Event Equivalence in Formal Models

• Event equivalence is a formal notion that identifies and classifies indistinguishable events, traces, or structures based on behavioral, observational, or statistical criteria.
• Algorithms such as event-flow graphs, folding techniques, and partition refinement enable efficient minimization and verification of complex event structures.
• Its application spans concurrency theory, program analysis, fuzzy DES supervision, and statistical model selection, underpinning scalable analysis in diverse systems.

Event equivalence is a foundational concept in formal modeling, verification, and statistical inference across computer science, control theory, and applied mathematics. It refers to the identification, classification, or minimization of event representations, systems, or paths that are indistinguishable under a given behavioral, observational, or statistical criterion. This notion underpins efficient analysis, model selection, minimization, and supervisory control in domains such as concurrency theory, discrete event systems, program verification, labeled event trees, and action models.

1. Formal Definitions and Structural Spectrum

Event equivalence manifests in diverse models as a formal equivalence relation on sets of events, traces, or system configurations. Principal frameworks include labeled prime event structures (PES), coherence spaces, elementary event structures (EES), fuzzy discrete event systems (FDES), action models in modal logic, and labeled event trees.

• Prime Event Structures (PES): A labeled PES is $(E, \leq, \sharp, \ell)$, where $E$ is a set of events, $\leq$ is a partial order (causality), $\sharp$ is a symmetric, hereditary conflict relation, and $\ell$ labels events. Configurations are finite, conflict-free, causally closed subsets of events. Event equivalence is determined by a spectrum of behavioral relations: interleaving trace and bisimulation, step trace and bisimulation, pomset-trace and bisimulation, and strong criteria such as hereditary history-preserving bisimulation (hhp-bisimilarity). Each relation captures increasingly fine distinctions between system executions (Gorla et al., 2019).
• Coherence Spaces and EES: Removing causality ($\leq$) or conflict ($\sharp$) collapses the spectrum: in coherence spaces, almost all equivalence notions coincide except for hhp-bisimilarity and full isomorphism; for finite EES, all causal-based equivalences coincide with isomorphism (Gorla et al., 2019).
• Fuzzy Discrete Event Systems (FDES): Simulation equivalence is defined via fuzzy simulation relations between fuzzy automata, characterized by max-min matrix operations. Two FDESs are simulation equivalent if each simulates the other via a fuzzy relation satisfying specific initial, marked state, and event conditions (Deng et al., 2016).
• Action Models: Equivalence between action models is captured by generalized action emulation, action-model bisimulation, and propositional action emulation. These relations operate over event spaces within modal logic frameworks and determine whether different action models induce the same transitions on all Kripke models (Li, 2022).
• Staged Labelled Event Trees: In statistical models, event equivalence refers to statistical equivalence classes of event trees—distinct trees that represent the same joint probability model via identical interpolating polynomials. These equivalence classes are efficiently characterized using algebraic methods such as the primary decomposition of monomial ideals (Görgen et al., 2017).

2. Algorithms and Computational Methodologies

A central goal in the study of event equivalence is efficient computation and minimization of models up to behavioral equivalence. Multiple algorithmic paradigms have been developed:

• Event-Flow Graphs (EFGs) for Path-Sensitive Program Analysis: Given a control-flow graph (CFG) and set of event nodes, equivalence is defined on CFG paths by their event traces. The EFG is constructed in linear time and each of its paths corresponds one-to-one with an equivalence class of CFG paths under event-equivalence, permitting drastic compression of the analysis space (Tamrawi et al., 2014).
• Minimization of Event Structures through Foldings: History-preserving bisimilarity allows for behavior-preserving quotients called foldings. An explicit lattice of folding-equivalences yields a unique, minimal quotient of a general or prime event structure. Foldings are characterized by conditions such as "no hidden conflict" and "closed under merging histories" in the prime case, and analogous conditions in asymmetric models (Baldan et al., 2019).
• Partition Refinement and Emulation Algorithms: In action model theory, partition refinement (Paige–Tarjan) reduces the event space up to bisimulation. For propositional action emulation and generalized action emulation, event equivalence can be decided by an iterative fixpoint algorithm over formula sets satisfying four key atom-set properties. Minimization under full equivalence is PSPACE-hard but computable by regularizing over canonical formulas and enforcing cover/separation constraints (Li, 2022).
• Algebraic Enumeration via Primary Decomposition: Statistical event equivalence among staged trees is captured by decomposing the monomial ideal of the interpolating polynomial. Minimal primes yield all nested representations, and recursive partitioning produces all statistically equivalent tree structures efficiently (Görgen et al., 2017).

