Stable Configuration Structures

• Stable Configuration Structures are defined as pairs (C, ℓ) capturing finite subsets of events that respect causal and concurrency relations.
• They underpin various process equivalences such as interleaving, step, and history-preserving bisimulations, advancing reversible computation analysis.
• Applications span both algebraic modeling of concurrent systems and topological studies of configuration spaces, highlighting stabilization phenomena.

A stable configuration structure is a foundational concept in the mathematical formalization of concurrent systems, particularly in the study of non-interleaving semantics, causality, and true concurrency. The theory of stable configuration structures provides an abstract, combinatorial framework for representing subsets of "events" that can coexist, respecting the causal and concurrency relations intrinsic to the system under consideration. This notion underlies both categorical approaches to concurrency (event structures, configuration structures, domain-theoretic models) and the algebraic and topological analysis of configuration spaces in geometry and topology.

1. Formal Definition and Properties

A stable configuration structure is typically defined as a pair $\mathcal{C} = (C, \ell)$, where:

• $C$ is a collection of finite subsets of a global event set $E$, called configurations.
• $\ell\colon E \rightarrow Act$ is a labeling map from events to an action alphabet.

The stability axioms are:

1. Rootedness: $\varnothing \in C$.
2. Connectedness: For any nonempty $X \in C$, there exists $e \in X$ such that $X \setminus \{e\} \in C$.
3. Closure under bounded unions: If $X,Y,Z \in C$ with $X \cup Y \subseteq Z$, then $X \cup Y \in C$.
4. Closure under bounded intersections: If $X,Y,Z \in C$ with $X \cap Y \subseteq Z$, then $X \cap Y \in C$.

A configuration $X \in C$ encodes a possible partial state of the system—specifically, a collection of events that have occurred together in some execution, constrained by the underlying causality and compatibility relations.

The causality and concurrency relations are extracted as follows:

• For $d, e \in X$, $d \leq_X e$ iff in every subconfiguration $Y \subseteq X$, $e \in Y \implies d \in Y$.
• $d$ and $e$ are concurrent in $X$ if neither $d <_X e$ nor $e <_X d$.

Stable configuration structures precisely characterize the configuration sets of stable event structures, and every such structure can be recovered from its configurations (Phillips et al., 2010).

2. Process Equivalences and Bisimulations

Stable configuration structures support the definition of several process equivalences reflecting different levels of observation power:

• Interleaving Bisimulation (IB): Only single-event forward transitions are matched.
• Step Bisimulation (SB): Allows matching concurrent steps—simultaneous execution of independent events (sets of pairwise concurrent events).
• Hereditary History-Preserving (HH) Bisimulation: Enriches SB further, where bijections preserving labels and causal order between configurations are maintained, and matching of forward and backward moves is required with preservation of "histories".

Reverse transitions substantially enrich the expressiveness:

• Reverse Interleaving Bisimulation (RB) and Reverse Step Bisimulation (RSB) allow both forward and backward (undo) transitions, capturing phenomena not visible in forward-only frameworks.
• In particular, RSB can be defined so that forward steps are redundant—it suffices to allow single forward events and arbitrary reverse homogeneous steps (simultaneous undo of multiple concurrent events with the same label). This observation enables a full characterization of RSB without complex forward step-matching machinery (Phillips et al., 2010).

A key result is that, under the absence of equidepth auto-concurrency (no two concurrent events with the same label at the same causal depth), reverse interleaving bisimulation (RB) coincides with HH-bisimulation (Phillips et al., 2010). Thus, in these regimes, reversed-based bisimulations are as discriminating as the richest history-tracking equivalence.

3. Examples, Variants, and Discriminating Power

Stable configuration structures serve as critical testbeds for analyzing process equivalence. Explicit process algebra examples confirm the separation of equivalences:

Processes	IB	SB	RSB	HH
$a \parallel b$ vs $a.b + b.a$	$\checkmark$	$\times$	$\times$	$\times$
$a \parallel a$ vs $a.a$	$\checkmark$	$\times$	$\times$	$\times$

Forward steps (as in SB) can distinguish between truly concurrent and sequential behaviors that interleaving semantics (IB) conflate. RSB, by incorporating reversibility and restricting attention to reverse homogeneous steps, further refines these distinctions without the full burden of HH state-tracking.

