taller @ concepción · self-assembly tiling (t;s;˝) t is a wang tileset with strengths: on each...
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Introducing freezing cellular automataTaller @ Concepción
G. Theyssier
(CNRS, CMM)
October, 2013
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Forewords
everything in 2D
work in progress
joint research with E. Goles
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Threshold rules
alphabet Q = 0,1
neighborhood N arbitrary
rule with threshold θ
F (x)(0,0) =
0 if x(0,0) = 0 and #
z ∈ N : xz = 1
< θ
1 else
limit fixed pointF∞(x) = lim
t→∞F t (x)
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Threshold rulesBootstrap percolation
fix 0 ≤ p ≤ 1Bernoulli distribution: 1 with probability pBootstrap percolation happens, denoted B(p), if
µ(x : F∞(x) = 1Z2
)= 1
Lemma: if p ≤ p′ then B(p)⇒ B(p′)
A central questionWhat is the critical probability infp : B(p)?
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Threshold rulesCriticality
3 behaviors for a threshold rule F :hypercritical finite seed of 1s
∃x finite s.t. F∞(x) is infinite
subcritical finite alliance of 0s
∃x co-finite s.t. F (x) = x
critical other cases
Proposition
hypercritical⇒ critical probability is 0
subcritical⇒ critical probability is 1
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Threshold rulesBootstrap percolation results
criticality is completely determinedGravner-Griffeath 1996simple geometrical/combinatorial conditions on N and θpolynomial algorithm
a large class of critical rules all behave the same way
Theorem (Duminil-Holroyd,2012)If F is critical with a “nice” N then critical probability is 0.
I don’t know...Is there a critical F with critical probability 6= 0?
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Threshold rulesBootstrap percolation results
criticality is completely determinedGravner-Griffeath 1996simple geometrical/combinatorial conditions on N and θpolynomial algorithm
a large class of critical rules all behave the same way
Theorem (Duminil-Holroyd,2012)If F is critical with a “nice” N then critical probability is 0.
I don’t know...Is there a critical F with critical probability 6= 0?
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Self-assembly tiling
(T , s, τ)
T is a Wang tileset with strengths: on each edge1 a label (matching condition)2 an integer (matching strength)
τ > 0 is the temperaturean assembly of tiles is τ -stable if separating it into 2 partsimplies removing a set of matching edges with totalstrength at least τs is a τ -stable finite assemblydynamics:
starting from sa→ a′ by adding a tile to a that preserves τ -stability
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Self-assembly tilingComputational power
general model Turing-universal in 2D (Winfree, 1998)temperature 1
Theorem (Cook-Fu-Schweller, 2011)Temperature 1 self-assembly is Turing-universal in 3D
A central questionWhat is the computational power of (2D) temperature 1?
Warning! asynchronous systemswhat is a valid simulation? what is universality?
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Self-assembly tilingSimulations and universality
BiblioDoty-Lutz-Patitz-Schweller-Summers-Woods-(Meunier-Theyssier)STACS 2010, FOCS 2012, SODA 2013
Definition1 block encoding
Z2 ↔ (mZ)× (pZ)
2 fuzz conditionempty cell↔ empty block (exception!)
3 requirements:same productions (through encoding)same dynamics (through encoding) not clear!
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Self-assembly tilingSimulations and universality
BiblioDoty-Lutz-Patitz-Schweller-Summers-Woods-(Meunier-Theyssier)STACS 2010, FOCS 2012, SODA 2013
Definition1 block encoding
Z2 ↔ (mZ)× (pZ)
2 fuzz conditionempty cell↔ empty block (exception!)
3 requirements:same productions (through encoding)same dynamics (through encoding) not clear!
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Self-assembly tilingSimulations and universality
BiblioDoty-Lutz-Patitz-Schweller-Summers-Woods-(Meunier-Theyssier)STACS 2010, FOCS 2012, SODA 2013
Definition1 block encoding
Z2 ↔ (mZ)× (pZ)
2 fuzz conditionempty cell↔ empty block (exception!)
3 requirements:same productions (through encoding)same dynamics (through encoding) not clear!
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Self-assembly tilingSimulations and universality
with a strong notion of “same dynamics”
Theorem (USA crew, 2012)There is an intrinsically universal tile set at temperature 2.
using only requirement of “same productions”
Theorem (USA crew + Meunier-Theyssier, 2013)No temperature 1 tile set can be intrinsically universal.
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Self-assembly tilingSimulations and universality
with a strong notion of “same dynamics”
Theorem (USA crew, 2012)There is an intrinsically universal tile set at temperature 2.
using only requirement of “same productions”
Theorem (USA crew + Meunier-Theyssier, 2013)No temperature 1 tile set can be intrinsically universal.
