Asynchronous Graph Rewriting with Consistent Space-Time Dynamics

Core Concepts

Asynchronous applications of local graph rewriting rules can produce well-determined space-time dynamics, provided the rules satisfy certain conditions such as commutativity, monotonicity, and port-decreasingness.

Abstract

The paper introduces a formalism for graph rewriting based on directed acyclic graphs (DAGs) of dependencies between vertices. It studies non-terminating graph rewriting models where local rules are applied asynchronously, and provides sufficient conditions for such asynchronous applications to produce well-determined events in the space-time unfolding of the graph.
The key ideas are:

Graphs represent space-like cuts of a space-time diagram, with vertices as events and edges as dependencies between events.
Local rules Ax can modify vertices u = t.x, but only if u is minimal (no longer awaiting information from others).
Weak consistency requires that the normal form of each space-time event be well-determined.
Full consistency (space-time determinism) requires that the state of each event (internal state and connectivity) be fully determined by its set of incoming ports.
Sufficient conditions for full consistency include:

Commutativity: Axy = Ayx for any positions x, y.
Time-increasingness: Ax can only increase or preserve time tags.
Monotonicity: the neighbourhood Nω(G) used by Aω must contain the neighbourhoods of all individual positions in ω.
Privacy: the neighbourhoods Nω(G) and Nω′(G) for disjoint sequences ω, ω′ can only intersect on border vertices.
Port-decreasingness: Ax must decrease the set of incoming private ports of any modified vertex.

The paper provides two examples illustrating these concepts: an asynchronous simulation of a dynamical system, and a model exhibiting time dilation effects reminiscent of general relativity.

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Key Insights Distilled From

Space-time deterministic graph rewriting

by Pablo Arrigh... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.05838.pdf

Deeper Inquiries

How can the concepts of space-time determinism and dynamical geometry developed in this work be extended to quantum or reversible settings of graph rewriting?

In the context of quantum or reversible settings of graph rewriting, the concepts of space-time determinism and dynamical geometry can be extended by incorporating the principles of quantum information theory and reversible computation. Quantum graph rewriting would involve the manipulation of quantum states and quantum operations on graphs, allowing for superposition and entanglement of states. This would require a framework that accounts for the probabilistic nature of quantum systems and the reversible nature of quantum operations.
In quantum graph rewriting, the notion of space-time determinism would need to be redefined to accommodate the probabilistic evolution of quantum states. Instead of deterministic outcomes, the focus would be on the probabilities of different states and the evolution of quantum superpositions over time. Dynamical geometry in this context would involve the manipulation of quantum information in a way that respects the principles of quantum mechanics, such as unitarity and measurement.
Extending these concepts to quantum or reversible settings would require a deeper understanding of quantum computation, reversible computation, and their application to graph rewriting. It would open up new avenues for exploring the dynamics of quantum systems on graphs and the implications for computational models based on quantum principles.

How can the concepts of space-time deterministic graph rewriting be extended to quantum or reversible settings of graph rewriting?

The practical implications and potential applications of space-time deterministic graph rewriting extend beyond the examples provided in the paper. Some of these implications and applications include:

Parallel Simulation of Dynamical Systems: Space-time deterministic graph rewriting offers a way to simulate dynamical systems in a parallel and asynchronous manner without the need for costly clock synchronization mechanisms. This can lead to more efficient and scalable simulations of complex systems, such as physical, biological, or social networks.

Distributed Computation: The framework of space-time deterministic graph rewriting can be applied to distributed computation scenarios where multiple agents or processes interact based on local rules. By ensuring space-time determinism, the system can maintain consistency and coherence even in asynchronous settings, improving the reliability and efficiency of distributed algorithms.

Quantum Computing: The principles of space-time determinism in graph rewriting can be adapted to quantum computing, where quantum states and operations play a crucial role. By incorporating quantum principles into the graph rewriting framework, it may be possible to develop quantum graph rewriting models that leverage the unique properties of quantum systems for computation.

General Relativity: The study of space-time deterministic graph rewriting could provide insights into the structure of space-time in the context of general relativity. By modeling space-time dynamics on graphs and ensuring determinism in the evolution of events, researchers may be able to explore the behavior of physical systems in curved spacetime and investigate the implications for gravitational interactions.

Overall, the practical implications of space-time deterministic graph rewriting extend to various fields, including computational modeling, distributed systems, quantum computing, and theoretical physics, offering new perspectives on complex systems and their dynamics.

Could the ideas presented here lead to new insights or formalisms in the study of general relativity and the structure of space-time?

The ideas presented in space-time deterministic graph rewriting have the potential to contribute to the study of general relativity and the structure of space-time in several ways:

Discrete Models of General Relativity: By formalizing space-time determinism in graph rewriting, researchers can develop discrete models that capture the essential features of general relativity. These models can provide insights into the behavior of physical systems in curved spacetime, gravitational interactions, and the dynamics of spacetime geometry.

Quantum Gravity: The framework of space-time deterministic graph rewriting can be extended to explore quantum aspects of gravity and the unification of quantum mechanics with general relativity. By incorporating quantum principles into the model, researchers may gain new insights into the quantum nature of spacetime and the gravitational field.

Causal Structures and Geometric Properties: The study of space-time determinism in graph rewriting can shed light on causal structures and geometric properties of space-time. By analyzing the evolution of events on graphs and ensuring consistency in space-time diagrams, researchers can investigate the causal relationships between events and the geometric properties of spacetime regions.

Computational Approaches to General Relativity: The formalisms developed in space-time deterministic graph rewriting can offer computational approaches to studying general relativity. By simulating spacetime dynamics on graphs and ensuring determinism in the evolution of events, researchers can explore the implications of general relativity in a computationally tractable manner.

In conclusion, the ideas presented in space-time deterministic graph rewriting have the potential to inspire new insights and formalisms in the study of general relativity, quantum gravity, and the structure of space-time, opening up new avenues for research and exploration in theoretical physics and computational modeling.