# The Design Multiverse: A Scientific Model for Design Evolution and Co-evolution

Ciprian Teodorov, Joeri Exelmans, Salvador Martinez, Sylvain Guérin, Hans Vangheluwe

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Ciprian Teodorov, Joeri Exelmans, Salvador Martinez, Sylvain Guérin, Hans Vangheluwe. The Design Multiverse: A Scientific Model for Design Evolution and Co-evolution. International Conference on Software Engineering, NIER Track, Apr 2026, Rio de Janeiro (Brazil), France. 10.1145/3786582.3786830 . hal-05503862

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# The Design Multiverse: A Scientific Model for Design Evolution and Co-evolution

Ciprian Teodorov

ENSTA | Institut Polytechnique de
Paris, Lab-STICC, UMR 6285

Brest, France

ciprian.teodorov@ensta.fr

Joeri Exelmans

University of Antwerp
Antwerp, Belgium

joeri.exelmans@uantwerpen.be

Salvador Martinez

IMT Atlantique,
Lab-STICC, UMR 6285

Brest, France

salvador.martinez@imt-atlantique.fr

Sylvain Guérin

IMT Atlantique,
Lab-STICC, UMR 6285

Brest, France

sylvain.guerin@imt-atlantique.fr

Hans Vangheluwe

University of Antwerp
and Flanders Make
Antwerp, Belgium

hans.vangheluwe@uantwerpen.be

## Abstract

Model-Based Systems Engineering lacks a formal model that treats design evolution, with its inherent branching, rework, and context-sensitivity, as a first-class citizen. Current approaches reduce rich design trajectories to repository artifacts, losing the semantics of design decisions. This paper calls for a scientific model for design evolution and introduces the Design Multiverse that frames design dynamics as a context-sensitive, nondeterministic transition system, while orthogonally separating observational link theories (e.g., conformance, refinement) for consistency reasoning. We demonstrate the model's explanatory power through co-evolution. Furthermore, we outline a research agenda grounded by Design Multiverse Experiments (DMX), which enable empirical validation and pave the way for scalable, history-aware modeling environments.

## CCS Concepts

* Software and its engineering → Designing software.

## Keywords

design evolution, scientific model, design multiverse, co-evolution

## ACM Reference Format:

Ciprian Teodorov, Joeri Exelmans, Salvador Martinez, Sylvain Guérin, and Hans Vangheluwe. 2026. The Design Multiverse: A Scientific Model for Design Evolution and Co-evolution. In *2026 IEEE/ACM 48th International Conference on Software Engineering (ICSE-NIER '26), April 12–18, 2026, Rio de Janeiro, Brazil*. ACM, New York, NY, USA, 5 pages. https://doi.org/10.1145/3786582.3786830

## 1 Introduction

Designing complex cyber-physical systems demand engineering practices that can cope with pervasive heterogeneity (multiple domains, tools, and artifacts), and reason explicitly about time (visions) and space (variants). Yet, most modeling environments and associated infrastructures still treat temporal and spatial variability

as external concerns, managed through repositories or ad hoc processes, rather than being first-class citizens within the modeling space itself [1–3, 31, 32, 35]. As a consequence, designers struggle to maintain coherence across evolving models, variants, and dependencies. Compounding the problem, the *decisions* that drive design transitions are seldom captured in a structured, machine-interpretable form: version control commit messages are 'after-the-fact text' that misses the semantics and internal structure of design rationales. Despite decades of MBSE research, we lack a *formal scientific model* of *design evolution*, one that natively accounts for branching, looping (including stuttering [23] and rework), and context-sensitive change across heterogeneous artifacts. In its absence, the community has largely converged on tool-oriented, repository-centric solutions (versioning, merging, product derivation) [1, 11, 14, 15, 26, 31], which are powerful yet lead to fragmented reasoning about design-time dynamics. Moreover, while prior work has advanced live modeling and co-evolution in specific settings [6, 7, 10, 16, 17, 36], these advances are not grounded in a unifying scientific model of design change. There is thus no common substrate on which to *state, test, and compare* hypotheses about how designs evolve, diverge, and reconverge in practice.

