Probabilistic Inclusion Operator

• The Probabilistic Inclusion Operator is a framework that generalizes traditional set inclusion to uncertain, fuzzy, and conditional domains.
• It employs methodologies such as fuzzy mask integration, probabilistic team semantics, preferential description logics, and coherence-based inference.
• Its applications span ensemble visualization, logical reasoning, and AI, providing robust measures for uncertainty quantification and computational efficiency.

A probabilistic inclusion operator is a formal mechanism for quantifying or asserting inclusion relations under uncertainty, extending the classical set-theoretic inclusion $A\subseteq B$ to probabilistic, fuzzy, or conditional domains. Such operators appear in ensemble visualization of scalar fields (as in the Probabilistic Inclusion Depth framework), in logics with probabilistic team semantics, in description logics with probabilistic typicality, and in coherence-based probabilistic inference for conditional events.

1. Formal Definitions and Variants

Several instantiations of probabilistic inclusion operators have been introduced, each tailored to the structure of the domain:

• Fuzzy Contours and Scalar Fields: Given fuzzy masks $u,v:\Omega\to[0,1]$ on a measurable domain $(\Omega,\mathcal{F},\mu)$, the operator $u\subset_p v$ is defined as

$$u \subset_p v := \mathbb{E}_{X\sim\pi_u}[v(X)] = \frac{1}{m(u)}\int_\Omega u(x)v(x)d\mu(x),$$

where $m(u)=\int_\Omega u(x)d\mu(x)$ and $\pi_u$ is the normalized measure induced by $u$. For indicator masks, this specializes to $\mu(A\cap B)/\mu(A)$, matching the continuous-subset operator of $\varepsilon$-Inclusion Depth (Wu et al., 17 Dec 2025).

• Probabilistic Team Semantics: The probabilistic inclusion atom in first-order logic, often realized as the marginal identity atom $x_1\dots x_k\approx y_1\dots y_k$, requires that the marginal distributions of these $k$-tuples coincide under a probabilistic team, that is,

$$\forall \vec a\in A^k: X_{x_1\dots x_k=\vec a} = X_{y_1\dots y_k=\vec a}$$

where $X_{x_1\dots x_k=\vec a}$ is the total mass for the assignment $\vec a$ to $x_1,\dots,x_k$ (Hannula et al., 2020).

• Preferential Description Logics: In $\mathcal{ALC}+\mathbb{T}$ with a probabilistic extension, the operator $(C)\sqsubseteq_p D$ expresses that "typically $C$'s are also $D$'s, and the probability of exception is $1-p$," providing a distributed probabilistic semantics based on selections over possible worlds (Giordano et al., 2020).
• Coherence-Based Conditional Probability: The Goodman–Nguyen relation for conditional events $A|H\subseteq E|K$ asserts that $AH\subseteq EK$ and $E^cK\subseteq A^cH$, capturing the logical relation that if $A|H$ is true, so is $E|K$, and if $E|K$ is false, so is $A|H$ (Gilio et al., 2013).

2. Key Properties and Theoretical Foundations

The defining properties of probabilistic inclusion operators are context-specific, but central attributes include:

• Directionality (Asymmetry): $u\subset_p v$ is not generally symmetric in $u$ and $v$.
• Normalization: $u\subset_p v=1$ if $v(x)=1$ $\mu$-a.e. where $u>0$; $u\subset_p v=0$ if $v(x)=0$ there (Wu et al., 17 Dec 2025).
• Linearity and Monotonicity: Linear in $v$, monotonic w.r.t. $v$, scale-invariance in $u$.
• Lipschitz Continuity: Robust to small perturbations in $u$ or $v$.
• Binary Specialization: Recovers classical inclusion for indicator functions.
• Coordinate Agnosticity: Invariant under bi-measurable domain transformations.

In logical frameworks, marginal identity atoms yield a short axiom system: reflexivity, symmetry, projection/permutation, and transitivity (Hannula et al., 2020). In the coherence setting, the Goodman–Nguyen relation is reflexive, antisymmetric, transitive, and monotonic with respect to quasi-conjunctions (Gilio et al., 2013).

