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Reversible causal graph dynamics
Pablo Arrighi1, Simon Martiel2, and Simon Perdrix3
1Aix-Marseille Univ., LIF, F-13288 Marseille Cedex 9, France
pablo.arrighi@univ-amu.fr,
2Univ. Nice-Sophia Antipolis, I3S, 06900 Sophia Antipolis, France
martiel@i3s.unice.fr
3CNRS, LORIA, Inria Project Team CARTE, Univ. de Lorraine, Nancy, France
simon.perdrix@loria.fr
Abstract. Causal Graph Dynamics extend Cellular Automata to arbi-
trary, bounded-degree, time-varying graphs. The whole graph evolves in
discrete time steps, and this global evolution is required to have a num-
ber of physics-like symmetries: shift-invariance (it acts everywhere the
same) and causality (information has a bounded speed of propagation).
We add a further physics-like symmetry, namely reversibility.
Keywords. Bijective, invertible, injective, surjective, one-to-one, onto,
Cayley graphs, Hedlund, Block representation, Lattice-gas automaton,
Reversible Cellular Automata.
Introduction
Cellular Automata (CA) consist in an array of identical cells, each of which
may take one in a finite number of possible states. The whole array evolves
in discrete time steps, and this global evolution is required to have a number
of physics-like symmetries: shift-invariance (it acts everywhere the same) and
causality (information cannot be transmitted faster than some fixed number of
cells per time step). CA are widely used to model spatially distributed computa-
tion (self-replicating machines, synchronization problems.. . ), as well as a great
variety of multi-agents phenomena (traffic jams, demographics.. . ). But their
origin lies in Physics, where they are commonly used to model waves or parti-
cles. In this context, it was natural to consider a further physics-like symmetry,
namely reversibility. The study of Reversible CA (RCA) was further motivated
by the promises of lower energy consumption in reversible computation. RCA
have turned out to have a beautiful mathematical theory, which relies on topol-
ogy or algebraic results to prove that 1/ the inverse of a CA is a CA [12], and
2/ that it admits a finite-depth reversible circuit decomposition [13].
CA model multi-agent systems, but under a fixed topology. There are many
situations, however, in which the notion of ‘who is next to whom’, also varies
in time (e.g. agents become physically connected, get to exchange contact de-
tails, move around, etc.). In the literature, several models (of physical systems,
self-replication, biochemical agents, economical agents, social networks. . . ) fea-
ture such neighbour-to-neighbour interactions with time-varying neighbourhood,
arXiv:1502.04368v1 [cs.DM] 15 Feb 2015
thereby generalizing CA for their specific sake. It is not until recently that CA
generalized to arbitrary, bounded-degree, time-varying graphs have been stud-
ied for their own sake, under the name of Causal Graph Dynamics (CGD). The
theoretical foundations of CGD have been laid in [1,3], including a first result
on reversibility (the inverse of a vertex-preserving CGD is a CGD). In this pa-
per, we improve this result, and prove that Reversible CGD admit a finite-depth
reversible circuit-like representation.
From a theoretical Computer Science viewpoint, the purpose of this paper is
therefore to generalize RCA theory to arbitrary, bounded-degree, time-varying
graphs. From a mathematical viewpoint, questions related to the bijectivity of
CA over certain classes graphs (more specifically, whether pre-injectivity implies
surjectivity for Cayley graphs generated by certain groups [10]) have received
some attention. This paper on the other hand provides a context in which to
study “bijectivity upon time-varying graphs”. This raises novel questions: Now
that the systems that support the information (i.e. the vertices) may be cre-
ated and deleted, does the bijectivity condition become vacuous? Or is it the
case that bijectivity will necessarily rigidify space (i.e. force the conservation of
each vertex)? We answer this question. From a theoretical physics viewpoint,
the question whether the reversibility of small scale physics (quantum mechan-
ics, micro-mechanical), can be reconciled with the time-varying topology of large
scale physics (relativity), is a topic of debate and constant investigation. This pa-
per provides a toy, discrete, classical model where reversibility and time-varying
topology coexist and interact.
1 Graphs
Basically, generalized Cayley graphs are the usual, connected, undirected, bounded-
degree graphs, but with five added twists:
•Edges are between ports of vertices, rather than vertices themselves, so that
each vertex can distinguish its different neighbours, via the port that con-
nects to it.
•There is a privileged vertex playing the role of an origin, so that any vertex
can be referred to relative to the origin, via a sequence of ports that lead to
it.
•The graphs are considered modulo isomorphism, so that only the relative
position of the vertices can matter.
•The vertices and edges are given labels taken in finite sets, so that they may
carry an internal state just like the cells of a Cellular Automaton.
•The labelling functions are partial, so that we may express our partial knowl-
edge about part of a graph. For instance it is common that a local function
may yield a vertex, its internal state, its neighbours, and yet have no opinion
about the internal state of those neighbours.
Notations. Let πbe a finite set, Π=π2be its square, and V=P(Π∗) the set
of languages over the alphabet Π. The operator ‘.’ represents the concatenation
of words and εthe empty word, as usual.
The vertices of the graphs (see Figure 1(a)) we consider in this paper are uniquely
identified by a name uin V. Vertices may also be labelled with a state σ(u) in
Σa finite set. Each vertex has ports in the finite set π. A vertex and its port are
written u:a.
An edge is an unordered pair {u:a, v :b}. Such an edge connects vertices uand
v; we shall consider connected graphs only. The port of a vertex can only appear
in one edge, so that the degree of the graphs is always bounded by |π|. Edges
may also be labelled with a state δ({u:a, v :b}) in ∆a finite set.
Formalization. Definitions 1 to 4 are as in [1]. The first two are reminiscent of
the many papers seeking to generalize Cellular Automata to arbitrary, bounded
degree, fixed graphs [17,7,11,10,9,22,15,21,20,6,16,8,18,19]. They are illustrated
by Figure 1(a).
Definition 1 (Graph). Agraph Gis given by
•An at most countable subset V(G)of V, whose elements are called vertices.
•A finite set π, whose elements are called ports.
•A set E(G)of non-intersecting two-element subsets of V(G)×π, whose
elements are called edges. In other words an edge eis of the form {u:a, v :b},
and ∀e, e0∈E(G), e ∩e06=∅ ⇒ e=e0.