The fundamental research question is: **How do we reason about design evolution for coupled design processes in the presence of heterogeneity?** We therefore seek a minimal, technology-agnostic theoretical framework that captures the evolution of heterogeneous design artifacts (e.g. models and metamodels) as a first-class dynamic object. This framework must abstract away language/tool specifics while preserving just enough structure to enable formal reasoning about (co-)evolution, repair, and history-aware consistency.

We propose the Design Multiverse: a scientific model of design dynamics, with link theories layered atop as observational semantics. Concretely, each design process leads to a multiverse whose dynamics are captured by an *input-driven, nondeterministic transition system* over design snapshots (named *slices*). The transition system admits loops (stuttering and rework) and overlapping successor sets (distinct decisions may converge) yielding a closed reflexive-transitive closure for formal analysis. Micro-steps can be grouped in macro-steps to reason at a useful level of abstraction. In other words, a multiverse is the set of universes (i.e. linear design

<figure><img src="image_2.png" /></figure>

This work is licensed under a Creative Commons Attribution 4.0 International License.
ICSE-NIER '26, Rio de Janeiro, Brazil

© 2026 Copyright held by the owner/author(s).

ACM ISBN 979-8-4007-2425-1/2026/04

https://doi.org/10.1145/3786582.3786830
trajectories), each universe is composed of discrete slices (design snapshots) with a total order relation induced by the design transitions among them. Orthogonally, *link theories*—e.g., conformance, usage, refinement, binding—are provided by a *parametric link environment* that interprets pairs (or tuples) of slices and enables consistency reasoning across evolving multiverses. This separation lets us change observational theories *without* altering the underlying dynamics, and vice versa.

To ground the approach and make it actionable at scale, we propose Design Multiverse Experiments (DMX) that will: instrument existing tools to collect micro-steps and context, apply abstractions offline, and evaluate link theories parametrically. DMX brings hypothesis-driven, reproducible experimentation to design evolution without over-burdening production workflows.

The term 'multiverse' intentionally evokes parallels with cosmological theories [12] where multiple universes coexist and evolve. While our model deals with design artifacts rather than physical realities, it shares the core structural concept: a branching space of possible states evolving under contextual constraints. However, where physical multiverses are often mutually inaccessible, design multiverses are explicitly comparable through parametric link theories, enabling formal reasoning about consistency and co-evolution across branching trajectories.

Concretely, this paper makes four contributions:

* We define the *Design Multiverse* as a compositional context-sensitive, nondeterministic transition system over slices, with loops, overlapping successor sets.

* We *separate dynamics from observational link theories* via a parametric link environment, enabling modular and reusable consistency analyses across multiple design multiverses.

* We demonstrate the model's explanatory power by re-framing *co-evolution*: when evolution in one multiverse breaks a link-scoped consistency relation, saturated reachability precisely characterizes (possibly multi-step, nondeterministic) repair trajectories in the other.

* We *outline a research agenda* for operationalization: scalable representations, visualization and querying of universes, semantic deltas across technologies, and empirical grounding via DMX and model federation [2].

Our proposal is intentionally *high-risk and foundational*: it shifts the focus from tool-specific mechanisms to a *scientific model* of design evolution. If successful, it will enable predictive and adaptive design reasoning, underpinning *history-aware* modeling environments that can explain, simulate, and guide the evolution of complex, heterogeneous systems in line with the INCOSE Systems Engineering Vision 2035 [19].

## 2 The Design Multiverse as a Scientific Model

We treat the *Design Multiverse* as a *scientific model of dynamics*: it specifies how design states (slices) may evolve under admissible inputs, with nondeterminism, stuttering, and rework. In contrast, *links* (e.g., conformance, usage, refinement) are *observational theories* *layered atop* the dynamics, used to *interpret* and *constrain* trajectories across slices. This separation yields modularity: we can swap, combine, or refine link theories without touching the core evolution semantics.

