T. E. Raptis © 2020


A PDF with all three parts is available here



In the previous parts we described how to make a hierarchical evaluation of certain symbolwise functions on binary integers that can be seen as abstractions of a variety of low level automata. We also showed some simple examples from combinatorics where arithmetic fractals and other interesting sequences were derived of who's the regularities are considered to be inherited from the multi-periodic nature of an underlying counter system endowed with a primitive form of self-similarity with a wreathing product structure.


What we shall deal with in this last part is more theoretical concerning the possibility of transferring the previous into a more abstract algebraic content where the set of all programs can be viewed from a different angle related to dynamical systems theory. We ask if it is possible to analyze the paths (history) of every computation as a whole in a way that would reveal the important role of the underlying structure inherited by the wreath product. The reader may want also to consult again the math appendix at the end of Part I where the evidence is presented on the origin of this effect.


It deserves to mention here previous efforts which were based on experiments with some modern versions of combinator calculus that recently appeared as a visualization of what became known as a Turing Tarpit and which also provided some evidence on a kind of fractal structure constraining the set of all outputs obtained this way. There may be also some association between the approach presented here and previous efforts in so called, Supercompilation as first introduced by Valentin Turchin to be incorporated in his version of the REFAL functional language. In order to make an abstract model out of the previously introduced Hamming Hierarchy we shall need to move to operator language.


Non-commuting multinomial expansions


Recall the previous construct of a sequence of Mersenne intervals SL and an associated sequence of SigmaL matrices representing Hamming spaces with a lexicographical ordering of all associated patterns out of a polynomial expansion is some alphabet base b over these same intervals in a 1-1 correspondence. We then talk of a Hamming hierarchy for simplicity denoting it with H. We now change the semantics of this scheme as follows: assuming a list of exactly b functions or automata realizing these functions as maps between input and output strings, we shall interpret each matrix as a Functional Look-Up Table (FLUT) for all possible successive compositions with elements drawn from each matrix column. A simple explicit form of all columns has already been given in Part I. To understand this in practice consider the following list of maps as say, M = [m0(x), m1(x), m0(x)] each with its own domain of dependence and co-domain of outputs but always in the natural numbers. Define a ternary alphabet and a list of possible applications of some length L as


   |0|à [0 0 ... 0] à m0( m0( ... m0( x )...))

   |1|à [1 0 ... 0] à m1( m0( ... m0( x )...))

   |2|à [2 0 ... 0] à m2( m0( ... m0( x )...))

   |3|à [0 1 ... 0] à m0( m1( ... m0( x )...))



There is now an additional difficulty due to the arbitrariness of domains and co-domains of these compositions which can only be treated with branching and this is treated with an additional construct in the next section where the full UMUX method will be developed. At the moment we assume the maps to be somehow valid on all integers by some internal consistency mechanism to simplify the discussion.


Such a scheme establishes a 1-1 correspondence of all possible paths of a computation that could be performed with a list of some elementary maps over all possible inputs as in product of the form H x H. Such entails a particular kind of product between the matrices for the symbolic path expressions and their associated intervals. Having examined the arithmetization of certain simple maps in Part II of this study we may also assume all such maps being strictly quantitative as m: N à N. With this simplification we may posit which in practice means that we only have to deal with a set product of all paths as symbolic words in a base b alphabet with all possible arithmetic inputs. If a list of maps of length b is sufficient for universal computation then the same correspondence will always be true independently of the existence of any fixed points or otherwise said, of some halting state assuming an unbounded hierarchy of intervals. The global output for such a procedure can be given as a hierarchy of 2D arrays or Universal Global Maps where each level of the hierarchy now represents a new step. Moreover, if an additional tag or special bit is preserved for any single entry to have been visited once per step, then the equivalent Boolean array indicating such a visit must by construction be a total permutation matrix. An interesting open question is then whether there do exist classes of universal FLUTs giving rise to a recursive, or even arithmetic fractal graphs out of the underlying transverse counter set as explained in the next paragraphs.


Let us now make this more systematic via the following algebraic trick. We can make use of the fact that functional composition is in general a non-commuting operation meaning that we are not in general allowed to write. Then we can directly assert that the LUT based construct is identical or isomorphic for any level L with the expansion of a multinomial of the form



where the symbols mi are to be interpreted as functional operators in general and {ij...k} is a word obtained from a column of the associated LUT. The only case where this could be reducible to a standard multinomial with many paths collapsing to the same multi-index space can be seen with an abstract analogy from linear algebra. Assuming then arbitrary linear maps over some vector space V via an equivalent list of matrices as, we can use the standard toolbox of linear algebra to verify that these can be made all to commute with each other if and only if they belong to the same conjugacy class. That is whenever there exists a set of linear transforms S for each and every one such that. In most applications where such a scheme would be used for really low level maps the above cannot hold true but perhaps in a small subset of rare occasions.