3. Computational Methods and Complexity

The evaluation strategy and computational complexity depend on the operator and data structure:

• PID for Fuzzy Contour Ensembles: Naïvely $O(N^2M)$ (for $N$ masks, $M$ voxels); PID-mean reduces to $O(NM)$ by comparing each member $u_i$ only to the mean mask $\bar u$. GPU-parallelization assigns ensemble members to CUDA thread blocks, parallelizes voxel-wise computation, and achieves $10$–$100\times$ speedups over CPU-based algorithms (Wu et al., 17 Dec 2025).
• Probabilistic Team Semantics: Satisfaction of inclusion atoms can be checked in PTIME via reduction to LP feasibility. Adding dependence atoms increases expressivity to NP-completeness (Hannula et al., 2020).
• Probabilistic DLs: TBox and ABox entailment under probabilistic typicality inclusions are both ExpTime-complete, requiring evaluation over $2^n$ worlds in the general case (Giordano et al., 2020).
• Coherence Algorithms: In the coherence setting, identifying the greatest quasi-conjunction subfamily $\mathcal{S}^*$ (entailing a given conditional via inclusion) is achieved through an iterative convex-hull/coherence checking algorithm with at most $n$ steps for $n$ conditionals (Gilio et al., 2013).

4. Illustrative Examples and Use Cases

Representative examples elucidate the behavior of probabilistic inclusion operators:

• Fuzzy Disks and Medical Masks: PID tracks spatial overlap of soft disks or U-Net–derived MRI segmentations, smoothly interpolating inclusion scores as contours shift or become less certain, and producing boxplot visualizations that avoid threshold artifacts (Wu et al., 17 Dec 2025).
• Binary vs. Fuzzy Sensitivity: PID eliminates ranking instability caused by hard thresholding and binary quantization, as demonstrated in scalar field ensembles and weather simulation data (Wu et al., 17 Dec 2025).
• Logical Distributions: In team semantics, marginal identity is illustrated by simple probabilistic assignments for variable pairs, capturing equality of distributions even with different underlying support (Hannula et al., 2020).
• Typicality in Description Logics: KBs with student-sport lover relationships use $(C)\sqsubseteq_p D$ to derive probabilistic entailments in canonical models and compute the probability a given ABox assertion follows (Giordano et al., 2020).
• Conditional Entailment via Inclusion: The greatest subfamily for p-entailment of $C|A$ in $\mathcal{F}=\{C|B,B|A\}$ is the full set, as the quasi-conjunction $C|B \wedge B|A \subseteq C|A$ holds (Gilio et al., 2013).

5. Comparison with Classical and Alternative Inclusion Operators

The probabilistic inclusion operator $\subset_p$ generalizes and often improves upon classical set-theoretic and binary inclusion measures:

Operator/Framework	Fuzziness Support	Threshold Dependencies	Computational Complexity	Outlier Robustness
Set Inclusion ($A\subseteq B$)	None	Full	N/A	All-or-nothing
$\varepsilon$-Inclusion (eID)	None	Hard segmentations	$O(NM)$	Sensitive to threshold
Contour Band Depth (CBD)	None	Hard segmentations	$O(N^2)$ or worse	Band enumeration only
PID ($\subset_p$)	Full (fuzzy)	None	$O(NM)$ (PID-mean)	Outlier-consistent
Fuzzy Dice/prob-IOU	Ensemble mean only	Mean thresholding	$O(NM)$	Outlier-sensitive

PID's pairwise construction improves outlier detection consistency in ensemble settings (e.g., Kendall's $\tau\approx0.96$ vs $0.88$ for mean-based measures), and robustly produces central-orderings unswayed by binary thresholding (Wu et al., 17 Dec 2025). In logic, probabilistic inclusion atoms extend inclusion dependencies but retain simplicity and low data complexity, in contrast to independence or full functional dependencies (Hannula et al., 2020).

6. Significance Across Research Domains

Probabilistic inclusion operators unify the treatment of inclusion in probabilistic, fuzzy, logical, and conditional frameworks, offering:

• Statistical Summaries in Visualization: Efficient summary statistics, such as contour boxplots from ensembles of probabilistic masks, are realized through PID (Wu et al., 17 Dec 2025).
• Formal Analysis of Probabilistic Dependencies: Probabilistic inclusion atoms define tractable logics for database and team semantics, precisely characterizing the expressive frontier of existential second-order logic with additive real arithmetic (Hannula et al., 2020).
• Nonmonotonic Reasoning and DL Extensions: Probabilistic inclusion axioms enrich typicality-based description logics, supporting distributed semantics with exact probability assignments for ABox entailments (Giordano et al., 2020).
• Coherence-Based Inference: The Goodman–Nguyen inclusion, together with quasi-conjunction, underpins rigorous assessment of conditional knowledge, p-consistency, and probabilistic entailment (Gilio et al., 2013).

The theoretical breadth and algorithmic advances enabled by probabilistic inclusion operators have established them as foundational constructs in modern statistical visualization, logic, and AI inference frameworks.