Full Negation Belief Transformation (FNBT)

• FNBT is an open-world fusion method based on Dempster–Shafer theory that transforms heterogeneous belief functions into a unified extended frame.
• It applies a full negation mechanism to convert local mass functions, eliminating essential conflicts and maintaining valid DST properties.
• FNBT enhances multi-source classification and decision-making, outperforming standard fusion techniques in accuracy and robustness.

Full Negation Belief Transformation (FNBT) is an open-world information fusion method built on Dempster-Shafer theory (DST) of evidence. It is designed for the case in which two or more sources produce belief or mass functions on heterogeneous frames of discernment, so that ordinary closed-world fusion assumptions no longer hold. In the formulation explicitly termed FNBT, the method introduces an open-world criterion, extends the relevant frames, applies a full negation mechanism to transform the mass functions, and then uses existing combination rules on the transformed evidence (He et al., 11 Aug 2025). The broader negation literature in DST also contains an earlier negation transformation for belief structures based on maximum uncertainty allocation, which maps singleton focal elements to their complements and pools nonsingleton mass into the whole frame (Deng et al., 2019).

1. Conceptual setting and scope

In DST, a frame of discernment $\Theta$ is a set of mutually exclusive and exhaustive hypotheses, and a mass function is defined on the power set $2^\Theta$. FNBT addresses the case in which evidence sources do not share a single common frame. The motivating examples are explicitly open-world: trained algorithms or data may originate from different regions or organizations, data silos may prevent joint retraining or raw-data sharing, and different sources may therefore recognize different subsets of hypotheses (He et al., 11 Aug 2025).

The canonical illustration is heterogeneous classification. One source may operate on $\Theta_A=\{\text{Setosa},\text{Versicolor}\}$, while another operates on $\Theta_B=\{\text{Versicolor},\text{Virginica}\}$. In such cases, projecting everything to the intersection of frames causes information loss, choosing one source’s frame as the fusion universe causes information distortion, and direct application of standard combination rules can mis-handle conflict because support for hypotheses absent from another source’s frame is treated as if it were ordinary closed-world incompatibility rather than structural heterogeneity (He et al., 11 Aug 2025).

FNBT therefore belongs to a specific subclass of belief transformation methods: it is not a generic negation operator for arbitrary logics, but a DST-based mechanism for converting heterogeneous-frame fusion into a transformed fusion problem on an extended effective frame. The formulation given in the FNBT paper states this directly: FNBT converts open-world, heterogeneous-frame fusion into a transformed closed-world fusion problem on an extended effective frame (He et al., 11 Aug 2025).

2. Open-world criterion and the failure of direct fusion

The FNBT framework introduces an open-world criterion based on essential conflict. Let $m_1$ and $m_2$ be mass functions on frames $\Theta_1$ and $\Theta_2$, let $F_1$ and $F_2$ be their sets of focal elements, and let $2^\Theta$0 be the set of essential conflict elements. A set $2^\Theta$1 is an essential conflict set if for every $2^\Theta$2, there exists a focal element $2^\Theta$3 such that $2^\Theta$4, and for the other source $2^\Theta$5, that focal element is disjoint from all focal elements of the other mass function: $2^\Theta$6 If $2^\Theta$7, the fusion problem satisfies the open-world criterion; if $2^\Theta$8, it is a closed-world fusion problem (He et al., 11 Aug 2025).

This criterion is stronger than a generic statement that “the sources conflict.” It isolates hypotheses that are supported by one source but structurally incompatible with the focal structure of the other source. The FNBT analysis associates two specific pathologies with direct use of Dempster’s rule when the open-world criterion holds. The first is plausibility absolutization: there exists some $2^\Theta$9 such that

$\Theta_A=\{\text{Setosa},\text{Versicolor}\}$0

but

$\Theta_A=\{\text{Setosa},\text{Versicolor}\}$1

A hypothesis that was plausible in at least one source can therefore become impossible after fusion (He et al., 11 Aug 2025).

