Randomness and Determination , from Physics and Computing towards Biology

1. RandomnessandDetermination, from Physics and Computing towards Biology Giuseppe Longo LIENS, CNRS – ENS, Paris http://www.di.ens.fr/users/longo

2. 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 isdeterministic 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

4. Deterministicunpredictability Classical (dynamical) deterministicunpredictability: 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/nα for all n ≥ 1), b. Chaotic = sensitivity, topological transitivity, density of periodic points… pure Mathematics (decreasing knowledge about trajectories, increasing ‘entropy’)  Randomness

5. Randomnessasdeterministicunpredictability Classical (epistemic) randomness is defined by deterministicunpredictability (short, long time) Examples: dies, coin tossing, a double pendulum, the Planetary System (Poincaré, 1890; Laskar, 1992)… finite (short and long) timeunpredictability (the dies, a SDS, ‘know’ where they go: along a geodetics, determined by Hamilton’s principle). Laplace: • infinitary demon: OK (over space-time continua); • determination  predictability (except singularities): Wrong!

6. Part I: Classical Dynamical Systems and Computing Dynamical vs. Algorithmic Randomness

7. Generic (point/trajectory) in Dynamics Objects are ‘generic’ in Physics: they are experimental and theoretical invariants (chose any falling body, gravitating planets…) A MethodologicalAim: 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 »

8. 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

9. 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 testis 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 effectiveintersection = to stay “eventually outside any test” (it passes all tests)

10. 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 containsno 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)

11. Dynamical random = algorithmic random (Hoyrup, Rojas Theses)

12. 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.

13. Towards Biology The Physical Singularity of Life Phenomenain terms of Dualities

14. The Physical Singularity of Life Phenomenain terms of Dualities • Physics: generic objects and specific trajectoires (geodetics) Biology: generictrajectories(compatible/possible) andspecific objects (individuation) [Bailly, Longo, 2006]

15. The Physical Singularity of Life Phenomenain terms of Dualities Physics: generic objects and specific trajectoires (geodetics) Biology: generictrajectories(compatible/possible) andspecific 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.

16. 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].

17. Randomness in Life Phenomena Physics: all within a given phase (reference) space (the possible states and observables). Biology:intrinsicindetermination 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?

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

19. 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, irreversibletime 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)

20. Thesis: IrreversibleTimeisRandomness(in Physical Theories)

21. Thesis 1: IrreversibleTimeimpliesRandomness(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.

22. Conversely: Randomnessimplies IrreversibleTime 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: measureproduces irreversible time, by a “before” and an “after”. In conclusion, in Physics, by the “structure of determination”: (irreversible) time and randomness are “related” (equivalent?)

23. 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)

24. 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...

25. Some references (more onhttp://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.