Space-time reversible graph rewriting

TL;DR

The paper defines space-time reversible graph rewriting by marrying asynchronous, non-deterministic schedules with a space-time deterministic framework. It introduces a rigorous name-and-time algebra, formal graph locality, and forward/backward neighborhood schemes, then provides three equivalent ways to characterize reversibility: axiomatic, constructive, and via causal rewrite systems. It proves that, under conditions like time-symmetry, two-two, or bounded neighborhoods, the inverse rule inherits commutativity and space-time determinism, ensuring a well-defined inverse. A concrete reversible time-dilation example demonstrates the framework’s capacity to model relativistic-like effects without information loss. Overall, the work lays groundwork for reversible, asynchronous discrete dynamics with potential applications in distributed systems and discretized physical theories.

Abstract

In the mathematical tradition, reversibility requires that the evolution of a dynamical system be a bijective function. In the context of graph rewriting, however, the evolution is not even a function, because it is not even deterministic -- as the rewrite rules get applied at non-deterministically chosen locations. Physics, by contrast, suggests a more flexible understanding of reversibility in space-time, whereby any two closeby snapshots (aka `space-like cuts'), must mutually determine each other. We build upon the recently developed framework of space-time deterministic graph rewriting, in order to formalise this notion of space-time reversibility, and henceforth study reversible graph rewriting. We establish sufficient, local conditions on the rewrite rules so that they be space-time reversible. We provide an example featuring time dilation, in the spirit of general relativity.

Figures (4)

• Figure 1: Induced subgraphs and borders.$(a)$ A graph $G$ and $(b)$ its induced subgraph $G_{\{x_2,x_3,x_7\}}$. Both graphs have borders, as shown by the dashed lines.
• Figure 2: Time-dilation and reversibility.$(a)$ The local operator acts by consuming the particle at $u$, thereby moving this particle to position $x$. It also flips the arrows pointing to $x$, and increments its timetag, in order to move the vertex from past $x$ to future $1.x$. $(b)$ Here we have the same behaviour except for the fact that dilation edges force to update twice as much vertices on the left of $x$ than on its right. $(d)$ These two rules together generate the trajectory of the particle passing by $u$ and $w$. However it seems hard to extend the trajectory of that passing by $w$. $(c)$ One solution to this problem is to create a fresh gray vertex in between $v$ and $1.x$ to store the information that was at $w$.
• Figure 3: Causal rewriting system for reversible time dilation. Whenever a cell (or half cell) is represented empty, the local rule does not change it. Gray chains are optional; if present they are of length $2^n$ and carry a particle at an odd position. In $(a)$ there are two optional dilation edges (blue dotted lines). In $(b)$ the chains $c_1'$ and $c_2'$ and the internal state $\sigma'$ are obtained by "cutting" the chain $c_3$ in two pieces so as to propagate its particles in a rectilinear manner. The chain $c_3'$ is obtained by merging $c_1$, $c_2$ conversely.
• Figure 4: Reversible time dilation example. In black we highlight two graphs : $H$ and one of its possible rewritings, $H'$. We start with one right moving particle on the left of the bar and a gray chain containing two particles on its right. After passing through the bar, the particle on the left ends up being stored in a gray vertex. In the other direction, the gray chain is split. Here one of its particles is stored in a black node, and an other in a new smaller gray chain, so that trajectories remain rectilinear. Note that we remove the empty gray.

Theorems & Definitions (65)

• definition 1: Name algebra
• definition 2: Renaming and renaming-invariance
• definition 3: Graphs
• definition 4: Closed subset of graphs
• definition 5: Cones and neighbourhood scheme
• definition 6: ${\mathcal{M}_{}}$-local operator
• definition 7: Mutex
• lemma 1: Neighbourhood schemes as mutex sets of cones
• definition 8: Causal rewrite system
• proposition 1: Local operators as a causal rewrite systems