The second pathology is assertion of closure. If $\Theta_A=\{\text{Setosa},\text{Versicolor}\}$2, then with Dempster’s combination result $\Theta_A=\{\text{Setosa},\text{Versicolor}\}$3, the plausibility of the consensus set

$\Theta_A=\{\text{Setosa},\text{Versicolor}\}$4

satisfies

$\Theta_A=\{\text{Setosa},\text{Versicolor}\}$5

and remains invariant under further combination with new evidence. In the language of the paper, fusion collapses onto the non-conflicting part and then becomes effectively uncorrectable (He et al., 11 Aug 2025).

These results delimit FNBT’s target problem. The method is not introduced as a general replacement for Dempster’s rule; it is introduced because ordinary combination on heterogeneous frames may erase plausibility or impose a spurious closed-world consensus once essential conflict is present.

3. Full negation mechanism

FNBT handles open-world heterogeneous fusion through four steps stated explicitly in the source formulation: detecting open-world cases, extending the frame, transforming each mass function using a full negation interpretation, and then applying ordinary combination rules on the transformed masses (He et al., 11 Aug 2025).

The first formal ingredient is the effective frame of each source: $\Theta_A=\{\text{Setosa},\text{Versicolor}\}$6 Only singleton hypotheses that actually appear in nonzero-mass focal sets are retained. The second ingredient is the extended effective frame: $\Theta_A=\{\text{Setosa},\text{Versicolor}\}$7 This $\Theta_A=\{\text{Setosa},\text{Versicolor}\}$8 is the common domain on which the transformed mass functions are defined (He et al., 11 Aug 2025).

The paper’s full negation interpretation is expressed by rewriting support for a set $\Theta_A=\{\text{Setosa},\text{Versicolor}\}$9 as support for the negation of everything outside $\Theta_B=\{\text{Versicolor},\text{Virginica}\}$0, relative to $\Theta_B=\{\text{Versicolor},\text{Virginica}\}$1. The source formula is

$\Theta_B=\{\text{Versicolor},\text{Virginica}\}$2

where $\Theta_B=\{\text{Versicolor},\text{Virginica}\}$3 denotes the negation of element $\Theta_B=\{\text{Versicolor},\text{Virginica}\}$4. Rewritten over the extended frame,

$\Theta_B=\{\text{Versicolor},\text{Virginica}\}$5

Here $\Theta_B=\{\text{Versicolor},\text{Virginica}\}$6 is the transformed mass function on $\Theta_B=\{\text{Versicolor},\text{Virginica}\}$7 (He et al., 11 Aug 2025).

The paper’s stated intuition is that if a source says “$\Theta_B=\{\text{Versicolor},\text{Virginica}\}$8 is supported,” FNBT interprets this as support for the complement of everything outside $\Theta_B=\{\text{Versicolor},\text{Virginica}\}$9, but within the correct extended universe $m_1$0. This is the specific sense in which the method is “full negation”: the transformation is not performed inside the source’s incomplete local frame, but inside a frame that includes the hypotheses needed to represent essential conflict (He et al., 11 Aug 2025).

After transformation, standard closed-world operators can be used: $m_1$1 The same paper distinguishes three decision strategies computed from the transformed masses, namely FNBT-Mass, FNBT-Bel, and FNBT-Pl, using the usual formulas

$m_1$2

The mechanism is thus explicitly designed to be compatible with existing DST decision functions rather than to replace them (He et al., 11 Aug 2025).

4. Formal properties

The FNBT paper proves three properties: mass function invariance, heritability, and essential conflict elimination (He et al., 11 Aug 2025).

Mass function invariance states that if $m_1$3 is a mass function on frame $m_1$4, and $m_1$5 is an essential conflict set, then after FNBT the transformed function $m_1$6 on the extended effective frame $m_1$7 satisfies

$m_1$8

The transformed object therefore remains a valid mass function under DST (He et al., 11 Aug 2025).