What is more interesting here is the fact that the primitive self-similarity which is inherited from the underlying multi-periodicity of the counters for each collection of symbols may influence the thus obtained 2D global maps. While it is difficult to give a precise quantitative measure of such an inheritance, it is possible to see where such an influence originates with the aid of an additional device. We can in fact study the formation of all words of indices in the summand above via an additional discrete "time" \tau governing the underlying counter system as say




Such is a type of 'transverse' inner flow which permeates all possible computational paths even though one may have to reach only a single one during a specific computation involving also some unavoidable branching (more on this in the next section.) Additionally, there is a possibility of generalizing this when dealing with such maps that may belong to the elements of some discrete group of symmetries. There is at least one other property that may be crucial whenever there is a representation allowing to make use of it. One such example comes straight from number theory where action over elements can be transferred to the action over their indices and it is known from the area of so called, Fibonacci Primes where we have the following unusual property for the action of the gcd function as. It would an obvious advantage if this could be generalized for a specific list of fundamental basis functions that can collapse computational paths as say. This is still an open research question at the moment.


The Universal Multiplexer (UMUX) method

Suppose one would like to use the previous method to analyze certain low level computations of which the global maps could reveal also possible side effects and in general allow for an alternative type of what is known as 'formal verification' at least at a primitive level. In order to deal with the general case one needs to find also ways to invoke branching in an effective way without breaking the algebraic structure in the original scheme of the previous section. A slight modification of the original then can be given with the aid of what is known as the fundamental theorem for structured programming. The essence of it is that we can cast any algorithmic expression of a well structured program into the standard format of a single loop with a large conditional inside. Loops can also be turned into singleton if...then conditionals using a counter and recursion. We may consider a particular normal or canonical form of the above whenever all operations inside any structured program can be adjoined with an equal set of appropriate disjoint Boolean characteristics or indicator functions over any interval of possible inputs. We already wrote about the need to refer to pairs of integers when working on the Hamming due to the imposition of ad hoc expansion lengths greater than their natural bit length. In analogy, we may talk here of a triplet representation requiring both { |x|, L|x|, \chi(|x|) } independently of its expansion to any arbitrary alphabet where, (|x|) is used to denote a property or even a set of properties as say, in a generic Boolean condition over various indicators as . Application of the above must be done with caution in special cases that appear incomplete as in a nested statement of the form

If Chi1 do f1

      if Chi2 do f2 endif


In such a case a transformation is required as and.

All the above can be nicely summarized as a total algebraic operator of the form

By a slight abuse of notation we interpret all compositions as a kind of "convolution" of a total (vectorized) set of inputs by a Boolean vector which "windows" out all invalid inputs from a span of m sub-domains according to its own indicator. This is basically the same as tagging all input data according to a quantifier which is also a function of itself this being a strongly non-linear operation. In practice this leads to a kind of multiplexing technique where each input is redirected to a different map. One may call this an algebraic pipelining method.

In most cases, where all maps are restricted inside certain sub-domains of validity, their co-domains may become disjoint at some point. In general not all paths available from the original matrices will be populated all the time. This can better be understood with a network diagram where all map outputs may have arrows to all others for as long as the relevant associated indicators allow that in sufficiently large input intervals. Collecting each path for all sub-domains at every step will leave a trace of words in alphabet b or their corresponding arithmetic indices. Depending on the nature of the maps the set of these indices may or may not cover the whole interval SL in successive steps and successive input intervals and whenever it does we may call the path space "ergodic" while many such maps are expected to be mostly stochastic.

Some generic properties follow immediately. Assume any list of indicators to form a perfect span of any interval SL by forcing at least one of the maps say m0 being the identity for completeness such that whenever the rest do not suffice to cover any interval completely. Under this condition their normalized summand becomes a convex set satisfying


where. It is also possible to apply a similar logic in order to derive a type of probability associated to each path as

where the notation again implies summing over all outputs filtered from the original input set reached through the same path.