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Freezing cellular automata
Q = 0, . . . ,n − 1 with natural order ≤N arbitrary neighborhoodF is freezing if
∀x , ∀z : F (x)z ≥ xz
limit fixed point always defined
F∞(x) = limt→∞
F t (x)
special case: F is monotonic wrt ≤special case: F is outer-multiset if invariant bypermutation of outer neighbors
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Freezing cellular automata
Q = 0, . . . ,n − 1 with natural order ≤N arbitrary neighborhoodF is freezing if
∀x , ∀z : F (x)z ≥ xz
limit fixed point always defined
F∞(x) = limt→∞
F t (x)
special case: F is monotonic wrt ≤special case: F is outer-multiset if invariant bypermutation of outer neighbors
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Freezing cellular automataExamples
threshold CA are monotonic outer-multiset freezing
self-assembly systems are freezing (asynchronous) CA
other examples (outer-multiset freezing):infection propagation (SIR models)life without death
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Freezing cellular automataPredictability
kind of poor dynamically (convergence to fixed-point)
...but P-complete for prediction problem,embed any 1D P-complete CAlife without death is P-complete (???)
...but prediction has low communication complexity
PropositionThe prediction problem of any freezing CA has communicationcomplexity in O(n log(n))
recall: for a general CA this can be Ω(n2)
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Freezing cellular automataPredictability
kind of poor dynamically (convergence to fixed-point)
...but P-complete for prediction problem,embed any 1D P-complete CAlife without death is P-complete (???)
...but prediction has low communication complexity
PropositionThe prediction problem of any freezing CA has communicationcomplexity in O(n log(n))
recall: for a general CA this can be Ω(n2)
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Freezing cellular automataPredictability
kind of poor dynamically (convergence to fixed-point)
...but P-complete for prediction problem,embed any 1D P-complete CAlife without death is P-complete (???)
...but prediction has low communication complexity
PropositionThe prediction problem of any freezing CA has communicationcomplexity in O(n log(n))
recall: for a general CA this can be Ω(n2)
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Freezing cellular automataBootstrap percolation
problem: probability that F∞(x) is uniformly n − 1?
generalized definition of criticality:hypercritical
finite seed generates infinitely many n − 1in any context
subcriticalfinite alliance of states 6= from n − 1resistant to any context
criticalother cases
PropositionEach property among hypercriticality, subcriticality andcriticality is undecidable for freezing CA
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Freezing cellular automataBootstrap percolation
problem: probability that F∞(x) is uniformly n − 1?
generalized definition of criticality:hypercritical
finite seed generates infinitely many n − 1in any context
subcriticalfinite alliance of states 6= from n − 1resistant to any context
criticalother cases
PropositionEach property among hypercriticality, subcriticality andcriticality is undecidable for freezing CA
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Freezing cellular automataBootstrap percolation
problem: probability that F∞(x) is uniformly n − 1?
generalized definition of criticality:hypercritical
finite seed generates infinitely many n − 1in any context
subcriticalfinite alliance of states 6= from n − 1resistant to any context
criticalother cases
PropositionEach property among hypercriticality, subcriticality andcriticality is undecidable for freezing CA
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Freezing cellular automataBootstrap percolation
PropositionThe general bootstrap percolation problem:
input: CA + Bernouilli distributionquestion: percolation with probability 1?
is also undecidable.
ConjectureHypercriticality, subcriticality and criticality are decidablefor monotonic outer-multiset freezing CA
Questions1 same problem with monotonic freezing CA?2 same problem with outer-multiset freezing CA?3 critical rules with non-zero critical probability?4 non-monotonicity of the percolation probability?
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Freezing cellular automataBootstrap percolation
PropositionThe general bootstrap percolation problem:
input: CA + Bernouilli distributionquestion: percolation with probability 1?
is also undecidable.
ConjectureHypercriticality, subcriticality and criticality are decidablefor monotonic outer-multiset freezing CA
Questions1 same problem with monotonic freezing CA?2 same problem with outer-multiset freezing CA?3 critical rules with non-zero critical probability?4 non-monotonicity of the percolation probability?
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Freezing cellular automataBootstrap percolation
PropositionThe general bootstrap percolation problem:
input: CA + Bernouilli distributionquestion: percolation with probability 1?
is also undecidable.
ConjectureHypercriticality, subcriticality and criticality are decidablefor monotonic outer-multiset freezing CA
Questions1 same problem with monotonic freezing CA?2 same problem with outer-multiset freezing CA?3 critical rules with non-zero critical probability?4 non-monotonicity of the percolation probability?
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Freezing cellular automataSimulation and universality
how rich are the dynamics of freezing CA?are freezing CA richer than self-assembly tilings?key point: max number of changes per cell
PropositionNo 1-change von Neumann freezing CA can simulate allfreezing CA.
Corollary: self-assembly tilings cannot simulate allfreezing behaviors
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Freezing cellular automataSimulation and universality
how rich are the dynamics of freezing CA?are freezing CA richer than self-assembly tilings?key point: max number of changes per cell
PropositionNo 1-change von Neumann freezing CA can simulate allfreezing CA.
Corollary: self-assembly tilings cannot simulate allfreezing behaviors
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Freezing cellular automataSimulation and universality
Conjectures1 there is a 2-changes von Neumann freezing CA universal
for all freezing CA2 there is a 1-change freezing CA of larger radius which is
universal for all freezing CA3 there is a 1-change von Neumann freezing CA universal
for all 1-change von Neumann freezing CA and it can bechosen as a self-assembly tiling
also: outer-multiset? monotonicity? 3D? clean definitionsof simulations? etc
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