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Sharpening Occam ’s razor with Quantum Mechanics

Sharpening Occam ’s razor with Quantum Mechanics. SISSA Journal Club. Matteo Marcuzzi. 8th April , 2011. Describing Systems. Clausius Ptolemaeus ( Ptolemy ). Niclas Koppernigck ( Copernicus ). Tyge Brahe ( Tychonis ). Describing Systems. Johannes Kepler. Describing Systems.

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Sharpening Occam ’s razor with Quantum Mechanics

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  1. SharpeningOccam’s razorwith Quantum Mechanics SISSA Journal Club Matteo Marcuzzi 8th April, 2011

  2. DescribingSystems ClausiusPtolemaeus (Ptolemy) NiclasKoppernigck (Copernicus) TygeBrahe (Tychonis)

  3. DescribingSystems Johannes Kepler

  4. DescribingSystems

  5. DescribingSystems AlgorithmicAbstraction

  6. DescribingSystems AlgorithmicAbstraction Same output

  7. DescribingSystems Solar system celestialobjects SunFlares PlanetOrography Meteorology People behaviour Compton Scattering Same output Differentintrinsic information!

  8. DescribingSystems Much more memoryrequired! OCCAM’S RAZOR Same output Differentintrinsic information!

  9. DescribingSystems N SpinChain ifeven Up parity ifodd 1spin-flip per second 0

  10. DescribingSystems N SpinChain ifeven Up parity ifodd 1spin-flip per second 0 1

  11. DescribingSystems N SpinChain ifeven Up parity ifodd 1spin-flip per second 0 1 0

  12. DescribingSystems N SpinChain ifeven Up parity ifodd 1spin-flip per second 0 1 0 1

  13. DescribingSystems N SpinChain ifeven Up parity ifodd 1spin-flip per second 0 1 0 1 0

  14. DescribingSystems N SpinChain ifeven Up parity ifodd 1spin-flip per second 0 1 0 1 0 1

  15. DescribingSystems N SpinChain ifeven Up parity ifodd 1spin-flip per second 0 1 0 1 0 1 0

  16. DescribingSystems N SpinChain ifeven Up parity ifodd 1spin-flip per second 0 1 0 1 0 1 0 1

  17. DescribingSystems N SpinChain ifeven Up parity ifodd 1spin-flip per second N bitsneeded 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

  18. DescribingSystems Hidden System N bits read x 1-bit only! return (x+1) mod 2 Statisticallyequivalent output 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

  19. ComputationalMechanics • Statisticalequivalence • Measureofcomplexity • Pattern identification

  20. ComputationalMechanics • Statisticalequivalence • Measureofcomplexity • Pattern identification

  21. ComputationalMechanics • Statisticalequivalence • Measureofcomplexity • Pattern identification

  22. ComputationalMechanics • Statisticalequivalence • Measureofcomplexity • Pattern identification ?

  23. ComputationalMechanics StochasticProcess Discrete Stationary Alphabet RandomVariables

  24. ComputationalMechanics StochasticProcess Discrete Stationary Alphabet RandomVariables Pasts Futures

  25. ComputationalMechanics StochasticProcess Discrete Stationary Alphabet RandomVariables Set ofhistories Set of future strings

  26. ComputationalMechanics StochasticProcess Discrete Stationary 000101000101110101101… Machine StatisticalEquivalence

  27. ComputationalMechanics StochasticProcess Discrete Stationary PartitionR …010100010 …01010101 Machine States …1100111

  28. ComputationalMechanics StochasticProcess Discrete Stationary PartitionR Machine States

  29. ComputationalMechanics StochasticProcess Discrete Stationary Rj Ri a Machine TransitionRates

  30. ComputationalMechanics StochasticProcess Discrete Stationary OCCAM POOL

  31. ComputationalMechanics A little information theory Shannonentropy Conditionalentropy Mutual information Excessentropy

  32. ComputationalMechanics PartitionR Wewanttopreserve information Machine Cannotdistinguishbetweenthem

  33. ComputationalMechanics PartitionR minimize Log(# states) Wewanttopreserve information Machine with the leastpossiblememory

  34. ComputationalMechanics PartitionR Statistical complexity minimize Wewanttopreserve information Machine with the leastpossiblememory

  35. ComputationalMechanics OCCAM POOL Optimalpartition Statistical complexity minimize Wewanttopreserve information with the leastpossiblememory

  36. ComputationalMechanics Optimalpartition if CausalStates ε Statistical complexity minimize Wewanttopreserve information ε-machine with the leastpossiblememory (unique)

  37. ComputationalMechanics: Examples 2-periodic sequence 2-periodic, endswith A initial state I B 2-periodic, endswith

  38. ComputationalMechanics: Examples 2-periodic sequence A initial state I transient recurrent B

  39. ComputationalMechanics: Examples 1D Isingmodel transfer matrix

  40. ComputationalMechanics: Examples 1D Next-nearest-neighboursIsing 2

  41. ComputationalMechanics: Examples 1D Next-nearest-neighboursIsing 3 2

  42. ComputationalMechanics: Examples 1D Next-nearest-neighboursIsing 3 2 1

  43. ComputationalMechanics: Examples 1D Next-nearest-neighboursIsing 3 2 1

  44. ComputationalMechanics: Examples 1D Next-nearest-neighboursIsing negligible

  45. ComputationalMechanics: Examples 1D Next-nearest-neighboursIsing period3 period1

  46. Sharpening the razorwith QM Statisticalcomplexity Excessentropy Ideal system

  47. Sharpening the razorwith QM ε ε ε-machines are deterministic

  48. Sharpening the razorwith QM ε

  49. Sharpening the razorwith QM ε fixed i,c unique j fixed j,c unique i ideal

  50. Sharpening the razorwith QM qε ε causal state Ri system state symbol “s” symbol state q-machinestates

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