Mereological Approach to PID

• Mereological PID is a framework that structures multivariate information decomposition using part–whole relationships inherent in complex systems.
• It leverages lattice structures, antichains, and Möbius inversion to identify unique, redundant, and synergistic information atoms.
• The approach unifies diverse decomposition schemes, scales to high dimensions, and clarifies higher-order interactions in systems like logic gates and networks.

A mereological approach to Partial Information Decomposition (PID) reconceptualizes the decomposition of multivariate information by structuring the analysis explicitly around the part–whole relationships that underlie the emergence, sharing, and loss of information in complex systems. Rather than relying solely on redundancy or unique information as primitives, the mereological framework leverages order-theoretic and logical structures—typically lattices of subsets, antichains, or parthood distributions—to formalize how "atomic" bits of information relate to possible groupings of sources. This approach not only grounds PID in general mathematical principles, such as Möbius inversion, but also unifies a diversity of decomposition schemes, scaling gracefully to high dimensions and complex interactions, and providing conceptual clarity on the interpretation of information atoms as irreducible parts, intersections, or emergent wholes.

1. Foundations: Part–Whole Relationships, Lattices, and Parthood

At its core, the mereological formulation of PID begins with the formalization of "part–of" relations, traditionally in the context of set inclusion but generalizable to any system where elements can be regarded as parts of a whole. A mereology on a finite system $S$ is defined as a co-rooted poset $(\mathcal{D}(S), \le)$ with $S$ as the unique greatest element, and a reflexive, antisymmetric, and transitive partial order representing part–whole relationships (Jansma, 17 Apr 2024).

In the context of PID, the elements of $\mathcal{D}(S)$ are typically subsets of source variables, antichains of such subsets, or logical statements about them. The fundamental object is a "parthood distribution," a monotone Boolean function $f:2^{[n]} \to \{0,1\}$ satisfying

• $f(\emptyset) = 0$, $f([n]) = 1$,
• $f(A) = 1 \implies f(B) = 1$ for all $B \supset A$,

where $[n]$ indexes the sources (Gutknecht et al., 2020). Each $f$ marks exactly where a given atomic piece of information is "present." The collection of all such $f$ forms the parthood lattice, which is isomorphic to the Williams–Beer antichain lattice and a lattice of logical statements under implication.

2. πD: Atoms, Consistency, and Möbius Inversion

PID seeks to decompose multivariate information measures (e.g., $I(T; S_1, ..., S_n)$) into non-overlapping pieces termed "atoms,” each corresponding to a particular pattern of being "part of" certain groupings of the sources. The consistency principle mandates that every observable information quantity is the sum of the corresponding atoms:

$I(T; A) = \sum_{f: f(A)=1} \Pi(f) \,,$

for all $A \subseteq [n]$. The redundancy associated with any antichain $\alpha$ is

$$I_\cap(\alpha) = \sum_{f: f(a)=1 \;\forall a \in \alpha} \Pi(f) \,.$$

By Möbius inversion on the redundancy lattice (the poset of antichains ordered by inclusion), the atoms $\Pi(f)$ are uniquely determined from the redundancies $I_\cap(-)$:

$$\Pi(f) = \sum_{g \preceq f} \mu(g, f) I_\cap(g) \,,$$

where $\mu$ is the Möbius function of the lattice (Gutknecht et al., 2020, Jansma, 17 Apr 2024). This approach not only ensures uniqueness of the decomposition but also provides a direct mapping between logical, set-theoretic, and information-theoretic views (Gutknecht et al., 2023).

3. Generalization: Logical Constraints and Families of Decompositions

All classical and novel forms of information decomposition fall under a unified mereological logic: every composite information measure is defined by formal logical constraints on parthood distributions—positive (requiring presence in some components), negative (requiring absence in others), or combinations thereof. For example:

• Redundancy: Atoms present in all specified source combinations,
• Synergy: Atoms present only when all specified combinations are present,
• Unique information: Atoms present in exactly one combination,
• Vulnerable information: Atoms at risk of being lost if any specified component is missing.

This logic allows systematic definition of not only classical PID measures but also novel quantities such as "vulnerable information," relevant in security or robustness analysis (Gutknecht et al., 2023). The mereological approach permits the construction of partner measures and dualities by exchanging positive and negative constraints, facilitating a taxonomy of all possible decompositions arising from monotone logical properties.