Uniqueness, irreducibility and factorizations

Taking a second look at the finally obtained expression of Z allows realizing that it works much as a prime factor say, like 2L or 5L and so on. The analogy is not superficial which can be understood when trying to compile by hand a really large program in the above spirit. It is quite possible that in a variety of cases it would be much preferable to decompose everything to really low level operations each of which can then be put into correspondence with its own UMUX with a set of associated operators as


Trying to match them would then lead to a product much like in a factorization as

On the other hand this implies that all causal paths created this way should belong to a subset of a more complete operator over a quaternary alphabet like

By what was said on the fundamental theorem, the two last expressions should be somehow equivalent that is there should always be a way to write things in a "prime" format. The only way that these two could be compatible is by additional compositions of indicators which lead us to the territory of the general Satisfiablity(SAT) problems such that the set of all possible paths is appropriately "clipped". When and how this is possible depends strongly on the inner structure of all SAT problems the solutions of which might also be influenced by the same wreath product structure underlying any Hamming hierarchy. That is, there will be cases where one can always factorize a total Z operator whenever the associated SAT problems "resonate" with specific subsets of paths. Equivalent, predicting what will be analyzed in detail in next sections there will be a stochastic transitions matrix of whom several entries will be zero corresponding to symbols of operators that should not "meet" thus effecting the same clipping.

Unexpectedly, this might help to shed some light to an apparently remote problem in physics, that of the so called "entangled" states where factorizability is a crucial issue at the cost of interpreting all particles states as being associated to the causal paths of a massively parallel execution of an underlying machine. Moreover, such a machine would be required to curry over a 'self-modifying' type of code at least with respect to the indicators which should also involve what is referred to as an act of 'observation' which of course is kind of a cyclic argument since any observer would be part of the machine! The mentioned act of clipping the set of causal paths also has the ability to present higher correlations when looking at the set of all possible outputs than if all were available to execution as if there were inner constraints. The same could look like a kind of "trickstery" where an observer without access to the true indicators may be led to believe to the existence of "invisible" or "hidden" interactions. Of course, many more things are to be added to such a construct regarding additional symmetries, etc., but we have already seen in Part I that the underlying wreath product structure already contains "for free" a sampled version of two important ones, specifically a discrete version of U(1) (to be found in all 'projectified' Hilbert spaces where a 'phase' cannot be measured)and the "boost"" sector of a Lorentz group.

Implementation issues

To use the above in any efficient manner without any need for a branch conditional we need an additional recipe given a selection-redirection mechanism for all input data x according to an auxiliary FLUT of indicators. To actually make use of this, we need a method for making arbitrary permutations of such tagged inputs from any interval across the hierarchy to be fed into an ordered list of maps that can be efficiently coded via a cell of anonymous functions using the standard Matlab conventions. There is one last tricky point on the inclusion of a zero arithmetic input to any map due to the double character of this symbol per se being simultaneously a possible input and an arithmetic indicator, the simplest solution to this being a double shift. As an example, for an input interval [0,...,2L - 1] and an indicator as (mod 2) - or any other - we should always take the set of valid inputs via y = ( Mod( x, 2 ).*( x + 1 ) ). This can then be directly fed to any string representing a FLUT member for evaluation as m(y( y >= 0 ) - 1 ) . One can either store function expressions on a file or write them at the top of a script.

As in the previous two installments, we are more interested in arithmetized primitive operations on patterns but for similicity we can use here an example of a well known case of branched function, the one giving rise to the Hailstone sequence for which we only have two complementary indicators making a full span of every interval (no need for an additional Id map). A simple example script showing the combined operations is shown below for the S8 interval

clc, close all, clear all

L = 8; % iteration length

F1 = { @(x)x/2  @(x)3*x+1 };

F2 = { @(x)1-mod(x, 2) @(x)mod(x, 2) };

base = numel( F1 ); S = base^L;

xin = 0:S-1; xout=zeros(1,S);

seq = ones(1, S); w = [];

for t = 1:S %reduce this to L or similar for less memory consumption if need be 

    for i = 1:base

        idx = F2{i}( xin ); y = idx.*(xin + 1) - 1 ;

        xout( find( idx ) ) = F1{i}( y( y >= 0 ) );  % automated sorting

        px(i, t) = sum( idx )/S ; % normalized indicator densities


    xin = xout; w = [w; idx]; % path traces


figure( 1 ), hold on, for i=1:base, plot( px( i, : ) ), end, grid, hold off

figure( 2 ), imagesc(1 - w), colormap gray, pause(0.1)


Figure 1 will show up all intermediate parts of this sequence along all initial conditions. The evolution of the indicator densities is shown in the next figure


The equilibration we see after a transient interval is due to the well known fact that all trajectories in this sequence finally converge towards a 2-periodic attractor, something which has been verified for extremely large intervals but not formally proven, also known as the Collatz conjecture. The second figure shows all paths followed by the binary indicators for all initial conditions as a 2D graph of a 2L x 2L array representing the original "paths-by-inputs" set.


