Composite Explanation in Complex Systems

• Composite explanation is a framework that integrates multiple mechanisms—ranging from Bayesian inference and genetic algorithms to alpha-blending and tensor operations—to decompose complex phenomena.
• It employs systematic decomposition and compositional integration to analyze data, ensuring robust and scalable interpretations across various scientific domains.
• Its applications span deep generative models, compositional semantics, micromechanical analyses, and composite Higgs models, demonstrating broad interdisciplinary utility.

A composite explanation integrates multiple constituent elements, mechanisms, or hypotheses into a structured whole to account for complex phenomena, systems, or datasets. In scientific contexts, composite explanation often refers to joint interpretations that reveal underlying structures—whether in generative image modeling, semantics, statistical reasoning, or theoretical physics—by decomposing observed phenomena into interacting parts and elucidating their combination mechanisms.

1. Composite Explanations in Probabilistic Graphical Models

In probabilistic reasoning, a composite explanation typically denotes a full assignment to the non-evidence variables in a Bayesian network, given observed evidence. The k-Most Plausible Explanation (k-MPE) problem formalizes this: given evidence $S_e$ in a set of variables $E$, one seeks the $k$ assignments $H_1,\dots,H_k$ to the remaining variables that maximize the posterior probability $P(H|S_e)$ (Wierzchoń et al., 2018).

Composite hypotheses are not restricted to Bayesian networks; similar concepts extend to Valuation-Based Systems (VBS), where interactions and uncertainty representations are generalized beyond probabilities, and the Dempster–Shafer framework, which operates over basic belief assignments and commonality functions. In VBS and D-S contexts, a composite explanation maximizes combined valuations or commonalities, respectively.

The identification of composite explanations in these frameworks is computationally NP-hard; genetic algorithms are therefore employed to explore the solution space, encoding candidate hypotheses as chromosomes and iteratively optimizing their fitness (joint probability or commonality). Chromosome genes corresponding to observed variables are fixed, while genes for non-evidence variables explore possible assignments. This architecture is robust across Bayesian, VBS, and D-S settings, provided combination and marginalization operations are defined.

2. Composite Explanations via Generative Models and Alpha-Blending

Composite explanation also arises in deep generative modeling, notably in composite generative adversarial networks (CGANs) wherein complex image structure is modeled through the sequential contribution of multiple generators (Kwak et al., 2016). In CGAN:

• The image is generated part by part, with each generator responsible for different aspects (e.g., background, face, details).
• A recurrent neural network processes independent noise vectors, producing hidden states for each generator.
• Each generator’s output—a per-pixel RGBA (RGB plus alpha opacity) image—is successively composited via alpha blending:
  • For $t=1$, $$O^{(1)}_{ij,RGB} = C^{(1)}_{ij,RGB} \cdot C^{(1)}_{ij,A}$$.
  • For $t>1$, $$O^{(t)}_{ij,RGB} = (1 - C^{(t)}_{ij,A}) \cdot O^{(t-1)}_{ij,RGB} + C^{(t)}_{ij,A} \cdot C^{(t)}_{ij,RGB}$$.
• Discriminator receives only fully opaque images, ensuring interpretable outputs.

Specialization to distinct image "parts" (e.g., background vs. foreground) emerges unsupervised, driven by adversarial training and augmented by an $\alpha$-regularization loss that forces all generators to contribute. This mechanism enables interpretable decomposition without explicit part annotations or supervision.

3. Compositional Explanations in Semantics and Cognitive Models

In semantic theory and cognitive modeling, compositional explanation rigorously describes how the meanings of complex expressions arise from syntactic and distributional interactions among their parts (Coecke et al., 2015). The compositional distributional model, for instance, illuminates phenomena like the “pet fish” paradox, wherein the typicality of “goldfish” for “pet fish” exceeds its typicality for either “pet” or “fish” in isolation.

Key aspects include:

• Each word is mapped to a vector or tensor in a semantic space, constructed via corpus statistics or attribute elicitation.
• Compositional operations are governed by grammatical roles, encoded as types in a pregroup grammar.
• For adjective–noun constructs, the adjective is modeled as a matrix acting on the noun:
  • If “pet” (adjective) is a matrix $\underline{pet}$ and “fish” a vector $|fish\rangle$, then “pet fish” is $\underline{pet} \times |fish\rangle$.
• The observed overextension (e.g., goldfish becomes a prototypical “pet fish”) arises naturally from matrix–vector composition and is quantifiable using cosine similarity.

Extensions of the framework model non-commutativity (“pet fish” ≠ “fishy pet”), inheritance failure, attribute emergence, and conjunction fallacy, giving a unified account for a wide range of compositional anomalies in concept formation.

