Randomness and Determination, from Physics and Computing towards Biology

1
Randomness and Determination, from Physics and
Computing towards Biology

• Giuseppe Longo
• LIENS, CNRS ENS, Paris
• http//www.di.ens.fr/users/longo

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Classical dynamical determinism and
unpredictability

• A physical system/process is deterministic when
  we have or we believe that it is possible to have
  a set of equations or an evolution function
  describing the process
• i.e. the evolution of the system is fully
  determined
• by its current states and by a law.
• Classical/Relativistic systems are State
  Determined Systems
• randomness is an epistemic issue

3
Classical dynamical determinism and
unpredictability

• Classical and Relativistic Physics are
  deterministic randomness is deterministic
  unpredictability (in chaotic systems)
• Quantum Mechanics is not deterministic
• (intrinsic/objective role of probabilities in
  constituting the theory the measure
  entanglement, no hidden variables)
• Recent survey/reflections Bailly, Longo, 2007,
  Longo, Paul, 2008
• Early confusion in Computing
• A non-deterministic Turing Machine is a
  classical deterministic device (ill-typed),
  unless a non-classical physical process (which
  one?) specifies/implements the branching

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Deterministic unpredictability

• Classical (dynamical) deterministic
  unpredictability
• a relation between
• a formal-mathematical system (equations,
  evolution functions)
• a physical process, measured by intervals (the
  access).
• By the mathematical system one cannot predict
  (over short, long time) the evolution of the
  physical process
• e. g. 1. describing/modelling 2. is non
  linear
• Mixing (a weak chaos) decreasing correlation of
  observables (Cn(fi, fj) ci,j/na for all
  n 1),
• b. Chaotic sensitivity, topological
  transitivity, density of periodic points pure
  Mathematics
• (decreasing knowledge about trajectories,
  increasing entropy)
• ? Randomness

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Randomness as deterministic unpredictability

• Classical (epistemic) randomness
• is defined by
• deterministic unpredictability (short, long time)
• Examples dies, coin tossing, a double pendulum,
  the Planetary System (Poincaré, 1890 Laskar,
  1992) finite (short and long) time
  unpredictability
• (the dies, a SDS, know where they go along a
  geodetics, determined by Hamiltons principle).
• Laplace
• infinitary demon OK (over space-time continua)
• determination ? predictability (except
  singularities) Wrong!

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Part I Classical Dynamical Systems and Computing

• Dynamical vs. Algorithmic Randomness

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Generic (point/trajectory) in Dynamics

• Objects are generic in Physics they are
  experimental and theoretical invariants (chose
  any falling body, gravitating planets)
• A Methodological Aim
• in a deterministic dynamical system (D,T,?)
•  Pick a generic point in D, at random
  (randomize)
• replaced by  pick a random (as generic)
  point in D
• Mathematically
•  a probabilistic property P holds for almost all
  points
• replaced by  the set of random points has
  measure 1 and P holds for all random points 

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Birkhoff randomness in Dynamical Systems

• Given (D, T, ?), dynamical system, a point x is
  generic (or typical, in the ergodic sense) if,
  for any observable f,
• Limn (f(x) f(T(x)) f(Tn(x)))/n ? f d?
• That is, the average value of the observable f
  along the trajectory
• x, T(x), Tn(x)
• (its time average)
• is asymptotically equal to the space average of
  f (i.e. ? f d?).
• A generic point is a (Birkhoff) random point for
  the dynamics.
• It is a purely mathematical and limit notion,
  within physico-mathematical dynamical systems, at
  asymptotic time.
• ? ML-randomness

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Algorithmic Randomness as strong undecidability

• Algorithmic randomness (Martin-Löf, 65 Chaitin,
  Schnorr.) (for infinite sequences in Cantor
  Space D 2?)
• Def. ?, measure on D, an effective tatistical
  test is
• an (effective) sequence Unn, with ?(Un) ? 2n
• I.e. a statistical test is an infinite decreasing
  sequence of effective open set in Cantors 2?
  (thus, it is given in Recursion Theory)
• Def. x is random if, for any statistical test
  Unn, x is not in ?nUn,
• (x passes all tests)
• Random not being contained in any effective
  intersection
• to stay eventually outside any test (it
  passes all tests)

