1. Theophanes E. Raptis Division of Applied Technologies NCSR DEMOKRITOS 2010 - 2014

2.  Kolmogorov Complexity A 24-bit color image of Mandelbrot set requiring 1.62 x 10^6 bits of storage space BUT less than a few Kbytes for storage of the original program that constructed the image! K(I) << Storage Space  Algorithmic Information Theory (Solomonoff, Chaitin)  Binary Lambda Calculus Let |s(I)| be the length of the bit-string of the above image and d(s) the equivalent shortest description on a Universal Language. Then K(I) ~ d(s). Alternatively, K(s) ~ min(|p|+|so|): T(p,so) -> s where T is a Universal Turing Machine, “p” is a ‘program’ string, “so” is an ‘input’ string (Length of T description is offset) Theorem :

3.  PROBLEM : K(s) non computable for arbitrary strings  Previous image stored as a PNG file takes no more than 0.5 x 10^6 bits (~58Kb). Passed through both RAR and Zip gives almost the same. (Compressed Format)  Ordinary compressors rely on statistics of digrams, trigrams, etc. Restricted by general results of Information and Coding Theory.  Most binary integers appear to be ‘incompressible’. Problem of ‘Capacity’ of the initial ‘constructing set’.  Powerful Fractal/Wavelet compressors are always ‘lossy’.  It appears that the only practical methods we have to assign randomness/complexity values to finite sequences is either through Entropic measures and/or through the max. achievable Compression ratio.  Alternative is given by ‘Automaticity’ : find p : T(p) -> s.  Continuous equivalent : ‘Reverse Engineering’ of Dynamical Systems (J. Bongard, H. Lipson PNAS 2006, Delgado-Eckert 2009 PLoS)

4.  Jurgen Schmidhuber, IDSIA “Is the entire past and future history of our universe describable by a finite sequence of bits, just like a movie stored on a compact disc, or a never ending evolution of a virtual reality determined by a finite algorithm…? Contrary to a widely spread misunderstanding, quantum physics, quantum computation and Heisenberg's uncertainty principle do not rule this out.” - ‘A computer scientist's view of life, the universe, and everything’. In C. Freksa, M. Jantzen, and R. Valk, editors, Foundations of Computer Science: Potential - Theory - Cognition, volume 1337, pages 201-208. Lecture Notes in Computer Science, Springer, Berlin, 1997. - ‘Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit’ International Journal of Foundations of Computer Science 13(4):587-612, 2002.  Fredkin-Zuse Thesis : Algorithmic Equivalence Classes for ‘Natural Laws’ – Finite Discrete Universes  Non-local Hidden Variable Theories cannot be excluded from local experiments. Universes as ‘TV-Screens’ w. hidden projection/image processing mechanism!  G. T’Hooft “Holographic” Paradigm – problems with dimensionality!  Kochen-Conway “Free Will Theorem” (Found. Phys. 2006) even more worrying!

5.  Need for new definitions  Correct answers often linked to correct questions  Any ‘Physically Admissible Observer’ (PAO) carries a Finite Memory Capacity and a Finite Resolution Sensory Capacity. (The second includes all possible means and instruments of observation)  Physical Theories that include a proper definition of an Observer should be termed “Endophysical” (Otto Roessler, 1992)  Need to study Complexity/Randomness/ Compressibility in successive approximations.

6.  What’s in a number? - An abstract picture/symbol, eg. {‘2’, ‘ 0 ’, ‘Д’,…} called the “Symbolic Alphabet”. - An algebraic value/ quantity assigned via the use of a special mapping from the Symbolic Alphabet to the integers known as the Polynomial Representation and given as n =, n E N where <s| = [S1, S2, …, Sn] is the set of coefficients |b> = [1, b, b^2, …, b^(n-1)] with b the number of symbols used in the range [0, b-1]. - An n-Dim. hyperplane corresponding to the 1-form (Dual is also possible under the change of coordinates ) - Any image of n-bits resolution in the interval [0, (2^n)-1]

