Towards the ultimate APL-TOE

ABSTRACT

This paper presents the results of more than 10 years of transdisciplinary work. The initial idea was: can the laws of Nature also been found of rebuilt, independently from theoretical research in Physics (on elementary particles and matter in general), also in the field of Computer Science i.e. Information Processing? Pressing a lemon reveals its juice and stones; if one “tortures” matter, the components of its (first electrons, neutrons and protons, then quarks and gluons at a lesser order of magnitude) may be detected. What will appear if one tries to compress algorithms instead of atoms? APL seemed to be the ideal candidate for such a systematical investigation that led to some intriguing results which first proved to be indeed strongly connected with the conventional laws of Physics, then might enlighten in a new way many apparently-independent observations and studies, in a variety of fields such as neural networks, natural-language and signal processing, fractal geometry, Biology and Genetics inter alia.

REFERENCES

• AM: A. Ambrbzy (ed.) Noise in Physical Systems, including 1/f-Noise, Biological Systems and Membranes, Akademia Kiad6, Budapest, Hungary, ISBN 963-05-5778-9, (1990).
• BE: N.V. Belov, Yu.G. Zagal'skaia, G.P. LiWinskaia & YuoK. Egorov-Tismenko, Atlas of the space groups 3f the cubic system, Nauka, Moskva, USSR (1980) {in Russian}.
• BL: D.G. Blair, The Detection of Gravitational Waves, Cambridge University Press, ISBN 0-521-35278-9, p. 46 (1991).
• BU: Armin Bunde , Shlomo Havlin, Fractals and disordered systems, Springer-Verlag New York, Inc., New York, NY, 1991
• CH: G. Chaitin, A Computer Gallery of Mathematical Physics, (in APL2), IBM Research Report, Yorktown Heights, NY, USA, cfr. pp. 46 & 52 (March 23rd, 1985).
• DA: L.S. Davis, Shape Representation, in: R.A. Vaughan (ed.), Pattern Recognition and Image Processing in Physics, Proc. of the 37th Scottish Univ. Summer Schools in Physics, Dundee, 1990, ISBN 0-7503-0121-X, p. 114-117 (1991).
• DG: P.G. de Gennes, Introduction to Polymer Dynamics, Cambridge University Press, ISBN 0-521-38172-X, (1990); and: Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca/London, ISBN 0-8014-1203-X (1979).
• DU: M.J. Dumontier, Genetics of the XOR-futlction on Boolean Matrices, accepted in APL-CAM Journal, Belgium (1992).
• FR: E.S. Fredkin, D.M. Gitman & Sh. M. Shvartsman, Quantum Electrodynamics with Unstable Vacuum, Springer, Berlin, ISBN 0-387-52935-7 p. 156 & sq. (1991).
• GR: J.B. Griffith, Colliding Plane Waves in General Relativity, Clarendon Press, Oxford, G.B., p. 468 & sq. (1991).
• HAH: Th. Hahn (ed.) International Tables for Crystallography, Vol. 4. "Space Group Symmetry", Reidel Publishing Company, Dordrecht/ Boston, ISBN 90-277-2280-03, fig. 2.8.1a p. 23; see also the references to M.J. Buerger's work and H. Arnold's contribution p.70 & sq. (1987).
• HAW: S.W. Hawking, Wormholes & Non-simply Connected Manifolds, in: Quantum Cosmology & Baby Universes, (S. Coleman & al., eds), World Scientific, Singapore, ISBN 981-02-0345-4, vol. 7, p. 245-266 (1991).
• HO: J. Horgan, Profile: Physicist John A. Wheeler, Questioning the "It from Bit", Scientific American, pp. 36-37, (June 1991).
• IS: ISO8485, ISO, Geneva, Switz., p. 91 (1989).
• JE: R.V. Jean, Mathematical Approach to Pattern and Form in Plant Growth (Phyllotaxy), J.Wiley & Sons, New York, ISBN 0-471-88357-3, pp. 143- 144 (growth matrix), (1984).
• KAL: L.H. Kauffmann, Knots and Physics, World Scientific, Singapore, ISBN 981-02-0343-8, cf. The Lie algebra of SL(2) p. 143, Eigenstructures, the M+ and M- Matrices, p. 388, Q-spin networks.., p. 459 (1991).
