Asymptotic Density of Properties in Cellular Automata Laurent Boyer - PowerPoint PPT Presentation

Limit sets of CA A tool to study long term behaviour of CA. ◮ For one given CA A , Definition (Limit set) def � A t ( Q Z ) Ω A = t ∈ N ”Configurations that may appear arbitrarily late in the evolution.” Examples : ◮ Ω MAX = { ω 1 ω } ∪ { ω 0 ω } ∪ { ω 1 · 0 ω } ∪ { ω 0 · 1 ω } ∪ { ω 1 · 0 ∗ · 1 ω } ◮ Ω JustGliders = ω { R , ∅} · { L , ∅} ω Definition (Nilpotency) def A ∈ Nil ⇔ Ω A = { c } ”The CA always converges to this single configuration.” Cellular Automata – Limit sets 9/32

Intrinsic simulation (1/2) Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) Mazoyer, Delorme, Rapaport, Ollinger, Theyssier (1998-2010) ◮ A simulation relation Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) Mazoyer, Delorme, Rapaport, Ollinger, Theyssier (1998-2010) ◮ A simulation relation Two ingredients : Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) Mazoyer, Delorme, Rapaport, Ollinger, Theyssier (1998-2010) ◮ A simulation relation Two ingredients : ◮ the sub-automaton relation ⊑ restriction of the local rule to a stable subset of Q Example : in JustGliders : { L , ∅} defines a sub-automaton, { L , R } doesn’t. Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) Mazoyer, Delorme, Rapaport, Ollinger, Theyssier (1998-2010) ◮ A simulation relation Two ingredients : ◮ the sub-automaton relation ⊑ restriction of the local rule to a stable subset of Q Example : in JustGliders : { L , ∅} defines a sub-automaton, { L , R } doesn’t. ◮ rescalings (spatio-temporal transforms) ◮ packing ◮ time cutting ◮ shifting Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) ◮ rescalings (spatio-temporal transforms) ◮ packing ◮ time cutting ◮ shifting Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) ◮ rescalings (spatio-temporal transforms) ◮ packing ◮ time cutting ◮ shifting Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) ◮ rescalings (spatio-temporal transforms) ◮ packing ◮ time cutting ◮ shifting Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) ↔ ◮ rescalings (spatio-temporal transforms) ◮ packing ◮ time cutting ◮ shifting Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) ↔ ◮ rescalings (spatio-temporal transforms) ◮ packing ◮ time cutting ◮ shifting Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) Mazoyer, Delorme, Rapaport, Ollinger, Theyssier (1998-2010) ◮ A simulation relation Two ingredients : ◮ the sub-automaton relation ⊑ restriction of the local rule to a stable subset of Q Example : in JustGliders : { L , ∅} defines a sub-automaton, { L , R } doesn’t. ◮ rescalings (spatio-temporal transforms) ◮ packing ◮ time cutting ◮ shifting Definition (Simulation) def ⇔ ⊑ up to spatio-temporal transform � ⊑ ”The simulator can emulate uniformly the behaviour of the simulated CA.” Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) Definition (Simulation) def ⇔ ⊑ up to spatio-temporal transform � ⊑ ”The simulator can emulate uniformly the behaviour of the simulated CA.” Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) Definition (Simulation) def ⇔ ⊑ up to spatio-temporal transform � ⊑ ”The simulator can emulate uniformly the behaviour of the simulated CA.” Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (1/2) � ⊑ Definition (Simulation) def ⇔ ⊑ up to spatio-temporal transform � ⊑ ”The simulator can emulate uniformly the behaviour of the simulated CA.” Cellular Automata – Simulations and universality 10/32

Intrinsic simulation (2/2) Definition (Universality) def U ∈ Univ ⇔ ∀A , A � ⊑ U ” U is able to emulate the behaviour of any other CA.” Cellular Automata – Simulations and universality 11/32

Intrinsic simulation (2/2) Definition (Universality) def U ∈ Univ ⇔ ∀A , A � ⊑ U ” U is able to emulate the behaviour of any other CA.” Theorem (N. Ollinger – 2003) There exists a universal CA. Cellular Automata – Simulations and universality 11/32

Intrinsic simulation (2/2) Definition (Universality) def U ∈ Univ ⇔ ∀A , A � ⊑ U ” U is able to emulate the behaviour of any other CA.” Theorem (N. Ollinger – 2003) There exists a universal CA. Remarks : ◮ Central notion in CA litterature, ◮ Stronger than Turing universality in CA, ◮ Elements of Univ are maximal elements in the preorder induced by � ⊑ . Cellular Automata – Simulations and universality 11/32

