by

Dr. Gerd Döben-Henisch

INM - Institut for New Media

andinm numerical magic gmbh

Daimlerstr. 32

D-60314 Frankfurt am Main

Tel: 069-941 963-10

Fax: 069-941 963-22

email: doeb@inm.de

This paper demonstrates a basic equation between 'semiotics' and 'computational semiotics'. The term 'semiotics' is here understood according to the writings of Charles Morris, and the term 'computational' is based on the concept of the 'Turing machine' as provided by Alan Matthew Turing. Taking the concept of a 'scientific theory' as the common point of reference, it is shown how the concept of the Turing machine and the concept of semiotics can be reconstructed uniformly within this framework. Finally, it is shown how one can construct a mapping between the concept of 'semiotic agent' as proposed by Morris and the concept of the Turing machine. The result is that everything that can be said about a semiotic agent within Morris's concept of semiotics can be stated in terms of the Turing machine concept.

- Turing was not a Semiotician
- Meta-Mathematics: Turing's first Playground
- Relating Turing to Semiotics
- Being Scientific
- Reconstructing Turing's Contribution within a Theory Concept
- Reconstructing Charles Morris's Contribution within a Theory Concept
- Comparing Turing and Morris
- The Vision of Intelligent Systems as Semiotic Systems
- References"

Connecting Alan Matthew Turing (1912-1954), the great British mathematician and logician, with semiotics is not a straightforward task.

Turing himself never mentioned semiotics explicitly and he never called himself in any sense a 'semiotician', nor did any of his colleagues thus identify him, nor any writer after his death.

Turing was an outstanding researcher active mainly in the field of mathematics, meta-mathematics, and logic, with a period of intensive work with calculating machines, and then, later on, with groundbreaking work in theoretical chemistry and in the philosophy of artificial machines. But not in semiotics (cf. Hodges 1988, 1994)

Despite this lack of an explicit historical relationship between Turing and semiotics, one can nevertheless show that such a relationship exists. Moreover, from the present vantage point, one would be inclined to say that Turing's contribution to semiotics is very fundamental indeed, if not the most fundamental to the history of semiotics.

To understand Turing's importance to semiotics, one must step back a bit and look at the larger perspective through the fields of meta-mathematics, computational machines, theory of science, semiotics and modern engineering.

During
his short life, Turing wrote many important papers (cf. the list of
Turing's publications in *References*). But the one most
relevant here is "On Computable Numbers, with an Application to
the Entscheidungsproblem*
*[decision problem]", written in 1936 and published at the
turn of 1936-37. In this paper he solved this long-debated problem of
meta-mathematics. Although he was not the only one to publish papers
on this topic in these years (cf. the diagram: 'TURINGSLIFE'), and
although his solution was preceded by a similar result achieved by
Alonzo Church (Church 1936, 1936a), it was Turing's paper that became
most widely accepted and which was the starting point for the famous
'Turing Machine' concept, a name introduced by Alonzo Church in his
review of Turing's paper in the *Journal of Symbolic Logic, *1937.

While Gödel was not convinced by Church's proof of 1936, he immediately accepted Turing's paper, and in his remark at Princeton in 1946 he stated that 'with this concept [i.e. Turing's computability], one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not dependent upon the formalism chosen' (cited in Davis 1965:83). And even Church was willing to accept Turing's conceptual approach as convincing when, in his review of Turing, he stated: "Of these methods [Turing's computability, general recursiveness, and lambda-definability], the first has the advantage of making evident immediately the identification effective in the ordinary (not explicitly defined) sense - i.e. without the necessity of proving preliminary theorems. The second and third have the advantage of being suitable for being incorporated within a system of symbolic logic.' (cf. Church 1937).

To
understand the importance of Turing's paper, in particular, one must
consider for a moment the essence of the *'Entscheidungsproblem'*,
which was so prominent at that time.

At
the turn of the 19^{th} and in the first quarter of the 20^{th}
century, meta-mathematics (theorizing about mathematics) reached a
new peak: the formalization of logic attained a new maturity which
allowed mathematicians increasingly to formalize their thinking *about*
mathematics and not just mathematics itself. And it was 'proof
theory' which most profited from this development.