These can be further analyzed via a direct run length analysis or use partial contractions along each dimension to locate possible sequences of interest. In the primitive example used here, there is a randomization of blocks due to the fact that one of the branches is actually used to eliminate all prime factors of 2. All paths could be made into simple alternating sequences as 0101... or 1010... sequences for even and odd positions respectively by just using a direct factorization of each new input at the first branch and simultaneously eliminating any power 2k without affecting convergence. Lastly, it should also be mentioned that the sorting method used in the script is not yet fully arithmetized while a better or more consistent method to the spirit of present analysis would be to utilize the 'viral' encoding presented in Part I so that all input data are formed via a concatenation with their index or 'pointer' while only the 2nd part is actually passed through every individual map m. While the method is presented with a symmetric expansion this may be a problem due to exponential increase of memory usage unless some symmetries or other interesting properties can be efficiently located within smaller intervals or using a segmentation method. A different probabilisitic approach is presented in a next section.

Branching via complexification

The approach explained in the previous section can be applied either to an interval or to a single input. A remaining question is the possibility of further formalizing it in a manner independent of the specifics of any programming dialect. Interestingly, an algebraic alternative in the implementation of the previous section would be simply a product like or a summand of powers like where the constant is to take care of the residual summand of all invalid inputs. The problem with any such expression is that no protection from side effects is offered for possible invalid inputs which are simultaneously fed unless there is a specific axiom of precedence which evaluates exponents first and nullifies any further evaluation apart for a single member each time but this is something to be added as an extra axiom hence these expressions do not appear self-sufficient.

One other road towards a safe and compact global branchless expression for any input appears to be an interpretation of the choice of branch as a "rotation" which entails at least one additional dimension. One can do so (always assuming only arithmetized integer inputs) by promoting any such into a complex pair using the set of b roots of unity as


with the exponent always in the range [0...b-1] being defined by the corresponding set of indicators as a product or a summand form

Notice that at least the first part allows connections with a special linear vector space via the transform where now k is a vector containing the logarithms of [1,...,b]. The overall operation of double exponentiation over the roots of unity is identical with a "tetration" generalized so as to include an intermediate mapping between every pair of exponents. It then becomes possible for any input datum to bind its own function via the auxilliary argand function as

While the above is not the easiest way to implement a structured algorithm it does offers some additional theoretical opportunities. In the simplest case of a binary choice as in the example of the previous section one can project paths in a particular geometric structure in the complex plane where the use of some phase like {-1, 1} or {-i, i} does not matter at the moment unless a particular choice makes more sense. The main theoretical question which is also one of the ultimate reasons for writing this part follows next.

The totality of causal paths

As already noticed in Part II of this series, the totality of causal paths may still hold some secrets at least for the general class of N à N fully arithmetized maps that can give rise to some fractal or otherwise multi-periodic sequences. This brings about a natural question on the biggest possible subset that could behave in an apparently constrained way including the set of all possible fixed points for those that finally halt. Another technical question is how to make this question precise in the UMUX language of the two previous sections. More general discussion of such an effect can be found in the last epistemological addendum.

Using the material from the previous two sections we may directly compare an arbitrary path composed from the decoding of a large word in the picture of the underlying transverse counter flow as

What this implies is a crossing of two "transverse" discrete times one being defined by t as the time of a specific computation on some valid path and the other as a discrete "free" counter time over all possible paths. At the moment we may also make the simplifying assumption that all functions f do not suffer from side effects so that the choice of indicators influencing n(xt-k) form a free span of a large interval. It is natural to ask to what extent and under what conditions would some properties of the underlying word {ij...k}get reflected in the associated sequence of indices n(xt-k) for some valid subset of paths under a given set of indicators. This in itself seems like a humongous problem since it concerns at the very least 2b unknown functions half of them as maps from N to N, the other half being purely Boolean. One could for instance define an inverse problem by picking up a number of sequences for the indices n(xt-k) with some interesting structure and try locating map lists for recreating the relevant paths.

