be given a marked name. The most important example is the term ‘energy’, which became a scientific term through the discovery of the corresponding conservation principle.

To sum up, processes of classification, pattern recognition, and concept formation can be man-aged mainly by operators fromV and R.

Processes of problem-sol6ingshow a permanent interplay of preliminary, tentative steps and valu-ations of the proposals generated in that way.

There must be a chance that a series of tentative steps can be totally discarded and a new attempt can be made from a different starting-point or with a new series of tentative steps (backtracking). In this context, it sometimes can make sense to temporarily store some information in more than one representation simultaneously (e.g. in the original and a condensed version, see Section 4.2.4). Problem-solving processes can essentially be built up by operators from V and T.

4.4. Fundamental operators acting in parallel For the sake of simplicity, parallel processes have not been addressed until now. LetX,Y,Z,… denote operators from any of the five classes introduced above (Section 4.2). If two operators Xand Yact in parallel this can be understood as an operator again9

, and that operator will be written asX+Y. By way of contrast, this compo-sition is always commutative: X+Y=Y+X. Of course, a concrete implementation of a system with parallel processing would require regulations concerning the relative independence of the com-ponents working in parallel and their possible interactions, but this would not contribute to the purpose of this paper. Some relevant aspects will be discussed in Section 5.4.

5. The algebras A and A% _{defined by classes of}

operators

5.1. Elementary properties of A

In order to study the algebraic structure the following classes of operators are introduced: A=L @ C @ V @ R

A%=A @ T

Both on Aand onA% _{two binary compositions}are defined: a composition o, which formerly was simply written by a juxtaposition of two operators (XY instead of XoY), and a composition + for 9_{Because the underlying system is finite only a finite number} of operators can be combined by ‘+’.

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the parallel execution of operators (Section 4.4). These composite operators are always defined. The classAwill 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 classesL,C,V,R separately that generally two operators are non -commutati6_e _{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: LioCk, LioVk, …,VioRk. In order to avoid boring legwork we only consider LioCk, LioVk, andLioRk. The three operators Ck,

Vk, andRkshare the property that their execution can be simplified by some new information (Li) coming in before they are started, whereas the belated arrival of the same information need not offer that advantage. Hence, in the general case LioCk, LioVk, and LioRk are noncommutative; similar arguments hold for the rest.

Both laws of associativity hold: (XoY)oZ=Xo(YoZ)

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 thatX, Y, andZact in parallel. Next, both laws of distribu-tivity hold:

(X+Y)oZ=(XoZ)+(YoZ), Zo(X+Y)=(ZoX)+(ZoY).

The reason for this lies in the definition of ‘+’. The mechanism can be seen from an example:

Let(X+Y)oZmean 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 ifX,Y,Zare 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 anACI-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 Aand 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).

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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 processes10_{. ‘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 Qand A.

Now Qforms a commutative semigroup for+

and a noncommutative semigroup for the juxta-position of operators, and hence a semiring. Both A and Q are additi6_{ely commutati}6_{e semirings}. Operators from an important subclass of Q, the projection operators, are idempotent (P2₌_P_). Similarly, operators in A are idempotent with respect to o: we haveXoX=X for all XA. The semigroups (A,+) and (A,o), and as well their counterparts inQ, 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 (Ti

=Ti), 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 G1 two vertices 6 _and 6 % _{are not connected by an edge,}

and thatT1 just inserts an edge between6and6 %. Then a second application of T1 will bring about no change, since an edge between6_and6 %_already

exists and multiple edges are excluded. As an example for the contrary, consider the graph G2 which has exactly one vertex of degree 1 (one edge leaving that vertex), and an operator T2 which introduces a new vertex that is connected with the unique vertex of degree 1. ThenT2can be applied repeatedly, always generating a graph different from the earlier ones, and T2

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 in6_{erses 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 ‘Geisteswis‘Naturwis-senschaften’ (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.

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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 allX 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 notationX−Yinstead of X+(−Y). Eq. (1) may mean that in two compo-nents of the systemsimultaneouslytwo 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 operatorX, is −Xuniquely defined? 3. What is the exact meaning of ‘simultaneously’? IfLi, Lk L and Li+Lk=¥ (or Lk= −Li), 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 Lk is the negate of the information received by Li. Thus, neither Li nor

Lkwill 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 reliable11_.