In proof theory (cf. the diagram 'HILBERT&CO) one investigates the method of arriving at a new true statement from a set of existing true statements (called 'axioms' and 'assumptions') and also at a set of rules that indicate how to generate ('infer') new true statements from previous ones. Any such sequence of assumptions and rule-generated new statements is called - with regard to the last statement generated - a proof. The generating rules are called 'inference rules', and a system of inference rules is called a logic.

In
reference to this framework, one can characterize the
*Entscheidungsproblem* as follows: Given a collection of
assumptions A stated in some logical system, and a statement S, is it
possible to decide on the basis of some universal computation method
whether S can or cannot be inferred from A? If such a universal
computational method existed, then could such a method be called a
'decision method' for this logic. For logical systems that are weaker
than 'first-order logic', several such procedures existed before
Church's and Turing's publications, but not for systems of logic that
are at least as strong as first-order logic (cf. Kneale, William/
Kneale, Martha 1962:724-737).

At that time, it was generally held that the whole process of generating a proof should be 'finitistic'. But there was no common agreement about the properties which would necessarily constitute a finitistic method. Mathematicians considered arithmetic to be an example of finitistic method. Other methods considered to be similar to the finitistic were Post's formal systems (beginning in the 1920s), Church's lambda-calculus (together with Kleene) (1932-36), and Herbrand-Gödel's General Recursive Functions (1934).

All of these systems (except Post's) themselves relied on complex concepts dependent on external explanations and whose finitistic character was not established beyond doubt.

Under
these circumstances, the advent of Turing's computational device,
later called a 'Turing machine', was a major event. Unlike all
others, but similar in some respects to Post, Turing primarily
analyzed the *pragmatic conditions* under which a *computing
person* calculates with numbers. His analysis ( cf.
the diagram 'TURINGMACHINE') revealed as minimal structure (i) a
*tape* with rectangular cells which can be filled with symbols,
representing the sheets of paper of a computing agent, (ii) a *machine*
with a finite number of conditions {q1, ..., qn} representing the
computing agent and (iii) a *relation between the tape and the
machine* based on the fact that 'at any given moment' only one
square of the tape is 'in the machine'; the content of this square
can be 'read' or can be 'written', and the tape can be 'moved' by one
square in one of two directions. Which symbol would be written at any
given moment and what move would occur depend solely on the state the
machine is 'actually in' (cf. Turing 1936-37:117, citation according
to the reprint in Davis 1965).

With these minimal elements of a symbol-processing system, Turing moved ahead of all that meta-mathematics had done so far: he no longer spoke solely about the formal expressions of a given system and their properties related to certain complex procedures of logical inference. Instead, he also took into account the operating logician or mathematician himself together with his pragmatic relationship to the symbolic expressions with which he was dealing in a specific case.

Clearly,
Turing himself originally had in mind only a person dealing with
numbers, but his model has far broader applications. A symbol on the
tape can represent anything - any segment of reality - including
properties of the computing agent himself; and the finite states
within the machine can also represent any variety of state, e.g.,
physiological states within a biological system or subjective states
within a phenomenologically described reality. Within these general
parameters, Turing's minimal computing structure constitutes a very
*general pragmatic agent model*. And it is this aspect of
Turing's approach that made his arguments more convincing than any
others.

Under
the operating conditions obtaining for any symbol-processing agent
(in philosophical terms this could be called a 'transcendental
argument' in a Kantian sense), Turing is not obliged to use complex
symbolic expressions stemming from other sources (as in the case of
Church, Kleene, Herbrand or Gödel). He has only to rely on this
minimal operating-agent structure, and anything more complex must be
shown to be reconstructable from these *a priori* conditions.
Turing's model implies that any symbol-processing agent can be formed
within the framework of his assumptions. And insofar as one can
assume that the physiological states of the human body (including the
brain!) are conceivable as a finite set of states - not yet proved *in
sensu stricto* - then Turing's model would also claim to cover the
instance of human agents. Thus Turing's approach is very broad, and
it is hard to conceive of an agent to be 'configured' whose behavior
could not be covered by Turing's model. (A widespread argument
challenging the explicatory power of Turing's model uses the opposing
concepts 'discrete' and 'continuous', stating that continuous
processes exist which cannot be comprehended within a discrete
machine. But the property of the 'continuousness' of a process
depends entirely on the way the process is observed and described;
and it is impossible - at least in principle - to exclude the
possibility of lowering the level of detail to a point at which any
'continuous' process is made up of a number of elementary parts
interacting in a certain manner. Thus the opposition of 'discrete'
and 'continuous' may amount to a mere artificial product of thinking
and not a 'real' property of 'nature', in which case Turing's
argument is not necessarily refuted.)