Core multiverse. A multiverse is a tuple $\mathcal{M} = \langle S, I, O, \Delta, \text{ctx}_I, \text{ctx}_O \rangle$ where $S$ is the set of *slices* (snapshots of all design artifacts and their internal references); $I$ is a set of *inputs* (contextual triggers, design decisions), containing a silent $\tau$ for stuttering; $O$ is a set of *outputs* (by-products of the design process that escape to the underlying context); $\text{ctx}_I: S \to 2^I$ yields the *admissible inputs* at each slice (the implicit multiverse around $s$: external dependencies, variants, metamodel, etc.); symmetrically $\text{ctx}_O: S \to 2^O$ labels the states with possible outputs; and $\Delta: S \times I \to \mathcal{P}_{\neq\emptyset}(S)$ is a *nondeterministic transition kernel*. A micro-step is $(s, i) \rightsquigarrow (o, s')$ when $i \in \text{ctx}_I(s)$, $o \in \text{ctx}_O(s)$ and $(o, s') \in \Delta(s, i)$. We allow overlapping successor sets: for $i \neq j$, $\Delta(s, i) \cap \Delta(s, j)$ may be non-empty (distinct inputs can converge to the same slice). Loops, including $(s, \tau) \rightsquigarrow (\emptyset, s)$, capture stuttering and rework. For simplicity, we assume a void output type and omit it, so a micro-step is $(s, i) \rightsquigarrow s'$. When inputs are fixed, the transition is saturated (the contextual input space is closed), we also drop the input component. We write $\rightsquigarrow^*$ for the reflexive-transitive closure; when applied to saturated transitions, this yields saturated reachability.

**Universes and Step-folding** A universe is any (finite or infinite) micro-step trace. To reason at a higher level of abstraction, analysts may define policies $P$ to fold micro-steps into macro-steps: $s \Rightarrow_P s'$ iff a contiguous subsequence of $\rightsquigarrow$ collapses under $P$ (e.g., same input vector, bounded time window, or until a consistency check passes).

**Composite slices and product evolutions.** Across multiverses $\mathcal{M}_1, \dots, \mathcal{M}_n$ (slice sets $S_k$), a *composite slice* is $\langle s_1, \dots, s_n \rangle \in \prod_k S_k$ whose inter-multiverse dependencies are *syntactically closed* within the tuple. Component micro-steps lift to product micro-steps via input vectors $\langle i_1, \dots, i_n \rangle$ with $i_k \in \text{ctx}(s_k)$. This supports multi-model, multi-team settings without collapsing autonomy.

**Parametric link environment.** We keep link theories *orthogonal* to dynamics via a *link environment* $\mathcal{L} = \langle \text{Types}, \text{Inst}, \llbracket \cdot \rrbracket \rangle$: Types is a set of link kinds $L$ (e.g., conformance $\chi$, usage, refinement, binding), $\text{Inst}(L; s_1, s_2)$ is the set of link instances between slices $s_1$ and $s_2$, and $\llbracket \cdot \rrbracket$ assigns truth to each instance under the appropriate semantics. Link-scoped consistency is defined *relative* to $\mathcal{L}$:

$$C_L^\mathcal{O}(s_1, s_2) \stackrel{\text{def}}{\rightleftharpoons} \forall \ell \in \text{Inst}(L; s_1, s_2) : \llbracket \ell \rrbracket \text{ holds.}$$

This parametric view enables analyzing the *same* evolution traces under *different* observational theories.

### 2.1 Design Multiverse Experiments with Existing Tools

To empirically ground the Design Multiverse and address scalability, we propose DMX: lightweight instrumentation of existing tools to observe micro-steps and context inputs, enabling offline evaluation of link theories. Rather than introducing new modeling infrastructures, DMX operates in *shadow mode*, avoiding disruption to production workflows.

