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The chanciness of time

John M. Myers, Hadi Madjid

TL;DR

This paper reframes temporal order in digital networks by abandoning global snapshots in favor of local event records organized into history graphs. It introduces partitioned history graphs as a robust representation of token-game plays, enabling concurrency to be characterized without a universal clock. The approach provides methods to compute concurrency, logical distance, and cycle-firing counts directly from local records, and demonstrates how network changes can be tracked without markings. The work advances a clock-free, structurally grounded model for understanding unpredictable network dynamics with potential implications for distributed computing and physics-inspired notions of time.

Abstract

Digital network failures stemming from instabilities in measurements of temporal order motivate attention to concurrent events. A century of attempts to resolve the instabilities have never eliminated them. Do concurrent events occur at indeterminate times, or are they better seen as events to which the very concept of temporal order cannot apply? Logical dependencies of messages propagating through digital networks can be represented by marked graphs on which tokens are moved in formal token games. However, available mathematical formulations of these token games invoke "markings" -- global snapshots of the locations of tokens on the graph. The formulation in terms of global snapshots is misleading, because distributed networks are never still: they exhibit concurrent events inexpressible by global snapshots. We reformulate token games used to represent digital networks so as to express concurrency. The trick is to replace global snapshots with "local snapshots." Detached from any central clock, a local snapshot records an action at a node during a play of a token game. Assemblages of local records define acyclic directed graphs that we call history graphs. We show how history graphs represent plays of token games with concurrent motions, and, importantly, how history graphs can represent the history of a network operating while undergoing unpredictable changes.

The chanciness of time

TL;DR

This paper reframes temporal order in digital networks by abandoning global snapshots in favor of local event records organized into history graphs. It introduces partitioned history graphs as a robust representation of token-game plays, enabling concurrency to be characterized without a universal clock. The approach provides methods to compute concurrency, logical distance, and cycle-firing counts directly from local records, and demonstrates how network changes can be tracked without markings. The work advances a clock-free, structurally grounded model for understanding unpredictable network dynamics with potential implications for distributed computing and physics-inspired notions of time.

Abstract

Digital network failures stemming from instabilities in measurements of temporal order motivate attention to concurrent events. A century of attempts to resolve the instabilities have never eliminated them. Do concurrent events occur at indeterminate times, or are they better seen as events to which the very concept of temporal order cannot apply? Logical dependencies of messages propagating through digital networks can be represented by marked graphs on which tokens are moved in formal token games. However, available mathematical formulations of these token games invoke "markings" -- global snapshots of the locations of tokens on the graph. The formulation in terms of global snapshots is misleading, because distributed networks are never still: they exhibit concurrent events inexpressible by global snapshots. We reformulate token games used to represent digital networks so as to express concurrency. The trick is to replace global snapshots with "local snapshots." Detached from any central clock, a local snapshot records an action at a node during a play of a token game. Assemblages of local records define acyclic directed graphs that we call history graphs. We show how history graphs represent plays of token games with concurrent motions, and, importantly, how history graphs can represent the history of a network operating while undergoing unpredictable changes.

Paper Structure

This paper contains 16 sections, 10 theorems, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Local records of interdependent events
  3. History graphs for networks of nodes
  4. Base graph of a partitioned history graph
  5. Token games
  6. Token game without any still moment for a global snapshot
  7. Local records of three token games on the Hex game graph
  8. Replacing markings by local records and partitioned history graphs
  9. History graphs for plays of live and safe token games
  10. Example
  11. Game graphs that support more than one live and safe token game
  12. Index increments and logical distance in token games
  13. Cycle firing counts
  14. History graphs to show changes in live and safe token games
  15. Results and discussion
  16. ...and 1 more sections

Key Result

Proposition 1

From one firing to the next firing of any node in a live and safe token game, the indices on local records for that node and its predecessor nodes all increase by 1.

Figures (13)

  • Figure 1: (a) Local records of event predecessors; (b) graphical portrayal of local records; (c) history graph assembled from local records.
  • Figure 2: Coarsening that combines $e_6$, $e_7$, and $e_8$ into the single event $e_{ *}$
  • Figure 3: (a) history graph with events partitioned into columns for nodes; (b) base graph (explained below).
  • Figure 4: The "Hex" game graph.
  • Figure 5: Play of a "Hex(3,3)" game with no still moment.
  • ...and 8 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Definition 4
  • Proposition 2
  • Proposition 3
  • proof
  • Lemma 1
  • ...and 10 more