How can we relate Turing's concept of computability, his 'Turing machine', to semiotics?

Is there a relationship of any kind?

To establish such a relationship between Turing's thinking and semiotics, one must first have a sufficiently clear concept of semiotics. But there is no clear-cut definition of semiotics. Clearly, semiotics refers primarily to some concept of a symbol or sign combined with methods on how to work with this concept. But up until now (cf., e.g., the excellent handbook of Nöth [1985]), we have a great wealth of ideas and concepts related to semiotics but no clear-cut and commonly accepted theory. Thus speaking about semiotics implies some ambiguity, and the position an author ultimately assumes will inevitably be determined by his own preferences.

This author strongly prefers to employ modern prerequisites for scientific theory as a conceptual framework for the discussion. And to support this 'bias', he prefers using the writings of Charles Morris (1901-1979) as a main point of reference for discussion of the connection between Turing and semiotics. No exclusion of other 'great fathers' of modern semiotics is intended (Peirce 1839-1914, but also de Saussure 1857-1913, von Uexküll 1864-1944, Cassirer 1874-1945, Bühler 1879-1963, and Hjelmslev 1899-1965, to mention only the most prominent). But Morris seems to be the best starting point for the discussion because his proximity to the modern concept of science makes him a sort of bridge between modern science and the field of semiotics.

To make explicit whether Turing has contributed to semiotics, and in which senses this contribution might be of importance, we must first offer a short description of semiotics in regard to the writings of Morris. But before doing so, we must introduce the concept of a scientific theory as the framework for our preparation for an encounter between Turing's concept of computability and Morris' concept of semiotics.

At
the time of the *Foundations* in 1938, his first major work
after his dissertation of 1925, Morris was already strongly linked to
the new movement of a 'science of the sciences', which was the focus
of several groups connected to the Vienna Circle, to the Society of
the History of the Sciences, to several journals and conferences and
congresses on the theme, and especially to the project of an
Encyclopedia of the Unified Sciences (cf., e.g., Morris, Charles W.
[1936]) ()see the diagram 'MORRISLIFE').

In
the *Foundations* he states clearly that semiotics should be a
science, distinguishable as pure and descriptive semiotics (cf.
Morris 1977: 17, 23), and that semiotics could be presented as a
deductive system (cf. Morris 1977: 23). The same statements appear in
his other major book about semiotics (Morris 1946), in which he
specifically declares that the main purpose of the book is to
establish semiotics as a scientific theory (M. 1946:28). He makes
many other statements in the same vein.

At the time of Morris's writings, the philosophy of science was dominated by what was later called the 'received view' (for a good summary of the main lines of the discussion up to 1967, see Suppe, 1977), which was propagated primarily by Rudolf Carnap.

Rooted in Machean neo-positivism with sensations viewed as data, and combined with the conventionalism of Poincaré, the key idea of the 'received view' was that a scientific theory must use mathematics - or formal logic - as its general language for the stating of onsistencies and general laws, but that the terms of this mathematical language must be interpreted by a clearly defined observational language, the descriptive terms of which are to be based exclusively on observable phenomena (cf. Suppe 1977:50ff).

During the 1950s and 1960s it became clear that this concept was unsatisfactory. One of the main reasons for this was the fact that the theoretical terms can at best be only partially interpreted by the observational language. After the historic conference in Urbana, Illinois, in March 1969, enthusiasm for the theoretical debates of philosophy of science cooled a bit, and right up to the present day there is no new single unifying notion on the form a scientific theory should take.

For
my proposal I will adopt ideas from the theory-concept urged by the
theoretical physicist Ludwig (1978, 1978b), by the structuralist view
as explained by Wolfgang Balzer (1982) as well as in Balzer,
Moulines, and Sneed (1987), and by the meta-theoretical
clarifications of Peter Hinst (1996). All these positions are rooted
in the structural concept of Bourbaki (1970). To distinguish my
proposal from the ones cited, I term my concept *minimal
theoretical framework *(MTF). The intention of the MTF is to allow
as many instances of scientific activity as possible to be subsumed
under this framework, including semiotics.

