Randomness and Determination, from Physics and Computing towards

Biology

Giuseppe Longo

LIENS, CNRS – ENS, Paris

http://www.di.ens.fr/users/longo

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

Classical dynamical determinism and unpredictability

1. Classical and Relativistic Physics are deterministic: randomness is deterministic unpredictability (in chaotic systems)

2. 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

Deterministic unpredictability

Classical (dynamical) deterministic unpredictability:

a relation between

1. a formal-mathematical system (equations, evolution functions…)

2. 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:

a. Mixing (a weak chaos) = decreasing correlation of observables: (|Cn(fi, fj)| ≤ ci,j/nα for all n ≥ 1),

b. Chaotic = sensitivity, topological transitivity, density of periodic points… pure Mathematics

(decreasing knowledge about trajectories, increasing ‘entropy’)

Randomness

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 Hamilton’s principle).

Laplace:

1. infinitary demon: OK (over space-time continua);

2. determination predictability (except singularities): Wrong!

Part I: Classical Dynamical Systems and Computing

Dynamical vs. Algorithmic Randomness

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 »

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

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 {Un}n, with (Un) 2n

I.e. a statistical test is an infinite decreasing sequence of effective open set in Cantor’s 2 (thus, it is given in Recursion Theory);

Def. x is random if, for any statistical test {Un}n, 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)

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-Turing’s 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)

Dynamical random = algorithmic random (Hoyrup, Rojas Theses)

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 Poincaré’s Theorem (deterministic unpredictability)

Algorithmic randomness is a strong form of (Gödel’s) undecidability Q.E.D.

Towards Biology

The Physical Singularity of Life Phenomenain terms of Dualities

The Physical Singularity of Life Phenomenain terms of Dualities

• Physics: generic objects and specific trajectoires (geodetics)Biology: generic trajectories (compatible/possible) and specific objects (individuation) [Bailly, Longo, 2006]

The Physical Singularity of Life Phenomenain terms of Dualities

• Physics: generic objects and specific trajectoires (geodetics)

Biology: generic trajectories (compatible/possible) 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: Gould’s 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.

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].

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 1. Physics: 2 forms of randomness (different probability measures)2. In Concurrency? In Computers’ Neworks? A lot of work…3. Biology: the sum of all forms? What can we learn from the

different forms of randomness and (in-)determination?

Physical time vs. RandomnessGeneral tentative approach to time as an irreversible

parameter (in Physical Theories)

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)

Thesis: Irreversible Time is Randomness(in Physical Theories)

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.

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?)

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)

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...

Some references (more on http://www.di.ens.fr/users/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.