• KAS: S.A. Kauffman, Boolean Functions and State Cycles (caption in the paper: "Antichaos and Adaptation"), Scientific American, Vol. 265, No 2, p. 80 (Aug. 1991) {see also ~ J. Theor. Biol. vol. 22, 437 (1969)}.
• KE: J. de Kerf, Fibonaccitis, APL XII Studiedag, Antwerp, Belgium (Nov. 1991), also ~ APL-CAM Journal, 13, 4, 942-977 (1991) {in Dutch}.
• KO: R. Kopelman, Diffusion-controlled Reaction Kinetics, from" D. Avnir (ed.) "The Fractal Approach to Heterogeneous Chemistry", ISBN 0-471-91723-0, p. 296 (1989).
• KR: E. Krause, Taxicab Geometry, Addison Wesley Publishing Company, USA (1975).
• LA1: G.A. Langlet, Paritons & Cognitons, Towards a New Theory of Information, APL-CAM Journal, 13, 3, 399-432 (1991).
• LA2: G.A. Langlet, New properties of the Paritons,... (Part II), APL-CAM Journal, 13, 3, 709-743 (1991).
• LA3: G.A. Langlet, Variations sur Sierpiflski, Pascal & Fibonacci, APL-CAM Journal, 13, 2, pp. 375-389 (1991) {in French}.
• LA4: G.A. Langlet, La Propagation Asym6trique de la Parit6..., BioMath, VOL XXIX, No 115, pp. 5-42, Paris (1992) {in French}.
• LA5: G.A. Langlet & M.V. Locquin, Presentation de Factotum, APL-XII, Antwerpen, Belgium (Nov. 1991) {in French}.
• LA6: G.A. Langlet, The Structures of Hybrids in the new Game of Life, with simple animated models {accepted in Biomath} (1992).
• LI: S. Lipschutz, Theory and Problems of Discrete Mathematics, McGraw-Hill Inc, New York, USA, ISBN 0-07-037981-5 (1976) & : Math6matiques discr~tes, cours et probl~mes, McGraw-Hill, Paris, ISBN 2-7042-1233-3 (1990).
• LO: S. Lovejoy, P. Gabriel & G.L. Austin & D. Schertzer, from: Fractal Aspects of Materials: Disordered Systems, M.R.S. extended abstracts, p. 205 (Nov. 1988).
• MA: V.G. Makhankov, Soliton Phenomenology, Kluwer, NL, ISBN 90-277-2830-5, 108 (1990).
• OT: E. Ott, Strange attractors and chaotic motions of dynamical systems, Rev. Mod. Phys., 53,655 (1981).
• PA: J K Park , K Steiglitz , W P Thurston, Soliton-like behavior in automata, Physica D, v.19 n.3, p.423-432, April 1986 [doi>10.1016/0167-2789(86)90068-0 ]
• PO: W. Poundstone, The Recursive Universe, Contemporary Books, Chicago, USA (1986).
• RE: Howard L. Resnikoff, The illusion of reality, Springer-Verlag New York, Inc., New York, NY, 1989
• RO: E.E. Rosinger, Non-Linear Partial Differential Equations (An Algebraic View of Generalized Solutions), North Holland Mathematic Studies, 164, ISBN 0-444-88700-8, Elsevier, NL (1990). {see the chapter devoted to Colombeau's algebra}.
• SAL: A. Salam, Unification of Fundamental Forces, Paul Dirac's Conf. (1968), "Difficulties in Quantum Electrodynamics"), Cambridge University Press, ISBN 0-521-37140-6, p. 143 (1990).
• SAM: Hanan Samet, The design and analysis of spatial data structures, Addison-Wesley Longman Publishing Co., Inc., Boston, MA, 1990
• SP: Thomas L. Springall , Gustav Tollet, A shading approach to non-convex clipping, ACM SIGAPL APL Quote Quad, v.20 n.4, p.369-372, July 1990
• SUS: L. Susskind, Critique of Coleman's Theory of the Vanishing Cosmological Constant. Same as {HAW}, p. 347-351.
• SUZ: D.T. Suzuki, A.J.F. Griffith, J.H. Miller & R.C. Lewontin, An Introduction to Genetic Analysis, Freeman & Co., New York/Oxford (1989) 750 p. chapter 22, Fig. 32, 34, 41 & text.
• STA: Dietrich Stauffer, 9 Cellular automata, Fractals and disordered systems, Springer-Verlag New York, Inc., New York, NY, 1991
• STE: I. Stewart, in Pour la Science, French Ed. of Scientific American, No 142, p. 103 (1989).