3. Behavioral and Observational Equivalence in Concurrency and Verification

Equivalence relations support crucial abstractions in concurrency theory and verification:

• Trace/Observational Equivalence: In relaxed memory models for C/C++ concurrency, trace equivalence can be refined to observational equivalence (OE), where behaviors are equivalent iff every read event reads the same value. Observational independence (OI), coupled with DPOR techniques, allows dynamic verification via OE, achieving substantial savings in trace exploration and enabling more comprehensive analysis of program behaviors (Singh et al., 2019).
• Model Minimization: In concurrency models, minimal quotients under hhp-bisimilarity preserve true-concurrent behavior while reducing event structure size. For action models, minimal event spaces up to bisimulation or emulation maintain indistinguishability on all underlying Kripke models (Baldan et al., 2019, Li, 2022).
• Path-Sensitive Program Analysis: By grouping CFG paths into event-equivalence classes, event-flow graphs reduce the state space nearly exponentially, supporting both scalability and human comprehensibility. Empirical evaluation on Linux kernel code shows >99% reduction in analyzed paths (Tamrawi et al., 2014).

4. Statistical Event Equivalence and Model Selection

In probabilistic inference and graphical modeling, event equivalence articulates the indistinguishability of models producing identical outcome distributions:

• Statistical Equivalence of Event Trees: Distinct staged trees can encode the same joint probability law. Algebraic decomposition of the support polynomial enables enumeration of all statistically equivalent trees, safeguarding against misinterpretation of one particular causal ordering. For real-world datasets, such as the Christchurch example, the equivalence class may contain multiple trees with interchangeable variable orderings (Görgen et al., 2017).
• Implications for Causal Discovery: Event equivalence generalizes Markov equivalence from Bayesian networks to staged trees, revealing that model selection and hypothesis testing must account for entire equivalence classes. This mitigates identifiability issues and strengthens the foundation for causal inference.
• Survival Analysis under Non-Proportional Hazards: Equivalence in time-to-event data analysis is operationalized by direct comparison of survival functions or hazard ratios at fixed time points or intervals, irrespective of proportional hazards assumptions. Parametric frameworks derive pointwise confidence bands and implement tests for non-inferiority or equivalence, proving robust in settings where classical log-rank or Cox tests fail (Möllenhoff et al., 2020).

5. Comparative Results, Expressiveness, and Theoretical Implications

The discriminating power of event equivalence strongly depends on structural features:

• Causality vs. Conflict: Removing causality leads to almost complete collapse of the spectrum of event equivalences; conflict removal in finite settings forces all causal-based equivalences to coincide with structure isomorphism but allows diversity in infinite systems. Thus, causality imparts stronger expressiveness and finer behavioral distinctions than conflict in concurrency models (Gorla et al., 2019).
• Simulation vs. Language Equivalence in FDES: Simulation-equivalence is strictly finer and more expressive than language-equivalence. Supervisory controllers designed for simulation equivalence achieve necessary and sufficient behavioral matches, whereas those for language equivalence may fail in expressive adequacy (Deng et al., 2016).
• Algorithmic Complexity: Deciding and minimizing event equivalence differs radically in complexity: polynomial-time algorithms exist for bisimulation quotienting, whereas propositional emulation and full equivalence minimization are PSPACE-complete or PSPACE-hard, reflecting sharp boundaries in tractability (Li, 2022).

6. Illustrative Examples and Applications

Event equivalence is illustrated in a range of settings:

Domain	Equivalence Criterion	Computational Approach
Concurrency theory	hhp-bisimilarity, interleaving trace	folding, lattice of equivalences (Baldan et al., 2019, Gorla et al., 2019)
Program analysis (CFGs)	event-trace equivalence	event-flow graph, linear-time algorithm (Tamrawi et al., 2014)
Fuzzy DES supervision	simulation-equivalence	max–min matrix simulation, fixpoint (Deng et al., 2016)
Action models	generalized action emulation	iteration over formula sets, partition refinement (Li, 2022)
Statistical modeling	polynomial/nested event-equivalence	primary decomposition, staged trees (Görgen et al., 2017)
Survival analysis	equivalence in time-to-event endpoints	parametric confidence bands (Möllenhoff et al., 2020)

In each case, event equivalence enables minimization, model selection, abstraction, and rigorous verification under the most discriminative behavioral or statistical semantics admitted by the modeling framework.

7. Significance and Future Directions

Event equivalence facilitates scalable verification, concise representation, and robust inference. Its algebraic, logical, and algorithmic frameworks bridge theoretical formalisms and applied methodologies, from concurrency and control to probabilistic modeling. Ongoing research in quantifying expressiveness, identifying algorithmically tractable subclasses, and understanding the interplay between structure (causality/conflict), labeling, and cardinality will further refine the granularity and effectiveness of event-equivalence-based analysis.

A plausible implication is that advances in event equivalence algorithms—such as efficient minimization under PSPACE-hard structural relationships or algebraic enumeration in complex statistical models—will continue to enhance automated verification, model selection, and inference in large-scale, high-dimensional systems.