The disposition of the MTF here will be somewhat informal, a rigorous treatment being beyond the scope of this paper.

The
starting point for any type of scientific exploration ( see
diagram 'SCIENTIFIC') is the *group of researchers*, e.g.,
semioticians or researchers interested in computational semiotics
(CS). That group is able to communicate in a *primary language L0*
(e.g. 'natural' English), which is embedded in a communicative
context.

This
group has a *pre-scientific view of their subject* of
investigation. (In the case of CS, this view is given through
'candidates for semiotic processes‘.)

The
group is able to define in their primary language certain *methods
of measurement*. Depending on which 'point of view' one takes to
view the 'world' one can distinguish several main 'types of
measurement‘:

- behavior-based (S-R := Stimulus Response Pairs)
- inner states of a system (N := [neuro-]physiological)
- correlations of behavior and physiology (S-N-R)
- phenomenological 'givens' (Ph)
- correlations between physiology and phenomenological data (N – Ph)
- correlations between behavior and phenomenological data (S-R-Ph)
- correlations between behavior and physiology and phenomenological data (S-N-R-Ph)

These
measurements *define the data*. The data must be represented in
some formal representation, a *data representation language L1*.

Furthermore,
the group must have some *axioms (rules)* articulated in a
*theory language L2* to represent certain assumed regularities/
patterns/ consistencies in the realm of the data. L1 could be a
subset of L2.

And
finally, the group must have some meta-rules describing operations on
data and axioms that permit logical *inferences*. These rules
are called *inference rules*
and are constituting a *logical inference concept*.

The
formal structures representing the theory are intended to 'grasp‘
the 'empirical content‘ of the subject through formal means. In
any case, this will be a form of *approximation*. The whole unit
of data and axioms we call a *theory*.

It
is possible, but not necessary, to set up an additional formal
structure with the aid of an additional formal language L3 to encode
the *intended meaning* of the theory in a model-theoretic
manner.

As
in many cases today, a *computational model *of the theory will
be necessary. One reason is the complexity of the subject, especially
if it has dynamic features. It is nearly impossible for a human brain
to handle complex dynamic structures effectively. In order to claim
those computational models as 'scientific‘ with regard to the
presupposed theory, one must establish a *mapping function between
theory and the computational model. *Insofar as such a mapping
function exists, one can use the computational model to *emulate
*the theory within this computational model and thereby *simulate*
certain features of 'reality' if the main theory is an empirical
theory. In case of *engineering*
one constructs also machines based on a theory but these machines are
not necessarily 'computational' machines in the sense of Turing.

A
theory without data would amount to a purely formal structure and
would thus be called a *pure theory* (in this sense any
mathematical structure would be considered as a pure theory). With
(empirical) data at hand, a pure theory becomes a *descriptive or
interpretive theory*. This is what science usually intends to be.
And a theory connected with a (computational) machine is called an
*applied theory*.

With this concept of a scientific theory at hand, we will reconstruct Turing's contribution within this framework as well as the position of Morris. And we shall see that this will enable us to evaluate the semiotic relevance of Turing's contribution.

In Turing's case, one can quite straightforwardly relate his paper of 1936-7 to the concept of a scientific theory (see diagram: 'TURINGMACHINE').

The
*group of researchers* is represented by Turing himself and by
the people with whom he was communicating his ideas. The *focus of
investigation* is given by real persons doing real computing work
- writing numbers on sheets of paper with pencils. The modes of
*measurement* have been restricted to normal perception; i.e.,
the subjective (= phenomenological) experience of an intersubjective
situation and the symbolic representation has been done in ordinary
English (=L1) as well as in a first-order language (=L1). But Turing
was not primarily interested in the construction of a descriptive
theory but in the elaboration of a *formal structure *that
represents all the features of the subject of investigation that 'are
of importance'. He represented this formal structure in a first-order
theory (=L2) which can/must be understood rather as a *pure*
theory than as an elaborated descriptive theory. Nevertheless, it is
very simple to convert his pure theory in many instances into an
interpreted theory.

