The question. Consider a network that evolves reversibly, according to nearest neighbours interactions. Can its dynamics create/destroy nodes?
Issue 1. Consider a network that evolves according to nearest neighbours interactions only. This means that the same, local causes must produce the same, local effects. If the neighbourhood of a node looks the same as that of a node , then the same must happen at and .
Therefore the names of the nodes must be irrelevant to the dynamics. By far the most natural way to formalize this invariance under isomorphisms is as follows. Let be the function from graphs to graphs that captures the time evolution; we require that for any renaming , . But it turns out that this commutation condition forbids node creation, even in the absence any reversibility condition—as proven in [ArrighiCGD]. Intuitively, say that a node infants a node through , and consider an that maps into some fresh . Then , which has no , differs from , which has a .
Issue 2. The above issue can be fixed by asking that new names be constructed from the locally available ones (e.g. from ), and that renaming available names (e.g. into ) through leads to renaming constructed ones ( into ) through . Then invariance under isomorphisms is formalized by requiring that for any renaming , there exists , such that . But it turns out that this conjugation condition, taken together with reversibility, still forbids node creation, as proven in [ArrighiIC]. Intuitively, say that a node infants two nodes and . Then should merge these back into a single node . However, we expect to have the same conjugation property that for any renaming , there exists , such that . Consider an that leaves unchanged, but renames into some fresh . What should do upon ? Generally speaking, node creation between and augments the naming space and endangers the bijectivity that should hold between the set of renamings of and the set of renamings of .
Issue 3. Both the above no-go theorems rely on naming issues. In order to bypass them, one may drop names altogether, and work with graphs modulo isomorphisms. Doing this however is terribly inconvenient. Basic statements such as “the neighbourhood of determines what will happen at ”—needed to formalize the fact the network evolves according to nearest-neighbours interactions—are no longer possible if we cannot speak of .
Still, because these are networks and not mere graphs, we can designate a node relative to another by giving a path from one to the other (the successive ports that lead to it). It then suffices to have one privileged pointed vertex acting as the origin, to be able to designate any vertex relative to it. Then, the invariance under isomorphisms is almost trivial, as nodes have no name. All we need is to enforce invariance under shifting the origin. If stands for with its origin shifted along path , then there must exist some successor function such that . But it turns out that this even milder condition, taken together with reversibility, again forbids node creation but for a finite number of graphs—as was proven in [ArrighiRC].
Intuitively, node creation between and augments the number of ways in which the graph can be pointed at. This again endangers the bijectivity that should hold between the sets of shifts and .
Three solutions and a plan. In [MeyerLGA], Hasslacher and Meyer describe a wonderful example of a nearest-neighbours driven dynamics, which exhibits a rather surprising thermodynamical behaviour in the long-run. This toy example is non-vertex-preserving, but also reversible, in some sense which is left informal.
The most direct approach to formalizing the HM example and its properties, is to work with pointed graphs modulo just when they are useful, e.g. for stating causality, and to drop the pointer everywhen else, e.g. for stating reversibility. This relaxed setting reconciles reversibility and local creation/destruction—it can be thought of as a direct response to Issue 3. Section 4 presents this solution.
A second approach is to simulate the HM example with a strictly reversible, vertex-preserving dynamics, where each ‘visible’ node of the network is equipped with its own reservoir of ‘invisible’ nodes—in which it can tap in order to infant an visible node. The obtained relaxed setting thus circumvents the above three issues. Section 5 presents this solution.
A third approach is to work with standard, named graphs. Remarkably it turns out that naming our nodes within the algebra of variables over everywhere-infinite binary trees directly resolves Issue 2. Section 6 presents this solution.
The question of reversibility versus local creation/destruction, is thus, to some extent, formalism-dependent. Fortunately, we were able to prove the three proposed relaxed settings are equivalent, as synthesized in Section 7. Thus we have reached a robust formalism allowing for both the features. Section 2 recalls the context and motivations of this work. Section 3 recalls the definitions and results that constitute our point of departure. Section 8 summarizes the contributions and perspectives. This paper is an extended abstract designed to work on its own, but the full-blown details and proofs are made available in the appendices.
