Sofic groups and cellular automata Silvio Capobianco Institute of - - PowerPoint PPT Presentation
Sofic groups and cellular automata Silvio Capobianco Institute of Cybernetics at TUT silvio@cs.ioc.ee 29 th Estonian Theory Days K ao January 29 3031, 2016 Joint work with Jarkko Kari (University of Turku) and Siamak Taati (Leiden
Silvio Capobianco Institute of Cybernetics at TUT silvio@cs.ioc.ee 29th Estonian Theory Days – K¨ ao January 29–30–31, 2016
Joint work with Jarkko Kari (University of Turku) and Siamak Taati (Leiden University)
Revision: February 1, 2016
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Cellular automata (CA) are models of parallel synchronous computation where the nodes of a regular grid take their next state according to the current state of a uniform neighborhood. The properties of the group underlying the grid are often linked to those of the CA defined on it. An important open problem asks whether injectivity of the global function alone implies existence of a CA for the reverse update. This is known to hold for a class so large, that no counterexamples are known! We discuss this class, and the proof of the conjecture in this context. We then propose a “dual” to the conjecture above, based on a property introduced and discussed in previous work.
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Let G be a group and S be a finite nonempty set. For E, M ⊆ G: EM = {x · y | x ∈ E, y ∈ M}, E −1 = {x−1 | x ∈ E}. A configuration is a function c : G → S. c, e ∈ SG are asymptotic if |{g ∈ G | c(g) = e(g)}| < ∞. A pattern is a function p : E → S with E ⊆ G, 0 < #E < ∞. B ⊆ G generates G if words over B ∪ B−1 represent all elements of G. The length of g ∈ G is the minimum length g of such a word. We set Dn = {g ∈ G | g ≤ n}. The Cayley graph of G w.r.t. B is the labeled graph Cay(G, B) = (G, E, B ∪ B−1) where (x, b, y) ∈ E if and only if x · b = y.
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A cellular automaton (CA) over a group G is a triple A = S, N, f where: S is a finite set of states with two or more elements. The neighborhood N = {ν1, . . . , νm} ⊆ G is finite and nonempty. f : Sm → S is the local update rule. The global transition function FA : SG → SG is defined by the formula FA(c)(g) = f (c(g · ν1), . . . , c(g · νm)) ∀g ∈ G Note that F is continuous. A pattern q : M → S is a preimage of p : E → S if EN ⊆ M and f (q(x · ν1), . . . , q(x · νm))) = p(x) ∀x ∈ E Fact: if every pattern has a preimage, so does every configuration.
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Let A be a CA on G with global function F. A is pre-injective if: whenever c, e ∈ SG are asymptotic and different it happens that F(c) = F(e) The Garden of Eden theorem Moore, 1962: Every surjective CA on Zd is pre-injective. Myhill, 1963: Every pre-injective CA on Zd is surjective.
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A cellular automaton with global function F is reversible if there exists a CA with global function H such that H ◦ F = F ◦ H = idSG . Fact: reversibility comes for free with bijectivity. Every injective CA on Zd is reversible. (Follows from the Garden of Eden theorem.) No injective, non-surjective CA is known!
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Let A be an injective CA on Zd. Then A is clearly pre-injective . . . . . . thus also surjective by the Garden of Eden theorem. A group G is surjunctive if every injective CA on G is surjective. Zd is surjunctive. Actually, every group where the Garden of Eden theorem holds is surjunctive. There do, exist, however, groups where the Garden of Eden theorem does not hold!
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A group G is amenable if it satisfies one of the following, equivalent conditions:
1 For every finite K ⊆ G and every ε > 0 there exists a finite,
nonempty F ⊆ G such that |{x ∈ F | xK ⊆ F}| ≥ (1 − ε)|F|.
2 There exists a finitely additive probability measure µ : P(G) → [0, 1]
such that µ(gA) = µ(A) for every g ∈ G and A ⊆ G. G is amenable iff every finitely generated H ≤ G is. Zd is amenable for every d ≥ 1. (Standard proofs use ultrafilters, compactness, etc.) Abelian groups are amenable. Reason: f.g. abelian groups “are” the Zd × H with H finite abelian. Any group with a free subgroup on two generators is not amenable. Reason: otherwise, 1 = 2.
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a b
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Bartholdi, 2010: Amenable groups are precisely those where the Garden
All amenable groups are surjunctive. Yet another proof that F2 is not amenable: The majority rule with first four neighbors on the free group on two generators is surjective, but not pre-injective. Are there any surjunctive, non-amenable groups?
