Hervé ZWIRN
CNRS & Paris Diderot University (LIED)
ENS Cachan (CMLA)
IHPST
Histoire et Philosophie de l’Informatique
IHPST le 22 juin 2017
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Determinism and Impredictibility
A few examples
Cellular automata : general framework (discret)
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Langton’s ant
Rules:
1. If the ant is on a black square, it turns right and moves forward one unit.
2. If the ant is on a white square, it turns left and moves forward one unit.
3. When the ant leaves a square, it inverts the color.
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Langton’s ant
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This is impredictible and nobody knows how to prove that.
The only way to know what happens is to run a simulation of the
10 000 steps of the ant and see what happens
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Rule 30
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Rule 110
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Rule 110
If you want to know what is the 1 000th line, you have to run
the simulation and go through the 999 previous steps.
There is no other way, no shortcut allowing to compute the
nth line without going through the (n-1) previous steps.
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Conway’s game of life
Any live cell with fewer than two or more than three live neighbours dies.
Any live cell with two or three live neighbours lives on to the next generation.
Any dead cell with exactly three live neighbours becomes a live cell.
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Conway’s game of life
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Conway’s game of life
Glider gun
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Conway’s game of life
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Computational Irreducibility
CIR
The behavior of the system can be found only by direct
simulation or observation: No general predictive procedure
is possible.
Wolfram S., Undecidability and intractability in theoretical physics, Vol 54, N 8. Phys. Rev. Letters, 1985
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Predicting :
• Knowing the result before the system
computing faster than the system
• finding shortcuts
•
Intuitively a system will be CIR if there is no other way to reach
the nth state than to go successively through the (n-1) previous
ones. There is no shortcut.
This is not a robust definition since we would like a process to be
CIR even if it is possible to reach the nth state by going through
(n-1) states that are “near” of the (n-1) previous states of the
system if it is still impossible to go directly to it.
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Giving a rigorous and robust definition is not easy because it is
necessary to precise formally which path of (n-1) steps is
acceptable as being near the real path followed by the system.
• H. Zwirn & J.P. Delahaye , Irreducibility and Computational Equivalence: Wolfram Science 10
Years After the Publication of A New Kind of Science, H. Zenil (Ed), Springer, 2013
• H. Zwirn, Computational Irredudicibility and Computational Analogy, Complex Systems, Vol
24, Issue 2, 2015
• H. Zwirn "Les systèmes déterministes simples sont-ils toujours prédictibles" in Complexité
et désordre, Grenoble Sciences Ed., 2015
• H. Zwirn “Emergence et irréductibilité computationnelle”, à paraitre dans « Complexité et
désordre : adaptation, localisation, dynamique », Editions Matériologiques, 2017
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Why is Computational irreducibility (CIR) interesting?
 From the algorithmic point of view :
oIs there any robust definition?
oDo any really CIR processes exist?
 Philosophical reasons:
o Understanding the behaviour of Complex Systems
o Understanding emergent phenomena
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CIR means impossible to speed-up
 Is every algorithm ’’speed able’’ ?
 The computation model matters
o Turing machines with k tapes (k ≥ 2)
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Some speed-up theorems:
problem of deciding if a string is a palindrome which is O(n2)
in the 1-tape Turing machines model and O(n) in the 2-tape
Turing machines model.
• The
•For any k-tapes Turing machine M operating in time f(n) there
exists a k'-tapes Turing machine M' operating in time f'(n)=f(n)+n
(where  is an arbitrary small positive constant) which simulates
M.
• Given any k-tape Turing machine M operating within time f(n),
it's possible to construct a 1-tape Turing machine M' operating
within time O(f(n)2) and such that for any input x, M(x)=M'(x).
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The computation model
•Turing machines with 3 symbols (0, 1, #) k ≥ 2 tapes and one way
write only tape which is used for output. We suppose that when the
computation ends, the result is the number written at the right end
of the output tape.
•Given a Turing machine M computing f(n) in time T(M(n)), let's