Cellular Automata (CA) constitute the most established model of computation that accounts for euclidean space: they are widely used to model spatially-dependent computational problems (self-replicating machines, synchronization…), and multi-agents phenomena (traffic jams, demographics…). But their origin lies in Physics, where they are constantly used to model waves or particles (e.g. as numerical schemes for Partial Differential Equations). In fact they do have a number of in-built physics-like symmetries: shift-invariance (the dynamics acts everywhere the same) and causality (information has a bounded speed of propagation). Since small scale physics is reversible, it was natural to endow CA with this other, physics-like symmetry. The study of Reversible CA (RCA) is further motivated by the promise of lower energy consumption in reversible computation. RCA have turned out to have a beautiful mathematical theory, which relies on a topological characterization in order to prove for instance that the inverse of a CA is a CA[Hedlund]—which clearly is non-trivial due to [KariRevUndec]. Another fundamental property of RCA is that they can be expressed as a finite-depth circuits of local reversible permutations or ‘blocks’ [KariBlock, KariCircuit, Durand-LoseBlock].
Causal Graph Dynamics (CGD) [ArrighiCGD, ArrighiIC, ArrighiCayleyNesme, MartielMartin, Maignan] are a twofold extension of CA. First, the underlying grid is extended to arbitrary bounded-degree graphs. Informally, this means that each vertex of a graph may take a state among a set , so that configurations are in , whereas edges dictate the locality of the evolution: the next state of a depends only upon the subgraph induced by the vertices lying at graph distance at most of . Second, the graph itself is allowed to evolve over time. Informally, this means that configurations are in the union of for all possible bounded-degree graph , i.e. . This leads to a model where the local rule is applied synchronously and homogeneously on every possible sub-disk of the input graph, thereby producing small patches of the output graphs, whose union constitutes the output graph. Figure 1 illustrates the concept.
CGD were motivated by the countless situations featuring nearest-neighbours interactions with time-varying neighbourhood (e.g. agents exchange contacts, move around…). Many existing models (of complex systems, computer processes, biochemical agents, economical agents, social networks…) fall into this category, thereby generalizing CA for their specific sake (e.g. self-reproduction as [TomitaSelfReproduction], discrete general relativity à la Regge calculus [Sorkin], etc.). CGD are a theoretical framework, for these models. Some graph rewriting models, such as Amalgamated Graph Transformations [BFHAmalgamation] and Parallel Graph Transformations [EhrigLowe, TaentzerHL], also work out rigorous ways of applying a local rewriting rule synchronously throughout a graph, albeit with a different, category-theory-based perspective, of which the latest and closest instance is [Maignan].
In [ArrighiRC, ArrighiBRCGD] one of the authors studied CGD in the reversible regime. Specific examples of these were described in [MeyerLGA, MeyerLove]. From a theoretical Computer Science perspective, the point was to generalize RCA theory to arbitrary, bounded-degree, time-varying graphs. Indeed the two main results were the generalizations of the two above-mentioned fundamental properties of RCA.
From a mathematical perspective, questions related to the bijectivity of CA over certain classes of graphs (more specifically, whether pre-injectivity implies surjectivity for Cayley graphs generated by certain groups [Bartholdi, CeccheriniEden, Gromov]) have received quite some attention. The present paper on the other hand provides a context in which to study “bijectivity of CA over time-varying graphs”. We answer the question: Is it the case that bijectivity necessarily rigidifies space (i.e. forces the conservation of each vertex)?
From a theoretical physics perspective, the question whether the reversibility of small scale physics (quantum mechanics, micro-mechanical), can be reconciled with the time-varying topology of large scale physics (relativity), is a major challenge. This paper provides a rigorous discrete, toy model where reversibility and time-varying topology coexist and interact—in a way which does allow for space expansion. In fact these results open the way for Quantum Causal Graph Dynamics [ArrighiQCGD] allowing for vertex creation/destruction—which could provide a rigorous basic formalism to use in Quantum Gravity [QuantumGraphity1, QuantumGraphity2].
