Theophanes E. Raptis Division of Applied Technologies NCSR DEMOKRITOS

 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 :

 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)

 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 Lecture Notes in Computer Science, Springer, Berlin, ‘Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit’ International Journal of Foundations of Computer Science 13(4): ,  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!

 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.

 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]

 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

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

 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

 B.The set of ‘Reflectors’ (mirror images): s = [ ] -> s*=[ ] (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.

 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.

 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.

 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

 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.

 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.

 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.

 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!

 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?

 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.

 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).

 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 = [ ]  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

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

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

 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

 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… …which contains its “world”! …iioooioooo iooiiiiooiiooi  iooiiiiiiio…

 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.

 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.

 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.

 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.  The above works even if connectivity changes with time in which case one can use

 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

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

 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.

 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.

 Λ eigenvalues  Eigen-energies

 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.

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

 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 … or … or …  Entropic ratio: τ/Ν max where τ stands for the sub- period inside the permutation.

 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.

 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!

 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?

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