The
'content' of his pure theory can be described as follows (cf. also
above): there is (i) a *tape* with rectangular cells
representing the sheets of paper of a computing agent which can be
filled with symbols, (ii) a *machine* with a finite number of
states {q1, ..., qn} representing the computing agent and (iii) a
*connection between the tape and the machine* that is based on
the fact that 'at any moment' only one square of the tape is 'in the
machine', and the content of this square can be 'read' and can be
'written'; and the tape can be 'moved' by one square in one of two
directions. Which symbol would be written at any moment and what move
would occur depends solely on the state the machine is 'actually in'
(cf. Turing 1936-37:117, citation according to the reprint in Davis
1965). The expression 'q_r = <S_i, a,b,
M,S_j>' can be read as follows: the state 'q_r' is given by the
quintupel '<S_i,
a,b, M,S_j>'
and 'S_i' is the identifying number of the state q_r, '
a' is
the symbol just read on the actual square of the tape, ' b'
is the symbol which has to be written on the actual square of the
tape, 'M' is the next move on the tape ('left', 'right', 'halt'), and
'S_j' is the number of the next state which has to be entered.

Turing was also fortunate in being one of the first people to work on 'real' embodiments of this very general concept in that during World War II he was involved in the development of the Colossus computing machine. After the war, he also headed the ACE project (Automatic Calculating Engine) at the National Physics Laboratory (NPL) for some time. It remains an open question how far the North American projects (von Neumann and others) have been influenced by Turing's concept: his paper was well known there, and it was much broader in scope than the one that was used in the very limited ENIAC computer. Thus Turing's pure theory was gradually becoming also an applied theory.

To reconstruct the contribution of Charles Morris within the theory concept is not as straightforward as one might think. Despite his very active involvement in the new science-of-the-sciences movement (cf. diagram 'MORRISLIFE'), and despite his repeated claims to handle semiotics scientifically, Morris did not provide any formal account of his semiotic theory. He never left the level of ordinary English as the language of representation. Moreover, he published several versions of a theory of signs which overlap extensively but which are not, in fact, entirely consistent with one another.

Thus
to speak about 'the' Morris theory would require an exhaustive
process of *reconstruction*, the outcome of which might be a
theory that would claim to represent the 'essentials' of Morris's
position. Such a reconstruction is beyond the scope of this paper.
Instead, I will rely on my reconstruction of Morris's *Foundations*
of 1938 (cf. Döben-Henisch 1998) and on his basic methodological
considerations in the first chapter of his 'Signs, Language, and
Behavior' of 1946 (for the following see diagram[s]) (cf.
diagram 'MORRIS´s Interpreter').

As
the *group of researchers,* we assume Morris and the people he
is communicating with.

As
the *domain of investigation,* Morris names all those processes
in which 'signs' are involved. And in his *pre-scientific view*
of what must be understood as a sign, he introduces several basic
terms simultaneously. The primary objects are distinguishable
organisms which can act as *interpreters [I]*. An organism can
act as an interpreter if it has internal states called *dispositions
[IS]* which can be changed in reaction to certain stimuli. A
*stimulus [S]* is any kind of physical energy which can
influence the inner states of an organism. A *preparatory-stimulus
[PS]* influences a response to some other stimulus. The source of
a stimulus is the *stimulus-object [SO]. *The *response [R]*
of an organism is any kind of observable muscular or glandular
action. Responses can form a *response-sequence [<r_1, ...,
r_n>], *whereby every singly intervening response r_i is
triggered by its specific *supporting* stimulus. The
stimulus-object of the first response in a chain of responses is the
*start object,* and the stimulus-object of the last response in
a chain of responses is the *goal object*. All
*response-sequences* with similar *start objects *and
similar *goal objects* constitute a *behavior-family [SR-FAM]*.
Based on these preliminary terms he then defines the characteristics
of a *sign* *[SGN] *as follows: 'If anything, A, is a
preparatory-stimulus which in the absence of stimulus-objects
initiating response-sequences of a certain behavior-family causes a
disposition in some organism to respond under certain conditions by
response-sequences of this behavior-family, then A is a sign.'
(Morris 1946:10,17). Morris stresses that this characterization
describes only the necessary conditions for the classification of
something as a sign (Morris 1946:12).

