Complex Belief Functions

• Complex belief functions are an extension of Dempster–Shafer theory that uses complex numbers to represent both magnitude and phase in uncertainty quantification.
• They preserve core properties such as normalization, monotonicity, and duality by computing belief and plausibility via the moduli of complex masses.
• They introduce advanced combination rules and a fractal-based entropy measure, enabling robust modeling of interference, cyclic behavior, and epistemic fusion.

Complex belief functions generalize Dempster–Shafer evidence theory by extending basic belief assignments from the real line to the complex plane, enabling richer representation of uncertainty—including phase, interference, and cyclicity. Developed through independent mathematical frameworks and motivated by needs in engineered and natural information systems, complex belief functions admit combination, decomposition, and uncertainty quantification analogous to their real-valued classical counterparts, but can express phenomena such as quantum-inspired effects, oscillatory support, and multidimensional epistemic fusion.

1. Formal Definition and Structure of Complex Belief Functions

Let $\Theta$ be a finite frame of discernment and $2^\Theta$ its power set. A complex basic belief assignment (CBBA) is defined as a mapping

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

subject to $m(\varnothing) = 0$ and the complex normalization condition $\sum_{A \subseteq \Theta} m(A) = 1$ (in $\mathbb{C}$) (Xiao, 2019, Xiao, 2019). Each focal mass $m(A)$ is written in polar form $r(A) e^{i\phi(A)}$ with modulus $r(A) \in [0,1]$ and phase $\phi(A) \in [-\pi, \pi)$. The normalization enforces

$2^\Theta$0

Belief and plausibility functions are generalized using the absolute-value or modulus of the complex mass: $2^\Theta$1 under the normalization $2^\Theta$2. For any $2^\Theta$3,

$2^\Theta$4

These functions coincide with the classical Dempster–Shafer belief/plausibility functions when all phases vanish (i.e., $2^\Theta$5), thus showing reduction to the standard theory (Xiao, 2019).

2. Algebraic Properties and Theoretical Guarantees

Complex belief functions, constructed as above, retain the main structural properties of classical belief functions but extend them to capture interference-like phenomena:

• Normalization: $2^\Theta$6, $2^\Theta$7; $2^\Theta$8, $2^\Theta$9.
• Monotonicity and Inclusion–Exclusion: For collections $m : 2^{\Theta} \longrightarrow \mathbb{C}$0,

$m : 2^{\Theta} \longrightarrow \mathbb{C}$1

• Möbius Inversion: Commitment degrees (moduli of the complex masses) are recovered from $m : 2^{\Theta} \longrightarrow \mathbb{C}$2 via

$m : 2^{\Theta} \longrightarrow \mathbb{C}$3

• Duality: $m : 2^{\Theta} \longrightarrow \mathbb{C}$4 (Xiao, 2019).

The extension to complex-valued masses preserves the capacity-theoretic foundation: complex belief functions are normalized, $m : 2^{\Theta} \longrightarrow \mathbb{C}$5-valued capacities and satisfy generalized monotonicity. This guarantees consistency with lower/upper probability theory in the appropriate real-valued limit (Smets, 2013).

3. Fusion and Combination: Generalized Dempster’s Rule

A foundational operation is the combination (orthogonal sum) of two independent belief sources. For complex CBBAs $m : 2^{\Theta} \longrightarrow \mathbb{C}$6, the generalized Dempster combination rule is: $m : 2^{\Theta} \longrightarrow \mathbb{C}$7

$m : 2^{\Theta} \longrightarrow \mathbb{C}$8

provided $m : 2^{\Theta} \longrightarrow \mathbb{C}$9. Unlike the real case, the modulus of $m(\varnothing) = 0$0 may exceed unity, allowing combination under strong or “oscillatory” conflict, and the only algebraic constraint is non-vanishing denominator (Xiao, 2019). If all phases vanish, the rule reduces exactly to classical DST.

No explicit orthogonal sum for arbitrary complex masses is given in (Xiao, 2019); the structure above is developed in (Xiao, 2019).