Subfamilies of CA (example 1 ) Cellular Automata – Syntactically defined subfamilies 12/32

Subfamilies of CA (example 1 ) ◮ Captive CA Definition (Captive CA) ∀ x 1 , x 2 , . . . , x k ∈ Q, def A ∈ K ⇔ δ A ( x 1 , x 2 , . . . , x k ) ∈ { x 1 , x 2 , . . . , x k } ◮ Introduced by G. Theyssier (2004), ◮ under some conditions most captive CA are universal (2005). Cellular Automata – Syntactically defined subfamilies 12/32

Subfamilies of CA (example 2 ) ◮ Multiset CA Definition (Multiset CA) for all permutation π : { 1 , . . . k } → { 1 , . . . k } , def A ∈ MS ⇔ δ A ( x 1 , x 2 , . . . , x k ) = δ A ( x π (1) , x π (2) , . . . , x π ( k ) ) ◮ Captures the idea of isotropy . ◮ Other interesting properties ( rescalings ...). Cellular Automata – Syntactically defined subfamilies 13/32

Cellular Automata Introduction Limit sets Simulations and universality Syntactically defined subfamilies Density of properties Context Our framework Densities among CA Link with Kolmogorov complexity Densities among subclasses Perspectives Density of properties – Context 14/32

Motivations and previous related work ◮ Goal: ◮ quantify properties of CA, ◮ precise properties of random CA. Density of properties – Context 15/32

Motivations and previous related work ◮ Goal: ◮ quantify properties of CA, ◮ precise properties of random CA. ◮ Previous related work : ◮ Dubacq, Durand, Formenti – 2001 ◮ used Kolmogorov complexity as a classification parameter, ◮ proved that some properties are rare. ◮ Theyssier – 2005 ◮ Studied density of universality among captive CA. Density of properties – Context 15/32

Motivations and previous related work ◮ Goal: ◮ quantify properties of CA, ◮ precise properties of random CA. ◮ Previous related work : ◮ Dubacq, Durand, Formenti – 2001 ◮ used Kolmogorov complexity as a classification parameter, ◮ proved that some properties are rare. ◮ Theyssier – 2005 ◮ Studied density of universality among captive CA. ◮ Our contribution : ◮ a unified framework to study density among CA or subfamilies, ◮ various results. Density of properties – Context 15/32

Objects and properties ◮ What objects ? Density of properties – Our framework 16/32

Objects and properties ◮ What objects ? We consider the set CA of triplets ( Q n , V k , δ ) for n , k ∈ N , with ◮ Q n = { 0 , 1 , . . . , n − 1 } ◮ V k centered and connected neighbourhood of size k ◮ δ any function ( Q n ) k → Q n Density of properties – Our framework 16/32

Objects and properties ◮ What objects ? We consider the set CA of triplets ( Q n , V k , δ ) for n , k ∈ N , with ◮ Q n = { 0 , 1 , . . . , n − 1 } ◮ V k centered and connected neighbourhood of size k ◮ δ any function ( Q n ) k → Q n 1. some restrictions � but no influence on results . Density of properties – Our framework 16/32

Objects and properties ◮ What objects ? We consider the set CA of triplets ( Q n , V k , δ ) for n , k ∈ N , with ◮ Q n = { 0 , 1 , . . . , n − 1 } ◮ V k centered and connected neighbourhood of size k ◮ δ any function ( Q n ) k → Q n 1. some restrictions � but no influence on results . 2. syntactical descriptions � but redundancy does not biaised results . Density of properties – Our framework 16/32

Objects and properties ◮ What objects ? We consider the set CA of triplets ( Q n , V k , δ ) for n , k ∈ N , with ◮ Q n = { 0 , 1 , . . . , n − 1 } ◮ V k centered and connected neighbourhood of size k ◮ δ any function ( Q n ) k → Q n 1. some restrictions � but no influence on results . 2. syntactical descriptions � but redundancy does not biaised results . We consider densities among CA or among subfamilies C ⊆ CA . Density of properties – Our framework 16/32

