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Ergodic properties of concurrent systems

Samy Abbes, Vincent Jugé

TL;DR

This work develops a probabilistic theory for concurrent systems built from trace monoids acting on finite state spaces. It proves a key spectral property for irreducible systems, ensures the existence and uniqueness of Markov measures on infinite trajectories via probabilistic valuations, and shows the Möbius matrix has a one-dimensional kernel, enabling a canonical cocycle-based normalization. The authors formulate an intrinsic ergodic theory, establishing a Strong Law of Large Numbers for broad test functions and introducing speedup as a universal measure of average concurrency. The approach blends combinatorial encoding (cliques, cylinders, and the DSC) with analytic tools (generating series, Möbius transforms) to yield robust ergodic and probabilistic results with wide applicability to concurrency models and their examples.

Abstract

A concurrent system is defined as a monoid action of a trace monoid on a finite set of states. Concurrent systems represent state models where the state is distributed and where state changes are local. Starting from a spectral property on the combinatorics of concurrent systems, we prove the existence and uniqueness of a Markov measure on the space of infinite trajectories relatively to any weight distributions. In turn, we obtain a combinatorial result by proving that the kernel of the associated Möbius matrix has dimension 1; the Möbius matrix extends in this context the Möbius polynomial of a trace monoid. We study ergodic properties of irreducible concurrent systems and we prove a Strong law of large numbers. It allows us to introduce the speedup as a measurement of the average amount of concurrency within infinite trajectories. Examples are studied.

Ergodic properties of concurrent systems

TL;DR

This work develops a probabilistic theory for concurrent systems built from trace monoids acting on finite state spaces. It proves a key spectral property for irreducible systems, ensures the existence and uniqueness of Markov measures on infinite trajectories via probabilistic valuations, and shows the Möbius matrix has a one-dimensional kernel, enabling a canonical cocycle-based normalization. The authors formulate an intrinsic ergodic theory, establishing a Strong Law of Large Numbers for broad test functions and introducing speedup as a universal measure of average concurrency. The approach blends combinatorial encoding (cliques, cylinders, and the DSC) with analytic tools (generating series, Möbius transforms) to yield robust ergodic and probabilistic results with wide applicability to concurrency models and their examples.

Abstract

A concurrent system is defined as a monoid action of a trace monoid on a finite set of states. Concurrent systems represent state models where the state is distributed and where state changes are local. Starting from a spectral property on the combinatorics of concurrent systems, we prove the existence and uniqueness of a Markov measure on the space of infinite trajectories relatively to any weight distributions. In turn, we obtain a combinatorial result by proving that the kernel of the associated Möbius matrix has dimension 1; the Möbius matrix extends in this context the Möbius polynomial of a trace monoid. We study ergodic properties of irreducible concurrent systems and we prove a Strong law of large numbers. It allows us to introduce the speedup as a measurement of the average amount of concurrency within infinite trajectories. Examples are studied.
Paper Structure (39 sections, 27 theorems, 130 equations, 7 figures)

This paper contains 39 sections, 27 theorems, 130 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Motivations and framework.
  3. Combinatorial encoding.
  4. Contributions and outline of the paper.
  5. Trace monoids and concurrent systems
  6. Trace monoids (cartier69viennot86diekert90dehornoy14)
  7. Algebraic description of traces.
  8. Combinatorial description of traces.
  9. Generalized normal form of traces and boundary at infinity of the trace monoid.
  10. Combinatorics of trace monoids; valuations.
  11. Möbius transform.
  12. Normal and visual cylinders.
  13. Memoryless measures and probabilistic valuations.
  14. Markov chain of cliques.
  15. Examples.
  16. ...and 24 more sections

Key Result

Proposition 2.1

For every generalized trace $\xi$ and every integer $n\geqslant0$, let $\xi_n$ denote the trace of height $n$ defined by $\xi_n=C_1(\xi)\cdots C_n(\xi)$.

Figures (7)

  • Figure 1: Digraph of cliques for the dimer model on four generators
  • Figure 2: A concurrent system with $X=\{0,1,2\}$ and $\mathcal{M}=\langle a,b,c\,|\,ab=ba\rangle$
  • Figure 3: Directed graph of state-and-cliques (DSC) for the previous example
  • Figure 4: The $\text{\normalfontDSC}^+$ of our running example. The single basic component is framed. Compare with the DSC depicted on Figure \ref{['fig:qwdqqwfgkojngfffff']}.
  • Figure 5: The trace $w^3$ with $w=031425$ represented as a heap of pieces with a cyclic base (identify half-pieces labeled '$5$' with the same altitude)
  • ...and 2 more figures

Theorems & Definitions (73)

  • Proposition 2.1
  • Remark 2.2: connecting Möbius transform and Möbius polynomial
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 63 more