7.  Let be thematrix with all rows <S| corresponding to the symbolic representation of the integers in the interval (also known as the “Factorial Design”)  There is a well ordered hierarchy of matrices  We will call the above the Lexicon matrices of order n in base b with being the Universal Lexicon in base b.  Properties : 1. All matrices are Self-similar : 2. Self-complementarity of binary Lexicons through 0 ->1/1 ->0 NOT-exchange

8.  Examples of binary Lexicons (white->0/black->1)

9.  A setwill be called an ‘Essential Subset’ iff its members are sufficient to reconstruct all of N through some appropriate procedure.  Eg. The set of primes constitute a sufficient set of constructors for N (Fundamental Theorem of Arithmetic)  Are there sufficient constructors for every member of the Lexicon hierarchy? ANSWER: more than one!  A.The set of ‘Cyclic Generators’ P being a cyclic permutation matrix

10.  B.The set of ‘Reflectors’ (mirror images): s = [011010101] -> s*=[101010110] (or simply invert the base vector) b = [1,b,…,b^(n-1)] -> b* = [b^(n-1),…,b,1]  where I is the invariant set which contains all “Palindromes” (Self-reflective sequences). It holds that #(I)= The essential subset is here a recursive function of the previous matrices.

11.  Consider the set of all Morphisms on the integers S M.  Consider where S C is the set of all such morphisms that act on an integer “symbol-wise” either through analytical form or via some program/algorithm.   Conjecture: All such morphisms inherit the fundamental self-similarity of the Lexicon Hierarchy  Intuitively obvious, very difficult to prove.

12.  FIIA: A Finite Inductive Inference Agent should be able to deduce a general law from any (n-1, n)  n+1 scheme.  Arithmetic Fractals formed by componentwise maps offer the possibility of constructing very efficient FIIA.  Fractality evident in exponential intervals over the base of any particular alphabet chosen.  Many noisy number theoretic functions can be deduced this way.

13.  Binary Sum-of Digits: [0…3]: {0,1,1,2} [0…7]: {0,1,1,2,1,2,2,3} [0…15]: ?  Iterated Sequence System:  Analytic Expansion:  y(τ 1, τ 2 ) is a periodic integer function

14.  Binary Sum-of-Divisors [2…5]: {0,1,1,2} [2…9]: {0,1,1,2,1,3,2,3} [2…17]: ?  Iterated Sequence System: Not known – under investigation (crypto apps etc.)  Analytic Expansion:  The above is known to have minima exactly at the prime numbers.

15.  Let be the “Trailing Zeros” function. [0…3]: {2,0,1,0} [0…7]: {3,0,1,0,2,0,1,0} [0…15]: ?  Iterated Sequence System (even pos. > 0):  Analytic Expansion:  Theorem: The exponents of the factorial decomposition of ν will be given by Tz(ν, b). Proof: Divisibility in every prime base.

16.  Radon – Hurwitz Sequence [1…4]: {1,3,1,7} [1…8]: {1,3,1,7,1,3,1,8} [1…16]: {1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,9}  Iterated Sequence System:  Analytic Expansion:  Connection with Lexicon Structure not yet understood. Associated with Bott’s periodicity in the Cayley-Dickson Hierarchy.

17.  CAND(x,y) / CXOR(x,y) as special cases  Use Global Encoding, turn both args in single integer to get a unique graph.  Final graphs are Arithmetic Fractals.  Can be approximated by ISS or analytic formulas.  Arbitrary CL circuits can also be analyzed the same way allowing further reduction of complexity.  Fractality allows proving theorems on certain algorithms without any computation at all!

19.  Canonical ISS: Seed Set S 0 over an alphabet, Update Rule O, Recursion Rule  Expanded Canonical Form: Nth order sequence given by the set of operator powers where exponents are given by the SoD of the recursion index.  Generalised ISS: Allow multiple update operators on multiple seed sets. Increasing complexity of sequences.  Further Research: Find a generic protocol to approximate pseudo-random sequences. Where is the boundary with true randomness?