Notably, certain classical "true concurrency" laws (such as the absorption law) hold for SB but not for RB or RSB (Phillips et al., 2010).

4. Connections to Geometric and Topological Structures

The concept of a stable configuration structure also appears in algebraic topology and manifold calculus, though with different technical incarnations. For configuration spaces of points on a manifold $M$, the set of all possible finite configurations underpins both functor calculus and the formulation of co-FI-space and FI-module structures. In this context, stability refers to the phenomenon that, in certain dimension and degree ranges, algebraic invariants such as homotopy groups, (co)homology groups, or representation types of symmetric groups stabilize as the number of points grows (Guès, 27 Mar 2025).

For instance, for manifolds $M$ with $\dim M \geq 3$, explicit linear stable ranges can be given for when the homotopy groups ${\pi_p (\mathrm{Conf}_n(M))}$ and their dual FI-modules stabilize, with precise dependence on $p$ and $d$: $$n \geq \left\lceil \frac{3(p-1)}{d-2} \right\rceil + 2$$ implies that the representational pattern of $\pi_p (\mathrm{Conf}_n(M))$ is stable (Guès, 27 Mar 2025). This justifies the assertion that such manifolds possess a "stable configuration structure" in the topological and homological sense.

5. Applications and Implications

Stable configuration structures unify:

• The algebraic modeling of concurrent and reversible computation, where they enable fine-grained process comparison, the definition of intricate bisimulations, and rigorous handling of causality.
• The algebraic topology of configuration spaces, facilitating explicit computation of stabilization thresholds for homotopy and homology groups, and the exploitation of functorial and representation-theoretic techniques (Guès, 27 Mar 2025).

Key technical consequences include:

• The reduction of complex geometric or system-theoretic stability questions to combinatorial and algebraic verification within the configuration structure framework.
• Efficient algorithms for checking local and global stability, based on combinatorial properties (axis-removal, run-count tests), in both discrete models (e.g., molecular networks (Bétermin et al., 2020)) and continuous configuration parameter spaces.

In concurrency theory, the design of reversible process calculi and operational semantics relies critically on the ability to trace, reverse, and compare event structures using the full apparatus of configuration structures and their associated bisimulations.

6. Generalizations and Ongoing Developments

Active research directions include:

• FI-module generalizations: Extension to FI$_G$-modules and orbit configuration spaces, broadening the class of "stable" configuration functors to include group actions and colored symmetries (Guès, 27 Mar 2025).
• Depth-respecting bisimulations: Finer invariants in reverse bisimulation that capture event depth and enable sharper correspondences with history-preserving equivalences (Phillips et al., 2010).
• Stable ranges in algebraic topology: Identification of linear and nonlinear stabilization thresholds in integral and rational cohomology, and their relation to manifold and functor calculus.
• Links to physical models: Translation of configuration stability criteria into molecular rigidity and material science contexts, as in angle-rigidity for energy-minimizing structures (Bétermin et al., 2020).

7. Summary Table: Core Features (Selected)

Feature	Pure concurrency (process algebra)	Geometric/topological context
Configuration objects	Event-sets with causality/concurrency	Ordered/unordered points in $M$
Stability notion	Bisimulation, reversibility, histories	Cohomological/homotopical stabilization
Structural axiomatics	Rooted, connected, bounded union/intersection	Functoriality over FI/FI$_G$
Main technical result	RSB = HH (under no equidepth auto-concurrency)	Linear stable ranges for $\pi_p, H^*$
Characterization of stable regime	Depth, labeling, concurrency classes	Explicit $n \geq f(p,d)$, representation stability

Stable configuration structures thus constitute a foundational mathematical language for stability phenomena in both concurrency theory and the topology of configuration spaces, facilitating a rich interplay between combinatorial, algebraic, and geometric methods (Phillips et al., 2010, Guès, 27 Mar 2025, Bétermin et al., 2020).