The graphs are all assumed to be connected, i.e. for any two u, v ∈V(G), there
exists v1, . . . , vn−1∈V(G)such that for all i∈ {0. . . n−1}, one has {vi:ai, vi+1 :
bi} ∈ E(G)with v0=uand vn=v.
Definition 2 (Labelled graph). A labelled graph is a triple (G, σ, δ), also
denoted simply Gwhen it is unambiguous, where Gis a graph, and σand δ
respectively label the vertices and the edges of G:
•σis a partial function from V(G)to Σ;
•δis a partial function from E(G)to ∆.
The set of all graphs with ports πis written Gπ. The set of all labelled graphs
with states Σ, ∆ and ports πis written GΣ,∆,π. To ease notations, we sometimes
write v∈Gfor v∈V(G).
We now want to single out a vertex. The following definition is illustrated by
Figure 1(b).
Definition 3 (Pointed graph). Apointed (labelled) graph is a pair (G, p)
with p∈G. The set of all pointed graphs with ports πis written Pπ. The set of
all pointed labelled graphs with states Σ , ∆ and ports πis written PΣ,∆,π.
Definition 4 (Isomorphism). An isomorphism Ris a function from Gπto Gπ
which is specified by a bijection R(.)from Vto V. The image of a graph Gunder
the isomorphism Ris a graph RG whose set of vertices is R(V(G)), and whose
set of edges is {{R(u) : a, R(v) : b}|{u:a, v :b} ∈ E(G)}. Similarly, the image
of a pointed graph P= (G, p)is the pointed graph RP = (RG, R(p)). When P
and Qare isomorphic we write P≈Q, defining an equivalence relation on the
set of pointed graphs. The definition extends to pointed labelled graphs.
In the particular graphs we are considering, the vertices can be uniquely distin-
guished by the paths that lead to them starting from the pointer vertex. Hence,
we might just as well forget about vertex names. The following definition is
illustrated by Figure 1(c).
Definition 5 (Generalized Cayley graphs). Let Pbe a pointed (labelled)
graph (G, p). The Generalized Cayley graph Xis the equivalence class of Pwith
respect to the equivalence relation ≈. The set of Generalized Cayley graphs with
ports πis written Xπ. The set of labelled Generalized Cayley graph with states
Σ, ∆ and ports πis written XΣ,∆,π .
These pointed graph modulo will constitute the set of configurations of the gen-
eralized Cellular Automata that we will consider in this paper.
u v
:a:b
(a)
u v
:a:b
(b)
:a:b
(c)
Fig. 1. The different types of graphs. (a) A graph. (b) A pointed graph.
(c) A pointed graph modulo or “Generalized Cayley graph”. These can also
be described as a language {ε, ab, ab.ba, . . .}and an equivalence relation with
equivalence classes corresponding to vertices: ˜ε={ε, ab.ba, . . .}and ˜
ab =
{ab, ab.ba.ab, . . .}
Paths and vertices. We mentioned that vertices can be uniquely distinguished
by the paths that lead to them starting from the pointer vertex. Let us make
this more precise.
Definition 6 (Path). Given a Generalized Cayley graph X, we say that αis
a path of Xif αis a sequence of ports aibisuch that, starting from the pointer,
it is possible to travel in the graph according to this sequence. More formally,
αis a path if there exists (G, p)∈Xand v1, . . . , vn∈V(G)such that for all
i∈ {0. . . n −1}, one has {vi:ai, vi+1 :bi} ∈ E(G), with v0=pand αi=aibi.
Notice that the existence of a path does not depend on the choice of (G, p)∈X.
Definition 7 (Equivalence of paths). Given a pointed graph modulo X, we
define the equivalence of paths relation ≡Xsuch that for all paths u, u0of X,
u≡Xu0if, starting from the pointer, uand u0lead to the same vertex of X. More
formally, u≡Xu0if there exist (G, p)∈Xand v1, . . . , v|u|, v0
1, . . . , v0
|u0|∈V(G)
such that for all i∈ {0. . . |u| − 1},i0∈ {0. . . |u0| − 1}, one has {vi:ai, vi+1 :bi} ∈
E(G),{v0
i0:a0
i0, v0
i0+1 :b0
i0} ∈ E(G), with v0=p,v0
0=p,ui=aibi,u0
i0=a0
i0b0
i0
and v|u|=v|u0|. We write ˜ufor the equivalence class of umodulo ≡X.
The vertices of a Generalized Cayley graph Xcan be designated by
•˜uthe set of all paths leading to this vertex starting from the pointer of X
(in this convention the pointer is ˜ε),
•or more directly by uan element of an equivalence class ˜u, i.e. a particular
path leading to this vertex starting from the pointer of X(in this convention
the pointer is ε).
In fact, given a Generalized Cayley graph we can reconstruct a pointed graph
(G(X),˜ε) according to these vertex naming conventions:
Definition 8 (Associated (pointed) graph). Let Xbe some Generalized
Cayley graph. Let P(X)be the pointed graph (G(X),˜ε), with G(X)such that:
•The set of vertices V(G(X)) is the set of equivalence classes modulo ≡Xof
the paths of X;
•The edge {˜u:a, ˜v:b}is in E(G(X)) if and only if u.ab is a path of Xand
u.ab ≡Xv, for all u∈˜uand v∈˜v.
We define the associated graph to be G(X). We define the associated pointed
graph to be P(X).
For convenience from now on ˜uand uwill no longer be distinguished. The latter
notation will be given the meaning of the former. I.e. we shall speak of a “vertex”
uin V(X) (or simply u∈X).
2 Operations
For a generalized Cayley graph (G, p) non-modulo (see [1] for details):
•the neighbours of radius rare just those vertices which can be reached in r
steps starting from the pointer p;
•the disk of radius r, written Gr
p, is the subgraph induced by the neighbours
of radius r+ 1, with labellings restricted to the neighbours of radius rand
the edges between them, and pointed at p.