4. Uncertainty Quantification: Fractal-Based Entropy

Quantifying total uncertainty for complex belief functions necessitates new entropy measures. The Fractal-based Complex Belief (FCB) entropy is designed for CBBAs. It reallocates the complex mass $m(\varnothing) = 0$1 of each focal set $m(\varnothing) = 0$2 among all nonempty sub-subsets $m(\varnothing) = 0$3 in proportion $m(\varnothing) = 0$4, as motivated by fractal-splitting arguments (Wu et al., 2023):

For nonempty $m(\varnothing) = 0$5,

$m(\varnothing) = 0$6

and the normalized support degree is

$m(\varnothing) = 0$7

The FCB entropy is then

$m(\varnothing) = 0$8

Properties:

• Recovers Shannon entropy when $m(\varnothing) = 0$9 is singleton-supported.
• Reduces to fractal-based entropy for real-valued BBAs.
• Is additive for products and monotone under negation.
• Decomposes as $\sum_{A \subseteq \Theta} m(A) = 1$0, with $\sum_{A \subseteq \Theta} m(A) = 1$1 (discord) measuring discord via the Complex Pignistic transformation and $\sum_{A \subseteq \Theta} m(A) = 1$2 (non-specificity) capturing ambiguity due to multi-element focal sets (Wu et al., 2023).

Numerical simulation on target detection shows FCB entropy is more sensitive to both discord and non-specificity compared to previous measures.

5. Calibration, Epistemic Uses, and Lattice Extensions

Complex belief functions subsume classical models of uncertainty and extend naturally to arbitrary lattices (0811.3373). For a finite lattice $\sum_{A \subseteq \Theta} m(A) = 1$3, a belief function $\sum_{A \subseteq \Theta} m(A) = 1$4 with $\sum_{A \subseteq \Theta} m(A) = 1$5, $\sum_{A \subseteq \Theta} m(A) = 1$6 induces

$\sum_{A \subseteq \Theta} m(A) = 1$7

with related commonality and plausibility functions. Dempster’s rule, Möbius inversion, and decomposition into simple support functions generalize to these non-Boolean domains. This allows representation of complex epistemic and hierarchical relationships, crucial for integrating logic-programming with belief models (Azzolini et al., 23 Jul 2025, Black et al., 2013).

In the context of probabilistic logic programming, belief functions provide an interval-valued (capacity-based) alternative to point-wise probabilities. Capacity Logic Programs (CaLPs) incorporate mass functions as first-class domains, propagate lower and upper probabilities as belief/plausibility, and yield interval-valued query probabilities, thus quantifying epistemic gaps and non-additivity of evidence (Azzolini et al., 23 Jul 2025).

6. Illustrative Phenomena and Applications

By embedding mass assignments in the complex plane, the phase degree of freedom enables complex belief functions to model interference patterns, cyclic hypothesis dependencies, and time-varying uncertainty:

• Signal-processing fusion: Models incorporating CBBA can explicitly represent phase-coherent uncertainty in multisensor arrays.
• Human-machine teaming: Phase enables representations of context-dependent oscillatory or cyclic shifts in expert opinions.
• Decision making: Quantum-inspired settings benefit from modeling interference and resonance between alternative hypotheses.

In canonical examples, distinct CBBA patterns with identical marginal moduli can be distinguished with FCB entropy, revealing subtler distinctions invisible to standard measures (Wu et al., 2023, Xiao, 2019). In probabilistic logic settings, belief intervals and capacities quantify epistemic uncertainty and distinguish knowledge-level uncertainty from statistical (aleatory) risk (Azzolini et al., 23 Jul 2025).

7. Outlook and Comparative Remarks

Complex belief functions unify and extend probability measures, Dempster–Shafer evidence theory, and capacity theory. Their algebraic and uncertainty-quantifying frameworks admit reduction to classical theory, support generalization to discrete and lattice-theoretic domains, and offer new expressivity for epistemic modeling.

Main distinctions relative to classical DST:

• Complex masses encode both support magnitude and phase, modeling oscillatory, periodic, or interference-rich evidence.
• Combination rules are extended to allow for arbitrary, potentially highly conflicting evidence, subject only to nonvanishing denominator.
• Entropy and uncertainty indexes such as FCB entropy are both more discriminative and naturally decomposable into discord and non-specificity, informed by multiscale (fractal) structure.

This approach is relevant wherever epistemic uncertainty exhibits cycles, interference, or context-dependency—areas not adequately addressed by real-valued evidence models (Xiao, 2019, Xiao, 2019, Wu et al., 2023).