3 In a nutshell : Reversible Causal Graph Dynamics
The following provides an intuitive introduction to Reversible CGD. A thorough formalization was given in [ArrighiCayleyNesme], and is reproduced in Appendix A [ArrighiIMRCGD].
Networks. Whether for CA over graphs [PapazianRemila], multi-agent modeling [Danos200469] or agent-based distributed algorithms [Chalopin], it is common to work with graphs whose nodes have numbered neighbours. Thus our ’graphs’ or networks are the usual, connected, undirected, possibly infinite, bounded-degree graphs, but with a few additional twists:
The set of available ports to each vertex is finite.
The vertices are connected through their ports: an edge is an unordered pair , where are vertices and are ports. Each port is used at most once: if both and are edges, then and . As a consequence the degree of the graph is bounded by .
The vertices and edges can be given labels taken in finite sets and respectively, so that they may carry an internal state.
These labeling functions are partial, so that we may express our partial knowledge about part of a graph.
The set of all graphs (see Figure 2) is denoted .
Compactness. In order to both drop the irrelevant names of nodes and obtain a compact metric space of graphs, we need ’pointed graphs modulo’ instead:
The graphs has a privileged pointed vertex playing the role of an origin.
The pointed graphs are considered modulo isomorphism, so that only the relative position of the vertices can matter.
The set of all pointed graphs modulo (see Figure 2) is denoted .
If, instead, we drop the pointers but still take equivalence classes modulo isomorphism, we obtain just graphs modulo, aka ’anonymous graphs’. The set of all anonymous graphs (see Figure 2) is denoted .
Operations over graphs. Given a pointed graph modulo , denotes the sub-disk of radius around the pointer. The pointer of can be moved along a path , leading to . We use the notation for i.e., first the pointer is moved along , then the sub-disk of radius is taken.
Causal Graph Dynamics. We will now recall their topological definition. It is important to provide a correspondence between the vertices of the input pointed graph modulo , and those of its image , which is the role of : [Dynamics] A dynamics is given by
a function ;
a map , with and .
Next, continuity is the topological way of expressing causality: [Continuity] A dynamics is said to be continuous if and only if for any and , there exists , such that
where denotes the partial map obtained as the restriction of to the co-domain , using the natural inclusion of into .
Notice that the second condition states the continuity of itself. A key point is that by compactness, continuity entails uniform continuity, meaning that does not depend upon —so that the above really expresses that information has a bounded speed of propagation of information.
We now express that the same causes lead to the same effects: [Shift-invariance] A dynamics is said to be shift-invariant if for every , , and ,
Finally we demand that graphs do not expand in an unbounded manner: [Boundedness] A dynamics is said to be bounded if there exists a bound such that for any and any , there exists and such that . Putting these conditions together yields the topological definition of CGD: [Causal Graph Dynamics] A CGD is a shift-invariant, continuous, bounded dynamics. Reversibility. Invertibility is imposed in the most general and natural fashion. [Invertible dynamics] A dynamics is said to be invertible if is a bijection. Unfortunately, this condition turns out to be very limiting. It is the following limitation that the present paper seeks to circumvent: [Invertible implies almost-vertex-preserving [ArrighiRC]] Let be an invertible CGD. Then there exists a bound , such that for any graph , if then is bijective. On the face of it reversibility is stronger a stronger condition than invertibility: [Reversible Causal Graph Dynamics] A CGD is reversible if there exists such that is a CGD. Fortunately, invertibility gets you reversibility: [Invertible implies reversible [ArrighiRC]] If is an invertible CGD, then is reversible. As a simple example we provide an original, general scheme for propagating particles on an arbitrary network in a reversible manner: [General reversible advection] Consider a finite set of ports, and let be the set of internal states, where: means ‘no particle is on that node’; means ‘one particle is set to propagate along port ’; means ‘one particle is set to propagate along port and another along port ’…. Let be a bijection over the set of ports, standing for the successor direction. Fig. 4 specifies how individual particles propagate. Basically, when reaching its destination, the particle set to propagate along the successor of the port it came from. Missing edges behave like self-loops. Applying this to all particles synchronously specifies the ACGD.