If the definition of −Li is to be unique then

−Limust be defined as the negate ofLi. 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 xB1, 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 −Xcan be identified if X C orX V. If Ci and −Ciare two processes of conclusion (performed by differ-ent compondiffer-ents) 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 Viand −Vi, two acts of valuation (of the same object) with oppo-site outcomes; −Vimeans that the valuationViis

notascribed to the object under consideration. On the other hand, it is impossible to define −Rior

−Ticorrespondingly 12_.

As an intermediate result we can state that operatorsXL,C, orVhave 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 interexter-nal 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 operationsLi and −Liare performed within 12_{It follows from noncommutativity that an inverse of} _X with respect toocannot be consistently defined forXL,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).

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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 betweenCiand−Ci(or betweenViand−Vi) 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 -neousif 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

6.1. The role of noncommutati6_ity

The noncommutativity of operators inA% (Sec-tions 4.2, 5.1 and 5.2) is connected with the very nature of information. The acquisition and the spreading of information are, in principle, irre-versible processes (if the risk of later forgetting is ignored), and the spoken word can no more be made unspoken. All nontrivial processes by which information is received, processed, or transmitted are entangled with time, temporal order, and history.

This can also be illustrated by the ‘internal structure’ of texts. The chapters or subchapters of a normal textbook cannot be rearranged in an inverse or randomized order without making the use of that book a mere horror. Incoherent mate-rial, such that the task of memorizing can be coped with only by rote-learning, can be arranged in a different order without an influence on the labour of learning. As a by-product, a measure of com-plexity can be derived: comcom-plexity can be measured

by the disturbances brought about by changing the order of operators from L (e.g. starting from the dissimilarities between XY and YX).

Problem-solving tasks can be chararacterized by their ‘degree of acceptable arbitrariness’. Some steps in a problem-solving process have ‘degrees of freedom’ (in solving a crossword-puzzle one can start anywhere), some steps or sequences of steps can be parallelized, but there are decisive steps which must utilize the intermediate results supplied by certain earlier operations, and hence in the latter case the temporal order of steps cannot be inverted. An extreme example is given by chess problems where it is a rule of the art that the solution is a unique sequence of steps (including their fixed order).

6.2. Perspecti6e notions in modern physics

Perspective notions and their peculiar features (Section 3.2) do not only play a role on a descriptive or epistemological level. By way of contrast, they surprisingly emerge in down-to-earth experimental physics and empirical astrophysics. Atmanspacher and Scheingraber (1990) have shown that the behaviour of ‘multimode continuous-wave dye lasers’ around instabilities can be understood and even quantitatively analysed on the basis of prag-matic information. Quite a similar analysis was performed by Kurths and his group for solar activity (Atmanspacher et al., 1992; Kurths et al., 1994).

The fact that a physical system can perform a transition from its present state to a specific differ-ent state can, in anthropomorphic terms, have a meaning for that system, and this meaning can be measured on the basis of pragmatic information. It is not so important that a description is also possible without any use of perspective notions — the crucial fact is that an analysis based upon perspective notions is possible (without loss of information), and that this is likely to open a path to further understanding13_.

13_{The propensity to exorcise terms with an} ‘anthropomor-phic touch’ in physics can be understood from the history and sociology of sciences. But in the light of modern knowledge such a restriction is no more tenable and can only turn out as a roadblocker.

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6.3. Time-windows and internal time

‘‘20th century’s philosophy has witnessed a steady but off-mainstream interest in concepts of time that can be covered by the notion of an ‘internal time’.’’ (Atmanspacher, 1998). As the main protagonists Bergson, Whitehead, Gebser, and Prigogine are pointed out14_{. In recent years,} some endeavour has been taken to analyse the concept of internal time also under the aspects of quantum theory and the endo – exo distinction (e.g. Matsuno, 1996; Atmanspacher, 1998; Gunji, 1998; Matsuno, 1998).