Heritability states that when $m_1$9, FNBT preserves the original mass functions and their fusion: $m_2$0

$m_2$1

and

$m_2$2

The source interpretation is straightforward: if the setting is actually closed-world, FNBT behaves exactly like standard DST and changes nothing (He et al., 11 Aug 2025).

Essential conflict elimination states that when $m_2$3, FNBT eliminates all essential conflicts between the transformed mass functions: $m_2$4 where $m_2$5 and $m_2$6 are the focal sets of the transformed masses. After transformation, focal elements from different sources are no longer fundamentally disjoint (He et al., 11 Aug 2025).

Taken together, these theorems characterize FNBT as a conservative extension of DST fusion machinery. It preserves the mathematical status of mass functions, reduces to the identity in the absence of essential conflict, and specifically targets the focal-set disjointness responsible for open-world failure modes.

5. Relation to earlier negation transformations in Dempster-Shafer theory

An earlier DST negation transformation is given in "On the negation of a Dempster-Shafer belief structure based on maximum uncertainty allocation" (Deng et al., 2019). There, the negation of a singleton $m_2$7 is interpreted as “anything except $m_2$8”: $m_2$9 and for a set $\Theta_1$0, the negation is defined as the union of the negations of all elements in $\Theta_1$1: $\Theta_1$2 Operationally, if $\Theta_1$3 is a singleton $\Theta_1$4, its negation is $\Theta_1$5; if $\Theta_1$6 is not a singleton, its negation becomes the whole frame $\Theta_1$7 (Deng et al., 2019).

The associated belief-structure transformation is

$\Theta_1$8

which simplifies to

$\Theta_1$9

Equivalently, singleton mass goes to $\Theta_2$0, while all nonsingleton mass is pooled into $\Theta_2$1 (Deng et al., 2019).

This earlier transformation proves several structural properties. If $\Theta_2$2 is any reasonable uncertainty measure on belief structures attaining its maximum at the vacuous belief structure $\Theta_2$3, then for $\Theta_2$4,

$\Theta_2$5

while for $\Theta_2$6,

$\Theta_2$7

It also proves that for $\Theta_2$8,

$\Theta_2$9

whereas for $F_1$0,

$F_1$1

In addition, the paper shows compatibility with Yager’s probabilistic negation

$F_1$2

when the belief structure is Bayesian and the set-valued negation is flattened back into probabilities (Deng et al., 2019).

This comparison suggests a useful distinction. The 2019 transformation is a negation operator for belief structures under a maximum uncertainty allocation principle, whereas the 2025 FNBT method is an open-world fusion procedure for heterogeneous frames that uses full negation relative to an extended effective frame. Both are negation transformations within DST, but they solve different formal problems: the former defines how a belief structure is negated; the latter uses negation to repair heterogeneous-frame fusion (Deng et al., 2019, He et al., 11 Aug 2025).

6. Related non-classical belief and negation frameworks

Negation-centered belief transformation also appears outside ordinary DST frame fusion. "Updating belief functions over Belnap--Dunn logic" develops belief and plausibility update in a setting explicitly designed for incomplete and contradictory information (Frittella et al., 2022). In that framework, Belnap-Dunn frame semantics defines negation by swapping positive and negative support: $F_1$3 The paper defines

$F_1$4

yielding the identity

$F_1$5

A central consequence is that after updating by a positive formula, belief in the negated formula need not be zero, because contradictory support can survive updating (Frittella et al., 2022).

A different line of work studies strong negation at the proof-theoretic level. "Strong Negation is Definable in 2Int" shows that in Wansing’s bilateral bi-intuitionistic logic $F_1$6, strong negation is definable via

$F_1$7

The result is proof-theoretic rather than a DST belief-revision result: the paper derives the bilateral introduction and elimination rules for strong negation from that formula and argues that negation can be represented as a structural transformation between proof and dual proof (Oddsson, 23 Jan 2025). The source explicitly states that it does not discuss Full Negation Belief Transformation by name, but it is conceptually relevant to negation-transformation ideas.