This
entire group of terms constitutes the *subject matter* of the
intended science of signs (= semiotics) as viewed by Morris (Morris
1946:17). And based on this, he introduces certain additional terms
for discussing this subject.

Already
at this fundamental stage in the formation of the new science of
signs, Morris has chosen 'behavioristics', as he calls it in line
with von Neurath, as the point of view that he wishes to adopt in the
case of semiotics. In the *Foundations* of 1938, he stresses
that this decision is not necessary (cf. Morris 1971: 21) and also in
his 'Signs, Language, and Behavior' of 1946, he explicitly discusses
several methodological alternatives ('mentalistic',
'phenomenological' [Morris 1946:30 and Appendix]), but he considers a
behavioristic approach more promising with regard to the intended
scientific character of semiotics.

From today's point of view, it would no longer be necessary to oppose these different approaches to one another, but as this was the method used by Morris at that time, we will follow his lead for a moment.

Morris
did not mention the problem of *measurement* explicitly. Thus
the modes of measurement are - as in the case of Turing - restricted
to normal perception; i.e., the subjective (= phenomenological)
experience of an intersubjective situation restricted to observable
stimuli and responses and the symbolic representation has been done
in ordinary English (=L1) without any attempt at formalization.

Clearly,
Morris did not limit himself to characterizing in basic terms the
subject matter of his 'science of signs' but introduced a number of
additional terms. Strictly speaking, these terms establish a
*structure* which is intended to shed some theoretical light on
'chaotic reality'. In a 'real' theory, Morris would have
'transformed' his basic characterizations into a formal
representation (as Turing did with his postulated computing person),
which could then be formally expanded by means of additional terms if
necessary. But he didn't. Thus we can put only some of these
additional terms into ordinary English to get a rough impression of
the structure that Morris considered important.

Morris
used the term *interpretant [INT] *for all interpreter
dispositions (= inner states) causing some response-sequence due to a
'sign = preparatory-stimulus'. And the goal-object of a
response-sequence 'fulfilling' the sequence and in that sense
*completing* the response-sequence Morris termed the *denotatum*
of the sign causing this sequence. In this sense one can also say
that a sign *denotes* something. Morris assumes further that the
'properties' of a denotatum which are connected to a certain
interpretant can be 'formulated' as a set of conditions which must be
'fulfilled' to reach a denotatum. This set of conditions constitutes
the *significatum* of a denotatum. A sign can trigger a
significatum, and these conditions control a response-sequence that
*can* lead to a denotatum, but *do not necessarily *do so:
a denotatum is not *necessary*. In this sense, a sign *signifies*
at least the conditions which are *necessary* for a denotatum
but not *sufficient* (cf. Morris 1946:17ff). A *formulated
significatum* is then to be understood as a formulation of
conditions in terms of other signs (Morris 1946:20). A formulated
significatum can be *designative* if it describes the
significatum of an existing sign, and *prescriptive* otherwise.
A *sign-vehicle [SV]* can be any particular physical event which
is a sign (Morris 1946:20). A set of similar sign-vehicles with the
same significata for a given interpreter is called a *sign-family*.

We will restrict our discussion of Morris to the terms introduced so far.

The fact that Morris did not translate these assumptions into a formal representation is a real drawback, because even these few terms contain a certain amount of ambiguity and vagueness.

Morris did not work out a computational model for his theory. At that time, this would have been nearly impossible for practical reasons. Besides, his theory was formally too weak to be used as a basis for such a model.

In order to take the next step and compare Morris with Turing, we will need at least an initial formalization of Morris's contribution. We will offer some ideas for such a formalization by presenting the first steps of such a process.

First, one must decide how to handle the 'dispositions' or 'inner states' (IS) of the interpreter I. From a radical behavioristic point of view, the interpreter has no inner states, but only stimuli and responses (cf. my discussion in Döben-Henisch 1998). Any assumptions about the system's possible internal states would be related to 'theoretical terms' within the theory which have no direct counterpart in reality. If one were to enhance behavioral psychology with physiology (including neurology) in the sense of neuropsychology, then one could identify internal states of the system with (neuro-)physiological states (whatever this would mean in detail). In the following, we shall assume that Morris would accept this latter approach. We shall label such an approach an S-N-R-theory or SNR-approach.

Within an SNR-approach, it is in principle possible to correlate an 'external' stimulus event S with a physiological ('internal') event S' as Morris intended: a stimulus S can exert an influence on some disposition D' of the interpreter I, or, conversely, a disposition D' of the interpreter can cause some external event R.

To work this out, one must assume that the interpreter is a structure with at least the following elements (cf. diagram 'SNR-MORPHISMs'):

I(x) iff x = <IS, <f_1, ..., f_n>, Ax>; i.e., an x is an Interpreter I if it has some internal states IS as objects (whatever these might be), some functions f_i operating on these internal states like f_i: pow(IS) ---> pow(IS), and some axioms stating certain general dynamic features. These functions we will call 'type I functions' and they are represented by the symbol 'f_I'.

By the same token, one must assume a structure for the whole environment E in which those interpreters may occur: E(x) iff x = <<I,S,R,O>, <p_1, ..., p_m>, Ax>. An environment E has as objects at least some interpreters I, something which can be identified as stimuli S or responses R, and, possibly, some additional objects O (without any further assumptions about the 'nature' of these different sets of objects). Furthermore, there must be different kinds of functions p_i, e.g.:

- (Type E Functions [f_E]:) p_i: pow({I,S,R,O}) ---> pow({I,S,R,O}), stating that there are functions which operate on the 'level' of the assumed environmental objects. A special instance of these functions would be functions of the type p_i*: pow(S) ---> pow(R);
- (Type E u I Functions [f_E u I]:) p_j: pow(S) ---> pow(IS_I), stating that there are functions which map environmental stimuli events as internal states of interpreters (a kind of 'stimulation'-function); and
- (Type I u E Functions [f_I u E]:) p_j: pow(IS_I) ---> pow(R), stating that there are functions which map the internal states of interpreters as environmental response events (a kind of 'activation'-function).

The
overall constraint for all of these different functions is depicted
in the diagram 'SNR-MORPHISMS'. This shows the basic equation f_E =
f_E u I **o **f_I **o **f_I u E, i.e. the mapping f_E of
environmental stimuli S into environmental responses R should yield
the same result as the concatenation of f_E u I (mapping
environmental stimuli S into internal states IS of an interpreter I),
followed by f_I (mapping internal states IS of the interpreter I onto
themselves), followed by f_I uE (mapping internal states IS of the
interpreter I into environmental responses R).

Even
these very rough assumptions make the functioning of the sign
somewhat more precise. A sign as a preparatory stimulus S2 'stands
for' some other stimulus S1 and this shall especially work in the
*absence* of S1. This means that if S2 occurs, then the
interpreter takes S2 'as if' S1 occurs. How can this work? We make
the assumption that S2 can only work because S2 has some 'additional
property' which encodes this aspect. We assume that the *introduction
of S2 for S1* occurs in a situation in which S1 and S2 occur 'at
the same time'. This 'togetherness' yields some representation
'(S1'', S2'')' in the system which can be 'reactivated' each time one
of the triggering components S1' or S2' occurs again. If S2 occurs
again and triggers the internal state S2', this will then trigger the
component S2'', which yields the activation of S1'' which in turn
yields the internal event S1'. Thus S2 -> S2' -> (S2'', S1'')
-> S1' has the same effect as S1 -> S1', and vice versa. The
*encoding property* is here assumed, then, to be a
representational mechanism which can be reactivated constantly.

We shall stop at this preliminary formalization. It would be no problem to integrate phenomenological data and phenomenologically based functions into such a formalization as well, but we will not take this up at this point. Instead, we shall continue comparing Turing and Morris directly.

We shall compare Morris's interpreter with Turing's computing device, the Turing machine, in the light of our preliminary formalization by establishing informally a kind of mapping between the two (cf. diagram TURING-MORRIS).

It is clear what we have in the Turing machine: a tape with symbols [SYMB] which are arguments of the machine table [MT] of a Turing machine as well as values; i.e., we have MT: pow(SYMB) ---> pow(SYMB).