A short remark may be added here starting from the approach outlined in Section 5.4. Within a single component of a system capable of parallel processing, a ‘duration’ or ‘length of an internal time interval’ can be defined opera-tionally by the property that the effects of two processes will be or will not be confronted with each other in order to check whether they can be accepted. In computer simulations, the lengths of internal time intervals can be approximately mea-sured ‘from outside’; furthermore, it can be stud-ied which will be the consequences of different styles of confrontation, and what will be the influence of different parameters upon internal time intervals.

7. Concluding remarks and outlook

In the words of Pattee (1993), ‘‘the quantum theory of measurement remains a confusing prob-lem that cannot be isolated from the existence of organisms since organisms are the ultimate ob-servers that perform the measurements’’. Accord-ing to the same author, the root of the measurement problem is the restriction of our thinking to the formal mathematical languages and the consequent necessity of making a sharp distinction between syntax and semantics (Pattee, 1993).

In this situation, we have all good reasons

to update our understanding of terms like ‘measurement’ and ‘observation’. It is not a necessary property of mathematics that we are forced to drop semantic aspects in mathe-matical modelling. Rather, there is a cultural bias leading to an inadequate modelling style governed by the tacit assumption that se-mantic aspects can not or should not be in-cluded. As it was demonstrated above (albeit without all technical details), we can indeed account for the peculiar requirements brought about by perspective notions (Section 3.2), which means nothing else but incorporating semantic aspects.

‘low-context measurement’and‘high-context mea -surement’. By neglecting the context we will return to the traditional understanding, and the customary low-context measurement is included as a special case in the more comprehensive con-cept.

The proposal displayed here can be tested by modelling and simulation on a standard PC. One of the early questions may concern the required minimum size of a model system with a specific performance, e.g. a minimal system which will be able to simulate noncommutativity of operators and irreversibility. Surprising effects can be expected, and probably connections with other fields of research will be found (e.g. associative networks and memories, pattern recognition, Artificial Life). It can be hoped that the present proposal will be a step towards a unifying description of traditionally separated phenomena.

Acknowledgements

The author wants to express his gratitude to two anonymous referees.

14_{For an overview of Bergson’s contribution see Lacey} (1995), for Whitehead see Atmanspacher (1998).

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Appendix. A simplified example for mathematical treatment

The following example starts from a simple situation in three-dimensional elementary geome-try. The outcome is known in before, such that it can be checked whether the mechanism proposed here will work as it is expected. It will be shown how two projection operators can act indepen-dently upon the same geometric object, thus gen-erating two distinct figures. Operators of a different kind, called preselectors, will represent the decision which one of the two projection operators is to be activated.

forming the initial phase of an observation (defin-ition phase), as well as the principle of two-step measurement(Sections 3.2 and 7) and the depen-dence of the result from the observer’s prior choice.

Fig. 3 shows a prism with triangular basis in a perspective drawing. If it is accepted that the prism exactly stands on the (x,y) -coordinate-plane of three-dimensional space, then its projec-tion onto the (x,y)-plane (ground-plan) is a

triangle, whereas its projection e.g. onto the(y,z) -plane (profile view) is a quadrangle. The same drawing can be understood as a graph with six vertices and nine edges, which is denoted as C6, since it is the complement ofC6, the cycle with six vertices. Then the two projections can be written as C3 and C4, the cycles with three or four ver-tices, respectively.

Any graph, G, (finite, undirected, without loops or multiple edges) can be represented by its adja-cency matrix A(G) with the elements aik=1 if there is an edge connecting the vertices 6_i_and 6_k_,

and aik=0 else. Now, let A be the adjacency matrix of the prism graph C6:

Á

Ã

Ä

0 0 1 1 1 0

0 0 0 1 1 1

1 0 0 0 1 1

1 1 0 0 0 1

1 1 1 0 0 0

0 1 1 1 0 0

Â

Ã

Å

The selector matrix S1,3,5 is defined as the (6, 6)-matrix with s11=s33=s55=1, and sik=0 else. Selector matrices of this kind are designed such that their multiplication15 _{by a square matrix} ex-actly selects those columns and rows indicated by the subscripts. In the concrete case this means

AS1,3,5=

Á

Ã

Ä

0 0 1 0 1 0

0 0 0 0 0 0

1 0 0 0 1 0

0 0 0 0 0 0

1 0 1 0 0 0

0 0 0 0 0 0

Â

Ã

Å

and this matrix, after deletion of all zero columns and zero rows, is exactly the adjacency matrix of the triangle graph C3(the subscripts 1, 3, 5 corre-spond to the labelling in Fig. 3). In the same way, A is multplied by a selector matrix S1,3,4,6, and

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AS1,3,4,6is a matrix which, again after omission of trivial columns and rows, is the adjacency matrix of the quadrangle graph C4.