Notice that the vertices of Gr
pcontinue to have the same names as they used
to have in G. For generalized Cayley graphs, on the other hand, the analogous
operation is:
Definition 9 (Disk). Let X∈XΣ,∆,π be a generalized Cayley graph and (G, ε)
its associated pointed graph. Let Xrbe f
Gr
ε. The generalized Cayley graph Xr∈
XΣ,∆,π is referred to as the disk of radius rof X. The set of disks of radius r
with states Σ, ∆ and ports πis written Xr
Σ,∆,π .
Definition 10 (Size). Let X∈XΣ,∆,π be a generalized Cayley graph. We say
that a vertex u∈Xhas size less or equal to r+ 1, and write |u| ≤ r+ 1, if
u∈Xr. We denote V(Xr
π) = SX∈Xr
πV(X).
It will help to have a notation for the graph where vertices are named relatively
to some other pointer vertex u.
eε
eaa
ebb
gbb.ac
eca
eda
gda.cb
:a
:a
:b
:b
:a
:c
:c
:a
:d
:a
:b
:a
:c
:b
:c
:beε
eaa
ebb
eca
eda
:a
:a
:b
:b
:c
:a
:d
:a
XX0
Fig. 2. A generalized Cayley graph and its disk of radius 0.Notice that the
equivalence classes describing vertices in X0are strict subsets of those in X,
eventhough their shortest representative is the same. For instance the path ca.cb
is in ˜
da in Xbut is not a path in X0, and thus does not belong to ˜
da in X0.
Definition 11 (Shift). Let X∈XΣ,∆,π be a generalized Cayley graph and
(G, ε)its associated pointed graph. Consider u∈Xor Xrfor some r, and con-
sider the pointed graph (G, u), which is the same as (G, ε)but with a different
pointer. Let Xube
^
(G, u). The generalized Cayley graph Xuis referred to as X
shifted by u.
The composition of a shift, and then a restriction, applied on X, will simply be
written Xr
u. Whilst this is the analogous operation to Gr
uover pointed graphs
non-modulo, notice that the shift-by-ucompletely changes the names of the
vertices of Xr
u. As the naming has become relative to u, the disk Xr
uholds no
information about its prior location u.
We may also want to designate a vertex vby those paths that lead to the
vertex urelative to ε, followed by those paths that lead to vrelative to u. The
following definition of concatenation coincides with the one that is induced by
the concatenation of words belonging to the classes uand v:
Definition 12 (Concatenation). Let X∈Xπbe a generalized Cayley graph
and (G, ε)its associated pointed graph. Consider u∈Xand v∈Xuor Xr
ufor
some r. Let (G0, ε)be the associated pointed graph of (Xu)v,Rbe an isomorphism
such that G0=RG, and u.v be R−1(ε). The vertex u.v ∈Xis referred to as u
concatenated with v.
According to Definition 11, G0and Gare isomorphic. Moreover, the restriction
of R−1to V(G0) is uniquely determined; hence the definition is sound.
It also helps to have a notation for the paths to εrelative to u.
Definition 13 (Inverse). Let X∈Xπbe a generalized Cayley graph and (G, ε)
its associated pointed graph. Consider u∈X. Let (G0, ε)be the associated pointed
graph of Xu,Rbe an isomorphism such that G0=RG, and ube R(ε). The vertex
u∈Xuis referred to as the inverse of u.
Notice the following easy facts: (Xu)v=Xu.v,u.u =ε. Notice also that the
isomorphism Rsuch that G(Xu) = RG(X) maps vto u.v.
3 Causality
This notion of causality extends the known mathematical definition of Cellular
Automata over grids and Cayley graphs. The extension will be a strict one for
two reasons: not only the graphs become arbitrary, but they can also vary in
time.
The main difficulty we encountered when elaborating an axiomatic definition
of causality from XΣ,∆,π to XΣ,∆,π , was the need to establish a correspondence
between the vertices of a generalized Cayley graph X, and those of its image
F(X). Indeed, on the one hand it is important to know that a given u∈Xhas
become u0∈F(X), e.g. in order to express shift-invariance F(Xu) = F(X)u0.
But on the other hand since u0is named relative to ε, its determination requires
a global knowledge of X.
The following analogy provides a useful way of tackling this issue. Say that
we were able to place a white stone on the vertex u∈Xthat we wish to
follow across evolution F. Later, by observing that the white stone is found at
u0∈F(X), we would be able to conclude that uhas become u0. This way of
grasping the correspondence between an image vertex and its antecedent vertex
is a local, operational notion of an observer moving across the dynamics.
Definition 14 (Dynamics). A dynamics (F, R•)is given by
•a function F:XΣ,∆,π →XΣ,∆,π;
•a map R•, with R•:X7→ RXand RX:V(X)→V(F(X)).
For all X, the function RXcan be pointwise extended to sets, i.e. RX:P(V(X)) →
P(V(F(X))) maps Sto RX(S) = {RX(u)|u∈S}.
The intuition is that RXindicates which vertices {u0, v0, . . .}=RX({u, v, . . .})⊆
V(F(X)) will end up being marked as a consequence of {u, v, ... ∈X} ⊆ V(X)
being marked. Now, clearly, the set {(X, S)|X∈XΣ ,∆,π, S ⊆V(X)}is isomor-
phic to XΣ0,∆,π with Σ0=Σ× {0,1}. Hence, we can define the function F0that
maps (X, S)∼
=X0∈XΣ0,∆,π to (F(X), RX(S)) ∼
=F0(X0)∈XΣ0,∆,π, and think
of a dynamics as just this function F0:XΣ0,∆,π →XΣ0,∆,π. This alternative
formalism will turn out to be very useful.
Definition 15 (Shift-invariance). A dynamics (F, R•)is said to be shift-
invariant if for every Xand u∈X,v∈Xu,
•F(Xu) = F(X)RX(u)
•RX(u.v) = RX(u).RXu(v).
The second condition expresses the shift-invariance of R•. Notice that RX(ε) =
RX(ε).RX(ε); hence RX(ε) = ε.
In the F0:XΣ0,∆,π →XΣ0,∆,π formalism, the two above conditions are equivalent
to just one: F0(X0
u) = F0(X0)RX(u).
Definition 16 (Uniform continuity). A dynamics (F, R•)is said to be con-
tinuous if for all m, there exists nsuch that for all X, X 0,X0n=Xnimplies
both
•F(X0)m=F(X)m.