denote by Rn,1, …, Rn,i, …, Rn,T(M(n)) the content of the output tape of
M during the computation of f(n) after 1 step of computation, …, i
steps of computation and T(M(n)) steps of computation.
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(E-Turing machine): A Turing machine Mf will be called a
E-Turing machine for f if:
(i) Mf computes every f(n)
(ii) during the computation of f(n), there exist increasing kn(i)
for i=1 to n-1, such that f(i) is written on the output tape Rn,kn(i)
at the right of the last symbol #
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E-Turing machine for f
n
n-1
kn-1 (1)
f(1)
f(1)
kn (1)
kn-1 (2)
f(2)
f(2)
kn (2)
f(n-2)
kn (n-2)
kn-1 (n-2)
f(n-2)
f(n-1)
kn (n-1)
kn-1 (n-1)
f(n-1)
f(n)
kn (n)
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Tentative definition (CIR):
A function f will be said CIR if and only if any Turing machine
computing every f(n), is a E-Turing machine for f.
Not a Robust Definition
Need for more sophisticated concepts
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(Asymptotically optimal Turing machine): We will say that a
Turing machine Mf* for f is an asymptotically optimal Turing
machine for f if for any other Turing machine M computing f:
T(Mf*(n)) = O(T(M(n)))
i.e. there are constants c > 0, n0 > 0 such that n > n0,
T(Mf* (n))  cT(M(n)).
Asymptotically, no other Turing machine computing f
computes faster than Mf*
We assume that it is the case
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(Efficient E-Turing machine): We will say that a E-Turing
machine Mfeff for f is an efficient E-Turing machine for f if for
any other E-Turing machine Mf for f:
T(Mfeff(n)) = O(T(Mf(n)))
i.e. there are constants c > 0, n0 > 0 such that n > n0,
T(Mfeff (n))  cT(Mf(n)).
Asymptotically, no other E-Turing machine for f computes
faster than Mfeff
We assume that it is the case
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A Turing Machine M will be said to be a
P-approximation of a E-Turing machine for f if and only
if there are a function F such that F(n)=O(T(Mf* (n)/n))
and a Turing machine P such that:
(i) on input n, M computes a result rn such that P
computes f(n) from rn in a number of steps F(n) and
halts.
(ii) during the computation, there exist increasing kn(i)
for i=1 to n-1, such that P computes f(i) from i and
Rn,kn(i) in a number of steps F(i) and halts.
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P-approximation of a E-Turing machine for f
n
n
P
f(1)
r1
kn (1)
f(2)
r2
kn (2)
f(n-1)
rn-1
kn (n-1)
rn
kn (n)
f(n)
O(T(Mf*(n)/n))
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Let M be a P-approximation of a E-Turing machine for f.
Computation of f(n) based on the P-approximation M:
The computation of f(n) done initially through M with input n and
continued when M has computed rn, by P which computes f(n) from
n and rn in a time F(n) and halts.
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Computation based on a P-approximation of a E-Turing machine for f
n
P
f(1)
r1
f(2)
r2
f(n-1)
rn-1
f(n)
O(T(Mf* (n)/n))
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Strongly CIR (resp CIR) function:
A function f(n) from N to N will be said to be strongly CIR (resp
CIR) if and only if for any Turing machine M computing every f(n)
there is a P-approximation of a E-Turing machine for f, M’, such
that for every n (resp. for infinitely many n), the computation of
f(n) by M is based on M’.
If a function is strongly CIR, for each n there is no other way to compute
f(n) than to compute before all the values f(i) for i<n (or values that are
near in the sense given in the definition of the approximation of a ETuring machine). There is no shortcut allowing to get directly the value of
f(n) without having computed before f(n-1) or a value that is near f(n-1)
and so forth for the previous values.
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Theorem
if f is CIR no Turing machine computing every f(n) can
compute f(n) faster than an efficient E-Turing machine
for f.
More precisely, if Mf is a Turing machine computing
every f(n) and if f is CIR then T(Mfeff(n)) = O(T(Mf(n))).
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Possible candidates:
• Langton’s ant
• Rule 110
• The number of configurations of index < n still alive after n steps in the
game of life
• The function f défined as :
• f(1) = the first digit of 
• f(n) = the digit of  after having skipped f(n-1) digits from the digit f(n-1)
Open problem:
Prove that any of the above possible candidates is CIR.
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 What is necessary for Emergence?

2 levels:


individual / collectif
micro / macro

Knowledge of the rules for the low level

Apparent irreducibility of the phenomenon
appearing at the upper level to the low level rules
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Objective (weak) emergence
Objective emergence should be independant of our human
capacities.
Non-epistemic criterion.
Emergence appears when the dynamics of the
low level is CIR
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