4 The anonymous solution
Having a pointer is essential in order to express causality, but cumbersome when it comes to reversibility. Here is the direct way to get the best of both worlds. [Anonymous Causal Graph Dynamics] Consider a function over . We say that is an ACGD if and only if there exists a CGD such that over naturally induces over . Invertibility, then, just means that is bijective. Fortunately, this time the condition is not so limiting, and we are able to implement non-vertex-preserving dynamics, as can be seen from this slight generalization of the HM example:
[Anonymous HM]. Consider the state space of Example 3 and alternate: 1. a step of advection as in Fig. 4, 2. a step of collision, where the collision is the specific graph replacement provided in Fig. 4. The composition of these two specifies the ACGD. So, ACGD feature local vertex creation/destruction. Yet they are clearly less constructive than CGD, as is no longer explicit. In spite of this lack of constructiveness, we still have [Anonymous invertible implies reversible] If an ACGD in invertible, then the inverse function is an ACGD. Proof outline. By Th. 7 the invertible ACGD can be directly simulated by an invertible IMCGD, see next. By Th. 5 the inverse IMCGD is also an IMCGD. Dropping the invisible matter of this inverse provides the CGD that underlines .
5 The Invisible Matter solution
Reversible CGD are vertex-preserving. Still, we could think of using them to simulate a non-vertex-preserving dynamics by distinguishing ‘visible’ and ‘invisible matter’, and making sure that every visible node is equipped with its own reservoir of ‘invisible’ nodes—in which it can tap. For this scheme to iterate, and for the infanted nodes to be able to create nodes themselves, it is convenient to shape the reservoirs as everywhere infinite binary trees. [Invisible Matter Graphs] Consider , and , assuming that and . Let be the infinite binary tree whose origin has a copy of at vertex , and another at vertex . Every can be identified to an element of obtained by attaching an instance of at each vertex through path . The hereby obtained graphs will be denoted and referred to as invisible matter graphs. We will now consider those CGD over that leave stable. In fact we want them trivial as soon as we dive deep enough into the invisible matter: [Invisible-matter quiescence] A dynamic over is said invisible matter quiescent if there exists a bound such that, for all , and for all in , we have . [Invisible Matter Causal Graph Dynamics] A CGD over is said to be an IMCGD if and only if it is vertex-preserving and invisible matter quiescent. Fortunately, we are indeed able to encode non-vertex-preserving dynamics in the visible sector of an invertible IMCGD:
[Invisible Matter HM]
Consider X as in Example 3 and extend it to Y. Alternate: 1. a step of advection as in Example 3 and 4, 2. a step of collision, where the collision is the specific graph replacement provided in Fig. 5. The composition of these two specifies the invertible IMCGD.
Notice how the graph replacement of Fig. 4—with the grey color taken into account—would fail to be invertible, due to the collapsing of two pointer positions into one.
Fortunately also, invertibility still implies reversibility: [Invertible implies reversible] If is an invertible IMCGD, then is an IMCGD. Proof outline. Intuitively this property is inherited from that of CGD over . Th. 3, however, relies on the compactness of , and as matter of fact is not compact. Still it admits a compact closure , over which IMCGD have a natural, continuous extension, see Appendix B of [ArrighiIMRCGD].
6 The Name Algebra solution
So far we worked with (pointed) graphs modulo. But named graphs are often more convenient e.g. for implementation, and sometimes mandatory e.g. for studying the quantum case[ArrighiQCGD]. In this context, being able to locally create a node implies being able to locally make up a new name for it—just from the locally available ones. For instance if a dynamics splits a node into two, a natural choice is to call these and . Now, apply a renaming that maps into and into , and apply . This time the nodes and get merged into one; in order not to remain invertible a natural choice is to call the resultant node . Yet, if is chosen trivial, then the resultant node is , when demands that this to be instead. This suggests considering a name algebra where . [Name Algebra] Let be a countable set (eg ). Consider the terms produced by the grammar together with the equivalence induced by the term rewrite systems
i.e. and are equivalent if and only if their normal forms and are equal.