20.  CHSH Inequality: Validity region is a disconnected set (Fractal Dust).  Leggett-Garg Inequality:  Connections may run deeper than we currently understand. Nuclear Emission Spectra also similar to Primes distribution.

21.  TSP: Let D be a distance matrix storing the travelling cost between cities i->j, |D ij |.  Finding a minimal cost shortest path requires checking sums over all possible paths.  Let b an alphabet base equal to the number of cities. There can only be sequences in the order b Lexicon that do not repeat the same symbol. We may call this property “Anti- clustering”.  TSP Global Solution Space ordered.  Natural languages also follow this property with rare exceptions to forming words.  We may build a correspondence of words w. minimal graphs for M2MLI (Machine2Machine Linguistic Interfaces).

22.  We define a set of functions  Let Jo be the sum of digits of a sequence.  The rest are defined with the aid of the ‘Cluster Vector’ which is equivalent to a polynomial. Example: s = [0110100010]  c = [-1,2,-1,1,-3,1,-1] Equivalent with |c| plus 1 bit for the alternating sign (s(1)).  We can now define J1  Cluster Dim., J3 = prod(c), J4  Auto-correlation of c (w. r. to circular shifts), J5 = c(1)+2c(2)+3c(3)+…, etc.  All J functions are self-similar reflecting the self-similarity of the primordial object

23. First index recursive over exponential intervals : [0,1]  [[0,1],1,2]  [[[0,1],1,2],1,2,2,3]… Second index reflects the Inherent Symmetry w.r. to 1/0 exchange

25. Horizontal/ Vertical indices in Global encoding. Example : convert to grayscale/ threshold to binary

26.  Lexicon constructs allows to have a unique, ‘democratic’ characterization of all binary sequences as a whole, in successive approximations  In this context there is no distinction between “images” and symbolic sequences.  “Dimensional Reduction” is always possible for any finite, discrete set. ‘Dimensionality’ is only an ‘intermediary’ for the convenience of an external Agent-Observer.  ‘Folding’-’Unfolding’ of a sequence in higher dimensional boxes allows compression of huge images

27.  Let A={T,F,M,S} be an ‘Agent’s’ tuple, M being the memory (storage array), F being a folding filter, S being the set of sensory faculties and T a processing unit.  Such an agent can always perform an arbitrary dimensional increase/decrease of an incoming symbolic sequence.  Are there Super-Agents that can produce their own ‘tape’? (Self-Reading Agents / Universal Constructors / ‘Dreaming Machines’!) The agent reading the tape… 0110101000001001111001100110011111110... …which contains its “world”! …iioooioooo iooiiiiooiiooi  iooiiiiiiio…

28.  Lemma: Every discrete N-dimensional subspace is essentially one dimensional.  The above can be proven in certain interesting cases given a minimal number of premises.  Assume an arbitrary agent interpreting external data with a maximum resolution δχ.  Assume a maximal interval in some arbitrary base such that.(For Planck scale, L 2 ~ 119)  Assume a generic process that can always be written as a functional composition of a linear input map and a nonlinear output map.  It can be shown that there always exist a 1-dim. representation of this process that automatically takes care of all possible correlations.

29.  Take Agent’s T as a composition where x stands for an arbitrary K-dim grid/array, f stands for a nonlinear part and h a linear “contraction” map.  Let all grid points take values on a finite alphabet on some maximal base b.  Let N(x,b) stand for a neighbourhood of arbitrary topology.  Let A(N) stand for an arbitrary Boolean Connectivity Matrix recording the correlations assumed in the application of f.  With the aid of an additional universal matrix we can always achieve a one dimensional rearrangement of f.

30.  Lay all values from the K-dim. chunk to an 1- D array by keeping bookmarks of the “neighbors” connectivity in the A(N) matrix.  For any chunk an connectivity matrix is required.  Fill in all non-zero rows of with the cyclic permutations of the base vector.

