Extrinsic or exophysical perception can be conceived as a hierarchical process, in which the system under observation and the experimenter form a two-level hierarchy. The system is laid out and the experimenter peeps at every relevant feature of it without changing it. The restricted entanglement between the system and the experimenter can be represented by a one-way information flow from the system to the experimenter; the system is not affected by the experimenter's actions. (Logicians might prefer the term meta over exo.)
Intrinsic or endophysical perception can be conceived as a non-hierarchical effort. The experimenter is part of the universe under observation. Experiments use devices and procedures which are realisable by internal resources, i.e., from within the universe. The total integration of the experimenter in the observed system can be represented by a two-way information flow, where ``measurement apparatus'' and ``observed entity'' are interchangeable and any distinction between them is merely a matter of intent and convention. Endophysics is limited by the self-referential character of any measurement. An intrinsic measurement can often be related to the paradoxical attempt to obtain the ``true'' value of an observable while - through interaction - it causes ``disturbances'' of the entity to be measured, thereby changing its state. Among other questions one may ask, ``what kind of experiments are intrinsically operational and what type of theories will be intrinsically reasonable?''
Imagine, for example, some artificial intelligence living in a (hermetic) cyberspace. This agent might develop a ``natural science'' by performing experiments and developing theories. It is tempting to speculate that also a figure in a novel, imagined by the poet and the reader, is such an agent.
Since in cyberspace only syntactic structures are relevant, one might wonder if concerns of this agent about its ``hardware basis,'' e.g., whether it is ``made of'' billard balls, electric circuits, mechanical relays or nerve cells, are mystic or even possible (cf. H. Putnam's brain-in-a-tank analysis [4]). I dont think this is necessarily so, in particular if the agent could influence some features of this hardware basis. One example is a hardware damage caused by certain computer viruses by ``heating up'' computer components such as storage or processors. I would like to call this type of ``back-reaction'' of a virtual reality on its computing agent ``virtual backflow interception'' (VBI). Intrinsic phenomenologically, VBI could manifest itself by some violation of a ``superselection rule;'' i.e., by some virtual phenomenon which violates the fundamental laws of a virtual reality, such as symmetry & conservation principles.
No attempt is made here to (re-)write a comprehensive history of related concepts; but a few hallmarks are mentioned without claim of completeness. Historically, Archimedes conceived ``points outside the world, from which one could move the earth.'' Archimedes' use of ``points outside the world'' was in a mechanical rather than in a metatheoretical context: he claimed to be able to move any given weight by any given force, however small. The 18'th century physicist B. J. Boscovich realised that it is not possible to measure motions or transformations if the whole world, including all measurement apparata and observers therein, becomes equally affected by these motions or transformations (cf. O. E. Rössler [2], p. 143). Fiction writers informally elaborated consequences of intrinsic perception. E. A. Abbot's Flatland describes the life of two- and onedimensional creatures and their confrontation with higher dimensional phenomena. The Freiherr von Münchhausen rescued himself from a swamp by dragging himself out by his own hair. Among contemporary science fiction authors, D. F. Galouye's Simulacron Three and St. Lem's Non Serviam study some aspects of artificial intelligence in what could be called ``cyberspaces.'' Media artists such as Peter Weibel create ``virtual realities'' or ``cyberspaces'' and are particulary concerned about the interface between ``reality'' and ``virtual reality,'' both practically and philosophically. Finally, by outperforming television & computer games, commercial ``virtual reality'' products might become very big business. From these examples it can be seen that concepts related to intrinsic perception may become fruitful for physics, the computer sciences and art as well.
Already in 1950 (19 years after the publication of Gödel's incompleteness theorems), K. Popper has questioned the completeness of self-referential perception of ``mechanic'' computing devices [5]. Popper uses techniques similar to Zeno's paradox (which he calls ``paradox of Tristram Shandy'') and ``Gödelian sentences'' to argue for a kind of ``intrinsic indeterminism.''
In a pioneering study on the theory of (finite) automata, E. F. Moore has presented Gedanken-experiments on sequential machines [6]. There, E. F. Moore investigated automata featuring, at least to some extend, similarities to the quantum mechanical uncertainty principle. In the book Regular Algebra and Finite Machines [30], J. H. Conway has developed these ideas further from a formal point of view without relating them to physical applications. Probably the best review of experiments on Moore-type automata can be found in W. Brauer's book Automatentheorie [] (in German).
D. Finkelstein [32,33] has considered Moore's findings from a more physical point of view, introducing an ``experimental logic of automata'' and the term ``computational complementarity.'' An illuminating account on endophysics topics can be found in O. E. Rössler's article on Endophysics [1], as well as in his book Endophysics (in German) [2]; O. E. Rössler is a major driving force in this area. Also H. Primas has considered endophysical and exophysical descriptions in various contexts [7].