Negation is also treated as an interpretation-transforming mechanism in semiring-based nonmonotonic logic programming. "A Unifying Framework for Semiring-Based Constraint Logic Programming With Negation" does not mention FNBT by name, but introduces a generalized negation-as-failure semantics

$F_1$8

together with AFT-based operators such as the immediate consequence operator $F_1$9, the ultimate approximator $F_2$0, the generalized Fitting-style approximator $F_2$1, and the stable operator $F_2$2 (Spaans et al., 21 Jul 2025). This suggests that the phrase “negation belief transformation” covers several distinct research programs: DST mass transformation, paraconsistent belief updating, bilateral proof-theoretic definability, and approximation-based nonmonotonic semantics.

7. Evaluation, counterexamples, and applications

The FNBT paper evaluates the method on real classification tasks using the UCI Iris, Seeds, and Wine datasets. The reported dataset characteristics are: Iris, 150 samples, 3 classes, 4 attributes; Seeds, 210 samples, 3 classes, 7 attributes; Wine, 178 samples, 3 classes, 13 attributes. Different attributes are treated as independent sources of information, and for Wine the experiments use SMOTE to mitigate class imbalance (He et al., 11 Aug 2025).

The experimental setup uses 10-fold cross-validation and reports Accuracy and Macro-F1. To create an open-world heterogeneous setting, the training data is split into two disjoint subsets with partially overlapping label sets; for labels $F_2$3, one source may use $F_2$4 and the other $F_2$5. For each test sample, $F_2$6 pieces of evidence are generated from training set $F_2$7 and another $F_2$8 from training set $F_2$9; within each subset, evidence is fused first using Dempster’s rule to obtain $2^\Theta$00 and $2^\Theta$01, and then $2^\Theta$02 and $2^\Theta$03 are fused using FNBT or baseline methods (He et al., 11 Aug 2025).

Among the three decision variants, FNBT-Pl performs best overall. The reported results are: on Iris, 93.33 $2^\Theta$04 4.22% accuracy and 93.18 $2^\Theta$05 4.39% macro-F1; on Seeds, 86.67 $2^\Theta$06 5.13% accuracy and 85.91 $2^\Theta$07 5.60% macro-F1; on Wine, 86.05 $2^\Theta$08 7.86% accuracy and 85.88 $2^\Theta$09 8.14% macro-F1. The baselines include Dempster’s rule, Yager’s rule, TBM, mGCR, and ETV-MSIF. On the Iris dataset, the reported accuracies are 33.33% for Dempster’s, 33.33% for Yager’s, 33.33% for TBM, 33.33% for mGCR, 66.67% for ETV-MSIF, and 93.33% for FNBT-Pl (He et al., 11 Aug 2025).

The paper also gives a worked treatment of Zadeh’s counterexample. With

$2^\Theta$10

and

$2^\Theta$11

the essential conflict set is

$2^\Theta$12

the effective frames are

$2^\Theta$13

and thus

$2^\Theta$14

The transformed masses are

$2^\Theta$15

and

$2^\Theta$16

After Dempster combination on transformed masses,

$2^\Theta$17

The paper interprets this as resolving Zadeh’s paradox because the dominant mass remains on $2^\Theta$18 rather than collapsing onto the weakly supported singleton $2^\Theta$19 (He et al., 11 Aug 2025).

The applications explicitly identified for FNBT are multi-source classification, distributed decision systems, cross-organization intelligence fusion, fault diagnosis, target recognition, recommendation systems, and settings with data silos and partial label overlap (He et al., 11 Aug 2025). The stated limitations are equally specific: the method is designed for cases in which an open-world criterion can be identified through essential conflict, it still depends on how mass functions are initially generated, and the reported experiments focus on classification tasks rather than a broader range of application domains (He et al., 11 Aug 2025).