Well-foundedness outline. The TRS was checked terminating and locally confluent using CiME, hence its confluence and the unicity of normal forms via Church-Rosser.
This is the algebra of symbolic everywhere infinite binary trees. Indeed, each element of can be thought of as a variable representing an infinite binary tree. The (resp. ) projection operation recovers the left (resp. right) subtree. The ‘join’ operation puts a node on top of its left and right trees to form another—it is therefore not commutative nor associative. This infinitely splittable/mergeable tree structure is reminiscent of Section 5, later we shall prove that named graphs arise by abstracting away the invisible matter.
No graph can have two distinct nodes called the same. Nor should it be allowed to have a node called and two others called and , because the latter may merge and collide with the former. [Intersectant] Consider in . Two vertices in and in are said to be intersectant if and only if there exists in such that . We then write . We also write if and only if there exists in such that .
[Well-named graphs.] We say that a graph is well-named if and only if for all in and in then implies and . We denote by the subset of well-named graphs.
We now have all the ingredients to define Named Causal Graph Dynamics. [Continuity] A function over is said to be continuous if and only if for any and any , there exists , such that for all , for all ,
[Renaming] Consider an injective function from to such that for any , and are not intersectant. The natural extension of to the whole of , according to
is referred to as a renaming. [Shift-invariance] A function over is said to be shift-invariant if and only if for any and any renaming , . Our dynamics may split and merge names, but not drop them: [Name-preservation] Consider a function over . The function is said to be name-preserving if and only if for all in and in we have that is equivalent to . [Named Causal Graph Dynamics] A function over is said to be a Named Causal Graph Dynamics (NCGD) if and only if is shift-invariant, continuous, and name-preserving. Fortunately, invertible NCGD do allow for local creation/destruction of vertices:
[Named HM example] Consider W with ports and labels as in Example 3. Alternate: 1. a step of advection as in Example 3 and 4, 2. a step of collision, where the collision is the specific graph replacement provided in Fig. 6 That the latter is an involution follows from the three equalities holding in . Fortunately also, invertibility still implies reversibility. [Named invertible implies reversible] If an NCGD in invertible, then the inverse function is an NCGD. Proof outline. By Th. 7 the invertible NCGD can be directly simulated by an invertible IMCGD , whose pointer mimics the behaviour of atomic names. Its inverse thus captures the full behaviour of over graphs including vertex names. By Th. 5 is continuous, and thus so is .
Previous works gave three negative results about the ability to locally create/destroy nodes in a reversible setting. But we just described three relaxed settings in which this is possible. The question is thus formalism-dependent. How sensitive is it to changes in formalism, exactly? We show that the three solutions directly simulate each other. They are but three presentations, in different levels of details, of a single robust solution.
In what follows is the natural, surjective map from to , which (informally) : 1. Drops the pointer and 2. Cuts out the invisible matter. Whatever an ACGD does to a , an IMCGD can do to —moreover the notions of invertibility match:
[IMCGD simulate ACGD]
Consider an ACGD. Then there exists an IMCGD such that for all but a finite number of graphs in , . Moreover if is invertible, then this is invertible.
Proof outline. Any ACGD has an underlying CGD . We show it can be extended to invisible matter, an then mended to make bijective, thereby obtaining an IMCGD. The precise way this is mended relies on the fact vertex creation/destruction cannot happen without the presence of a local asymmetry—except in a finite number of cases. Next, bijectivity upon anonymous graphs induces bijectivity upon pointed graphs modulo.
Similarly, whatever an IMCGD does to a , a ACGD can do to : [ACGD simulate IMCGD] Consider an IMCGD. Then there exists an ACGD such that . Moreover if is invertible, then this is invertible. Proof outline. The ACGD is obtained by dropping the pointer and the invisible matter.
In what follows, if is a graph in , then is the graph obtained from by attaching invisible–matter trees to each vertex, and naming the attached vertices in according to Fig. 7.
[IMCGD simulate NCGD] Consider an NCGD. There exists such that for all , is a bijection from to . This induces an IMCGD via
is the path between and in , where is obtained by following path from in .
Moreover if is invertible, then this is invertible.
