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 –30–31, 2016 Joint work with Jarkko Kari (University of Turku) and Siamak Taati (Leiden University) Revision: February 1, 2016 S. Capobianco Sofic groups and CA January 29–30–31, 2016 1 / 22
Introduction 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. S. Capobianco Sofic groups and CA January 29–30–31, 2016 2 / 22
Configurations and patterns over groups 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 ∈ S G 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 D n = { 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 . S. Capobianco Sofic groups and CA January 29–30–31, 2016 3 / 22
Cellular automata over groups 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 = { ν 1 , . . . , ν m } ⊆ G is finite and nonempty. f : S m → S is the local update rule. The global transition function F A : S G → S G is defined by the formula F A ( c )( g ) = f ( c ( g · ν 1 ) , . . . , c ( g · ν m )) ∀ g ∈ G Note that F is continuous. A pattern q : M → S is a preimage of p : E → S if E N ⊆ M and f ( q ( x · ν 1 ) , . . . , q ( x · ν m ))) = p ( x ) ∀ x ∈ E Fact: if every pattern has a preimage, so does every configuration. S. Capobianco Sofic groups and CA January 29–30–31, 2016 4 / 22
Pre-injectivity and the Garden of Eden theorem Let A be a CA on G with global function F . A is pre-injective if: whenever c , e ∈ S G are asymptotic and different it happens that F ( c ) � = F ( e ) The Garden of Eden theorem Moore, 1962: Every surjective CA on Z d is pre-injective. Myhill, 1963: Every pre-injective CA on Z d is surjective. S. Capobianco Sofic groups and CA January 29–30–31, 2016 5 / 22
Reversible cellular automata A cellular automaton with global function F is reversible if there exists a CA with global function H such that H ◦ F = F ◦ H = id S G . Fact: reversibility comes for free with bijectivity. Every injective CA on Z d is reversible. (Follows from the Garden of Eden theorem.) No injective, non-surjective CA is known! S. Capobianco Sofic groups and CA January 29–30–31, 2016 6 / 22
Surjunctive groups Let A be an injective CA on Z d . 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. Z d 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! S. Capobianco Sofic groups and CA January 29–30–31, 2016 7 / 22
Amenable groups 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. Z d is amenable for every d ≥ 1. (Standard proofs use ultrafilters, compactness, etc.) Abelian groups are amenable. Reason: f.g. abelian groups “are” the Z d × H with H finite abelian. Any group with a free subgroup on two generators is not amenable. Reason: otherwise, 1 = 2. S. Capobianco Sofic groups and CA January 29–30–31, 2016 8 / 22
The free group F 2 on two generators C b B a A D S. Capobianco Sofic groups and CA January 29–30–31, 2016 9 / 22
Amenable groups and the Garden of Eden theorem Bartholdi, 2010: Amenable groups are precisely those where the Garden of Eden theorem holds. All amenable groups are surjunctive. Yet another proof that F 2 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? S. Capobianco Sofic groups and CA January 29–30–31, 2016 10 / 22
Residually finite groups A group is residually finite if the intersection of all its subgroups of finite index is trivial. Call c ∈ S G 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 S G . Examples: Z d is residually finite for every d ≥ 1. Reason: if n > | x | > 0 then x �∈ ( n Z ) d , which has index n d in Z d . Free groups are residually finite. Reason: nontrivial words of length n induce nontrivial permutations of n + 1 objects. S. Capobianco Sofic groups and CA January 29–30–31, 2016 11 / 22
Residually finite groups are surjunctive Suppose G is residually finite. Injective CA are bi jective on periodic configurations. Reason: only finitely many configurations have any given period. Fix c and let c n → c be made of periodic configurations: Each c n has a (periodic) preimage e n . By continuity, every limit point of { e n } n ≥ 0 is a preimage of c . Corollary: F 2 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. S. Capobianco Sofic groups and CA January 29–30–31, 2016 12 / 22
Sofic groups 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 V 0 ⊆ V of vertices such that: 1 For every v ∈ V 0 , the disk of radius r with center v in the graph is isomorphic to the disk D r of radius r of the group as a labeled graph. 2 | V 0 | ≥ ( 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 . S. Capobianco Sofic groups and CA January 29–30–31, 2016 13 / 22
Residually finite groups are sofic 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 ∩ D r = { 1 G } . g ∈ G gHg − 1 is normal in G , and is of finite index The subgroup K = � if so is H . The corresponding quotient homomorphism φ is injective on D r . Then Cay ( G / K , φ ( B )) , with V 0 = G / K , is an ( r , ε ) -approximation of G whatever ε > 0 is! S. Capobianco Sofic groups and CA January 29–30–31, 2016 14 / 22
Amenable groups are sofic Weiss, 2000: Let G be an amenable group. Fix r ≥ 0 and ε > 0. Choose F as by definition of amenable group with K = D r . Let G = ( F , E , B ) be the subgraph of Cay ( G , B ) induced by F . Then G with V 0 = { x ∈ F | xD r ⊆ F } is an ( r , ε ) -approximation of G . S. Capobianco Sofic groups and CA January 29–30–31, 2016 15 / 22
Non-sofic groups None known! S. Capobianco Sofic groups and CA January 29–30–31, 2016 16 / 22
Sofic groups are surjunctive 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 : F A ( S G ) → S G is the restriction of a CA. Let r 0 ≥ 1 be such that ◮ both A and H are defined with D r 0 as neighborhood, and ◮ some patterns p : D r 0 → S are not reachable by A . Take a ( 5 r 0 , ε ) -approximation of G . Lemma: there exists V 2 ⊆ V 0 such that ◮ the disks of radius 2 r 0 centered in the points of V 2 are pairwise disjoint, and ◮ | V 2 | / | V 0 | ≥ 1 / ( 2 | D 2 r 0 + 1 | ) . S. Capobianco Sofic groups and CA January 29–30–31, 2016 17 / 22
Sofic groups are surjunctive (cont.) The local update rule of A induces a function φ from S V to S V , where V is the set of points of V that have a neighborhood isomorphic to D r 0 . Surely V 0 ⊆ V , hence by injectivity | φ ( S V ) | ≥ | S | | V 0 | ≥ | S | ( 1 − ε ) | V | . But by non-surjectivity, � | V 2 | | φ ( S V ) | ≤ � | S | | D r 0 | − 1 · | S | | V 0 | − | V 2 | · | D r 0 | Because of the estimate from the lemma, this leads to a contradiction if ε is small enough. S. Capobianco Sofic groups and CA January 29–30–31, 2016 18 / 22
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