•dom Rm
X0⊆V(X0n), dom Rm
X⊆V(Xn)and Rm
X0=Rm
X.
where Rm
Xdenotes the partial map obtained as the restriction of RXto the
codomain F(X)m, using the natural inclusion of F(X)minto F(X).
In the F0:XΣ0,∆,π →XΣ0,∆,π formalism, the two above conditions are equivalent
to just one: F0uniformly continuous.
We need one third, last condition:
Definition 17 (Boundedness). A dynamics (F, R•)from XΣ,∆,π to XΣ,∆,π is
said to be bounded if there exists a bound bsuch that for all X, for all w0∈F(X),
there exist u0∈im RXand v0∈F(X)b
u0such that w0=u0.v0.
The following is the definition of causality:
Definition 18 (Causal dynamics). A dynamics is causal if it is shift-invariant,
uniformly continuous and bounded.
Actually, uniform continuity can be weakened into continuity, due to the com-
pactness of XΣ,∆,π , see [2].
An example of causal dynamics is the inflating grid dynamics illustrated in
Figure 3. In the inflating grid dynamics each vertex gives birth to four distinct
vertices, such that the structure of the initial graph is preserved, but inflated.
The graph has maximal degree 4, and the set of ports is π={a, b, c, d}, vertices
and edges are unlabelled.
4 Locality
Causal Graph Dynamics change the entire graph in one go. The word causal there
refers to the fact that information does not propagate too fast. Local operations,
on the other hand, act just in one bounded region of the graph, leaving the rest
unchanged. We introduce the following locality definition:
Definition 19 (Local dynamics). A dynamics (L, S•)is r0-local if it is uni-
formly continuous and bounded, and there exists r0such that for all Xand
u0∈L(X)with |u0|> r0, there exists u∈Xsuch that we have both:
•L(X)0
u0=X0
u,
• ∀v∈X0
u, SX(u.v) = u0.v.
Lemma 1 (Bounded inflation). If (L, S•)is r0-local, then for all sthere exists
s0such that for all Xand v∈X, if |v| ≤ s, then |SX(v)| ≤ s0.
7→
Fig. 3. The inflating grid dynamics. Each vertex splits into 4 vertices. The
structure of the grid is preserved. Ports are omitted here.
Proof. Suppose the contrary: X(s0) has some |v(s0)| ≤ ssuch that |SX(v)|> s0.
Since XΣ,∆,π is compact [2], X(s0) admits a subsequence which converges to some
limit X, in the sense that X(s0
k)k=Xk. For this particular X, for any s0, there is
some |v(s0)| ≤ ssuch that |SX(v)|> s0. This is because we can choose kso that
s0
k≥s0and ksuperior to the radius needed to determine L(X)s0
k=L(X(s0
k))s0
k,
so that |SX(v)|=|SXk(v)|=|SX(sk)k(v)|=|SX(sk)(v)|> s0
k≥s0. Thus, there
exists a point of v∈Xhas |SX(v)|>∞, which is a contradiction.
Lemma 2. If (L, S•)is r0-local, for all t, for all u0∈L(X)with |u0|> r0+t+1,
there exists u∈Xwith SX(u) = u0such that we have:
•L(X)t
u0=Xt
u,
• ∀v∈Xt
u, SX(u.v) = u0.v.
Proof. Take such a u0and consider usuch that u0=SX(u).
[First •] Since |u0|> r0+t+ 1, we have that for all v∈L(X)t
u0,|u0.v|> r0.
Hence, by r0-locality of L, there exists x∈Xsuch that SX(x) = u0.v and such
that L(X)0
u0.v =X0
x, i.e. the vertex vin L(X)t
u0, in terms of its internal states
and edges, is the same as the vertex xin X. Now, say there exists |z|= 1 such
that w=v.z ∈L(X)t
u0, i.e. there is an edge between vand v.z in L(X)t
u0. Again
since |SX(x)|> r0, the r0-locality yields u0.v.z =SX(x).z =SX(x.z), i.e. the
edge between vand v.z v in L(X)t
u0is the same as that between xand x.z in
X. Consider v1. . . vk=vwith k≤tand |vi|= 1. A similar argument starting
from u0and following these edges shows that xis at distance tof uin X, and
thus x.z is at distance t+ 1 of uin X. So the vertices x,x.v and their edge do
appear in Xt
u.
[Second •] Again take w∈Xt
u=L(X)t
u0. Consider w1. . . wk=wwith k≤t+ 1
and |wi|= 1. Since |u0|> r0+t+ 1 > r0, the r0-locality applies and yields
SX(u.w1) = SX(u).w1=u0.w1. Similarly, since |u0.w1. . . wi|> r0+t+ 1 −i>r0,
the r0-locality applies and yields SX(u.w1. . . wi.wi+1) = SX(u.w1. . . wi).wi+1 =
u0.w1. . . wi.wi+1. Eventually SX(u.w) = u0.w.
Local operations do not need to act just over the region surrounding the
origin. We may also shift them to act over the region surrounding some vertex
u.
Definition 20 (Shifted dynamics). Consider a dynamics (L, S•)and some
u∈Π∗. We define Luto be the map X7→ (L(Xu))SXu(u)if u∈X, and the
identity otherwise. We define Su,X to be the map v7→ SXu(u).SXu(u.v)if u∈X,
and the identity otherwise. We say that (Lu, Su,•)is (L, S•) shifted at u.
We may wish to apply a series of local operations at different positions ui, i.e.
a circuit. However, applying a local operation may change the graph and hence
vertex names, hence some care must be taken.
Definition 21 (Product). Consider a local dynamics (L, S•)and Xa gen-
eralized Cayley graph in its domain we define the product QL(X)as the limit
when rgoes to infinity of:
Y(r) = Y
i∈[1,...,|V(Xr)|]
Lu0
i(X)
where {u1, u2, ...}=V(Xr),u0
1=u1,u0
2=Su0
1,X (u2),u0
3=Su0
2,Lu0
1(X)(u3),...
5 Invertibility and almost-vertex-preservingness
The following definition imposes invertibity in a natural fashion.
Definition 22 (Invertible dynamics). A dynamics (F, R•)is said to be in-
vertible if Fis a bijection over XΣ,∆,π.