The terms ``intrinsic'' and ``extrinsic'' appear in the author's studies on intrinsic time scales in arbitrary dispersive media [8,9,10]. There, the intrinsic-extrinsic concept has been re-invented (probably for the 100'th time, and, I solemnly swear) independently. It is argued that, depending on dispersion relations, creatures in a ``dispersive medium'' would develop a theory of coordinate transformation very similar to relativity theory. Another proposal by the author was to consider a new type of ``dimensional regularisation'' by assuming that the space-time support of (quantum mechanical) fields is a fractal [11]. In this approach one considers a fractal space-time of Hausdorff dimension D = 4-e, with e << 1, which is embedded in a space of higher dimension, e.g., \Bbb Rn ³ 4. Intrinsically, the (fractal) space-time is perceived ``almost'' as the usual fourdimensional space.
Besides such considerations, J. A. Wheeler [12], among others, has emphasised the role of observer-participancy. In the context of what is considered by the Einstein-Podolsky-Rosen argument [13] as ``incompleteness'' of quantum theory, A. Peres and W. H. Zurek [14,15] and J. Rothstein [16] have attempted to relate quantum complementarity to Gödel-type incompleteness.
In what follows, the intrinsic-extrinsic concept will be made precise in an algorithmic context, thereby closely following E. F. Moore [6]. The main reason for the algorithmic approach is that algorithmic universes (or, equivalently, formal systems) are the royal road to the study of undecidability. The intrinsic-extrinsic concept will be applied to investigate computational complementarity and intrinsic indeterminism both in the algorithmic context.
(i) only the input and output terminals of the automaton are accessible. The experimenter is allowed to perform experiments via these interfaces in the form of stimulating the automaton with input sequences and receiving output sequences from the automaton. The experimenter is not permitted to ``open up'' the automaton, but
(ii) the transition and output table (diagram) of the automaton (in its reduced form) is known to the experimenter (or, if you prefer, is given to the experimenter by some ``oracle'').
The most important problem, among others, is the distinguishing problem: it is known that an automaton is in one of a particular class of internal states: find that state.
In the first kind of experimental situation, only a single copy of the automaton is accessible to the experimenter. The second type of experiment operates with an arbitrary number of automaton copies. Both cases will be discussed in detail below.
If the input is some predetermined sequence, one may call the experiment a preset experiment. If, on the other hand, (part of) the input sequence depends on (part of) the output sequence, i.e., if the input is adapted to the reaction of the automaton, one may call the experiment an adaptive experiment. We shall be mostly concerned with preset experiments, yet adaptive experiments can be used to solve certain problems with automaton propositional calculi.
Research along these lines has been pursued by S. Ginsburg [17], A. Gill [18], J. H. Conway [30] and W. Brauer [].
In the first kind of Gedankenexperiment, only one single automaton copy is presented to the experimenter. The problem is to determine the initial state of the automaton, provided its transition and output functions are known (distinguishing problem). In a typical experiment, the automaton is ``feeded'' with a sequence of input symbols and responds by a sequence of output symbols. An input-output analysis then reveals information about the automaton's original state.
Assume for the moment that such an experiment induces a state transition of the automaton. I.e., after the experiment, the automaton is not in the original initial state. In this process a loss of potential information about the automaton's initial state may occur. In other words: certain measurements, while measuring some particular feature of the automaton, may make impossible the measurement of other features of the automaton. This irreversible change of the automaton state is one aspect of the ``observer-participancy'' in the single-automaton configuration. (This is not the case for the multi-automaton situation discussed below, since the availability of an arbitrary number of automata ensures the possibility of an arbitrary number of measuring processes.)
In developing the intrinsic concept further, the automaton and the experimenter are ``placed'' into a single ``meta''-automaton. If the experimenter reacts mechanically, this can be readily achieved by simulating both the original finite deterministic ``black box'' automaton as well as the experimenter and their interplay by a universal automaton. One can imagine such a situation as one subprogram checking another subprogram, also including itself. For an illustration, see Fig. 1.
Picture Omitted
In certain cases it is necessary to iterate this picture in the following way. Suppose, for instance, the experimenter attempts a complete intrinsic description. Then, the experimenter has to give a complete description of his own intrinsic situation. In order to be able to model the own intrinsic viewpoint, the experimenter has to introduce an or system which is a replica of its own universe. This amounts to substituting the ``meta''-automaton for the automaton in Fig. 1. Compare also a drawing by O. E. Rössler [3], Fig. 2, where `` » '' stands for the interface, which is denoted by the symbols ``\rightleftarrows '' throughout this article.
ch-roessl.ps
Yet, in order to be able to model intrinsic viewpoint of a new experimenter in this new universe, this new experimenter has to introduce another system which is a replica of its own universe, ¼, resulting in an iteration ad infinitum. One may conjecture that an observer in a hypothetical universe corresponding to the ``fixed point'' or ``invariant set'' of this process has complete self-comprehension; see Fig. 3.