For example, probes in EMF-based editors can log design events such as “rename attribute X to Y” with stable identifiers and timestamps. These traces allow us to replay histories and inject hypothetical changes, such as metamodel evolutions, to evaluate repair paths, their cost, and frequency.
DMX instrumentation can leverage existing telemetry frameworks [38] (e.g., OpenTelemetry) to capture design events nonintrusively. An offline analysis service applies domain-specific folding policies (group the micro-steps of a refactoring) to derive macro-steps and uses specialized (e.g., conformance, usage) link environments to detect consistency breaks and repairs. These can leverage directly emerging debugging infrastructure for branching traces [15, 25, 27, 28]. These tools, particularly recent extensions supporting input/output operations [37], provide the analytics and exploration primitives needed to empirically ground our model. This approach enables hypothesis-driven studies with repeatability, metrics, and falsifiability. Crucially, DMX requires no new modeling infrastructures and it shifts heavy multiverse analyses out of the critical path, alleviating scalability concerns while providing a scientifically grounded substrate for design evolution studies.

## 2.2 Is the Design Multiverse a Scientific Model?

We emphasize that the Design Multiverse extends beyond a conceptual framework and aspires to function as a scientific model, in the sense articulated by Kuhn [22]: it provides a formal substrate to explain design evolution phenomena (Sec. 3), generate predictions, and support empirical testing (Sec. 2.1). Its mathematical structure as a nondeterministic, context-sensitive transition system with orthogonal link theories gives it precision and generality. Its explanatory scope covers branching, looping, reconvergence, and co-evolution across heterogeneous artifacts. Crucially, the Design Multiverse supports falsifiable hypotheses via DMX. This blend of formal rigor and empirical testability sets it apart from current approaches [17] and validates it as a scientific model, albeit an early-stage one still requiring empirical grounding and refinement, as outlined in Sec. 4. We therefore present this as a working hypothesis, open to critical scrutiny and future investigation.

## 3 Explain a Design Evolution Phenomena: Co-evolution

Let $M_1, M_2$ be multiverses and $\mathcal{L}$ a link environment (Sec. 2). Consider slices $s_1 \in M_1$ and $s_2 \in M_2$. We study, Figure 1, how an evolution on one side may break link-scoped consistency and trigger counter-evolution on the other.

### 3.1 Definition (Co-evolution)

Suppose $s_2 \rightsquigarrow s'_2$ in $M_2$. A co-evolution in $M_1$ is any $s_1 \rightsquigarrow s'_1$ such that:

$$C_L^\mathcal{L}(s_1, s_2) \wedge \neg C_L^\mathcal{L}(s_1, s'_2) \Rightarrow \exists s'_1 : s_1 \rightsquigarrow s'_1 \wedge C_L^\mathcal{L}(s'_1, s'_2).$$

This formulation allows nondeterministic, multi-step repairs and does not assume acyclicity.

Caveats. The existential quantifier expresses a property to be checked or measured, not a guarantee: repairs may not exist. In practice, we (i) search for repairs [34]; (ii) minimize a cost $J$ (edits, semantic distance, time); and (iii) report repairability rates (Sec. 4). A policy predicate admissible($\pi$) can constrain edit sequences to avoid trivial repairs (e.g., deleting the entire model). This makes repairability metrics meaningful and configurable.

<figure><img src="image_3.png" /><figcaption>Figure 1: Co-evolution across two coupled design multiverses, inspired by [17]. A client requirement change triggers a meta-model evolution ($v1 \rightsquigarrow v2$ - blue arrow top left), breaking conformance with the design slice ($\neg C_L^\mathcal{L}(s1, v2)$ - red hollow arrow). Automatic repair fails, leading to a split: (a) report bug, keep $s1$; (b) create degraded service $s2$. Under fixed context, relaxing multiplicity yields $v3$, enabling a new consistent slice $s3$ ($C_L^\mathcal{L}(s3, v3)$ hols - green thick arrow), which is advertised.</figcaption></figure>