With the abbreviations T=S1,3,5 and Q=

S1,3,4,6 this can be written as AT=TA$C3

AQ=QA$C4,

where $denotes that a matrix (if necessary, after deleting zero columns and rows) is the adjacency matrix of a graph.

As an intermediate result, we can state that (within the framework of this example) two differ-ent projections of the same object can be repre-sented by suitable selector matrices. In the following second step, it will be shown how a choice among these two possible projections can be made.

The preselector matrices P1 and P2 are defined by

P1=

1 0

0 0

P2=

0 0 0 1

For an easy handling of block matrices the Kro-necker product of two matrices, denoted by , is used. It can be explained by the standard text-book example (where X is any matrix and the right-hand side is a block matrix):

1 2

3 4

1X 2X 3X 4X

This binary composition is noncommutative. For technical reasons, two copies of the original ma-trix A are needed, arranged in a special block matrix:

1 0

0 1

DA=

A O

O A

where O is a block of zeros of the right size. Now the matrix products with the required properties can be written:

T O

O Q

n

(DA)$C3 (2)

T O

O Q

n

(DA)$C4 (3) Both in Eq. (2) and in Eq. (3) we find a repertoire of two possible selection operators (T and Q), each of which represents a specific selection (and hence a valuation). But Eqs. 2 and 3 are distinguished by their individual preselectors — P1 or P2 — and it is governed by that preselector which one of the two available selectors will be activated. This can be seen in analogy to the observer’s prior decision in the definition phase, but also to distinct previous assumptions leading to different inter-pretations of the same object. Starting from the geometric object in Fig. 3, two processes can be triggered such that either C3 or C4 will be ‘read out’.

In a further step, P1andP2can be replaced by a parameter-depending preselector P(a) (0 5 a

51) defined by

P(a)=

a 0

0 1−a

If P(a) is inserted into Eqs. 2 or 3 then different

‘intensities’aand(1−a)are assigned toTandQ,

respectively, and the result is a ‘combined projec-tion’ consisting ofC3with intensityaandC4with intensity (1−a), or, with some freedom in

notation:

aBC₃\+(1−a)BC₄\,

which corresponds to the co-existence of two al-ternative interpretations or of two complementary forms of appearance.

References

Adamatzky, A., 1999. Pathology of collective data: automata models. Submitted to: Appl. Math. Comput.

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infor-mation and dynamical instabilities in a multimode continu-ous-wave dye laser. Can. J. Phys. 68, 728 – 737.

Atmanspacher, H., Kurths, J., Scheingraber, H., Wacker-bauer, R., Witt, A., 1992. Complexity and meaning in nonlinear dynamical systems. Open Syst. Inf. Dyn. 1, 269 – 289.

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Suppl. 69, 113 – 158.

Flanagan, O., 1995. Consciousness. In: Honderich, T. (Ed.), The Oxford Companion to Philosophy. Oxford University Press, Oxford, pp. 152 – 153.

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Gabbay, D.M., Smets, P. (Eds.), 1998. Belief Change. Kluwer, Dordrecht.

Gernert, D., 1996. Pragmatic information as a unifying con-cept. In: Kornwachs, K., Jacoby, K. (Eds.), Information — New Questions to a Multidisciplinary Concept. Akademie – Verlag, Berlin, pp. 147 – 162.

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X+¥=¥+X=X (for allX 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 notationX−Yinstead of X+(−Y). Eq. (1) may mean that in two compo-nents of the systemsimultaneouslytwo 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):

If the definition of −Li is to be unique then

notascribed to the object under consideration. On the other hand, it is impossible to define −Rior

−Ticorrespondingly 12_.

12_{It follows from noncommutativity that an inverse of} _X

with respect toocannot be consistently defined forXL,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).

(2)

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

6.1. The role of noncommutati6_ity

by the disturbances brought about by changing the order of operators from L (e.g. starting from the dissimilarities between XY and YX).

