Behavior Structure Integration

• Behavior structure establishment is a formal process that unifies system structure and dynamic behavior using coherent models and rules.
• It leverages approaches like Structure–Behavior Coalescence, process algebraic metamodels, and graph-based models to ensure consistency across diverse views.
• This integrated framework aids in system verification, simulation, and reusability while enhancing predictive analytics in fields such as robotics and organizational science.

Behavior structure establishment refers to the rigorous, formally grounded process of defining, integrating, and representing the interdependence of structural and behavioral aspects in a system, model, dataset, or organization. Across systems engineering, data modeling, organizational science, robotics, and behavioral analytics, this concept enables the unification of “what exists” (structure) with “what happens and in what order” (behavior), providing a foundation for analysis, verification, simulation, and control.

1. Foundations and Core Formalisms

The challenge in behavior structure establishment arises from historical tendencies to treat structure and behavior as distinct, often unconnected modeling artifacts. Classical definitions (e.g., “an integrated whole embodied in its components and their interrelationships”) typical of Systems Definition 1.0, yield structural diagrams or data schemas, but leave behavior (e.g., workflows, dynamic rules, temporal sequences) floating in parallel, risking inconsistency and loss of semantic integration (Chao, 2021).

Modern approaches explicitly aim for a unified semantic space by introducing:

• Structure–Behavior Coalescence (SBC): Integrates both aspects within the same Process Algebra, tying component composition to behavior via common formal constructs: channels, guards, prefixes, and labeled transition systems. SBC-PA provides a labeled transition graph as the single source of truth (Chao, 2021).
• Process Algebraic Metamodels: C-M-SBC-PA extends the SBC paradigm to provide an integrated metamodel for SysML, allowing every block (structural) and state machine (behavioral) diagram to be a projection of the same formal LTS, ensuring total consistency and eliminating view multiplicity (Chao, 2021).
• Category-Theoretic Systems Theory: The behavioral approach models a system as an injective map $s: B \hookrightarrow U$, with behaviors $B$ as a subspace of possible signal trajectories $U$. The gluing (pullback) of systems equates structure and behavior at a categorical level (Adam et al., 2019).
• Graph-Based Behavior Structures: In behavioral analytics, the Behavioral Molecular Structure (BMS) model treats each behavior as a labeled graph over atomic attributes, capturing both attribute co-occurrence and relation edges. This graph representation vastly increases expressive power over flat, attribute-based models (Wang et al., 2023).

2. Methodological Frameworks for Establishment

A. Systems Engineering: SBC and C-M-SBC-PA

Establishing behavior structure proceeds in a strict, staged methodology:

1. Component and Actor Identification: List all system components ($I$) and environmental actors ($B$), defining candidate interaction points (channels).
2. Channel Signature Specification: Formally define allowed communication signatures $KS \subseteq A \times \Omega$, with parameterized input/output directionality (Chao, 2021, Chao, 2021).
3. Interaction Relation Construction: Specify all possible interactions $\Delta \subseteq E \times K \times I$, distinguishing between environment-to-component (type 1) and component-to-component (type 2) interactions.
4. Prefix and State Expression Encoding: For each process, express permitted transitions as guarded prefixes and build state expressions using a formal grammar (BNF), constructing the component's interaction transition graph (ITG).
5. Composition: Compose ITGs using parallel and sequential operators; check that all behavioral transitions are structurally anchored.
6. Iterative Consistency Checking: Iterate until the global LTS captures both structural topology and all permitted behaviors, achieving coalescence (Chao, 2021, Chao, 2021).

B. Thinging Machine (TM) Method

TM-based modeling imposes three distinct levels:

• Static (Structure): Models machines as “thimacs” and delineates possible flows (create, process, release, transfer, receive) without explicit time.
• Dynamic (Events): Instantiates time-bound events, organizing them in operational hierarchies as submachines.
• Behavioral (Chronology): Enumerates permissible event traces (ordered sequences), where permissible behavior is the set of all traces compatible with the underlying static and event constraints (Al-Fedaghi, 2020).

The key principle is clear separation, precise event definition, and systematic enumeration of allowed event sequences.

C. Temporal Logic Over Organization Structure

In multi-agent or organizational contexts, each structural entity (role, group, link) is associated with its own set of temporal-logic (e.g., LTL) behavioral constraints. Rigorous inter-level “glue” is imposed to guarantee that lower-level behaviors (role-level LTL) collectively satisfy higher-level group or organization-wide properties via explicit logical implications (Jonker et al., 2021).

D. Function-Behavior-Structure Framework

Here, abstraction, expected behavior, and structure are linked via explicit refinement mappings. For each level of abstraction, behavior is derived, analyzed, and compared to requirements. Cross-level consistency is enforced by explicit refinement rules:

$$\begin{align*} F' & = r_F(D, F) \ Be' & = r_B(Be, F') \ S' & = r_S(S, Be') \ D' & = r_D(D, S') \end{align*}$$

Discrepancies at any level—abstract or refined—trigger systematic reformulation, ensuring the refined structure delivers the required behavior (Diertens, 2013).