Objects and properties ◮ What objects ? We consider the set CA of triplets ( Q n , V k , δ ) for n , k ∈ N , with ◮ Q n = { 0 , 1 , . . . , n − 1 } ◮ V k centered and connected neighbourhood of size k ◮ δ any function ( Q n ) k → Q n 1. some restrictions � but no influence on results . 2. syntactical descriptions � but redundancy does not biaised results . We consider densities among CA or among subfamilies C ⊆ CA . ◮ Which properties ? Any subset P ⊆ CA . Density of properties – Our framework 16/32

Enumeration CA is infinite = ⇒ asymptotic densities, Density of properties – Our framework 17/32

Enumeration CA is infinite = ⇒ asymptotic densities, ◮ Which enumerations of CA ? Every possible enumeration � meaningless results. Density of properties – Our framework 17/32

Enumeration CA is infinite = ⇒ asymptotic densities, ◮ Which enumerations of CA ? Every possible enumeration � meaningless results. But a natural possibility: Density of properties – Our framework 17/32

Enumeration CA is infinite = ⇒ asymptotic densities, ◮ Which enumerations of CA ? Every possible enumeration � meaningless results. But a natural possibility: ◮ pack CA by size ( n , k ), def def CA n , k = { ( Q n , V k , δ ) } and C n , k = C ∩ CA n , k Density of properties – Our framework 17/32

Enumeration CA is infinite = ⇒ asymptotic densities, ◮ Which enumerations of CA ? Every possible enumeration � meaningless results. But a natural possibility: ◮ pack CA by size ( n , k ), def def CA n , k = { ( Q n , V k , δ ) } and C n , k = C ∩ CA n , k ◮ and consider the proportions = # ( C n , k ∩ P ) def D n , k ( C , P ) # ( C n , k ) C n , k elements of size ( n , k ) of the family C , P a property. Density of properties – Our framework 17/32

Paths among sizes D n , k ( C , P ) has no canonical limit, ◮ How to consider successive sizes ( n , k ) ? Density of properties – Our framework 18/32

Paths among sizes D n , k ( C , P ) has no canonical limit, ◮ How to consider successive sizes ( n , k ) ? Definition (Paths) ⇔ ρ : N → N 2 injective def ρ path Density of properties – Our framework 18/32

Paths among sizes k D n , k ( C , P ) has no canonical limit, ◮ How to consider successive sizes ( n , k ) ? Definition (Paths) ⇔ ρ : N → N 2 injective def ρ path n CA 2 , 5 Density of properties – Our framework 18/32

Paths among sizes k D n , k ( C , P ) has no canonical limit, ◮ How to consider successive sizes ( n , k ) ? 1 0 Definition (Paths) ⇔ ρ : N → N 2 injective def ρ path n Density of properties – Our framework 18/32

Paths among sizes k D n , k ( C , P ) has no canonical limit, 7 3 5 ◮ How to consider successive sizes ( n , k ) ? 2 1 0 6 Definition (Paths) 4 ⇔ ρ : N → N 2 injective def ρ path n Density of properties – Our framework 18/32

Paths among sizes k D n , k ( C , P ) has no canonical limit, ◮ How to consider successive sizes ( n , k ) ? Definition (Paths) k 0 ⇔ ρ : N → N 2 injective def ρ path n 0 n def ◮ ρ ( n 0 , k 0 ) -path ⇔ ρ ( N ) ⊆ N n 0 × N k 0 def = N \ { 0 , . . . , x − 1 } N x Density of properties – Our framework 18/32

Paths among sizes k D n , k ( C , P ) has no canonical limit, ◮ How to consider successive sizes ( n , k ) ? Definition (Paths) k 0 ⇔ ρ : N → N 2 injective def ρ path n 0 n def ◮ ρ ( n 0 , k 0 ) -path ⇔ ρ ( N ) ⊆ N n 0 × N k 0 def ◮ ρ ( n 0 , k 0 ) -surjective ⇔ ρ ( N ) = N n 0 × N k 0 def = N \ { 0 , . . . , x − 1 } N x Density of properties – Our framework 18/32

Paths among sizes k D n , k ( C , P ) has no canonical limit, ◮ How to consider successive sizes ( n , k ) ? Definition (Paths) k 0 ⇔ ρ : N → N 2 injective def ρ path n 0 n def ◮ ρ ( n 0 , k 0 ) -path ⇔ ρ ( N ) ⊆ N n 0 × N k 0 def ◮ ρ ( n 0 , k 0 ) -surjective ⇔ ρ ( N ) = N n 0 × N k 0 def = N \ { 0 , . . . , x − 1 } ◮ We may consider N x ◮ every possible size (with surjective path) Density of properties – Our framework 18/32