Proof outline. The names can be understood as keeping track of the splits and mergers that have happened through the application of to , as in Fig. 6. uses this to build a bijection from to , following conventions as in Fig. 5.
[NCGD simulate IMCGD] Consider an IMCGD. Then there exists an NCGD such that for all graphs in , . Moreover is invertible if and only if is invertible.
Proof outline. Each vertex of can be named so that the resulting graph is well-named. Then is used to construct the behaviour of over names of vertices. As does not merge nor split vertices, preserves the name of each vertex.
Thus, NCGD are more detailed than IMCGD, which are more detailed than ACGD. But, if one is thought of as retaining just the interesting part of the other, it does just what the other would do to this interesting part—and no more.
Summary of contributions. We have raised the question whether parallel reversible computation allows the local creation/destruction of nodes. Different negative answers had been given in [ArrighiCGD, ArrighiIC, ArrighiRC] which inspired us with three relaxed settings: Causal Graph Dynamics over fully-anonymized graphs (ACGD); over pointer graphs modulo with invisible matter reservoirs (IMCGD); and finally CGD over graphs whose vertex names are in the algebra of ‘everywhere infinite binary trees’ (NCGD). For each of these formalism, we proved non-vertex-preservingness by implementing the Hasslacher-Meyer example [MeyerLGA]—see Examples 4,5, 6. We also proved that we still had the classic Cellular Automata (CA) result that invertibility (i.e. mere bijectivity of the dynamics) implies reversibility (i.e. the inverse is itself a CGD)—via compactness—see Theorems 4,5, 6. The answer to the question of reversibility versus local creation/destruction is thus formalism-dependent to some extent. We proceeded to examine the extent in which this is the case, and were able to show that (Reversible) ACGD, IMCGD and NCGD directly simulate each other—see Theorems 7, 7, 7, 7. They are but three presentations, in different levels of details, of a single robust setting in which reversibility and local creation/destruction are reconciled.
Perspectives. Just like Reversible CA were precursors to Quantum CA [SchumacherWerner, ArrighiUCAUSAL], Reversible CGD have paved the way for Quantum CGD [ArrighiQCGD]. Toy models where time-varying topologies are reconciled with quantum theory, are of central interest to the foundations of theoretical physics [QuantumGraphity1, QuantumGraphity2]—as it struggles to have general relativity and quantum mechanics coexist and interact. The ‘models of computation approach’ brings the clarity and rigor of theoretical CS to the table, whereas the ‘natural and quantum computing approach’ provides promising new abstractions based upon ‘information’ rather than ‘matter’. Quantum CGD [ArrighiQCGD], however, lacked the ability to locally create/destroy nodes—which is necessary in order to model physically relevant scenarios. Our next step will be to apply the lessons learned, to fix this.
This work has been funded by the ANR-12-BS02-007-01 TARMAC grant and the STICAmSud project 16STIC05 FoQCoSS. The authors acknowledge enlightening discussions Gilles Dowek and Simon Martiel.
Appendix A Formalism
This appendix provides formal definitions of the kinds of graphs we are using, together with the operations we perform upon them. None of this is specific to the reversible case; it can all be found in [ArrighiCayleyNesme] and is reproduced here only for convenience.
Let be a finite set, , and some universe of names.
[Graph non-modulo] A graph non-modulo is given by
An at most countable subset of , whose elements are called vertices.
A finite set , whose elements are called ports.
A set of non-intersecting two element subsets of , whose elements are called edges. In other words an edge is of the form , and .
A partial function from to a finite set ;
A partial function from to a finite set ;
The graph is assumed to be connected: for any two , there exists , such that for all , one has with and .
The set of graphs with states in and ports is written .
We single out a vertex as the origin: [Pointed graph non-modulo] A pointed graph is a pair with . The set of pointed graphs with states in and ports is written .
Here is when graph differ only up to names of vertices: [Isomorphism] An isomorphism is a function from to which is specified by a bijection from to . The image of a graph under the isomorphism is a graph whose set of vertices is , and whose set of edges is . Similarly, the image of a pointed graph is the pointed graph . When and are isomorphic we write , defining an equivalence relation on the set of pointed graphs. The definition extends to pointed labeled graphs. (Pointed graph isomorphism rename the pointer in the same way as it renames the vertex upon which it points; which effectively means that the pointer does not move.)