Recall that, in general, CGD are allowed to evolve the graph, not only by
changing internal states and edges, but also by creating or deleting vertices.
Since invertibility imposes information-conservation, one may wonder whether
invertible CGD are still allowed to create or delete vertices. Indeed, they are, as
show by Figure 5. One notices, however, that the RHS of this example features
shift-equivalent vertices.
Definition 23 (Shift-equivalent vertices). Let X∈XΣ,∆,π and let u, v ∈
V(X). We say that uand vare shift-equivalent if Xu=Xv. This equivalence
relation is denoted u≈v. A graph is called asymmetric if it has only trivial (i.e.
of size one) shift-equivalence classes.
One can show that the shift-equivalence classes of generalized Cayley graph
must always have the same size. Intuitively, if you consider two shift-equivalent
vertices, you can travel along a given path starting from either of the them, this
will give you two distinct equivalent vertices.
X
ε
u
v
•u
Xu
u
ε
u.v L
L(Xu)
SXu(u)
ε
SXu(u.v)
Lu
L(Xu)SXu(u.v)
ε
SXu(u)
•SXu(u.v)
Fig. 4. Shift-invarience of a shifted dynamics Lu. In the bottom graph Lu(X),
former vertex vhas name SXu(u).SXu(u.v).
:a:a
Fig. 5. The turtle dynamics has the two above generalized Cayley graphs to
oscillate between one another. The two vertices of the RHS are shift-equivalent,
pointing the graph upon one or the other does not change the graph.
Lemma 3 (Shift-equivalence classes isometry). Let X∈XΣ,∆,π be a graph.
If C1⊆V(X)and C2⊆V(X)are two shift-equivalence classes of X, then
|C1|=|C2|.
Proof. Consider two equivalent and distinct vertices uand vin X. Consider a
path w. The vertices u.w and v.w are distinct and equivalent. More generally,
if we have nequivalent distinct vertices v1, ..., vn, any vertex u=v1.w will be
equivalent to v2.w, ..., vn.w and distinct from all of them, hence the equivalence
classes are all of the same size.
Shift-symmetry is fragile, however, and can be destroyed by adding a few
vertices to it:
Definition 24 (Primal extension). Given a finite graph X∈XΣ,∆,π where
|π|>1such that Xhas kshift-equivalence classes of size nwith k, n 6= 1,
consider the following transformation:
–If Xhas a free port: connect p−k.n new vertices in a line to this free port,
where pis the second smallest prime number greater than k.n
–If Xhas no free port: Xhas at least one cycle. Remove an edge of this cycle,
and do the same construction as above.
We denote as 2Xthis new graph.
Lemma 4 (Properties of primal extensions). Any primal extension 2Xis
asymmetric and |2X| − |X|=o(|X|)when |X|tends toward infinity, where |X|
is the size of the graph X.
Proof. As 2Xhas a prime number of vertices, by lemma 3, its has either one
single equivalence class of maximal size or only trivial equivalence classes. As the
2.operation adds at least 2 vertices and that these vertices have different degree
(1 for the last vertex on the line, and 2 for its only neighbor), 2Xcontains at
least two non equivalent vertices, hence the first result. Moreover, according to
the prime number theorem, the quantity |2X|−|X|is of the order of log(|X|),
hence the second result.
This primal extension construction is the key ingredient to prove that invertible
CGD cannot exploit symmetries of a graph to create or delete vertices. Indeed, by
considering a graph whose size is changed through the application of an invertible
CGD and considering its primal extension, we can contradict the continuity the
of the invertible CGD.
Lemma 5 (Invertible preserves shift-equivalence classes). Let (F, R•)be
a shift-invariant dynamics over Xπ, such that Fis a bijection. Then for all graph
X, for any u, v in Xwe have u≈vis equivalent to RX(u)≈RX(v).
Proof. u≈vexpresses Xu=Xv, which by bijectivity of Fis equivalent to
F(Xu) = F(Xv) and hence F(X)RX(u)=F(X)RX(v). This in turn is expressed
by RX(u)≈RX(v).
Using this fact and the primal extension construction, one can show that the
cases of node creation and deletion in invertible CGD are all of finitary nature,
i.e. they can no longer happen for large enough graphs. Intuitively, this is be-
cause according to the previous Proposition, creation or deletion of vertices must
respect to the shift-symmetries of the graph. But these are global properties, and
thus cannot be exploited by an invertible CGD.
Lemma 6 ((Finitely)-almost-vertex-preserving). Let (F, R•)be an invert-
ible causal graph dynamics over XΣ,∆,π , such that there exists a bound p, such
that for all finite graph X, if |X|> p then RXis bijective. Then Xinfinite
implies RXbijective.
Proof. •[RXinjective ]. By contradiction. Take Xinfinite such that there is
u6=vand RX(u) = RX(v). Without loss of generality we can take u=ε, i.e.
v6=εand RX(v) = ε. By continuity of R•, there exists a radius r, which we can
take larger than |v|and p, such that RX=RXr. Then RXr(v) = RX(v) = ε, thus
RXris not injective in spite of Xrbeing finite and larger than p, a contradiction.
•[RXsurjective ]. By contradiction. Take Xinfinite such that there is v0in
F(X) and v0/∈im RX. By boundedness and shift-invariance, we can assume
that there exists bsuch that |v0|< b. By continuity of R•, there exists a radius
r, which we can take larger than p, such that the images of RXand RXrcoincide
over the disk of radius b. Then, v0/∈im RXimplies v0/∈im RXr, thus RXris
not surjective in spite of Xrbeing finite and larger than p, a contradiction.
Theorem 1 (Invertible implies almost-vertex-preserving). Let (F, R•)
be a causal graph dynamics over XΣ,∆,π, such that Fis a bijection. Then there
exists a bound p, such that for all graph X, if |X|> p then RXis bijective.
Proof. Let us prove this result for all finite graphs. Then, using lemma 6, the
general result will follow.
By contradiction, let us assume that there exists a sequence of finite graphs
(X(n))n∈Nsuch that |X(n)|diverges and such that for all n,RX(n)is not bijec-
tive. As this sequence is infinite, we have that one of the two following cases is
verified an infinite number of time:
•RX(n)is not surjective,
•RX(n)is not injective.
