Space-time deterministic graph rewriting

TL;DR

The paper addresses non-terminating, asynchronous graph rewriting by introducing space-time determinism as a robust alternative to confluence. It defines a formal DAG-based framework with colored, time-tagged port graphs, neighbourhood schemes, and local rules, and establishes conditions (time-increasing, commutative, port-decreasing with an extensive, monotonous, private neighbourhood) under which all asynchronous evolutions yield a consistent space-time diagram. Through a particle-system example, a synchronous CA simulation, and a time-dilation toy model, it demonstrates that space-time determinism can be achieved and even emulate relativistic-like effects in a discrete setting. The work has potential implications for efficient, clock-free parallel simulations of dynamical and physical systems and opens avenues toward discrete models of general relativity and quantum-inspired extensions.

Abstract

We study non-terminating graph rewriting models, whose local rules are applied non-deterministically -- and yet enjoy a strong form of determinism, namely space-time determinism. Of course in the case of terminating computation it is well-known that the mess introduced by asynchronous rule applications may not matter to the end result, as confluence conspires to produce a unique normal form. In the context of non-terminating computation however, confluence is a very weak property, and (almost) synchronous rule applications is always preferred e.g. when it comes to simulating dynamical systems. Here we provide sufficient conditions so that asynchronous local rule applications conspire to produce well-determined events in the space-time unfolding of the graph, regardless of their application orders. Our first example is an asynchronous simulation of a dynamical system. Our second example features time dilation, in the spirit of general relativity.

Figures (12)

• Figure 1: Confluence yet inconsistencies. The local rule transports right-moving particles to the right and left-moving particles to the left, without interactions. $(a)$ We start with a left-moving particle in $7$ and a right-moving particle in $1$. $(b)\& (c)$ The point of collision between the two particles is not well defined, it depends on the evaluation strategy. Here $A_{71}G$ is short for $A_7(A_1 G)$. $(d)$. Still, the system is confluent, as the divergent configurations can be both evolve into the last.
• Figure 2: Induced subgraph and borders (a)(b): We consider a graph $G$ and its induced subgraph $G_{\{x_2,x_3,x_7\}}$. Graphs have borders, as shown pointed by the dashed lines for $(a)$$G$ and $(b)$$G_{\{x_2,x_3,x_7\}}$. Interior of a set (c): We consider a set of positions $X = \{x_i \mid x_i\notin \{x_5,x_0\}\}$. Positions in this set are either interior ($X^-$ in dark blue--e.g. $t_3.x_3$) or in the boundary ($X\setminus X^-$ in cyan).
• Figure 3: Action of a local rule $A_{(-)}$ centered on a vertex $u_1 = t_1.x_1$. It affects the vertices of ${ \mathbb Z}.\mathcal{N}_{x}(G)$ that are circled dark blue & cyan. Dark blue vertices (e.g. $u_1,u_4$) can be almost arbitrarily modified whereas cyan vertices must have their names and external edges preserved. Internal states $\Sigma = \{0,1\}$ are represented by white and black. One can set $u_1'$, $u_2'$, and $u_7'$ to $3.u_1$, $1.u_2$, and $4.u_7$ respectively for instance.
• Figure 4: The local rule for the particle system.$A_x$ acts by consuming the internal state $i=\sigma^r_G(v)$ of vertex $v$ (and symmetrically with $j=\sigma^l_G(w)$), thereby moving those particles at $x$ if they are present. It also updates ports from $a,b$ to $a',b'$, flips the arrows pointing to $x$, and increments its timetag, in order to move the vertex from past $u=t.x$ to future $u'= 1.u = (t+1).x$. Dashed edges and vertices do not influence the local rule.
• Figure 5: Particle system example (a): In black we highlight just the graphs $G$ and $G"$ belonging to the space-time diagram $\mathcal{M}_A(G)$. The local rule here moves the left-side particle towards the right and the right-side particle towards the left. Note how the problem raised by Fig. \ref{['fig : faster than light']} is solved. The only point in space-time which can contain both particles is $u$. States depend on the cut (b)(c): Both graphs $H$ and $H'$ belong to the same space-time diagram $\mathcal{M}_A(H_0)$ with $H_0$ containing a right-moving particle in $u$. They both contain vertex $v$. In $H'$, the particle that was present in $H$ has moved to point $w$. The state associated to $v$ thus depends on the way it is cut, which is captured by its set of incoming ports e.g. $\{a',b'\}$ versus just $\{a'\}$.

Theorems & Definitions (32)

• Definition 1: Positions, ports, states, names
• Definition 2: Graphs, Past
• Definition 3: (Induced) subgraph and boundaries.
• Definition 4: Neighbourhood scheme
• Definition 5: Extensivity
• Definition 6: Local rule
• Remark 7
• Definition 8: Space-time diagram
• Definition 9: Consistency
• Definition 10: Time-increasing commutative local rules