Paths among sizes k D n , k ( C , P ) has no canonical limit, ◮ How to consider successive sizes ( n , k ) ? Definition (Paths) 0 1 2 3 4 5 6 7 ⇔ ρ : N → N 2 injective def ρ path n def ◮ ρ ( n 0 , k 0 ) -path ⇔ ρ ( N ) ⊆ N n 0 × N k 0 def ◮ ρ ( n 0 , k 0 ) -surjective ⇔ ρ ( N ) = N n 0 × N k 0 def = N \ { 0 , . . . , x − 1 } ◮ We may consider N x ◮ every possible size (with surjective path) ◮ or particular paths e.g. if ρ n = π 1 ◦ ρ or ρ k = π 2 ◦ ρ is upperbounded) Density of properties – Our framework 18/32

Density of properties Definition (Density of P among C following ρ :) � � # C ρ ( i ) ∩ P def d ρ ( C , P ) = lim if the limit exists. � C ρ ( i ) � # i →∞ ”The limit of the proportion along the path.” Density of properties – Our framework 19/32

Density of properties Definition (Density of P among C following ρ :) � � # C ρ ( i ) ∩ P def d ρ ( C , P ) = lim if the limit exists. � C ρ ( i ) � # i →∞ ”The limit of the proportion along the path.” Remarks : 1. not always defined Density of properties – Our framework 19/32

Density of properties Definition (Density of P among C following ρ :) � � # C ρ ( i ) ∩ P def d ρ ( C , P ) = lim if the limit exists. � C ρ ( i ) � # i →∞ ”The limit of the proportion along the path.” Remarks : 1. not always defined 2. non-cumulative density. Density of properties – Our framework 19/32

Density of properties Definition (Density of P among C following ρ :) � � # C ρ ( i ) ∩ P def d ρ ( C , P ) = lim if the limit exists. � C ρ ( i ) � # i →∞ ”The limit of the proportion along the path.” Remarks : 1. not always defined 2. non-cumulative density. def 3. P negligible along ρ ⇔ d ρ ( CA , P ) = 0 Density of properties – Our framework 19/32

Density of properties Definition (Density of P among C following ρ :) � � # C ρ ( i ) ∩ P def d ρ ( C , P ) = lim if the limit exists. � C ρ ( i ) � # i →∞ ”The limit of the proportion along the path.” Remarks : 1. not always defined 2. non-cumulative density. def 3. P negligible along ρ ⇔ d ρ ( CA , P ) = 0 Proposition Density is path-independent in the surjective case. Density of properties – Our framework 19/32

One example Density of properties – Densities among CA 20/32

One example ◮ Quiescent CA def A ∈ Quies ⇔ ∃ x ∈ Q A , δ A ( x , x , . . . , x ) = x Density of properties – Densities among CA 20/32

One example ◮ Quiescent CA def A ∈ Quies ⇔ ∃ x ∈ Q A , δ A ( x , x , . . . , x ) = x � n � 1 − 1 D n , k ( CA , Quies ) = 1 − n Density of properties – Densities among CA 20/32

One example k ◮ Quiescent CA def A ∈ Quies ⇔ ∃ x ∈ Q A , δ A ( x , x , . . . , x ) = x � n � 1 − 1 n D n , k ( CA , Quies ) = 1 − n Which yields to the following densities ◮ d ρ ( CA , Quies ) = 1 − 1 e if lim i →∞ ρ n ( i ) = + ∞ Density of properties – Densities among CA 20/32

One example k ◮ Quiescent CA def A ∈ Quies ⇔ ∃ x ∈ Q A , δ A ( x , x , . . . , x ) = x � n � 1 − 1 n D n , k ( CA , Quies ) = 1 − n Which yields to the following densities ◮ d ρ ( CA , Quies ) = 1 − 1 e if lim i →∞ ρ n ( i ) = + ∞ n 0 ) n 0 if lim i →∞ ρ n ( i ) = n 0 ◮ d ρ ( CA , Quies ) = 1 − (1 − 1 Density of properties – Densities among CA 20/32