•[RX(n)not surjective]. There exists a vertex v0/∈im RX(n). Without loss of
generality, we can assume that |v0|< b where bis the bound from the bounded-
ness property of F. Consider the graph Y(n) = F−1(2F(X(n))). Using uniform
continuity of F−1and R•, and the fact that |X(n)|is as big as we want, we
have that there exists an index nand a radius rsuch that Y(n)r=X(n)rand
Rb
Y(n)r=Rb
X(n)r. As F(Y(n)) is asymmetric by construction, v0∈im Rb
Y(n)r
which contradicts v0/∈im RX(n).
•[RX(n)not injective]. There exists two vertices u, v ∈X(n) such that RX(n)(u) =
RX(n)(v) and u6=v. Without loss of generality, we can assume that u=εas
Fis shift-invariant. According to lemma 5, we have that ε≈v. Moreover, us-
ing the uniform continuity of R•, we have that, as RX(n)(v) = RX(n)(ε) = ε,
there exists a radius l, which does not depend on n, such that |v|< l . Let
us consider the graph 2X(n). In this graph, εand vare not shift-equivalent
and thus, R2X(n)(ε)6=R2X(n)(v) . By continuity of R•, we have that there
exists a radius r > l such that R0
2X(n)r=R0
X(n)rfor a large enough n, hence
R0
2X(n)r(v) = R0
X(n)r(v) = ε, which contradicts R2X(n)(ε)6=R2X(n)(v).
In [3] it is shown that the inverses of vertex-preserving invertible CGD are
also invertible.
Definition 25 (Vertex-preserving invertible). A shift-invariant dynamics
(F, R•)is vertex-preserving invertible if Fis a bijection and for all Xwe have
that RXis a bijection.
More precisely, we know from [3] that if (F, R•) is a vertex-preserving invert-
ible shift-invariant dynamics, then (F−1, S•) is a shift-invariant dynamics, with
SY= (RF−1(Y))−1. We also know that if (F, R•) is a vertex-preserving invert-
ible causal dynamics, then it is reversible, i.e. (F−1, S•) is also a causal dynam-
ics, with SY= (RF−1(Y))−1. The above theorem now shows that the vertex-
preservingness assumption was almost without loss of generality. Thus, it is easy
to extend the result to any invertible CGD.
Corollary 1 (Invertible implies reversible). Let (F, R•)be an invertible
causal graph dynamics over Xπ,∆,Σ . Then there exists a S•such that (F−1, S•)
is a causal graph dynamics, i.e. (F, R•)is reversible.
Proof. We must construct S•. For |F(X)|=|X|> p, we know that RXis bijec-
tive and we let SF(X)=R−1
X. For |X| ≤ p, we proceed as follows.
We write ˜ufor the shift-equivalence class of u. For all v0∈F(X), we make
the arbitrary choice SF(X)(˜
v0) = v, where vis such that RX(v)≈v0. For
this X, we have enforced ≈-compatibility. In order to enforce shift-invariance,
we must make consistent choices for SF(X)u0. This is obtained by demanding
that SF(X)u0(g
u0.v0) = u.v. Indeed, this accomplishes shift-invariance because
SF(X)u0(v0) = SF(X)u0(u0.u0.v0) = ε.v0=v0implying the equality: SF(X)(u0.v0) =
u.v =SF(X)(u0).SF(X)u0(v0). Moreover, SF(X)u0is itself shift-invariant because:
SF(X)u0.v0(w0) = SF(X)u0.v0(v0.v0.w0) = ε.w =wand SF(X)u0(v0) = vimply-
ing that SF(X)u0(v0.w0) = v.w =SF(X)u0(v0).SF(X)u0.v 0(w0) , and ≈-compatible
because v0≈w0implies SF(X)u0(v0) = SF(X)u0(w0), and thus SF(X)u0(v0)≈
SF(X)u0(w0).
Continuity of the constructed S•is due to the continuity of R•and the finiteness
of p.
Shift-invariance of (F−1, S•) follows from ≈-compatibility of S•and shift-invariance
of (F, R•), because F−1(F(X)0
u) = Xvwhere vis such that RX(v)≈u0, hence
F−1(F(X)0
u) = XSF(X)(u0).
Thus, invertible CGD are in fact Reversible CGD (RCGD).
6 Block representation
A famous result on RCA [13], is that these admit a finite-depth, reversible circuit
form, with gates acting only locally. The result carries through to Quantum
CA [5], whose proof technique inspired the following [4]. First, we show that
conjugating a local operation with an RCGD still yields a local operation.
Proposition 1. If (F, R•)is an RCGD and (L, S•)is local, then (L0, T•)is
local, with L0=F−1◦L◦Fand TX(u)=(RF−1(L(F(X)))−1(SF(X)(RX(u))).
Proof. Boundedness and uniform continuity by composition. Next, suppose: Lis
local, r0is such that for all X, Y if Xr0=Yr0then F−1(X)0=F−1(Y)0(given
by uniform continuity of F−1), r2bFis such that for all X, Y if Xr2bF=Yr2bF
then F(X)2bF=F(Y)2bF(given by uniform continuity of F), bF−1is the bound
given by the bounded inflation lemma applied on F−1,bLis the bound given by
the boundedness of Land rLthe radius of locality of L. In the two following
points, we chose a radius r0as follow :
r0=bF−1(rL+2+max(r0,2bF, r2bF))
Consider |u0|> r0.
[First •] Let us show that there exists u∈Xsuch that L0(X)0
u0=X0
u. By
definition of F−1, there exists w∈LF (X) such that RF−1
LF (X)(w) = u0. By
bounded inflation of F−1, we have |w|> rLand thus by locality of L, there
exists w0∈F(X) such that SF(X)(w0) = w. Finally by reversibility of F
there exists u∈Xsuch that RF
X(u) = w, and thus u0=TX(u). Notice
that we have that |SF(X)RF
X(u)|> r0+rL+ 2. Using lemma 2 with t=
r0, we have: LF (X)r0
SF(x)RF
X(u)=F(X)r0
RF
X(u)=F(Xu)r0. By definition of r0,
F(Xu)r0=LF (X)r0
SF(x)RF
X(u)implies X0
u=F−1(LF (X)r0
SF(x)RF
X(u))0, which
leads by shift-invariance of F−1to X0
u=F−1(LF (X)r0)0
RF−1
LF (X)SF(x)RF
X(u)Hence
X0
u=L0(X)0
u0.