4. Composite Explanations in Unified Theories of Composites

Composite explanation in micromechanics addresses the modeling of matrix composite materials with complex, possibly random, periodic, or deterministic microstructures (Buryachenko, 16 Mar 2025). The Additive General Integral Equation (AGIE) provides an exact, universal formulation:

• For a composite subject to a body force $b(x)$ of compact support, the AGIE calculates the field perturbations due to each inclusion, integrating their effects via one- and two-point density functions.
• Iterative solution schemes ensure mutual field consistency (“quasicrystalline” constraint) across all inclusions.
• The Representative Volume Element (RVE) concept is redefined: a domain $R$ is representative if the computed effective operators stabilize outside $R$ for all drives $b(x)$ with $\text{supp}\,b \subset R$.
• This RVE framework filters data for machine learning models, enabling clean, surrogate operator learning (e.g., neural operators $u(x)\approx\Psi[b(\cdot)](x)$) free from sample size and edge effects.
• The approach supports arbitrary linear/nonlinear, local/nonlocal, coupled multiphysics models and deterministic structures, subsuming classical and modern micromechanics.

Composite explanation here unifies analytic, statistical, and data-driven predictions of composite behavior from microscale to macroscale, spanning random, periodic, and deterministic topologies and arbitrary phase laws.

5. Composite Explanation in Particle Physics and Higgs Sector Models

Composite explanations feature in high-energy physics within composite Higgs models, which interpret the observed Higgs boson and related phenomena via strong-sector dynamics and symmetry breaking (Barducci et al., 2013, Carmona et al., 2016). In such models:

• The Higgs doublet arises as a pseudo–Nambu–Goldstone boson (pNGB) from global symmetry breaking (e.g., SO(5)/SO(4)).
• The effective Higgs potential, mass, and couplings are determined by loop contributions from gauge and fermion composite partners.
• Observed LHC signal strengths (e.g., $R_{\gamma\gamma}$, $R_{WW}$) are explained by composite operator mixing, modified coupling strengths ($\kappa_V$, $\kappa_f$), and resonant spectra. Benchmark scenarios demonstrate improved fits relative to the Standard Model (Barducci et al., 2013).
• In flavor-safe composite Higgs models, the explanation for anomalies such as $R_K$ in B meson decays is achieved by embedding right-handed leptons into extended representations of the global symmetry group (e.g., 14 of SO(5)), which naturally induce lepton-flavor universality violation and parametrically enhanced Higgs-mass corrections consistent with experimental constraints (Carmona et al., 2016).

6. Principles, Mechanisms, and Generalizations of Composite Explanations

Composite explanations in scientific modeling share several foundational principles:

• Decomposition: Breaking down complex phenomena into constituent parts or mechanisms, each modeled or assigned explicative responsibility.
• Composition: Integrating these parts via well-defined mathematical or algorithmic operations (tensor contraction, alpha-blending, operator summation, etc.).
• Emergent interpretation: Allowing unsupervised or data-driven specialization of parts (e.g., image layers in generative models, conceptual attributes in semantics).
• Generalization: Applying the composite framework across domains (statistical reasoning, language, physics, materials science) with appropriate formalism adjustments (probability, belief functions, algebraic symmetries, continuum operators).
• Scalability and modularity: Supporting block-based, independent module development for software and analytical frameworks (Buryachenko, 16 Mar 2025).

A plausible implication is that composite explanation serves as a foundational methodology for advancing interpretable, scalable, and domain-agnostic scientific modeling and reasoning, particularly when complexity precludes monolithic or non-compositional approaches.

7. Illustrative Examples and Applications

Select examples illustrating composite explanation include:

• Generating interpretable images part by part in CGAN, enabling foreground/background separation without supervision (Kwak et al., 2016).
• Modeling semantic typicality (“pet fish” paradox) via compositional distributional semantics (Coecke et al., 2015).
• Solving k-MPE problems in high-dimensional Bayesian, VBS, and Dempster–Shafer systems via genetic algorithm–based optimization (Wierzchoń et al., 2018).
• Predicting microstructural composite response and constructing surrogate nonlocal operators using AGIE, RVE, and data-driven machine learning (Buryachenko, 16 Mar 2025).
• Accounting for LHC Higgs boson signal anomalies and lepton flavor universality violations within composite Higgs models leveraging multi-representation embeddings and operator mixing (Barducci et al., 2013, Carmona et al., 2016).

These examples evidence the pervasiveness and utility of composite explanation as an interpretive and generative paradigm across mathematical, empirical, and computational sciences.