[Pointed graphs modulo] Let be a pointed (labeled) graph . The pointed graph modulo is the equivalence class of with respect to the equivalence relation . The set of pointed graphs modulo with ports is written . The set of labeled pointed Graphs modulo with states and ports is written .
a.2 Paths and vertices
Vertices of pointed graphs modulo isomorphism can be designated by a sequence of ports in that leads, from the origin, to this vertex.
[Path] Given a pointed graph modulo , we say that is a path of if and only if there is a finite sequence of ports such that, starting from the pointer, it is possible to travel in the graph according to this sequence. More formally, is a path if and only if there exists and there also exists such that for all , one has , with and . Notice that the existence of a path does not depend on the choice of . The set of paths of is denoted by . Paths can be seen as words on the alphabet and thus come with a natural operation ‘’ of concatenation, a unit denoting the empty path, and a notion of inverse path which stands for the path read backwards. Two paths are equivalent if they lead to same vertex: [Equivalence of paths] Given a pointed graph modulo , we define the equivalence of paths relation on such that for all paths , if and only if, starting from the pointer, and lead to the same vertex of . More formally, if and only if there exists and such that for all , , one has , , with , , , and . We write for the equivalence class of with respect to .
It is useful to undo the modulo, i.e. to obtain a canonical instance of the equivalence class . [Associated graph] Let be a pointed graph modulo. Let be the graph such that:
The set of vertices is the set of equivalence classes of ;
The edge is in if and only if and , for all and .
We define the associated graph to be .
Notations. The following are three presentations of the same mathematical object:
a graph modulo ,
its associated graph
the algebraic structure
Each vertex of this mathematical object can thus be designated by
an equivalence class of , i.e. the set of all paths leading to this vertex starting from ,
or more directly by an element of an equivalence class of , i.e. a particular path leading to this vertex starting from .
These two remarks lead to the following mathematical conventions, which we adopt for convenience:
and are no longer distinguished unless otherwise specified. The latter notation is given the meaning of the former. We speak of a “vertex” in (or simply ).
It follows that ‘’ and ‘’ are no longer distinguished unless otherwise specified. The latter notation is given the meaning of the former. I.e. we speak of “equality of vertices” (when strictly speaking we just have ).
a.3 Operations over pointed Graphs modulo
Sub-disks. For a pointed graph non-modulo:
the neighbours of radius are just those vertices which can be reached in steps starting from the pointer ;
the disk of radius , written , is the subgraph induced by the neighbours of radius , with labellings restricted to the neighbours of radius and the edges between them, and pointed at .
For a graph modulo, on the other hand, the analogous operation is: [Disk] Let be a pointed graph modulo and its associated graph. Let be . The graph modulo is referred to as the disk of radius of . The set of disks of radius with states and ports is written .
[Size] Let be a pointed graph modulo. We say that a vertex has size less or equal to , and write , if and only if .
Shifts just move the pointer vertex:
Let be a pointed graph modulo and its associated graph.
Consider or for some , and consider the pointed graph , which is the same as but with a different pointer. Let be . The pointed graph modulo is referred to as shifted by .
Appendix B IMCGD : compactness & reversibility
Notations. In the rest of the paper, ranges over arbitrary elements of and their natural identification in , as given by Definition 5. Let be a word in , we use as a shorthand notation for , i.e. the graph but pointed at within the nearest attached tree. Let be a word, then means either or the empty word .
The main result of this subsection is that, although is not a compact subset of by itself, IMCGD can be extended continuously over the compact closure of in .
Indeed is not a compact subset of , for instance the sequence , pointing ever further into the invisible matter, has no convergent subsequence in but has one in .
[Closure] The compact closure of in , denoted , is the subset of elements of such that, for all , there exists a in satisfying .
[Visible starting paths] Consider in with visible, and in . Then can be decomposed as , with in , and in . When is non-empty, is invisible. Moreover, for any in , we have that are in , and if and only if .