• STO: T. Stonier, Information and the Internal Structure of the Universe, Springer, London, G.B. ISBN 0-387-19599-8 See p. 34,83, 111-4 & e.g. Appendix A p. 121 & 131 (1990).
• STR: A. Strominger, Baby Universes, same as {HAW}, pp. 269-346.
• TH: R. Thom, Private communication (1992).
• VO: R.F. Voss in: "The Science of Fractal Images" (Peitgen & Saupe, eds.), Springer, New York, ISBN 0-387-96608-0, p. 31,39-40, fig. 1.9 (1988).
• VM: I.G. Voigt-Martin, H. Durst et al. Microstructure of a Discotic Polymer as revealed by electron diffraction and High-Resolution Imaging, Angewandte Chemie, Ed. in English, 28, 3, p. 323 (March 1989).
• WE: S. Weinberg, Conference to Paul Dirac's memory, Cambridge, GB in Feynman & Weinberg: The Character of Physical Law, Cambridge University Press (1967).
• WES: G. Weisbuch. Dynamique des Systi~mes Complexes, CNRS, Paris (1989) {translation into English" "Complex Dynamic Systems", Addison-Wesley, New York (1991)}.
• WIL: S. Wilson (ed.) The Effects of Relativity in Atoms, Molecules, and the Solid State, Plenum Press, New York, ISBN 0-306-43888-7 (1991).
• WIN: A.T. Winfree, The Geometry of Biological Time, Springer, N.Y., ISBN 0-387-9373-7 fig. 5, p.233 & fig. 10, p.239 (1980).
• WO: S. Wolfram, Theory and Applications of Cellular Automata, World Scientific, Singapore, ISBN 9971-50-123-6, p.1 & fig. 5 p. 13 (1986).
• DO: V.V. Dodonov & V.I. Man'ko, Integral Systems, Solid State Physics and Theory of Phase Transitions, Pt. 2, Proc. of the XViIIth Int. Coll. on Group Theoret. Methods in Physics, Moscow, Russia, June 1990, Nova Science, New York, USA, ISBN 1-56072-038-7 (1991).
• EV: R.A. Evarestov & V.P. Smirnov, Application of Site Symmetry in the Solid State Theory, in {DO} p. 81-4 (1991). Conclus. 6 magnon symmetry.
• KON: S.E. Konstein & M.A. Vasiliev, Higher Spin Superalgebras... in {DO}, p.233-6 (1991).
• KOS: O.S. Kosmachev, On the Projection Operators of the Irreducible Representation of Symmetric Groups, in {DO} pp. 287-289 (1991).
• KUB: Y.A. Kubyshin, J.M. Mourao & I.P. Volubujev, Graphic Schemes of Root Ordering in simple Lie algebras, in {DO} pp. 261-4 (1991).
• KUR: E.V. Kurmyshev, S.A. Dirov & A.V. Naumov, On some possibilities for description of associative media by Spin-Glass models, in {DO} pp. 147 (1991).
• PI: R.I. Pimenov, New Spacetimes - Anisotropic & SemiRiemannean, in {DO} pp. 257-60 (1991). "Anisotropy Degenerated" leading to Maxwell theory.
• SA: A.V. Samokhin, Symmetries and Equivalence of Differential Equations, in {DO} pp. 69-72 (1991); {leading to the Kortveg-de Vries equation}.
• YU: V.I. Yukalov, Group of Transformations for continuous Iteration, in {DO}, pp. 297-300 (1991); {a new method using self-similar transforms}.
• ZA: G.M. Zaslavsky, R.Z. Sagdeev, D.A. Usikov & A.A. Chernikov, Weak Chaos and Quasi- Regular Patterns, Cambridge Nonlinear Science 1, (Prof. Chirikov & al. ed.), Cambridge University Press GB, ISBN 0-521-373174-4 (1991). See, inter alia, the Kolmogorov-Arnold theorem (KAM theory), p. 28, the non-trivial effects of discretization, p. 53-5, the stochastic spinning of satellites, p.55-7, dynamic patterns, p.148 & 170, phyllotaxis & 5-fold symmetry, p. 224 & sq. nl

INDEX TERMS

Primary Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: APL

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on matrices

  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS

General Terms:
Algorithms, Languages, Theory

Keywords:
APL, Fibonacci, T.O.E., automata, binary algebra, chaos, dynamical systems, fractals, genetics, holography, integrals, parity, periodic systems, propagation, symmetry, topology