### 3.2 Examples (instantiating $\mathcal{L}$)

Model/metamodel ($L = \chi$; conformance). Let $M_1$ host successive model slices and $M_2$ host metamodel slices. The link environment provides conformance instances Inst($\chi$; $s_{model}, s_{mm}$) and semantics $[\chi(a, b)]$ (type satisfaction). At a working composite slice ($s_{model}, s_{mm}$), assume $C_\chi^\mathcal{L}$ holds. If the metamodel evolves $s_{mm} \rightsquigarrow s'_{mm}$ (e.g., a class split, attribute rename, constraint tightening) and conformance fails, $\neg C_\chi^\mathcal{L}(s_{model}, s'_{mm})$, we seek a model co-evolution $s_{model} \rightsquigarrow s'_{model}$ such that $C_\chi^\mathcal{L}(s'_{model}, s'_{mm})$. Folding can absorb micro-migrations (rename, move, pull-up/push-down) and constraint re-evaluations. Because metamodel changes can be non-invertible or ambiguous (e.g., class split without a canonical target), automatic repair may be impossible; our definition does not presuppose existence. Instead, it supports mixed-initiative search, cost minimization (e.g., minimal edit or least semantic drift), and explicit reporting of non-repairable cases. This generalizes classical model/metamodel co-evolution by (i) making design trajectories first-class, (ii) allowing loops/rework and convergent paths, and (iii) parameterizing conformance in $\mathcal{L}$.

Usage of Commercial Off-The-Shelf (COTS) components ($L = \text{use}$; dependency compatibility). Let $M_2$ host a COTS/library multiverse and $M_1$ a project design. The link environment provides usage instances (e.g., API calls, configuration bindings, behavioral assumptions) and their semantics $[\text{use}(\cdot)]$ (e.g., type and protocol compatibility). When the COTS evolves $s_{cots} \rightsquigarrow s'_{cots}$ (security patch, API change, license update), assumptions in the project slice $s_{proj}$ may be invalidated, yielding $\neg C_{\text{use}}^\mathcal{L}$. A project co-evolution $s_{proj} \rightsquigarrow s'_{proj}$ aims to restore $C_{\text{use}}^\mathcal{L}(s'_{proj}, s'_{cots})$ via API adaptation,
dependency re-resolution, configuration/ABI changes, or introduc-
ing shims. Compared to metamodel evolution, COTS is often black-
box, asynchronous, and coarse-grained (semantic changes without
syntactic cues; sudden deprecations; license/security constraints),
so link evaluation may rely on partial oracles (e.g., type checks + re-
gression tests). Again, existence of repairs is empirical: we measure
repairability and minimal cost under chosen step-folding policies.

Co-evolution depends on (i) the *context* `ctx(s)` (available meta-model/COTS versions, analysis results); (ii) the *observational power* of `L` (what link instances and semantics are available), and (iii) the *folding policy* (what micro-steps collapse, over which windows, with what heuristics), different choices lead to different $\Rightarrow^*$ relations and co-evolution outcomes. We therefore rely on DMX to instrument existing tools, *observe* micro-steps and context, *replay* what-if scenarios (e.g., injected COTS upgrades), and *evaluate* repairability and costs offline (Sec. 4).

## 3.3 Model Federation for Co-evolution DMX

These examples illustrate how link-scoped consistency can be broken and repaired across evolving multiverses. To operationalize such reasoning in heterogeneous design environments, model federation technologies [2] offer a promising substrate for implementing the Design Multiverse. Specifically, tools such as OpenFlexo [5] can support DMX by enabling: 1) slice loading from heterogeneous design multiverses into an unified *co-existence model federation*; 2) capturing of inter-slice relationships within a dedicated *link-model federation*; and 3) exposing these links to an orthogonal *link theory federation* for semantic interpretation. This layered approach aligns with the separation of dynamics and observational semantics proposed here, and provides a concrete pathway for analyzing co-evolution phenomena across diverse tools and domains without disrupting existing workflows.

