D Numbers Theory: Complex Belief Functions

• D Numbers Theory is a framework that generalizes Dempster-Shafer theory by employing complex-valued basic belief assignments to capture oscillatory and phase-dependent uncertainties.
• It introduces a generalized combination rule and relaxed conflict management, allowing effective fusion of evidence even under high or phase-induced conflicts.
• Novel entropy measures, like Fractal-Based Complex Belief entropy, quantify uncertainty through self-similar mass redistributions, enhancing applications such as sensor fusion and uncertainty-aware machine reasoning.

D Numbers Theory, often referred to in the literature as Complex Evidence Theory (CET) or the theory of complex belief functions, generalizes classical Dempster-Shafer evidence theory by modeling mass assignments as complex-valued functions. This framework fundamentally enhances the expressivity of uncertainty models by incorporating oscillatory phenomena, phase-dependent uncertainty, and richer fusion mechanisms—addressing scenarios where real-valued approaches are limited, particularly in the presence of time-varying or interference-prone evidence. D Numbers Theory relaxes the restrictive real-valued additivity constraints, supports novel entropy functionals respecting complex-valued uncertainties, and links naturally to applications in epistemic uncertainty quantification, sensor fusion with periodic/cyclic data, and uncertainty-aware machine reasoning.

1. Mathematical Foundations of Complex Mass Assignments

Let $\Theta$ denote a finite frame of discernment, and $2^\Theta$ its power set. D Numbers Theory replaces the standard real-valued basic belief assignment (BBA) $m:2^{\Theta}\to[0,1]$ with a complex basic belief assignment (CBBA)

$m:2^{\Theta}\longrightarrow\mathbb{C}$

subject to

$$m(\varnothing)=0, \qquad \sum_{A\subseteq\Theta} m(A)=1,$$

with each $m(A)$ expressible in polar form

$$m(A)=\mathbf{m}(A)\exp(i\theta(A)),\quad \mathbf{m}(A)\in[0,1],\;\;\theta(A)\in[-\pi,\pi],$$

or equivalently $m(A)=x_A + i\,y_A$ with $|m(A)|\in[0,1]$. This generalization preserves core normalization while allowing each focal element a phase factor that can represent oscillation, periodicity, or interference, unattainable in the real-valued regime (Xiao, 2019, Xiao, 2019, Wu et al., 2023).

2. Generalized Combination Rule and Relaxed Conflict Handling

Given two independent CBBAs $m_1$ and $m_2$ over $\Theta$, the generalized Dempster's rule of combination is defined as

$m(C) = \frac{1}{1-K}\sum_{\substack{A,B\subseteq\Theta\A\cap B=C}} m_1(A)m_2(B), \qquad m(\varnothing)=0,$

where the (generally complex) conflict coefficient is

$K = \sum_{\substack{A,B\subseteq\Theta\A\cap B=\varnothing}} m_1(A)m_2(B) \in \mathbb{C}.$

The only requirement is $1-K\neq0$, relaxing the classical $K<1$ condition. The consequence is tolerance of high-conflict and phase-induced interference scenarios; purely imaginary or non-vanishing interference effects can be explicitly captured by the complex structure of $K$ (Xiao, 2019). In the real case, this formula reduces exactly to Dempster's rule under $K<1$.

For example, with $\Theta=\{\theta_1,\theta_2\}$, explicit numerical calculations exhibit nontrivial complex masses and confirm that the summed CBBA is normalized in $\mathbb{C}$ with resultant $m(\theta_1)+m(\theta_2)+m(\Theta)=1$.

3. Generalized Belief and Plausibility Functions

To extend the classical belief/plausibility framework, define the commitment degree for each $A\subseteq\Theta$

$$\operatorname{Com}(A) = \frac{|m(A)|}{\sum_{\varnothing\neq B\subseteq\Theta} |m(B)|},$$

and set

$$\mathrm{Bel}(A)=\sum_{B\subseteq A} \operatorname{Com}(B),\qquad \mathrm{Pl}(A)=1-\mathrm{Bel}(A^c)=\sum_{B\cap A\neq\varnothing} \operatorname{Com}(B).$$

These retain core properties: normalization, monotonicity, and duality via complementation. When all phases vanish, one recovers exactly the classical Dempster-Shafer formulas (Xiao, 2019).