31.  Special care is to be taken for different types of boundary conditions if needed. (P.B.C by just a one step shift of all rows to the left.)  The matrices obtained that way all belong to the class of “Circulant Matrices”, a special variety of Toeplitz matrices with special properties. http://en.wikipedia.org/wiki/Circulant_matrix http://en.wikipedia.org/wiki/Circulant_matrix  The above works even if connectivity changes with time in which case one can use

32.  The overall dynamics in discrete time can be given with x being a 1-dim. vector as  The 1 st part can be diagonalised due to the general property of circulant matrices as  Eigenvalues are polynomials of roots of unity.  Factorization leads to with

33.  Turning this into polar form leads to  In each factor matrix magnitudes R are constant. Rewrite dynamics as  Diagonal matrices commute so

34.  Introduce new vectors {V,U} such that  Evidently, U is a hidden Unitary Quantum Propagator! There will be some Hamiltonian such that  What does it all means?  Assume a discrete sampling of an initial wavefunction corresponding to the DFT of the discrete N-points lattice initial condition.  Evolve the physical system for N steps.

35.  Take the evolved state as the intermediate variable by a “weak measurement” and project using Inverse DFT (preferably start with )  Restart the physical system from a new init. condition given by the new lattice variable.  A trajectory of an arbitrary discrete automaton defines a special partition of the phase space of a quantum system.  Any such automaton can also be made reversible with redundant bits that make f invertible.

36.  Λ eigenvalues  Eigen-energies

37.  Lets assume a cubic grid L 3 as a 3-D matrix.  Dimensional reduction with the C projection matrix saves the correlations in a higher alphabet of the associated 1-D representation.  At the limit of “global coupling” all the information of any other matrix element has been absorbed in a single number.  Even the 1-D representation becomes trivial as all projected numbers belong to the same class of Cyclic Permutations of the number digits.  Instead of the DFT we may now represent the whole L 3 length 1-D array with a special dynamical system.

38.  Cyclic Permutation Map  All Permutations over any exponential subinterval of the integers also self-similar  bla

39.  Only 3 classes of Input – Output pairs possible A. Automorphic: Permutation Invariant. Memory cost: #(Bits) to record group generator + Random permutation vector. B. C. Group  C. Group: #(Bits) double A. C. Group  Many Groups: # Increasing Dispersion (Entropy increase).  Subclasses: periodicity measures inside each permutation class. Consider the sequences 101010… or 010101.. 110110… or 0001000…  Entropic ratio: τ/Ν max where τ stands for the sub- period inside the permutation.

40.  Take an arbitrary map f defining a dynamical system apart from the “addressing” scheme given by the linear contraction of neighbors.  Define as the “Cyclicity Entropy” to be measured over all input-output pairs after dimensional reduction of f in a 1-D graph.  The ratio allows you to prove the analogue of an H-Theorem (Entropy increase) for the particular map. The opposite may also prove self-organization properties.

41.  One has the isomorphy  Certain interesting properties arise after reintorducing the DFT as  The above has the strange consequence that a quantum evolution becomes equivalent to an inhomogeneous scaling of the trajectory of the permutation map!

42.  One is forced to conclude that the following scheme is isomorphic to the original dynamics hf y y’  Similar perspective appears in the work of Diedriek Aerts ( “The One and the Many: Towards a Unification of the Quantum and Classical Description of One and Many Physical Entities.”, Doctoral dissertation, Brussels Free University.) Also David Bohm’s “Implicate Order”.  Could our universe be like a system of Prayer Wheels preserving the holographic structure?

43.  http://holometer.fnal.gov/ http://holometer.fnal.gov/  Fist attempt to measure the Holographic white Noise at Fermilab during 2015.  fluctuations in the light of a single attometer.  Mathematics know no ends, but all worlds are finite! Thank you all.