Since such is too vague a question, it is preferable to resort to a more detailed analysis of the kind of dynamics represented by the algebrization of structured algorithms. The first to ask would be the overall density of paths as |Z|/bL for any total state space. For a simple case of a dynamical system with only binary branching which does not behave as a random number generator this might represent a very small fraction the larger the expansion length L becomes. It is then natural to ask for the opposite extreme case. To study this as well as any intermediate it is necessary to introduce the hierarchy of stochastic transition matriceswhere all maps are iterated as f, f(2),..., f(n) and their contribution to each subinterval is used to define the relevant probabilities of falling into one of the b subdomains in the total co-domain thus defining a new symbol in each relevant path. For all extreme cases where \rho à (1/b)1 where 1 the all ones matrix any paths should cover completely the full rooted b-ary tree of depth L in which case we may call the action of such a system across the hierarchy of all intervals SL a fine distillation over any interval. There are two other limiting cases whenever \rho à I or J the first being the identity matrix and the second the anti-diagonal exchange matrix or left - right flip of I. The first represents a limit where each sub-domain has been isolated while the second represents a b-periodic exchange leading to cyclically alternating symbolic sequences for the relevant paths. This analysis has nothing to do with halting states that may appear before such ultimate convergence takes place.

A relatively trivial example can be given again with the aid of the Hailstone sequence where the iterative pairing of f0(x) = x/2 with its native indicator (mod 2) and its complement gives a result we have already seen in Part II in the case of the exponent of the first order factor in the factorization of any integer in any SL. In any such case we take zero as a fixed point. The second map f1(x) = 3x + 1 is a diverging one as =. Since the second term in the summand is always even only the factorization of x affects the final probabilities and the pattern is similar to the f0 case the essential observation being that both cancel out in all of SL after a number of steps given by the maximal interval length as in the multi-periodic sequence given in Part II. This inevitably leads to the last limiting case of the alternating J matrix which can be shown with the below script using a test interval S10

clc, close all, clear all

L = 10; N = 5 % Interval exponent and iteration length

F1 = { @(x)x/2  @(x)3*x+1 };

F2 = { @(x)1-mod(x, 2) @(x)mod(x, 2) };

base = numel( F1 );

S = base^L;

x = 0:S-1;

    y = repmat( x, base, 1 );

for i=1:base

    for image=1:N*L  

        y(i, :) = F1{i}( y(i, :) );

        mat( i, 1, image ) = sum( F2{i}( y(i, :) ) )/S;

        mat(i, 2, image) = 1 - mat( 1, i, image );



x=1:image; plot( x, squeeze( mat(1, 1, :) ),'.', x, squeeze( mat(1, 2, :) ),'r.',...

    x, squeeze( mat(1, 2, :) ),'g.', x, squeeze( mat(1, 2, :) ),'m.' )


The resulting figure shows the output being indeed of an alternating nature.


What really happens here is that a transition appears after the 10th image the reason being that at this point all zero blocks in the binary pattern of any initial condition in S10 signifying the presence of a factor 2C with C in [0, 10] have been exhausted. The code can immediately verify this for a sequence of intervals higher than S10.

Substitution Systems

Generic substitution systems present a different problem regarding their hierarchical structure due to certain ambiguities and structural differences. Some arithmetized substitution methods were presented in Part II. Leaving aside various technicalities regarding confluence and normalization one may start from a universal extension of the original Thue grammar which forms the 0 level of the Chomsky hierarchy of languages assuming no difference between random or sequential rule application.

Let then be a standard (RT) LUT for replacement rules in the sense of wL à wR with standing for the equivalent numerical value of each word and kL,R stand for the equivalent string lengths for each word where action starts from an initial "axiom" string. Some care is required in that these may not necessarily coincide with the natural length l(w) as defined in the previous parts due to the possible presence of zeros at the end. An important property can be used to classify all such systems via a parameter given by the ratios of their effective lengths, i = 1,..., N as "contractive" if all , "conservative" if all \lambdai = 1 and "expanding" if there is at least one \lambdai > 1. For the first two cases, we may still talk of endomorphisms inside each matrix for each interval SL of all possible initial axioms while the last one may act as a surjection on some higher interval for some steps even it finally converges on the original or smaller length. Especially for the conservative case it is evident that we can write an equivalent arithmetic formula of any replacement for the integer values of all axioms as

where k(x, xL) denotes a special function for finding the lowest symbol where replacement is to take place. We can show that all these obtain a canonical form due to the special properties of any matrix. Interestingly this case can be recast as a discrete derivative where Dx is used to denote a vector of signed integers replacing the original RT. Additionally the powerset of all possible subwords coincides with the set of all proper submatrices with L the total axiom string length and by simple counting over powers of some base we get a total of bL - 1 subwords that could be taken form an additional arithmetic interval. Thus one should in principle be able to define a total 2N+1 dimensional state space for a global map of all such possible axioms and subwords pairs as rules. To better show this in relation with the previous classification we may use the below figure for a simple case by choosing columns and rows for any possible RT left and right words respectively


To recast any of the above in the originally proposed multinomial form Z one should first obtain by some method a set of indicators forming a proper span of any interval SL. While it seems possible to associate any pattern locator with an indicator and the replacement with the arithmetized map m there are additional difficulties regarding the originally assumed convexity of the set of all indicators in any interval.

To better understand the necessary decomposition we should further analyze the structure of the lowest symbol pointing function that we denoted as k(x, xL) above. Suppose for instance a rule like '100' à '01' that requires finding all such instances for all strings across any interval SL fixing b to the binary case. But we do know that each segment of any length k < L will inevitably repeat periodically across all columns of any and if we set a first pointer at the beginning of any string than that period would be simply 23 for the example given. From the symmetries endowed due to the wreath product structure the only difference for the same rule pattern in higher positions will be a set of shifts and dilations of the original sequence at position 1 by some period multiplier. Thus we may fully characterize any indicator beforehand thus totally avoiding the need to separately applying an individual filter as some regexp finder to locate individual patterns. A Matlab oneliner for locating all instances of a substring can be given as

w=zeros(n, 2^n);for i=0:2^n,w( strfind( fliplr( dec2bin(i, n) ), substr ), i+1 ) = 1;end

imagesc( w ), colormap gray

For instance, setting substr = '101' and substr = '0101' for n = 7 gives two Boolean arrays as in the figures below where each white dot signifies the initial position of each substring while certain overlaps inevitably appear for strings containing periodicities in their total patterns. Each row of the below arrays stands for an individual total indicator over the interval.


There is now a certain ambiguity on the simultaneous handling of such overlaps which can only be ignored under the adoption of a precedence axiom where the first occurrence is the first to be substituted. This of course also assumes confluence but we may ignore technicalities here. Implementing this actually means scanning each large string for no more than substrings with replacements at every nk + 1 new valid position. Using the notation of Part II for arithmetized string manipulations, a generic update valid for all (non-conservative) cases in all of an SL interval can be given using also the "pulse" function of Part I as

where now k(x, xL, i) is any of the matrices obtained for all possible instance of any subword xL. Despite the regularities of the shifts and the periods pi of each row the general case presents difficulties in locating exact numeric formulas due to the high dimensionality which is still an open question.

Agents in the Warehouse: An epistemological addendum

"This dewdrop world is a dewdrop world, and yet, and yet . . ."

Kobayashi Issa

Some philosopher the author does not recall once asked "How a planet does know the law it has to follow?" Of course in the case of an algorithm or a program we can ascertain with certainty that any such do not need to know anything - they are purely mechanical. However, if a set of computational paths appear to have certain similarities as well as the thus obtained chain of outputs then it could be that people setting up an effective computational process from different and in some metrical or other sense "remote" initial conditions, strange correlations between such processes and their obtained results could be observed. As for possible physical implications, it may be worthy to repeat our original story in the introduction of Part I in a somewhat different format.

Suppose you live outside some large warehouse with trucks coming and going all the time leaving and taking packets. There is also always a large irregular pile of packets outside by the end of each day which if it could be precisely counted it would give you the integer 345 of whose the binary form is [1 0 0 1 1 0 1 0 1 0] while if measured by the end of next day it would give you 1021 expanded as [1 0 1 1 1 1 1 1 1 1]. How easy would it be to know that this pair hides a single transition of the famous universal rule 110 in the Wolfram's classification of elementary cellular automata with periodic boundary conditions not to mention keep doing so for each and every day!

Apart from this being a somewhat contrived, if old fashioned method of some agents communicating with hidden signals, it could take a very different meaning if what was to be measured was a gauge indicator in some lab giving a sequence of measurements like 0.345.., 0.1021.. etc. And of course we may nowadays have achieved immense accuracies with advanced sensors plus theoretical predictions but trying to measure the gravitational constant G with arbitrary precision would immediately reveal part of the problem! The only conclusion is that apart from the theoretical foundations of computational structures any other assertion remains undecidable but not in the ordinary formal way of any axiomatic system. It just seems out of the possibilities of any current empirical science to verify such assertions like those included in the famous Church-Turing Conjecture, unless some purely theoretical "trap" could be set up of which some prediction would lead to a major inconsistency.