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Algorithmic randomness and undecidability

• Algorithmic randomness a purely computational
  notion (a lot of work by Chaitin, Calude Gacs,
  Vyugin, Galatolo).
• An (infinite) algorithmic-random sequence
  contains no infinite effectively generated (r.e.,
  semidecidable) subsequence.
• Thus Algorithmic randomness is (strictly)
  stronger than undecidability (non r.e.,
  Gödel-Turings sense)
• there exist non rec. enum. sequences which are
  not algorithmically random
• (e.g. x1 e1 x2 e2 x3 x algo-random, e
  effective)
• Note there is no randomness in finite time
  sequential computing!
• At most uncompressibility (finite Kolmogoroff
  complexity)

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Dynamical random algorithmic random (Hoyrup,
Rojas Theses)
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Dynamical random algorithmic random (Hoyrup,
Rojas Theses)

• Given a mixing (weakly chaotic) dynamics (D, T,
  ?), with good computability properties (the
  metric, the measure are effective), then
• Main Theorem
• A point x in D is generic (Birkhoff random) for
  the dynamics iff it is (Schnorr) algorithmically
  random.
• Note at infinite time
• Dynamical randomness (a la Birkhoff) derives from
  Poincarés Theorem (deterministic
  unpredictability)
• Algorithmic randomness is a strong form of
  (Gödels) undecidability Q.E.D.

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Towards Biology

• The Physical Singularity of Life Phenomenain
  terms of Dualities

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The Physical Singularity of Life Phenomenain
terms of Dualities

• Physics generic objects and specific
  trajectoires (geodetics)
• Biology generic trajectories (compatible/possibl
  e) and specific objects (individuation)
  Bailly, Longo, 2006

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The Physical Singularity of Life Phenomenain
terms of Dualities

• Physics generic objects and specific
  trajectoires (geodetics)
• Biology generic trajectories (compatible/possibl
  e) and specific objects (individuation)
  Bailly, Longo, 2006
• Physics energy as operator Hf, time as
  parameter f(t, x)
• Biology time as operator, energy as parameter
• Time given by (speed of) entropy production by
  all irreversile processes it acts as an operator
  on a state function (bio-mass density)
• Applications both in phylogenesis (long-time
  Goulds curb) and ontogenesis (short-time
  scalling factors in allometry)
• F. Bailly, G. Longo. Biological Organization and
  Anti-Entropy,
• in J. of Biological Systems, Vol. 17, n.1, 2009.

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Randomness in Life Phenomena

• Recall in Computing and Physics
• 1. For infinite sequences
• (Birkhof) dynamical randomness algorithmic
  randomness
• 2. In finite time
• determistic unpredictability ? (quantum)
  indetermination and randomness
• (epistemic vs. intrinsic Bell inequalities)
• Yet, in infinite time, they merge (semi-classical
  limit)!
• T. Paul, 2008.

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Randomness in Life Phenomena

• Physics all within a given phase (reference)
  space (the possible states and observables).
• Biology intrinsic indetermination due to change
  of the phase space, in phylogenesis
  (ontogenesis?)
• A proper notion of biological randomness, at
  finite short/long time?
• Due to the entanglement of the two physical
  notions?
• Randomness Physics/Computing/Biology
• Physics 2 forms of randomness (different
  probability measures)
• In Concurrency? In Computers Neworks? A lot of
  work
• Biology the sum of all forms? What can we learn
  from the different forms of randomness and
  (in-)determination?