6.2. Perspecti6e notions in modern physics

13_{The propensity to exorcise terms with an}

‘anthropomor-phic touch’ in physics can be understood from the history and sociology of sciences. But in the light of modern knowledge such a restriction is no more tenable and can only turn out as a roadblocker.

(3)

6.3. Time-windows and internal time

7. Concluding remarks and outlook

In this situation, we have all good reasons

bias leading to an inadequate modelling

style governed by the tacit assumption that se-mantic aspects can not or should not be in-cluded. As it was demonstrated above (albeit without all technical details), we can indeed account for the peculiar requirements brought about by perspective notions (Section 3.2), which means nothing else but incorporating semantic aspects.

Exactly this leads to the necessity of a two-step proceedingortwo-step measurement, as contrasted to single-step measurementin the traditional ‘con-text-free cases’ (also see the Appendix). Of course, those situations of measurement or observation are not really context-free, but the existing context is separated by tacit assumptions. Perhaps we can express this by distinguishing between ‘low-context measurement’and‘high-context mea -surement’. By neglecting the context we will return to the traditional understanding, and the customary low-context measurement is included as a special case in the more comprehensive con-cept.

The proposal displayed here can be tested by modelling and simulation on a standard PC. One of the early questions may concern the required

minimum size of a model system with a

specific performance, e.g. a minimal system which will be able to simulate noncommutativity of operators and irreversibility. Surprising effects can be expected, and probably connections with other fields of research will be found (e.g. associative networks and memories, pattern recognition, Artificial Life). It can be hoped that the present proposal will be a step towards a unifying description of traditionally separated phenomena.

Acknowledgements

The author wants to express his gratitude to two anonymous referees.

14_{For an overview of Bergson’s contribution see Lacey}

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Appendix. A simplified example for mathematical treatment

This illustrates acts of valuation and selection. Furthermore, the preselection operators and the requirement of concatenated operators (see Eqs. 2 and 3) demonstratethe obser6_er_’_{s pre}6_{ious decision} forming the initial phase of an observation (defin-ition phase), as well as the principle of two-step measurement(Sections 3.2 and 7) and the depen-dence of the result from the observer’s prior choice.

Á

Ã

Ä

0 0 1 1 1 0

0 0 0 1 1 1

1 0 0 0 1 1

1 1 0 0 0 1

1 1 1 0 0 0

0 1 1 1 0 0

Â

Ã

Å

The selector matrix S1,3,5 is defined as the (6, 6)-matrix with s11=s33=s55=1, and sik=0 else. Selector matrices of this kind are designed such that their multiplication15 _{by a square matrix} ex-actly selects those columns and rows indicated by the subscripts. In the concrete case this means

AS1,3,5=

Á

Ã

Ä

0 0 1 0 1 0

0 0 0 0 0 0

1 0 0 0 1 0

0 0 0 0 0 0

1 0 1 0 0 0

0 0 0 0 0 0

Â

Ã

Å

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AS1,3,4,6is a matrix which, again after omission of trivial columns and rows, is the adjacency matrix of the quadrangle graph C4.

With the abbreviations T=S1,3,5 and Q=

S1,3,4,6 this can be written as

AT=TA$C3

AQ=QA$C4,

where $denotes that a matrix (if necessary, after deleting zero columns and rows) is the adjacency matrix of a graph.

The preselector matrices P1 and P2 are defined by

P1=

1 0

0 0

P2=

0 0 0 1

1 2

3 4

1X 2X 3X 4X

This binary composition is noncommutative. For technical reasons, two copies of the original ma-trix A are needed, arranged in a special block matrix:

1 0

0 1

DA=

A O

O A

where O is a block of zeros of the right size. Now the matrix products with the required properties can be written:

T O

O Q

n

(DA)$C3 (2)

T O

O Q

n

In a further step, P1andP2can be replaced by a parameter-depending preselector P(a) (0 5 a

51) defined by

P(a)=

a 0

0 1−a

If P(a) is inserted into Eqs. 2 or 3 then different ‘intensities’aand(1−a)are assigned toTandQ, respectively, and the result is a ‘combined projec-tion’ consisting ofC3with intensityaandC4with intensity (1−a), or, with some freedom in notation: