October 28, 2011 § Leave a comment
At first sight it may sound like a bad joke, indeed.
Turing not only provided many important theoretical insights on computing , including the Universal Turing Machine (UTM), he and his group in Bletchley Park also created a working prototype, which had been employing the theoretical results .
In order to clarify what non-Turing computing could be, we first have to inspect a bit closer how Turing-computing is defined. On Wikipedia one can find the following explanation in standard language:
With this encoding of action tables as strings it becomes possible in principle for Turing machines to answer questions about the behavior of other Turing machines. Most of these questions, however, are undecidable, meaning that the function in question cannot be calculated mechanically. For instance, the problem of determining whether an arbitrary Turing machine will halt on a particular input, or on all inputs, known as the Halting problem, was shown to be, in general, undecidable in Turing’s original paper. Rice’s theorem shows that any non-trivial question about the output of a Turing machine is undecidable.
A universal Turing machine can calculate any recursive function, decide any recursive language, and accept any recursively enumerable language. According to the Church-Turing thesis, the problems solvable by a universal Turing machine are exactly those problems solvable by an algorithm or an effective method of computation, for any reasonable definition of those terms.
One could add firstly that any recursive algorithm can be linearized (and vice versa). Secondly, algorithms are defined as procedures that produce a defined result after a finite state of time.
Here is already the first problem in computational theory. What is a result? Is it a fixed value, or would we accept a probability density or even a class of those (like Dirac’s delta) also as a result? Even non-deterministic Turing machines yield unique results. The alternative of an indeterminable result sounds quite counter-intuitive and I suppose that it indeed can not be subsumed under the classical theory of computability. It would simply mean that the results of a UTM are only weakly predictable. We will return to that point a bit later.
Another issue is induced by problem size. While analytic undecidability causes the impossibility for the putative computational procedure to stop, sheer problem size may render the problem as if being undecidable. Solution spaces can be really large, beyond 102000 possible solutions. Compare this to the estimated 1080 atoms of visible matter in the whole universe. Such solution spaces are also an indirect consequence of Quine’s principle of underdetermination of an empirical situation, which results in the epistemological fact of indeterminacy of any kind of translation. We will discuss this elsewhere (not yet determined chapter…) in more detail.
From the perspective of an entity being searching through a such a large solution space it does not matter very much, whether the solution space is ill-defined or vast, from the perspective of the machine controller (“user”) both cases belong to the same class of problems: There is no analytic solution available. Let us now return the above cited question about the behavior of other entities. Even for the trivial case that the interactee is a Turing machine, the question about the behavior is undecidable. That means that any kind of interaction can not be computed using an UTM, particularly however those between epistemic beings. Besides the difficulties raised by this for the status of simulation, this means that we need an approach, which is not derived or included in the paradigm established by the Church-Turing thesis.
The UTM as the abstract predecessor of today’s digital computers is based on the operations of writing and deleting symbols. Before an UTM can start to work, the task to be computed needs to be encoded. Once, the task has been actually encoded, including the rules necessary to accomplish the computation, everything that happens is just an almost material moving of semantically empty graphemes. (We avoid here to call the 0 and 1 “symbols,” since “symbol” is a compound concept, hence it could introduce complications to our investigation here.) During the operations of the UTM, the total amount information is constantly decreasing. Else, an UTM is not only initially completely devoid of any meaning, it remains semantically empty during the whole period it works on the task. Any meaning concerning the UTM remains forever outside the UTM. This remains true even if the UTM would operate at the speed of light.
Note, that we are not discussing the architecture of an actual physical computing device. Everybody uses devices that are built according von Neumann architecture. There are very few (artificial) computers on this earth not following this paradigm. Yet, it is unclear why DNA-computers or even quantum computers should not fall in this category. These computers’ processing is different from an instance that computes based on traditional logics, physically realized as transistors. Yet, the von Neumann architecture does not make any proposal about the processor except that there need to be one. Such fancy computers still need persistent storage, a bus system, encoding and decoding devices.
As said, our concern is not about the architecture, or even more trivial, about different speed of calculation. Hence, he question of non-Turing computing is also not a matter of accuracy. For instance, it is sometimes claimed that a UTM can simulate an analog neural net with with arbitrary accuracy. (More on that later!) The issue at stake has much more to do with the role of encoding, the status of information and being an embodied entity than with the question of how to arrange physical components.
Our suggestion here is that any kind of computer could be probably used in a way that it changes into a non-Turing computer. In order to deal with this question we have to discuss first the contemporary concept of “computation.”
To get clear about the concept of “computation” does not include the attempt to find an answer to the question “What is computation?”, as for instance Jack Copeland did . Such a question can not be included in any serious attempt of getting clear about it, precisely because it is not an ontological question. There are numerous attempts to define computation, then to invoke some intuitively “clear” or otherwise “indisputable” “facts”, only in order to claim an ontological status of the respective proposal. This of course is ridiculous, at least nowadays after the Linguistic Turn. Yet, the conflation of definitory means and ontic status is just (very) naive metaphysics, if not to say esoterism in scientifically looking clothes. The only thing we can do is to get clear about possible “reasonable” ways of usage of the concepts in question.
In philosophy of mind and cognitive science, and thus also for our investigation of machine-based epistemology, the interest in getting clear about computation is given by two issues. First, there is the question, whether, and, if yes, to what extent, the brain can be assigned “a computational interpretation.” To address this question we have to clarify what “computing” could mean and whether the concept of “brain” could match any of the reasonable definitions for computing. Second, as a matter of fact we know before any such investigation that we, in order to create a machine able to follow epistemological topics, have at least to start with some kind of programming.The question here is simply how to start practically. This concerns methods, algorithms, or machine architectures. A hidden but important derivative of this question is about the possible schemes of differentiation of an initial artifact, which indeed is likely to be just a software running on a contemporary standard digital computer.
These questions that are related to the mind are not in the focus of this chapter. We will return to them elsewhere. First, and that’s our interest here, we have to clarify the usage of the concept of computation. Francesco Nir writes :
According to proponents of computationalism, minds are computers, i.e., mechanisms that perform computations. In my view, the main reason for the controversy about whether computationalism is accurate in its current form, or how to assess its adequacy is the lack of a satisfactory theory of computation.
It is obvious that not only the concepts of computation, brain and mind are at stake and have to be clarified, but also the concept of theory. If we would follow a completely weird concept about “theory,” i.e. if our attempts would try to follow an impossible logical structure, we would have no chance to find appropriate solutions for those questions. We even would not be able to find appropriate answers about the role of our actions. This, of course, is true for any work; hence we will discuss the issue of “theory” in detail in another chapter. Similarly, it would be definitely to limited to conceive of a computer just as a digital computer running some algorithm (all of them are finite by definition).
The history of of computation as an institutionalized activity starts in medieval ages. Of course, people performed calculation long before. The ancient Egypts even used algorithms for problems that can’t be written in a closed form. In classics, there have been algorithms to calculate pi or square roots. Yet, only in medieval ages the concept of “computare” got a definite institutional, i.e. functional meaning. It referred to the calculation of the future Easter dates. The first scientific attempts to define computation start mainly with works published by Alan Turing and Alonzo Church, which then was later combined into the so-called Church-Turing-Thesis (CTT).
The CTT is a claim about effectively computable functions, nothing more, nothing less. Turing found that everything which is computable in finite time (and hence also on a finite strap) by his a-machine (later called Turing machine), is equivalent to the λ-calculus. As an effect, computability is equaled with the series of actions a Turing machine can perform.As stated above, even Universal Turing Machines (UTM) can’t solve the Halting-problem. There are even functions that can’t be decided by UTM.
It has been claimed that computation is just the sequential arrangement of input, transformation, and output. Yet, as Copeland and Nir correctly state, citing Searle therein, this would render even a wall into a computer. So we need something more exact. Copeland ends with the following characterization:
“It is always an empirical question whether or not there exists a labelling of some given naturally occurring system such that the system forms an honest model of some architecture-algorithm specification. And notwithstanding the truism that ‘syntax is not intrinsic to physics’ the discovery of this architecture-algorithm specification and labelling may be the key to understanding the system’s organisation and function.”
The strength of this attempt is the incorporation of the relation between algorithm and (machine) architecture into the theory. The weakness is given by the term “honest,” which is completely misplaced in the formal arguments Copeland builds up. If we remember that “algorithm” means “definite results in finite time and space” we quickly see that Copeland’s concept of computation is by far too narrow.
Recently, Wilfried Sieg tried to clarify the issues around computation and computability in a series of papers [5,6]. Similarly to Nir (see above), he starts his analysis writing:
“To investigate calculations is to analyze symbolic processes carried out by calculators; that is a lesson we owe to Turing. Taking the lesson seriously, I will formulate restrictive conditions and well motivated axioms for two types of calculators, namely, for human (computing) agents and mechanical (computing) devices. 1 My objective is to resolve central foundational problems in logic and cognitive science that require a deeper understanding of the nature of calculations. Without such an understanding, neither the scope of undecidability and incompleteness results in logic nor the significance of computational models in cognitive science can be explored in their proper generality.” 
Sieg follows (and improves) largely an argument originally developed by Robin Gandy. Sieg characterizes it (p.12):
“Gandy’s Central Thesis is naturally formulated as the claim that any mechanical device can be represented as a dynamical system satisfying the above principles.”
By which he meant four limiting principles that prevent that everything is regarded as a computer. He then proceeds:
I no longer take a Gandy machine to be a dynamical system 〈S, F〉 (satisfying Candy’s principles), but rather a structure M consisting of a structural class S of states together with two kinds of patterns and operations on (instantiations of) the latter;”
[decorations by W.Sieg]
What is a dynamical system for Sieg and Gandy? Just before (p.11), Sieg describes it as follows:
“Gandy’s characterization […] is given in terms of discrete dynamical systems 〈S, F〉, where S is the set of states and F governs the system’s evolution. More precisely, S is a structural class, i.e., a subclass of the hereditarily finite sets H F over an infinite set U of atoms that is closed under ∈- isomorphisms, and F is a structural operation from S to S, i.e., a transformation that is, roughly speaking, invariant under permutations of atoms. These dynamical systems have to satisfy four restrictive principles.”
[decorations by W.Sieg]
We may drop further discussion of these principles, since they just add further restrictions. From the last two quotes one can see two important constraints. First, the dynamical systems under considerations are of a discrete character. Second, any transformation leads from a well-defined (and unique) state to another such state.
The basic limitation is already provided in the very first sentence of Sieg’s paper: “To investigate calculations is to analyze symbolic processes carried out by calculators;” There are two basic objections, which lead us to deny the claim of Sieg that his approach provides the basis for a general account of computation
Firstly, from epistemology it is clear that there are no symbols out in the world. We even can’t transfer symbols in a direct manner between brains or minds in principle. We just say so in a very abbreviative manner. Even if our machine would work completely mechanically, Sieg’s approach would be insufficient to explain a “human computor.” His analysis is just and only be valid for machines belonging (as a subclass) to the group of Turing machines that run finite algorithms. Hence, his analysis is also suffering from the same restrictions. Turing machines can not make any proposal about other Turing machines. We may summarize this first point by saying that Sieg thus commits the same misunderstanding as the classical (strong) notion of artificial intelligence did. Meanwhile there is a large, extensive and somewhat bewildering debate about symbolism and sub-symbolism (in connectionism) that only stopped due to exhaustion of the participants and the practical failure of strong AI.
The second objection against Sieg’s approach comes from Wittgenstein’s philosophy. According to Wittgenstein, we can not have a private language . In other words, our brains can not have a language of thinking, as such a homunculus arrangements would always be private by definition. Searle and Putnam agree on that in rare concordance. Hence it is also impossible that our brain is “doing calculations” as something that is different from the activities when we perform calculation with a pencil and paper, or sand, or a computer and electricity. This brings us to an abundant misunderstanding about what computer really do. Computers do not calculate. They do not calculate in the same respect as we our human brain does not calculate. Computers just perform moves, deletions and—according to their theory—sometimes also an insertion into a string of atomic graphemes. Computers do not calculate in the same way as the pencil is not calculating while we use it to write formulas or numbers. The same is true for the brain. What we call calculation is the assignment of meaning to a particular activity that is embedded in the Lebenswelt, the general fuzzy “network”, or “milieu” of rules and acts of rule-following. Meaning on the other hand is not a mental entity, Wilhelm Vossenkuhl emphasizes throughout his interpretation of Wittgenstein’s work.
The obvious fact that we as humans are capable of using language and symbols brings again the question to the foreground, which we addressed already elsewhere (in our editorial essay): How do words acquire meaning? (van Fraassen), or in terms of the machine-learning community: How to ground symbols? Whatsoever the answer will be (we will propose one in the chapter about conditions), we should not fallaciously take the symptom—using language and symbols—as the underlying process, “cause”, or structure. using language clearly does not indicate that our brain is employing language to “have thoughts.”
There are still other suggestions about a theory of computation. Yet, they either can be subsumed to the three approaches as discussed here, provided by Copeland, Nir, and Sieg, or they the fall short of the distinction between Turing computability, calculation and computation, or the are merely confused by the shortfalls of reductionist materialism. An example is the article by Goldin and Wegner where they basically equate computation with interaction .
As an intermediate result we can state that that there is no theory of computation so far that would would be appropriate to serve as a basis for the debate around epistemological and philosophical issues around our machines and around our mind. So, how to conceive of computation?
Computation: An extended Perspective
Any of the theories of computation refer to the concept of algorithm. Yet, even deterministic algorithms may run forever if the solution space is defined in a self-referential manner. There are also a lot of procedures that can be made to run on a computer, which follow “analytic rules” and never will stop running. (By “analytic rules” we understand an definite and completely determined and encoded rule that may be run on an UTM.)
Here we meet again the basic intention of Turing: His work in  has been about the calculability of functions. In other words, time is essentially excluded by his notion (and also in Sieg’s and Gandy’s extensions of Turing’s work). It does not matter, whether the whole of all symbol manipulations are accomplished in a femto-second or in a giga-second. Ontologically, there is just a single block: the function.
Here at this pint we can easily recognize the different ways of branching off the classical, i.e. Turing-theory based understanding of computation. Since Turing’s concept is well-defined, there are obviously more ways to conceive of something different. These, however, boil down to three principles.
- (1) referring to (predefined) symbols;
- (2) referring to functions;
- (3) based on uniquely defined states.
Any kind of Non-Turing computation can be subsumed to either of these principles. These principles may also be combined. For instance, algorithms in the standard definition as given first by Donald Knuth refer to all three of them, while some computational procedures like the Game of Life, or some so-called “genetic algorithms” (which are not algorithms by definition) do not necessarily refer to (2) and (3). We may loosely distinguish weakly Non-Turing (WNT) structures from strongly Non-Turing (SNT) structures.
All of the three principles vanish, and thus the story about computation changes completely, if we allow for a signal-horizon inside the machine process. Immediately, we would have myriads of read/write devices working all to the same tape. Note, that the situation does not actualize a parallel processing, where one would have lots of Turing machines, each of them working on its own tape. Such parallelism is equivalent to a single Turing machine, just working faster.Of course, exactly this is intended in standard parallel processing as it is implemented today.
Our shared-tape parallelism is strikingly different. Here, even as we still would have “analytic rules,” the effect of the signal horizon could be world-breaking. I guess exactly this was the basis for Turing’s interest in the question of the principles of morphogenesis . Despite we only have determinate rules, we find the emergence of properties that can’t be predicted on the basis of those rules, neither quantitatively nor, even more important, qualitatively. There is not even the possibility of a language on the lower level to express what has been emerging from it. Such an embedding renders our analytic rules into “mechanisms.”
Due to the determinateness of the rules we still may talk about computational processes. Yet, there are no calculations of functions any more. The solution space gets extended by performing the computation. It is an empirical question to what extent we can use such mechanisms and systems built from such mechanisms to find “solutions.” Note, that such solutions are not intrinsically given by the task. Nevertheless, they may help us from the perspective of the usage to proceed.
A lot of debates about deterministic chaos, self-organization, and complexity is invoked by such a turn. At least the topic of complexity we will discuss in detail elsewhere. Notwithstanding we may call any process that is based on mechanisms and that extends the solution space by its own activity Proper Non-Turing Computation.
We have now to discuss the concept of Non-Turing Computation (NTC) more explicitly. We will yet not talk about Non-deterministic Turing Machines (NTM), and also not about exotic relativistic computers, i.e. Turing machines running in a black hole or its vicinity. Note also that as along as we would perform in an activity that finally is going to be interpreted as a solution for a function, we still are in the area defined by Turing’s theory, whether such an activity is based on so-called analog computers, DNA or quantum dots. A good example for such a misunderstanding is given in . MacLennan  emphasizes that Turing’s theory is based on a particular model (or class of models) and its accompanying axiomatics. Based on a different model we achieve a different way of computation. Despite MacLennan provides a set of definitions of “computation” before the background of what we labels “natural computation,” his contribution remains too superficial for our purposes (He also does not distinguish between mechanistic and mechanismic).
First of all, we distinguish between “calculation” and “computation.” Calculating is completely within the domain of the axiomatic use of graphemes (again, we avoid using “symbol” here). An example is 71+52. How do we know that the result is 123? Simply by following the determinate rules that are all based on mathematical axioms. Such calculations do not add anything new, even if a particular one has been performed the first time ever. Their solution space is axiomatically confined. Thus, UTM and λ-calculus are the equivalent, as it holds also for mathematical calculation and calculations performed by UTM or by humans. Such, the calculation is equivalent to follow the defined deterministic rules. We achieve the results by combining a mathematical model and some “input” parameters. Note that this “superposition” destroys information. Remarkably, neither the UTM nor its physical realization as a package consisting from digital electronics and a particular kind of software can be conceived as a body not even metaphorically.
In contrast to that by introducing a signal horizon we get processes that provoke a basic duality. On the one hand they are based on rules, which can be written down explicitly; they even may be “analytic.” Nevertheless, if we run these rules under the condition of a signal horizon we get (strongly) emergent patterns and structures. The description of those patterns or structures can not be reduced to the descriptions of the rules (or the rules themselves) in principle. This is valid even for those cases, where the rules on the micro-level would indeed by algorithms, i.e. rules delivering definite results in finite time and space.
Still, we have a lot of elementary calculations, but the result is not given by the axioms according to which we perform these calculations. Notably, introducing a signal horizon is equivalent to introduce the abstract body. So how to call calculations that extend their own axiomatic basis?
We suggest that this kind of processes could be called Non-Turing Computation, despite the fact that Turing was definitely aware about the constraints of the UTM, and despite the fact that it was Turing who invented the reaction-diffusion-system as a Non-UTM-mechanism.
The label Non-Turing Computation just indicates that
- – there is a strong difference between calculations under conditions of functional logics (λ-calculus) and calculations in an abstract and, of course, also in a concrete body, implied by the signal horizon and the related symmetry breaking; the first may be called (determinate) calculation, the latter (indeterminate) computation
- – the calculations on the micro-level extend the axiomatic basis on the macro-level, leading to the fact that “local algorithmicity” does not not coincide any longer with its “global algorithmicity”;
- – nevertheless all calculations on the micro-level may be given explicitly as (though “local”) algorithms.
Three notes are indicated here. Firstly, it does not matter for our argument, whether in a real body there are actually strict rules “implemented” as in a digital computer. The assumption that there are such rules plays the role of a worst-case assumption. If it is possible to get a non-deterministic result despite the determinacy of calculations on the micro-level, then we can proceed with our intention, that a machine-based epistemology is possible. At the same time this argument does not necessarily support either the perspective of functionalism (claiming statefulness of entities) or that of computationalism (grounding on “algorithmic framework”).
Secondly, despite the simplicity and even analyticity of local algorithms an UTM is not able to calculate a physical actualization of a system that performs non-Turing computations. The reason is that it is not defined in a way that it could. One of the consequences of embedding trivial calculations into a factual signal horizon is that the whole “system” has no defined state any more. Of course we can interpret the appearance of such a system and classify it. Yet, we can not claim anymore that the “system” has a state, which could be analytically defined or recognized as such. Such a “system” (like the reaction-diffusion systems) can not be described with a framework that allows only unique states, such as the UTM, nor can a UTM represent such a system. Here, many aspects come to the fore that are closely related to complexity. We will discuss them over there!
The third note finally concerns the label itself. Non-Turing computation could be any computation based on a customizable engine, where there is no symbolic encoding, or no identifiable states while the machine is running. Beside complex systems, there are other architectures, such like so-called analog computers. In some quite justifiable way, we could indeed conceive the simulation of a complex self-organizing system as an analog computer. Another possibility is given by evolvable hardware, like FPGA, even as the actual programming is still based on symbolic encoding. Finally, it has been suggested that any mapping of real-world data (e.g. sensory input) that are representable only by real numbers to a finite set of intensions is also non-Turing computation .
What is the result of an indeterminate computation, or, in order to use the redefined term, Non-Turing computation? We are not allowed to expect “unique” results anymore. Sometimes, there might be several results at the same time. A solution might be even outside of the initial solution space, causing a particular blindness of the instance performing non-Turing computations against the results of its activities. Dealing with such issues can not be regarded as an issue of a theory of calculability, or any formal theory of computation. Formal theories can not deal with self-induced extension of solution spaces.
The Role of Symbols
Before we are going to draw an conclusion, we have to discuss the role of symbols. Here we have, of course, to refer to semiotics. (…)
keywords: CS Peirce, symbolism, (pseudo-) sub-symbolism, data type in NTC as actualization of associativity (which could be quite different) network theory (there: randolation)
Our investigation of computation and Non-Turing-Computation brings a distinction of different ways of actualization of Non-Turing computation.Yet, there is one particular structure that is so different from Turing’s theory that it can not even compared to it. Naturally, this addresses the pen-ultimate precondition of Turing-machines: axiomatics. If we perform a computation in the sense of strong rule-following, which could be based even on predefined symbols, we nevertheless may end up with a machine that extends its own axiomatic basis. For us, this seems to be the core property of Non-Turing Computation.
Yet, such a machine has not been built so far. We provided just the necessary conditions for it. It is clear that mainly the software is missing for an actualization of such a machine. If in some near future such a machine would exist, however, this also would have consequences concerning the status of the human mind, though rather undramatic ones.
Our contribution to the debate of the relation of “computers” and “minds” spans over three aspects. Firstly, it should be clear that the traditional frame of “computationalism,” mainly based on the equivalence to the UTM, can recognized as an inappropriate hypothesis. For instance, questions like “Is the human brain a computer?” can be identified as inadequate, since it is not apriori clear what a computer should be (besides falling thereby into the anti-linguistic trap). David King asked even (more garbageful) “Is the human mind a Turing machine?”  King concludes that :
“So if we believe that we are more than Turing machines, a belief in a kind of Cartesian dualist gulf between the mental and the physical seems to be concomitant.”
He arrives at that (wrong) conclusion by some (deeply non-Wittgensteinian) reflections about the actual infinite and Cantor’s (non-sensical) ideas about it. It is simply an ill-posed question whether the human mind can solve problems a UTM can’t. Mode of the problems we as humans deal with all the day long can not be “solved” (within the same day), and many not even represented to a UTM, since this would require definite encoding into a string of graphemes. Indeed, we can deal with those problems without solving them “analytically.” King is not aware about the poison of analyticity imported through the direct comparison with the UTM.
This brings us to the second aspect, the state of mechanisms. The denial of the superiority or let it even be the equality of brains and UTMs does not mount to the acceptance of some top-down principle, as King suggests in the passage cited above. UTMs, as any other algorithmic machine, are finite state automata (FSA). FSA, and even probabilistic or non-deterministic FSA, are totalizing the mechanics such that they become equivalent to a function, as Turing himself clearly stated. Yet, the brain and mind could be recognized as something that indeed rests on very simple (material) mechanisms, while these mechanisms (say algorithms) are definitely not sufficient to explain anything about the brain or the mind. From that perspective we could even conclude that we only can build such a machine if we fully embrace the transcendental role of so-called “natural” languages, as it has been recognized by Wittgenstein and others.
The third and final aspect of our results finally concerns the effect of these mechanisms onto the theory. Since the elementary operations are still mechanical and maybe even finite and fully determined, it is fully justified to call such a process a calculation. Molecular operations are indeed highly determinate, yet only within the boundaries of quantum phenomena, and not to forget the thermal noise on the level of the condition of the possible. Organisms are investing a lot to improve the signal-noise-ratios up to a digital level. Yet, this calculation is not a standard computation for two reasons: First, these processes are not programmable. They are as they are, as a matter of fact and by means of the factual matter. Secondly, the whole process is not a well-defined calculation any more. There is even no state. At the borderlines between matter, its formation (within processes of interpretation themselves part of that borderline zone), and information something new is appearing (emerging?), that can’t be covered by the presuppositions of the lower levels.
As a model then—and we anyway always have to model in each single “event” (we will return to that elsewhere)—we could refer to axiomatics. It is a undeniable fact that we as persons can think more and in more generality than amoebas or neurons. Yet, even in case of reptiles, dogs, cats or dolphins, we could not say “more” anymore, it is more a “different” than a “more” that we have to apply to describe the relationships between our thinking and that of those. Still, dogs or chimpanzees did not develop the insight of the limitations of the λ-calculus.
As a conclusion, we could describe the “Non-Turing computation” with regard to the stability of its own axiomatic basis. Non-Turing computation extends its own axiomatic basis. From the perspective of the integrated entity, however, we can call it differentiation, or abstract growth. We already appreciated Turing’s contribution on that topic above. Just imagine to imagine images like those before actually having seen them…
There are some topics that directly emerge from these results, forming kind of a (friendly) conceptual neighborhood.
- – What is the relation between abstract growth / differentiation and (probabilistic) networks?
- – Part of the answer to this first issue is likely given by the phenomenon of a particular transition from the probabilistic to the propositional, which also play a role concerning the symbolic.
- – We have to clarify the notion “extending an axiomatic basis”. This relates us further to evolution, and particularly to the evolution of symbolic spaces, which in turn is related to category theory and some basic notions about the concepts of comparison, relation, and abstraction.
- – The relationship of Non-Turing Computation to the concepts of “model” and “theory.”
- – Is there an ultimate boundary for that extension, some kind of conditional system that can’t be surpassed, and how could we speak about that?
-  Alan M. Turing (1936), On Computable Numbers, With an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Series 2, Volume 42 (1936), p.230-265.
-  Andrew Hodges, Alan Turing.
-  B. Jack Copeland (1996), What is Computation? Synthese 108: 335-359.
-  Nir Fresco (2008), An Analysis of the Criteria for Evaluating Adequate Theories of Computation. Minds & Machines 18:379–401.
-  Sieg, Wilfried, “Calculations by Man and Machine: Conceptual Analysis” (2000). Department of Philosophy. Paper 178. http://repository.cmu.edu/philosophy/178
-  Sieg, Wilfried, “Church Without Dogma: Axioms for Computability” (2005). Department of Philosophy. Paper 119. http://repository.cmu.edu/philosophy/119.
-  Wilhelm Vossenkuhl, Ludwig Wittgenstein. 2003.
-  Ludwig Wittgenstein, Philosophical Investigations §201; see also the Internet Encyclopedia of Philosophy
-  Goldin and Wegner
-  Alan M. Turing (1952), The Chemical Basis of Morphogenesis. Phil.Trans.Royal Soc. London. Series B, Biological Sciences, Vol.237, No. 641. (Aug. 14, 1952), pp. 37-72.
-  Ed Blakey (2011), Computational Complexity in Non-Turing Models of Computation: The What, the Why and the How. Electronic Notes Theor Comp Sci 270: 17–28.
-  Bruce J MacLennan (2009), Super-Turing or Non-Turing? Extending the Concept of Computation. Int. J. Unconvent. Comp., Vol 5 (3-4),p.369-387.
-  Thomas M. Ott, Self-organised Clustering as a Basis for Cognition and Machine Intelligence. Thesis, ETH Zurich, 2007.
-  David King (1996), Is the human mind a Turing machine? Synthese 108: 379-389.