E. Data Structure Modeling: Behaverse Data Model

In the domain of behavioral data, BDM provides a strict schema for mapping raw events into structured entities (trials, activities, sessions), with explicit extraction logic (task-patterns) and entity-relationship linking. Naming conventions, metadata, and built-in consistency checks guarantee that structural organization and temporal/behavioral provenance are tightly coupled and reproducible (Defossez et al., 2020).

3. Expressive Power and Theoretical Analysis

Structure-based representations, especially graph-based models, far exceed pure attribute-based or even vector-based models in expressive power for behaviors:

• Comparatively, a vector encoder on $n$ attributes with $k$ values each can distinguish $k^n$ patterns; but a graph with $n$ nodes and $k$ edge types distinguishes up to $k^{n(n-1)}$ patterns—growing super-exponentially in $n$ (Wang et al., 2023).
• Implication: Only when the structural interrelations of behavioral “atoms” are explicitly represented does the model capture the full complexity of real-world behaviors.

This property is essential in domains such as crime detection, where the ability of BMS to represent fine-grained patterns among categorical attributes results in measurable gains in downstream classification task performance.

4. Integration and Consistency Across Views

Modern frameworks use a single unified transition graph or process algebra as the underlying semantics, guaranteeing that:

• All views (block/structure, state machines, activities/processes) are projective images of the same formal model.
• Cross-view Consistency: No semantic drift occurs; state machines, component networks, and activity flows are definitionally consistent (Chao, 2021, Chao, 2021).
• Compositionality: For a composed system, projections onto any subsystem recover the respective local views; projections are formally distributive (Chao, 2021).

This eliminates the problem of mutually inconsistent diagrams or models, a persistent issue in traditional multi-view modeling approaches.

5. Emergence, Adaptation, and Evolution in Behavioral Structures

Dynamical and evolutionary frameworks model how behavior structures emerge, adapt, and stabilize:

• Agent-Based and Evolutionary Game Dynamics: Individual strategy update rules and local payoffs lead to global emergent standards (e.g., moralist, immoralist clusters in spatial games), with explicit phase diagrams mapping parameter regions to collective behavior types (Helbing et al., 2010).
• System Identification and Sensitivity: Composite maps $h(x) = m(s(x))$ encapsulate the transformation from system state to emergent collective behavior, with Lyapunov exponents or entropy as quantifiers of stability vs. chaos (Park, 2019).
• Oscillator Models for Coordination: Behavioral stability or variability is characterized by the interplay of internal component dynamics and external cyclical influences, tracked by entropy or other macro-indices, but always rooted in a compositional structural map (Park, 2019).

6. Verification, Validation, and Implementation

Across all frameworks, formal establishment of behavior structure enables:

• Automated Validation: Ensuring behavioral constraints, chronological event properties, and structural pathways are mutually satisfiable; tools can traverse the transition or event graph, or check logical entailment in LTL (Defossez et al., 2020, Jonker et al., 2021).
• Model Checking: The integrated transition graph or LTL-based constraints facilitate standard verification methods to check deadlock-freedom, coverage of all required behaviors, and absence of unintended behaviors (Chao, 2021, Chao, 2021).
• Reusability and Interoperability: Standardized structure-behavior mappings allow datasets or models to be merged, extended, and reused in heterogeneous environments or across research groups (Defossez et al., 2020).

7. Illustrative Case Studies

• Automated Teller Machine (ATM): Complete structure and behavior unified as parallel ITGs, with channels mediating all allowed handshakes and every behavioral step attached to a physical component (Chao, 2021).
• Vending Machine (SysML): Orthogonal ITGs for all subcomponents (customer, coin logic, etc.) combine into a global ITG, with all SysML views extractable via projection operators (Chao, 2021).
• Organizational Factory Model: LTL constraints for all roles, groups, interlinks, and the full organization; end-to-end “gluing” of behavioral properties enforced via logical implication (Jonker et al., 2021).
• Crime Incident Analysis (BMS): Each incident mapped to a behavioral graph of 19 fields; BMS embeddings yield top-tier classification accuracy in detecting crime types, outperforming both vector and tree-based models (Wang et al., 2023).
• Robot Soccer Control (NimbRo-OP): Layered inhibition-based arbitration for high-level behaviors, finite state queues for low-level motion sequencing—together yielding a robust integrated behavior structure for real-time control (Allgeuer et al., 2018).

---

The establishment of behavior structure thus constitutes a technical, deeply formalized process whereby systems of all kinds—physical, informational, social, or cybernetic—are rendered as integrated wholes. Theoretical foundations, methodological rigor, formal and categorical semantics, universal verification capability, and practical downstream gains all depend on the meticulous, conceptually unified treatment of structure and behavior as a singular entity. This principle is now implemented in advanced systems engineering, data science, multi-agent modeling, and behavioral analytics (Chao, 2021, Chao, 2021, Wang et al., 2023, Al-Fedaghi, 2020, Jonker et al., 2021, Diertens, 2013, Defossez et al., 2020, Adam et al., 2019, Park, 2019, Allgeuer et al., 2018, Helbing et al., 2010).