[Second •] Consider uas above and v∈X0
u.
TX(u.v) = RF−1
LF (X)SF(x)(RF
X(u.v))
=RF−1
LF (X)SF(x)(RF
X(u).RF
Xu(v)) using shift-invariance of F
=RF−1
LF (X)SF(x)(RF
X(u)).RF
Xu(v) because |SF(X)RF
X(u)|> rL+ 2bF+ 2
=RF−1
LF (X)(SF(x)(RF
X(u))).RF−1
LF (X)SF(X)(RF
X(u)) (RF
Xu(v)) using shift-invariance of F−1
=TX(u).RF−1
LF (X)SF(X)(RF
X(u)) (RF
Xu(v))
We will now show that: v=RF−1
LF (X)SF(X)(RF
X(u)) (RF
Xu(v)). Since |RF
Xu(v)|<2bF
by bounded inflation of F, it is enough to show:
RF−1
LF (X)SF(X)(RF
X(u))
2bF(RF
Xu(v)) = v.
By definition of r2bF, we have that: if Xr2bF=Yr2bFthen RF−1(X)2bF=
RF−1(Y)2bF. Let us show that LF (X)r2bF
S2bF
F(X)(RF
X(u)) =F(Xu)r2bF. By apply-
ing lemma 2 with t=r2bF,LF (X)r2bF
S2bF
F(X)(RF
X(u)) =F(X)r2bF
RF
X(u)which, by shift-
invariance of F, is equal to F(Xu)r2bF. As a consequence,
RF−1
LF (X)SF(X)(RF
X(u))
2bF(RF
Xu(v)) = RF−1
F(Xu)
2bF(RF
Xu(v)) = RF−1
F(Xu)(RF
Xu(v)) = v
by definition of RF−1
•
Second, we give ourselves a little more space so as to mark which parts of
the graph have been update, or not.
Definition 26 (Marked generalized Cayley graphs). Consider the set of
generalized Cayley graphs XΣ,π with labels in Σ, and ports in π. Let Σ0=Σ×
{0,1}and π0=π×{0,1}. We define the set of marked generalized Cayley graphs
XΣ0,π0to be the subset of XΣ0,π0such that:
–for all u∈Xwith σX(u) = (x, a)and {u: (i, b), v : (j, c)} ∈ X, we have
a=c.
–for all v∈Xwith {u: (i, b), v : (j, c)} ∈ Xand {u0: (i0, b0), v : (j, c0)} ∈ X,
we have u=u0.
Definition 27 (Mark operation). Given a a label in Σ0or of a port in π0,
we define the mark operation µ(.), as flipping the bit in the second component.
Given a graph Xin XΣ0,π0, we define the mark operation, µ:XΣ0,π0→XΣ0,π0
as follows:
•σµ(X)(ε) = µ(σX(ε))
•For all i, j ∈π0,{ε:µ(i), ε :µ(j)} ∈ µ(X)if {ε:i, ε:j} ∈ X.
•For all v∈Xwith v6=εand i, j ∈π0,{ε:i, v :j0} ∈ µ(X)if {ε:i, v :j} ∈ X.
and leaving the rest of the graph Xunchanged.
Notice that the set of marked graphs XΣ0,π0is but the subset of XΣ0,π0obtained
by as closure of µ, and shifts, upon Xπ×{0},Σ×{0}.
Remark: The set XΣ0,π0is a compact subset of Xπ0,Σ0,∆ .
It turns out that any RCGD admits an extension that allows for these marks.
Definition 28 (Reversible extension). Let F:Xπ,Σ,∆ →Xπ,Σ,∆ be an
RCGD. We say that F0:XΣ0,π0→XΣ0,π0is a reversible extension of Fif
F0is an RCGD, and:
•For all X∈XΣ×{0},π×{0},F0(X) = F(X).
•For all X∈XΣ×{1},π×{1},,F0(X) = X.
•For all |X| ≤ pand X /∈XΣ×{0},π×{0},F0(X) = X, where pis that of
Theorem 1.
Proposition 2 (Reversible extension). Suppose F:Xπ,Σ,∆ →XΣ ,π is an
RCGD. Then it admits a reversible extension F0:XΣ0,π0→XΣ0,π0.
Proof. See appendix A.
In order to obtain our circuit-like form for RCGD, we will proceed by re-
versible, local updates.
Definition 29 (Conjugate mark). Given a reversible extension F0:XΣ0,π0→
XΣ0,π0, we define the conjugate mark K:XΣ0,π0→XΣ0,π 0to be the function:
K=F0−1◦µ◦F0.
Notice that by Proposition 1, the local update blocks are local operations. More-
over, since they are defined as a composition of invertible dynamics, so they are.
In order to represent the whole of an RCGD, it suffices to apply these local
update blocks at every vertex.
Theorem 2 (Reversible localizability). Suppose F:Xπ,Σ ,∆ →XΣ,π is an
RCGD. Then, there exists psuch that for all X∈Xπ,Σ,∆, if |X|> p we have:
F(X)=(Yµ)(YK)(X)
where K=F0−1µF 0for F0:XΣ0,π0→XΣ0,π0a reversible extension of F.
Proof. Take X∈Xπ,Σ,∆. We have:
(Yµ)(YK)(X)=(Yµ)(YF0−1µF 0)(X)
= (Yµ)F0−1(Yµ)F0(X)
= (Yµ)(Yµ)F(X)
=F(X).
Notice that the cases |X|< p are finite and Fis bijective, thus it just permutes
those cases. Thus, this theorem generalizes the block decomposition of reversible
cellular automata, which represents any reversible cellular automata as a circuit
of finite depth of local permutations. Here, the mark µand its conjugate Kare
the local permutations. The circuit is again of finite depth, a vertex uwill be
attained by all those Kthat act over Xr0
u, where r0is the locality radius of K.
Therefore, the depth is less than |π|r0.