In the interpreter [I], we have stimuli [S] which are 'arguments' of the interpreter function f_E that yields responses [R] as values, i.e., f_E: pow(S) ---> pow(R). The interpreter function f_E can be 'refined' by replacing it by three other functions: f_EI: pow(S) ---> pow(IS_I), f_I: pow(IS_I) ---> pow(IS_I), and f_IE: pow(IS_I) ---> pow(R) so that

f_E
= f_EI **o** f_I **o** f_IE.

(cf. the diagram 'SNRMORPHISM').

Now, because one can subsume stimuli S and responses R under the common class 'environmental events EV', and one can represent any kind of environmental event with appropriate symbols SYMB_ev, with SYMB_ev as a subset of SYMB, one can then establish a mapping of the kind

SYMB
<--- SYMB_ev <----> EV <---- S **u** R.

What then remains is the task of relating the machine table MT to the interpreter function f_E. If the interpreter function f_E is taken in the 'direct mode' (the case of pure behaviorism), without the specializing functions f_E u I etc., we can establish directly a mapping

MT <---> f_E.

The argument for this mapping is straightforward: any version of f_E can be directly mapped into the possible machine table MT of a Turing machine, and vice versa. In the case of an interpreted theory, the set of 'interpreted interpreter functions' will very probably be a true subset of the set of possible functions.

If
one replaces f_E by f_E = f_E u I **o** f_I **o** f_I u E, then
we must establish a mapping of the kind

MT
<---> f_E = f_E u I **o** f_I **o** f_I u E.

The
compound function f_E u I **o** f_I **o** f_I u E operates on
environmental states EV and on the internal states IS of the
interpreter; i.e., the 'tape' of the Turing machine must be extended
'into' the interpreter.

The machine table MT of the Turing machine is 'open' to any interpretation of what 'kind of states' can be used to 'interpret' the general formula. The same holds true for Morris. He explicitly left open which concrete states should be subsumed under the concept of an internal state. The 'normal' attitude (at least today) would be to subsume 'physiological states'. But Morris pointed out (Morris 1946:30) that this might not be so; for him it was imaginable also to subsume 'mentalistic' states. And as has been pointed out above, it would not be difficult formally to integrate 'phenomenalistic' states. Thus, what all of these possible different interpretations have in common is to posit that states exist which can be identified as such and therefore can be symbolically represented. From this it follows that we can establish a mapping of the kind

SYMB <--- IS.

Because the tape - but not the initial argument and the computed value - of a Turing machine can be infinite, one must assume that the number of distinguishable internal states IS of the interpreter that are not functions is also 'finite'. This assumption makes sense: though it cannot be formally proved, it cannot be disproved either.

What
remains is the mapping between MT and the compound interpreter
function f_E u I **o** f_I **o** f_I u E. The only constraint
on a possible interpretation of the states of a machine table MT is
the postulate that the number of machine table states must be finite.
In the case of the compound interpreter function f_E u I **o** f_I
**o** f_I u E (which can be considered as a synthesis of many
small partial functions), it also makes sense to assume that the
number is finite. It is thus fairly straightforward to map any
machine table into a function f_E u I **o** f_I **o** f_I u E
and vice versa.

Thus we have reached the conclusion that an exhaustive mapping between Turing's Turing machine and Morris's interpreter is possible. This is a far-reaching result. It enables every semiotician working within Morris's semiotic concept to use the Turing machine to introduce all of the terms and functions needed to describe semiotic entities and processes. As a by-product of this direct relationship between semiotics 'à la Morris' and the Turing machine 'à la Turing', the semiotician has the additional advantage that any of his/her theoretical constructs can be used directly as a computer program on any computational machine. Thus 'semiotics' and 'computational semiotics' need no longer be separately interpreted, because what they both signify and designate is the same. Thus, one can (and should) claim that

semiotics = computational semiotics.

This equation makes of semiotics not only a potential player on the international science team but, even more, opens up the possibility that semiotics might be one of the scientific forerunners in the field of 'intelligent systems'. Whether semiotics can establish itself in the future as an original science based on the characterizing concept of the term 'sign' will strongly depend on new definitions of what a sign must be within such a formal framework. The short discussion above, occasioned by Morris's characterizations, shows that such a definition is far from trivial. It implies a complex mechanism of conditions and functions which must be worked out sufficiently clearly, an undertaking which has yet to be attempted.

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