## 4 Future Plans

To transition the Design Multiverse from theory to practice building on DMX we propose a progressive research agenda. **Feasibility and foundations** through small-scale experiments and theoretical analysis: 1) *Empirical grounding*: Apply the Design Multiverse to historical design traces and live modeling environments to validate its ability to capture branching, rework, and co-evolution, leveraging insights from co-evolution studies[16]. 2) *Tool Prototyping*: Develop a model federation by instrumenting existing tools (e.g., the Capella EMF-based editor) with lightweight DMX probes. 3) *Theory Refinement*: Consolidate the theory based on the patterns observed and formalize key design multiverse metrics. **Core challenges:** 1) Develop techniques for *efficiently managing and querying large multiverses* ($> 10^6$ slices), including state-space reduction, indexing, and abstraction strategies. 2) Explore and formalize various *link theories* for reasoning about consistency, refinement, and conformance across coupled multiverses. 3) Investigate *user interaction models* that facilitate exploration and analysis of design multiverses. **Advanced capabilities:** 1) Develop algorithms for *predictive reasoning about design evolution*, leveraging historical multiverse data to forecast future trajectories. 2) Integration with mainstream modeling and versioning tools for *seamless adoption in real-world workflows*. 3) Conduct *large-scale empirical studies* to

evaluate the impact on design quality, efficiency, and collaboration.
By following this roadmap, we aim to establish the Design Multi-
verse as a foundational model for understanding and improving
design evolution in complex engineering contexts.

## 5 Related Work

Cimino et al. [9] propose a 'Digital Multiverse' for managing Digital Twin variants across viewpoints, a spatial approach to heterogeneity. Our proposal addresses the orthogonal challenge of temporal evolution, providing formal foundations for reasoning about design trajectories rather than managing co-existing model configurations.

Like Institutions [13], we separate core dynamics from observational theories, but we apply this to temporal evolution rather than static semantics. Unlike Graph Transformation [30] or Abstract State Machines [8], we abstract from specific update mechanisms to focus on the phenomenology of design trajectories (branching, rework, and co-evolution) enabling reasoning at the level of design decisions rather than syntactic edits.

Dynamic I/O Automata [4] share our emphasis on evolving system structure, but they address runtime reconfiguration of computational systems, whereas the Design Multiverse addresses design-time evolution of system specifications. Where DIOA models how a system's interface changes during execution (runtime adaptation), we model how the space of possible design decisions evolves during development (design evolution).

The Design Multiverse shares its state-transition foundation with TLA [23], B-Method [18], and Abstract State Machines [8], but differs fundamentally in purpose and scope. Where these methods excel at specifying and verifying system behavior our model addresses the evolution of the design artifacts themselves. In these formalisms, transitions represent atomic system actions; in the Design Multiverse, they represent design decisions that modify the system's specification. Their focus is on what the system does; our focus is on how the design of the system changes. While a TLA specification might define the correct behavior of a consensus algorithm [24], our model captures the historical process of designing, refactoring, and evolving that algorithm's models.

Model federation [2] allows the users to preserve their usual practice and tools, while an additional autonomous model, the *federation model* is introduced to encode the dependencies, the interoperability semantics, and new integrated services. Many model federation approaches have been proposed, including Vitruvius [20], Epsilon [21], Reactive Links [29], comprehensive systems [33], and OpenFlexo [5]. While these approaches offer some support for navigating alternatives and artifact interoperability, they lack a common formal substrate that integrates *temporal* and *spatial* evolution into a single scientific model of design that allows reasoning. Nevertheless, as stated in Sec. 3, they can serve as technological layer for DMX and result analysis in the presence of heterogeneity.

## Acknowledgments

This work was partially supported by Agence de l'innovation de
défense via Centre Interdisciplinaire d'Etudes pour la Défense et
la Sécurité under Grant AUGMENT (2025), and by ANR under
Grant ANR-23-CE25-0011 (https://anr.fr/Project-ANR-23).
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