7 Conclusion
Summary of results. We have studied Reversible Causal Graph Dynamics, thereby
extending Reversible Cellular Automata results to time-varying, Generalized
Cayley graphs. Generalized Cayley Graphs are arbitrary bounded-degree net-
works, with a pointed vertex serving as the origin, and modulo renaming of
vertices. Some of these graphs have shift-equivalent vertices. We have shown
that if a Causal Graph Dynamics (CGD) is invertible, then it preserves shift-
equivalence classes. This in turn entails almost-vertex-preservingness, i.e. the
conservation of each vertex but for big enough graphs. Combining this with
earlier results on vertex-preserving invertible CGD, we have shown that the in-
verse of a CGD is a CGD. Next, we have investigated whether these Reversible
CGD can be implemented by small, invertible, local operations. A local opera-
tion is one which acts upon a subdisk of the graph, leaving the rest unchanged.
Our result shows that any Reversible CGD can be simulated by another, which
proceeds via a bounded-radius, invertible, commutative updates, followed by a
0-radius, invertible, marking operation.
Future work. These are generalization of theoretical Computer Science results
[12,13]. From a mathematical point of view, however, they show that invertibility
is a very strong constraint on space-varying dynamics: beyond some finitary
cases, information conservation implies conservation of the systems that support
this information. Still, this cannot forbid that some ‘dark matter’ which was
there at all times, could now be made visible. We plan to follow this idea in a
subsequent work. We also wish to explore the quantum regime of these models,
as similar results where given for Quantum Cellular Automata over fixed graphs
[5,4], whose methods have inspired the block representation theorem of this
paper. Such results would be of interest to theoretical physics, in the sense of
discrete time versions of [14].
Acknowledgements
This work has been funded by the ANR-12-BS02-007-01 TARMAC grant, the
ANR-10-JCJC-0208 CausaQ grant, and the John Templeton Foundation, grant
ID 15619. The authors acknowledge enlightening discussions with Bruno Mar-
tin and Emmanuel Jeandel. This work has been partially done when PA was
delegated at Inria Nancy Grand Est, in the project team Carte.
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A Proof of Proposition 3
The next two definitions are standard, see [6,16] and [1]. Basically, we need a
notion of union of graphs, and for this purpose we need a notion of consistency
between the operands of the union:
Definition 30 (Consistency). Let X∈Xπbe a generalized Cayley graph. Let
Gbe a labelled graph (G, σ, δ), and G0be a labelled graph (G0, σ0, δ0), each one
having vertices that are pairwise disjoint subsets of V(X). The graphs are said
to be consistent if:
(i) ∀x∈G∀x0∈G0x∩x06=∅ ⇒ x=x0,
(ii) ∀x, y ∈G∀x0, y0∈G0∀a, a0, b, b0∈π({x:a, y :b} ∈ E(G)∧ {x0:a0, y0:b0} ∈
E(G0)∧x=x0∧a=a0)⇒(b=b0∧y=y0),
(iii) ∀x, y ∈G∀x0, y0∈G0∀a, b ∈π x =x0⇒δ({x:a, y :b}) = δ0({x0:a, y0:b})
when both are defined,
(iv) ∀x∈G∀x0∈G0x=x0⇒σ(x) = σ0(x0)when both are defined.
The consistency conditions aim at making sure that both graphs “do not dis-
agree”. Indeed: (iv) means that “if Gsays that vertex xhas label σ(x), G0should
either agree or have no label for x”; (iii) means that “if Gsays that edge ehas
label δ(e), G0should either agree or have no label for e”; (ii) means that “if G
says that starting from vertex xand following port aleads to yvia port b,G0
should either agree or have no edge on port x:a”.
Condition (i) is in the same spirit: it requires that Gand G0, if they have a vertex
in common, then they must fully agree on its name. Remember that vertices of
Gand G0are disjoint subsets of V(X). If one wishes to take the union of Gand
G0, one has to enforce that the vertex names will still be disjoint subsets of V(X).
Definition 31 (Union). Let X∈Xπbe a generalized Cayley graph. Let G
be a labelled graph (G, σ, δ), and G0be a labelled graph (G0, σ0, δ0), each one
having vertices that are pairwise disjoint subsets of V(X).S. Whenever they are
consistent, their union is defined. The resulting graph G∪G0is the labelled graph
with vertices V(G)∪V(G0), edges E(G)∪E(G0), labels that are the union of the
labels of Gand G0.
Definition 32 (Upper and lower projections). Let Gbe a graph in G(XΣ0,π0).
We define ↓
G(resp. ↑
G) the lower (resp. upper) projection of Gas the set of the
connected component obtained after removing all marked vertices (resp. all non-
marked vertices without used marked ports).
Lemma 7 (Characterization of connected components). Given Gin G(XΣ0,π0),
the elements of the sets ↓
Gand ↑
Gare of the form u.Y with u∈˜
Gand Y∈
G(XΣ0,π0).
Proposition 3 (Reversible extension). Suppose F:Xπ,Σ,∆ →XΣ ,π is an
RCGD. Then it admits a reversible extension F0:XΣ0,π0→XΣ0,π0.
Proof. Let us construct such a reversible extension F0. Let pbe that of Theorem
1. For all |X| ≤ pand X /∈XΣ×{0},π×{0}, we let F0(X) = X. The rest supposes
|X|> p.
Given L:XΣ,π →XΣ,π , we define L?as the function G◦F. Now for all
X∈XΣ0,π0, we define F0(X) as the equivalence class modulo isomorphism of
the following graph pointed on ε:
[
C∈↑
G
C
∪
[
u.Y ∈↓
G
u.F ?(˜
Y)
where G=G(X). Notice that if G∈GΣ×{0},π×{0}then ↑
Gis empty and ↓
G
contains a single connected component ε.G (the graph itself), thus F0computes
F. On the other hand, if G∈GΣ×{1},π×{1}then ↓
Gis empty and ↑
Gcontains
ε.G only, thus F0computes the identity. Hence this F0is a good candidate for
being a reversible extension of F. It remains now to check that F0is causal,
vertex-preserving and reversible.
[Causal] Shift-invariance, boundedness and continuity follow directly from the
shift-invariance, boundedness and continuity of both Fand the identity. [Reversible]
Replace Fby F−1in the previous definition.