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Physical time vs. RandomnessGeneral tentative
approach to time as an irreversible parameter (in
Physical Theories)
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Physical time vs. RandomnessPreliminary Remarks

• 1. There is no irreversible time in the
  mathematics of classical mechanics
  (Euler-Lagrange, Newton-Laplace... equations are
  time-reversible also a linear field has reverse
  determination).
• 2. Classically, irreversible time appears in
• 2.1 Deterministic chaos, where randomness is
  unpredictability (an action at finite time -
  short/long decreasing knowledge)
• 2.2 Thermodynamics increasing entropy
  (dispersion of trajectories, diffusion of a gas,
  of heath along random paths)
• Notes underlying a diffusion (e.g. energy
  degradation) there is always a random path
• 2.1 and 2.2 dispersion of trajectories (entropy
  increases in both)

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Thesis Irreversible Time is Randomness(in
Physical Theories)
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Thesis 1 Irreversible Time implies
Randomness(in Physical Theories)

• By the previous argument
• Classical Physics the arrow of time is related
  (implies) randomness (by deterministic
  unpredictability and random walks in
  thermodynamics), in finite (not asymptotic) time.
• But also, in Quantum Physics
• t and -t may be interchanged in Schrödinger
  equation, as -i is equivalent to i (time may be
  reversed)
• Irreversible time appears at the (irreversible)
  act of measure, which gives probability values
  (intrinsic randomness, to the theory)
• Thus, if one wants (irreversible) time, one has
  randomness.

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Conversely Randomness implies Irreversible Time

• Classical Physics Randomness is (deterministic)
  unpredictability
• But, unpredictability concerns predicting, thus
  the future, in time (decreasing knowledge or no
  inverse map).
• An epistemic issue, both in Dynamics and
  Thermodynamics (increasing entropy)
• Similarly, the intrinsic randomness in Quantum
  Physics, concern the irreversible act of measure,
  irreversible in time
• measure produces irreversible time, by a
  before and an after.
• In conclusion, in Physics, by the structure of
  determination
• (irreversible) time and randomness are related
  (equivalent?)

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What about Biology?

• Life phenomena include
• 1 - Irreversible thermodynamic processes (with
  their irreversible time)
• But also
• 2.1 Darwinian Evolution (increasing phenotypic
  complexity, Gould number of tissue
  differentiations, of connections in networks)
• 2.2 Morphogenesis (embryogenesis and its
  opposite disorganization - death)
• Evolution and Morphogenesis are setting-up of
  organization (the opposite of entropy and its
  internal random processes)
• Death is tissue disorganization and includes the
  randomness in thermodynamic processes (entropy
  increase)

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The double irreversibility of Biological Time

• Evolution, morphogenesis and death are strictly
  irreversible, but their irreversibility is
  proper, it adds on top of the physical
  irreversibility of time (thermo-dynamical)
• It is due to a proper observable biological
  organization (integration/regulation between
  different levels of organization in an organism)
• This observable anti-entropy
• F. Bailly, G. Longo. Biological Organization and
  Anti-Entropy, in J. of Biological Systems, Vol.
  17, n.1, 2009.
• One reason for an intrinsic, proper Biological
  Randomness...

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Some references (more on http//www.di.ens.fr/user
s/longo )

• Bailly F., Longo G. Mathématiques et sciences de
  la nature. La singularité physique du vivant.
  Hermann, Visions des Sciences, Paris, 2006.
• M. Hoyrup, C. Rojas, Theses, June, 2008 (see
  http//www.di.ens.fr/users/longo )
• Bailly F., Longo G., Randomness and Determination
  in the interplay between the Continuum and the
  Discrete, Mathematical Structures in Computer
  Science, 17(2), pp. 289-307, 2007.
• Bailly F., Longo G.  Extended Critical
  Situations, in J. of Biological Systems, Vol.
  16, No. 2, 1-28, 2007.
• F. Bailly, G. Longo. Biological Organization and
  Anti-Entropy, in J. of Biological Systems, Vol.
  17, n.1, 2009.
• G. Longo. From exact sciences to life phenomena
  following Schrödinger on Programs, Life and
  Causality, lecture at "From Type Theory to
  Morphological Complexity A Colloquium in Honour
  of Giuseppe Longo," to appear in Information
  and Computation, special issue, 2008.
• G. Longo, T. Paul. The Mathematics of Computing
  between Logic and Physics. Invited paper,
  Computability in Context Computation and Logic
  in the Real World, (Cooper, Sorbi eds) Imperial
  College Press/World Scientific, 2008.