Picture Omitted
Of course, in general this observer cannot be a finite observer: a complete description would only emerge in the limit of infinite iterations (cf. K. Popper's ``paradox of Tristram Shandy''). Finite observers cannot obtain complete self-comprehension.
The second kind of experiment operates with an arbitrary number of automaton copies. One automaton is a copy of another if both automata are isomorphic and if both are in the same initial state. With this configuration the experimenter is in the happy condition to apply as many input sequences to the original automaton as necessary. In a sense, the observer is not bound to ``observer-participancy,'' because it is always possible to ``discard the used automaton copies,'' and take a ``fresh'' automaton copy for further experiments. For an illustration, see Fig. 4.
Picture Omitted
In the foregoing section, important features of the extrinsic-intrinsic concept have been isolated in the context of finite automata. A generalisation to arbitrary physical systems is straightforward. The features will be summarised by the following definition. (Anything on which experiments can be performed will be called system. In particular, finite automata are systems.)
An intrinsic quantity is
associated
with an experiment
(i) performed on a single copy of
the system,
(ii) with the experimenter being part of
the system.
An extrinsic
quantity, denoted by a tilde sign ``[ \tilde]'' is
associated with an experiment
(i)
utilising, if necessary, an arbitrary number of copies of
the system,
(ii) with the experimenter not being part of the system.
One may ask whether, intuitively, the extrinsic point of view might be more appropriately represented by, stated pointedly, the application of a ``can-opener'' for the ``black box'' to see ``what's really in it.'' Yet, while the physical realisation might be of some engineering importance, the primary concern is the phenomenology (i.e., the experimental performance of the system) and not how it is constructed. In this sense, the technological base of the automaton is irrelevant. For the same reason, i.e., because this is irrelevant to phenomenology, it is not important whether the automaton is in its minimal form.
The requirement that in the extrinsic case an arbitrary number of system copies is available is equivalent to the statement that no interaction takes place between the experimenter and the system. (The reverse information flow from the observed system to the experimenter is necessary.) This results in a one-way information flow in the extrinsic case:
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The definition applies to physical systems as well as to logics and (finite) automata. Automaton worlds provide an ideal ``playground'' for the study of certain algorithmic features related to undecidability, such as ``computational complementarity'' and ``intrinsic indeterminism.'' The extrinsic-intrinsic problem is the interrelation between extrinsic and intrinsic entities.
The investigation is based on the construction of primitive experimental statements or propositions. Then the structure of these propositions will be discussed, thereby defining algebraic relations and operations between the propositions. Although specific classes of finite automata will be analysed, these considerations apply to universal computers as well. (Finite automata can be simulated on universal computers.)
Consider propositions of the form
``the automaton is in state aj''
``the automaton is in state aj or in state am or in state al ¼ .''
``the automaton is in some internal state''
The element 0 is given by the empty set Æ (or {}). This corresponds to a proposition which is false (by definition the automaton has to be in some internal state):
``the automaton is in no internal state''
The class of all propositions and their relations will be called automaton propositional calculus and denoted by \frak A. Each particular outcome which, if defined, has the value TRUE or FALSE, shall be called ``event.'' In this sense, an automaton propositional calculus, just as the quantum propositional calculus, is obtained experimentally. It consists of all potentially measurable elements of the automaton reality and their logical structure, with the implication as order relation.
The elementary propositions can be conveniently constructed by a
partitioning of automaton states generated from the input-output
analysis of the automaton as follows:
Let
w = s1 s2 ¼sk be a sequence of input symbols,
| (1) |
| (2) |
| (3) |
| (4) |
| (5) |
Let pi be propositions
of the
form ``the automaton is in state ai.''