4. Quantifying Uncertainty: Fractal-Based Complex Entropy

Quantification of uncertainty in CET leverages the Fractal-Based Complex Belief entropy $\mathbb{E}_{FCB}$ (Wu et al., 2023). This is constructed by a two-step fractal redistribution of mass (reflecting the self-similar, recursive splitting inherent to evidence transfer) followed by a Shannon-style entropy calculation on the normalized support degrees. The formal steps:

• Fractal Redistribution: For each $A\subseteq\Theta$,

$$\mathbb{M}_F(A) = \frac{\mathbb{M}(A)}{2^{|A|}-1} + \sum_{B\supset A,\,|B|>|A|} \frac{\mathbb{M}(B)}{2^{|B|}-1}$$

• Support degree: $$\mathrm{Com}_F(A)=\frac{|\mathbb{M}_F(A)|}{\sum_{C\subseteq\Theta}|\mathbb{M}_F(C)|}$$
• FCB entropy:

$$\mathbb{E}_{FCB}(\mathbb{M}) = -\sum_{A\subseteq\Theta} \mathrm{Com}_F(A)\log \mathrm{Com}_F(A)$$

Key properties are non-negativity, boundedness $[0, \log(2^n-1)]$, additivity for independent frames, subadditivity otherwise, and monotonic increase under exponential negation. Unlike classical entropies, FCB entropy distinguishes otherwise unresolved intersecting-focal-set patterns and is sensitive to phase interference (Wu et al., 2023).

5. Expressivity, Theoretical Properties, and Applications

D Numbers Theory confers the following advances:

• Oscillatory and Phase-Dependent Uncertainty: The phase factors $e^{i\theta(A)}$ model interference between pieces of evidence, seasonal/cyclic uncertainties, or synchronization (e.g., in sensor fusion with periodic signals or phase-dependent confidences) (Xiao, 2019).
• Complex Conflict Mitigation: The imaginary part of $K$ may counteract or mediate high real-valued conflicts, offering greater flexibility for evidence fusion and alleviating dampening effects known in traditional rules (Xiao, 2019).
• Generalized Decision Support: Pignistic transformations and probabilistic logic integration (via capacity logic programs) exploit belief/plausibility intervals derived from CBBAs, yielding epistemically robust decisions in uncertain or hierarchically structured settings (Azzolini et al., 23 Jul 2025). For instance, neural classifier outputs distributed across internal and leaf nodes can be interpreted as complex BBAs, leading to credible intervals for coarse- and fine-grained predictions.

A concrete example with $\mathbb{M}(\{x_1\})=0.2+0.3i$, $\mathbb{M}(\{x_2\})=0.5-0.1i$, $\mathbb{M}(\{x_1,x_2\})=0.3-0.2i$ demonstrates how CPBT yields redistributed probabilities to singletons, while fractal redistribution allows computation of FCB entropy, sensitive both to magnitude and phase.

6. Connections, Limitations, and Computational Aspects

CET encompasses standard DST as a special case, preserving Smets–Shafer decomposition, belief/plausibility conjugacy, and key axioms (0811.3373). D Numbers Theory is distinct from lower-probability or interval-based frameworks (e.g., credal sets), as it directly leverages complex amplitudes at the mass-functional level. The computational complexity for full FCB entropy is $O(n2^n)$, which is exponential in $|\Theta|$, necessitating approximation or focal-set pruning for large domains (Wu et al., 2023).

Classical combination is recovered when masses are real, and application of generalized rules to real-valued masses exactly reconstructs standard DST (Xiao, 2019). A further implication is the capacity to distinguish epistemic from aleatory uncertainties in joint probabilistic–belief logic systems (Azzolini et al., 23 Jul 2025).

7. Summary and Significance

D Numbers Theory generalizes the Dempster-Shafer framework by endowing mass functions with complex structure, relaxing restrictive combination constraints, and formalizing entropy measures that intrinsically reflect interference and self-similar redistribution of uncertainty. Its mathematical infrastructure is compatible with real-valued DST, but uniquely captures phase-dependent, oscillatory, and interference phenomena, positioning the framework as a promising tool for advanced uncertainty quantification in data fusion, inference under epistemic ambiguity, and reasoning in systems exhibiting periodic or cyclic information patterns (Xiao, 2019, Wu et al., 2023, Xiao, 2019, Azzolini et al., 23 Jul 2025).