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A group is residually finite if the intersection of all its subgroups of finite index is trivial. Call c ∈ SG periodic if its stabilizer st(c) = {g ∈ G | λ x . c(g · x) = c} has finite index. Then G is residually finite iff periodic configurations are dense in SG. Examples: Zd is residually finite for every d ≥ 1. Reason: if n > |x| > 0 then x ∈ (nZ)d, which has index nd in Zd. Free groups are residually finite. Reason: nontrivial words of length n induce nontrivial permutations of n + 1 objects.
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Suppose G is residually finite. Injective CA are bijective on periodic configurations. Reason: only finitely many configurations have any given period. Fix c and let cn → c be made of periodic configurations: Each cn has a (periodic) preimage en. By continuity, every limit point of {en}n≥0 is a preimage of c. Corollary: F2 is surjunctive. Conjecture: (Gottschalk, 1973) All groups are surjunctive. This only needs to be proves for finitely generated groups. Reason: A is injective, or surjective, if and only if it is so on the subgroup generated by the neighborhood.
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Let G be a group generated by a finite symmetric set of generators B. An (r, ε)-approximation of G is a finite labeled graph (V , E, B) together with a subset V0 ⊆ V of vertices such that:
1 For every v ∈ V0, the disk of radius r with center v in the graph is
isomorphic to the disk Dr of radius r of the group as a labeled graph.
2 |V0| ≥ (1 − ε)|V |.
A finitely generated group G is sofic if it has an (r, ε)-approximation for every r ≥ 0 and ε > 0. The notion of soficness does not depend on the choice of B.
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Weiss, 2000: Let G be a residually finite group and B a finite set of generators. Fix r ≥ 0. Take H ≤ G of finite index such that H ∩ Dr = {1G}. The subgroup K =
g∈G gHg−1 is normal in G, and is of finite index
if so is H. The corresponding quotient homomorphism φ is injective on Dr. Then Cay(G/K, φ(B)), with V0 = G/K, is an (r, ε)-approximation
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Weiss, 2000: Let G be an amenable group. Fix r ≥ 0 and ε > 0. Choose F as by definition of amenable group with K = Dr. Let G = (F, E, B) be the subgraph of Cay(G, B) induced by F. Then G with V0 = {x ∈ F | xDr ⊆ F} is an (r, ε)-approximation of G.
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Weiss, 2000: Let G be sofic and let A = S, N, f be an injective, non-surjective CA on G. The left inverse map H : FA(SG) → SG is the restriction of a CA. Let r0 ≥ 1 be such that
◮ both A and H are defined with Dr0 as neighborhood, and ◮ some patterns p : Dr0 → S are not reachable by A.
Take a (5r0, ε)-approximation of G. Lemma: there exists V2 ⊆ V0 such that
◮ the disks of radius 2r0 centered in the points of V2 are pairwise
disjoint, and
◮ |V2|/|V0| ≥ 1/(2|D2r0+1|).
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The local update rule of A induces a function φ from SV to SV , where V is the set of points of V that have a neighborhood isomorphic to Dr0. Surely V0 ⊆ V , hence by injectivity |φ(SV )| ≥ |S||V0| ≥ |S|(1−ε)|V |. But by non-surjectivity, |φ(SV )| ≤
|V2| · |S||V0|−|V2|·|Dr0| Because of the estimate from the lemma, this leads to a contradiction if ε is small enough.
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A cellular automaton A = S, N, f over a group G is post-surjective if: for every c, e ∈ SG with FA(e) = c and every c ′ ∈ SG asymptotic to c, there exists e ′ ∈ SG asymptotic to e with FA(e ′) = c ′ Post-surjective CA are surjective. Reversible CA are post-surjective. The right-hand XOR is surjective, but not post-surjective.
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Capobianco, Kari, and Taati, 2015: pre-injective, post-surjective CA on surjunctive groups are reversible Capobianco, Kari and Taati, to be submitted for Automata 2016: post-surjective CA on sofic groups are pre-injective Conjecture: Every post-surjective CA is pre-injective.
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We tweak Weiss’ proof of surjunctivity of sofic groups: Suppose G is sofic and A is post-surjective, but not pre-injective. Take two configurations c, e with same image, differing only on Dm. For an r ≥ 0 large enough, depending on m and a value determined by post-surjectivity, consider an (r, ε)-approximation of G. That choice of r leads to the following contradiction: By post-surjectivity, the local rule of A induces a surjective function φ : SV → SV0. But by non-pre-injectivity, |φ(SV )| < |S||V0| for ε > 0 small enough.
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Conclusions: Post-surjectivity was introduced as a “dual” to pre-injectivity. But on sofic groups, it is equivalent to reversibility! We thus have a “pseudo-dual” of Gottschalk’s conjecture: every post-surjective CA is pre-injective Also, another link between group theory and CA theory. Future work: Are there any CA that are post-surjective, but not pre-injective? Are there non-sofic groups?
Any questions?
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