The proposition
| (6) |
The proposition
| (7) |
The complement
| (8) |
| ||||||||||
A partial order relation pj\preceq pm, or
| (9) |
| 1 | 2 | 3 | |
| d1 | 1 | 1 | 1 |
| d2 | 2 | 2 | 2 |
| d3 | 3 | 3 | 3 |
| o1 | 1 | 0 | 0 |
| o2 | 0 | 1 | 0 |
| o3 | 0 | 0 | 1 |
Input of 1, 2 or 3 steers the automaton into the respective state. At the same time, the output of the automaton is 1 only if the guess is a ``hit,'' i.e., if the automaton was in that state. Otherwise the output is 0. After the measurement, the automaton is in a definite state, i.e., the state corresponding to the input symbol. If the guess has not been a ``hit,'' the information about the initial automaton state is lost. Therefore, the experimenter has to decide before carrying out the measurement which one of the following hypotheses should be tested (in short-hand notation, ``{1}'' stands for ``the automaton is in state 1'' et cetera): { 1 } = Ø{ 2,3 },{ 2 } = Ø{ 1,3 },{ 3 } = Ø{ 1,2 }. Measurement of either one of these three hypotheses (or their complement) makes impossible measurement of the other two hypotheses.
No input, i.e., the empty input string Æ, identifies all three
internal automaton states. This corresponds to the trivial information
that the automaton is in some internal state.
Input of the symbol 1 (and all sequences of symbols starting with 1)
distinguishes between the hypothesis {1} (output ``1'') and the
hypothesis
{2,3} (output ``0'').
Input of the symbol 2 (and all sequences of symbols starting with 1)
distinguishes between the hypothesis {2} (output ``1'') and the
hypothesis
{1,3} (output ``0'').
Input of the symbol 3 (and all sequences of symbols starting with 1)
distinguishes between the hypothesis {3} (output ``1'') and the
hypothesis
{1,2} (output ``0'').
The
propositional calculus is thus
defined by the partitions
| ||||||||||||||||||||||||
The obtained intrinsic propositional calculus in many ways resembles the lattice obtained from photon polarisation experiments or from other incompatible quantum measurements. Consider an experiment measuring photon polarisation. Then, three propositions of the form ``the photon has polarisation pf1,'' (i = 1,2,3), cannot be measured simultaneously for the angles f1 ¹ f2 ¹ f3 (mod p). An irreversible measurement of one direction of polarisation would result in a state preparation, making impossible measurement of the other directions of polarisation, and resulting in a propositional calculus of the ``Chinese latern'' form MO3.
The propositional calculi \frak Fi of all Mealy-type automata with i internal states can be constructed by combinatorical arguments [41]. Fig. 7 shows \frak F4, the Hasse diagrams of generic intrinsic propositional calculi of Mealy automata up to 4 states.
Let an orthomodular lattice be a lattice satisfying the
orthomodular law, and let a Hilbert lattice be the lattice
of all closed subspaces of a Hilbert space, with the ``infimum''
operator defined by the intersection of subspaces, the ``supremum''
operator defined by the closure of the linear span of subspaces and
the orthocomplement defined by the orthogonal subspace.
Any finite
(``finite''
means
that the lattice has a finite number of elements)
orthomodular
lattice is isomorphic (1-1 translatable) to some finite (lattice)
automaton
propositional calculus. I.e.,
| (14) |
| (15) |
An actual proof of these statements is too technical and will be given elsewhere [41]. It makes use of the fact that every orthomodular lattice is a pasting of its maximal Boolean subalgebras, also called blocks [25,42]. These blocks can be elegantly represented by sets of partitions of automata states, because ``at face value,'' every automaton state partition v(¼) with n elements generates a Boolean algebra 2n. If one identifies these Boolean algebras with blocks, the set of automaton state partitions V represents a complete family of blocks of the automaton propositional calculus.
It is not entirely unreasonable to speculate about logico-algebraic structures of automaton universes in general. To put it pointedly, one could ask ``how would creatures embedded in a universal computer perceive their universe?'' The lattice-theoretic answer might be as follows. Let \frak Fi stand for the family of all intrinsic propositional calculi of automata with i states. From the point of view of logic, the intrinsic propositional calculi of a universe generated by universal computation is the limiting class limn® ¥\frak Fn of all automata with n ® ¥ states. Since \frak F1 Ì \frak F2 Ì \frak F3 Ì ¼ Ì \frak Fi Ì \frak Fi+1 Ì ¼, this class ``starts with'' the propositional calculi represented by Fig. 7, p. pageref.
It is tempting to speculate that we live in a computer generated universe. But then, if the ``underlying'' computing agent were universal, there is no a priori reason to exclude propositional calculi even if they do not correspond to an orthomodular subalgebra of a Hilbert lattice. I.e., to test the speculation that we live in a universe created by universal computation, we would have to look for phenomena which correspond to automaton propositional calculi not contained in the subalgebras of some Hilbert space - such as, for instance, the one represented by Fig. 8, p. pageref, which was obtained from the state partition {{{1}, {2}, {3, 4}}, {{1}, {2, 4}, {3}}, {{1, 2},{3}, {4}}, {{1, 3}, {2}, {4}}}.