One example k ◮ Quiescent CA def A ∈ Quies ⇔ ∃ x ∈ Q A , δ A ( x , x , . . . , x ) = x � n � 1 − 1 n D n , k ( CA , Quies ) = 1 − n Which yields to the following densities ◮ d ρ ( CA , Quies ) = 1 − 1 e if lim i →∞ ρ n ( i ) = + ∞ n 0 ) n 0 if lim i →∞ ρ n ( i ) = n 0 ◮ d ρ ( CA , Quies ) = 1 − (1 − 1 ◮ d ρ ( CA , Quies ) is not defined if lim i →∞ ρ n ( i ) does not exists. Density of properties – Densities among CA 20/32

Density of nilpotency Density of properties – Densities among CA 21/32

Density of nilpotency Theorem Nil is negligible among CA following any (2 , 1) -path. Density of properties – Densities among CA 21/32

Density of nilpotency Theorem Nil is negligible among CA following any (2 , 1) -path. Lemma (gluing) ⇒ + + Density of properties – Densities among CA 21/32

Density of nilpotency Theorem Nil is negligible among CA following any (2 , 1) -path. Lemma (gluing) ⇒ + + + specific combinatorial arguments for each case. Density of properties – Densities among CA 21/32

Intuitions (1/2): Fixed neighbourhood “With increasing number of states, Nil is negligible.” Density of properties – Densities among CA 22/32

Intuitions (1/2): Fixed neighbourhood “With increasing number of states, Nil is negligible.” ◮ Consider the graph of uniform configurations ( Q n , G A ): ◮ Q n the alphabet def ⇔ δ A ( x k A ) = y ◮ ( x , y ) ∈ G A Density of properties – Densities among CA 22/32

Intuitions (1/2): Fixed neighbourhood “With increasing number of states, Nil is negligible.” ◮ Consider the graph of uniform configurations ( Q n , G A ): ◮ Q n the alphabet def ⇔ δ A ( x k A ) = y ◮ ( x , y ) ∈ G A Density of properties – Densities among CA 22/32

Intuitions (1/2): Fixed neighbourhood “With increasing number of states, Nil is negligible.” ◮ Consider the graph of uniform configurations ( Q n , G A ): ◮ Q n the alphabet def ⇔ δ A ( x k A ) = y ◮ ( x , y ) ∈ G A ◮ Two properties : ◮ A ∈ Nil = ⇒ ( Q n , G A ) is a tree, Density of properties – Densities among CA 22/32

Intuitions (1/2): Fixed neighbourhood “With increasing number of states, Nil is negligible.” ◮ Consider the graph of uniform configurations ( Q n , G A ): ◮ Q n the alphabet def ⇔ δ A ( x k A ) = y ◮ ( x , y ) ∈ G A ◮ Two properties : ◮ A ∈ Nil = ⇒ ( Q n , G A ) is a tree, ◮ the map A �→ G A is balanced. Density of properties – Densities among CA 22/32

Intuitions (1/2): Fixed neighbourhood “With increasing number of states, Nil is negligible.” ◮ Consider the graph of uniform configurations ( Q n , G A ): ◮ Q n the alphabet def ⇔ δ A ( x k A ) = y ◮ ( x , y ) ∈ G A ◮ Two properties : ◮ A ∈ Nil = ⇒ ( Q n , G A ) is a tree, ◮ the map A �→ G A is balanced. ◮ “trees are asympotically negligible among functionnal graphs”... Density of properties – Densities among CA 22/32

Intuitions (2/2): Fixed state set “With increasing neighbourhood, Nil is negligible.” Density of properties – Densities among CA 23/32

Intuitions (2/2): Fixed state set “With increasing neighbourhood, Nil is negligible.” def Periodic subshifts: ∀ u ∈ Q ∗ n , Σ u ⇔ ω u ω Density of properties – Densities among CA 23/32

Intuitions (2/2): Fixed state set “With increasing neighbourhood, Nil is negligible.” def Periodic subshifts: ∀ u ∈ Q ∗ n , Σ u ⇔ ω u ω ◮ A ∈ Nil = ⇒ A (Σ u ) �⊆ Σ u p p v v v v A u u u u u k ◮ Transitions u ∗ �→ x are constrained , ◮ Combining those constraints makes it possible to conclude.. Density of properties – Densities among CA 23/32

Link with Kolmogorov Complexity def ”K ( u ) ⇔ | shortest algorithmical description of u | ” def u c -random ⇔ K ( u ) ≥ l − c . Density of properties – Link with Kolmogorov complexity 24/32