First consider with visible. Clearly is in and is the minimal path to . Clearly also, can be minimally decomposed into with in , and when is non-empty, is invisible.
The same holds in the closure. Indeed consider in with visible. Let and pick in such that . By definition of we have that is a shortest path from to in if and only if it is one in . Therefore the form of the decomposition of , and its invisibility when is not empty, carry through to . So does the existence of . Finally, we have that implies , which implies , due to the tree structure of the invisible matter in . ∎
[Invisible starting paths] Consider in with invisible, and in . Then can be decomposed as , in which case is visible, or as , in which case is invisible—with in , and in . Moreover, any in , are also in , and we have that if and only if .
Same proof scheme as in Lemma B.1. ∎
Proposition (Closure of visible).
Consider in with visible. Then is in .
Consider visible in . By the first part of Lemma B.1 the shortest path from to is of the form with in and in . But this needs be the empty word, otherwise would be invisible. Therefore visible nodes form a -connected component, call it . By the second part of Lemma B.1 each vertex of has, in , an invisible matter tree attached to it—and no other invisible matter due again to the first part of Lemma B.1. Finally, there is no other invisible matter in altogether, because is connected. ∎
[Finite invisible root] Consider in . If has no visible matter, then, for all , there exists a unique word in such that . As a consequence, is the unique word in such that is in . Moreover, if , then is a suffix of .
Consider in . Pick in such that . Since has no visible vertex, with , and we can take to be the suffix of length of , and the complementary prefix, such that . is included in the invisible matter tree rooted in , hence .
For uniqueness, notice that for any two words of length , implies .
Since is the only word of length in to represent a valid path of , its prefix of length is the only word of length in to represent a valid path of , which we know is . ∎
Proposition (Closure of invisible).
is in and has no visible matter if and only if there exists a sequence of path in such that suffix of , , and
i.e. is the non-decreasing union of the .
First notice that is a sub-graph of . Indeed, by definition of , the vertex in is the root of a copy of , thus is a subgraph of . Shifting this statement by , is a sub-graph of . Thus it makes sense to speak about their non-decreasing union.
Next, for any , . So if is a graph of with no point in the visible matter, then
Reciprocally, any such non-decreasing union is equal to , for any graph of seen as an element of , thus it belongs to . ∎
Let be a graph in with no point in the visible matter and let be the sequence of such that and suffix of . being totally determined by the sequence , we can write . The sequence , growing for the suffix relation, can be identified with an infinite word of , id est an infinite word with an end but no beginning.
Based on the previous results we have
Now that we know what the closure of looks like, we can try to extend IMCGD to it.
Consider a continuous and shift-invariant dynamics over . We have that is invisible matter quiescent if and only if can be continuously extended to by letting and for any in .
Notice how, for all and , we have that .
. Let be a continuous, shift-invariant and invisible matter quiescent dynamics. Take a left-infinite word in . Continuity of over states that for all there is an such that . By invisible matter quiescence there is a such that for all , . Combining these, . Hence, if we extend to by , we get , and so remains continuous. Similarly, continuity of over states that for all there is an such that . Again combining it with invisible matter quiescence, . Hence, if we extend to by , we get , and so remains continuous.
We can thus continuously extend by setting and .
. Reciprocally, no longer assume invisible matter quiescence, and suppose instead that , when extended by and , is continuous over . Since is compact, is uniformly continuous (by the Heine–Cantor theorem). Take such that for all , in , and =1, we have . Such a exists by Lemma 3 of [ArrighiCayleyNesme]. Take such that, for all , we have
We prove, by recurrence, that is the bound for invisible matter quiescence. Indeed, our recurrence hypothesis is that for all in and in , we have . The hypothesis holds for , because a consequence of shift-invariance is that for any . Suppose it holds for some . Take in . We have . Since and, we have for any left-infinite in . By the choice of and , . Putting things together, we have . ∎
An IMCGD can be extended into a vertex-preserving invisible matter quiescent CGD over .
Consider an IMCGD. Extend it to by setting and