4. Synergy-First and Backbone Constructions

Synergy-first decompositions invert the classical approach by directly quantifying those parts of the whole that are irreducible to any proper subcollection—a definition that aligns naturally with mereological principles. Formally, for a state $x = (x_1, ..., x_k)$ and entropy $h(x)$, the 1-synergy is

$$h_1^{\mathrm{syn}}(x) = \min_{i} [h(x) - h(x^{-i})] = \min_{i} h(x_i | x^{-i}) \,,$$

and the general $\alpha$-synergy is

$$h_\alpha^{\mathrm{syn}}(x) = \min_{a: |a| = \alpha} [h(x) - h(x^{-a})] = \min_{a: |a| = \alpha} h(x^a | x^{-a}) \,,$$

which quantifies the information lost upon removal of any $\alpha$ parts (Varley, 13 Feb 2024). The resulting hierarchy of cumulative synergy measures leads, via finite differencing, to partial synergy atoms $\partial h_\alpha(x)$ along a total order. These atoms are interpreted as information that only emerges at a given scale $\alpha$—i.e., requiring at least $k-\alpha+1$ parts to survive, giving a graded backbone from most fragile (minimal, pure synergy) to most robust. In contrast to the combinatorial explosion in the PID lattice, the synergy-backbone has only $k$ atoms, with associated practical benefits in scalability and interpretability.

This backbone construction generalizes readily to Kullback-Leibler divergence ($D_{KL}$), total correlation, and single-target mutual information, since all are linear in entropy or conditional entropy terms.

5. Mereology and Higher-Order Structure: Möbius Unification

The choice of mereology—i.e., the precise partial order or lattice structuring the system—fundamentally determines the notion of higher-order interaction, whether in information theory, statistical mechanics, or network science. Standard inclusion–exclusion on the Boolean lattice yields (potentially signed) interaction information, while the antichain partial order gives a Williams–Beer PID (and its nonnegative decomposition under suitable redundancy assignment) (Jansma, 17 Apr 2024).

Changing the underlying mereology yields alternative decompositions:

• Antichain lattice (redundancy): Classical PID atoms,
• Powerset lattice: Interaction information, cumulants,
• Partition poset: Ursell functions, Green’s function expansion in physics,
• Subgraph or subcomplex posets: Higher-order network or topological invariants.

A single Möbius inversion captures all these cases:

$\Delta_F(Q) = \sum_{P \leq Q} \mu(P,Q) F(P) \,,$

where $F$ is any extensive observable (entropy, mutual information, energy, etc.) and $\mu$ is the Möbius function of the chosen mereology (Jansma, 17 Apr 2024). Rota's Galois connection theorem provides a means to coarse-grain or refine a decomposition in a principled way.

6. Mereological PID and Emergent Systems: Interpretational and Practical Implications

Within this paradigm, the unique role of synergy atoms as "emergent wholes" is made explicit: they quantify those informational or structural properties which are destroyed by the loss of any part, corresponding to irreducible, high-order interdependencies, akin to the emergence of new behavior in physical or biological systems. Redundancy atoms capture overlaps—parts common to multiple sources or collections—while unique atoms register idiosyncratic contributions.

The mereological framework provides interpretational clarity and computational tractability, unifying practical data-analytic methods (e.g., random sampling for subset minimization (Varley, 13 Feb 2024)), formal axiomatic generalizations (shared exclusions (Schick-Poland et al., 2021)), and connections to logical inference, object persistence, and network integration (Fields et al., 2018). Crucially, the approach also accommodates the stochastic/deterministic boundary—identifying when atoms may become negative or undefined due to overlapping identity and offering routes for further conceptual refinement (Chicharro et al., 2017).

7. Illustrative Examples

A consistent outcome across examples is the interpretation of logic gates. In the XOR gate, all information resides in the pure synergy atom, and redundancy vanishes, while the AND gate demonstrates a more intricate mixture of redundancy and synergy, including negative atoms at higher-order scales under mereological decompositions (Varley, 13 Feb 2024). In network contexts, the synergy-backbone reveals how global integration and communicability depend on collective stabilities not captured by pairwise connections.

System	Synergy Atom (∂h₁)	Redundancy Atoms	Remarks
XOR (3 variables)	All information	0	Pure synergy
AND (3 variables)	Most in ∂h₁, some in ∂h₂/∂h₃	Nonzero	Mixture, including negative higher atoms
Networks	Synergy at high α	–	Global integration tied to collective synergy

The formalism correctly identifies the locus and scale of irreducible contributions in these and broader applications (Varley, 13 Feb 2024, Jansma, 17 Apr 2024).

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The mereological approach to PID organizes the atomization of information according to part–whole logic, order theory, and Möbius inversion, delivering a powerful, conceptually transparent, and computationally efficient framework for analyzing higher-order interdependencies in complex systems. It bridges information theory with formal logic, algebra, and network science, with implications for the understanding of emergence, robustness, and decomposition across a wide spectrum of scientific disciplines (Varley, 13 Feb 2024, Jansma, 17 Apr 2024, Gutknecht et al., 2023, Gutknecht et al., 2020).