Boolean Zeta Transforms

• Boolean Zeta Transforms are linear operators on functions defined over power sets that sum values over all subsets, establishing a key framework in combinatorics.
• They enable precise inversion via the Möbius transform, playing a crucial role in evidence theory, signal processing, and combinatorial enumeration.
• Focal point reduction exploits sparse support in functions to restrict the computational domain and dramatically reduce the cost compared to full lattice methods.

A Boolean zeta transform is a linear operator acting on real-valued functions defined on the power set of a finite set, ordered by inclusion. In the context of the Boolean lattice $(2^X, \subseteq)$, the zeta transform and its inverse, the Möbius transform, provide essential tools for information processing tasks such as mass function fusion in evidence theory, combinatorial enumeration, and signal processing. Recent research has shown that these transforms can be dramatically accelerated by exploiting the structure of functions with sparse or concentrated support, through the notion of focal points and their join-closure, enabling restricted-domain computation without loss of exactness (Chaveroche et al., 2020).

1. Classical Boolean Zeta and Möbius Transforms

Let $X$ be a finite set with $|X| = n$ elements, and $2^X$ its power set ordered by inclusion. For any function $f: 2^X \to \mathbb{R}$, the zeta transform produces $g: 2^X \to \mathbb{R}$: $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$ The inverse operation is the Möbius inversion, given by: $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$ These can be equivalently formulated via the incidence algebra: $g = f * \zeta, \qquad f = g * \mu,$ where $\zeta(A, B) = 1_{A \subseteq B}$ and $X$0.

The zeta transform is a pointwise sum over all sub-objects, while Möbius inversions combine information with alternating signs over supersets, recovering the original function. These transforms are central in enumeration, probability, and Dempster–Shafer theory, among other domains.

2. Focal Points and Join-Closure in General Posets

Let $X$1 be a finite poset, and $X$2 a function. The support of $X$3 is

$X$4

The join-closure (or focal-point closure) of $X$5 is

$X$6

where $X$7 denotes the least upper bound (join) in $X$8, assumed to exist for all $X$9.

Define

$|X| = n$0

as the focal points of $|X| = n$1. Both the zeta transform and its inverse can be performed strictly inside $|X| = n$2. If $|X| = n$3 and $|X| = n$4 is extended by zero outside $|X| = n$5, then, for $|X| = n$6,

$|X| = n$7

and Möbius inversion uses a restriction of the full Möbius function $|X| = n$8 to $|X| = n$9. Thus,

$2^X$0

without ever referencing elements outside $2^X$1, establishing $2^X$2 as the smallest required computational domain (Chaveroche et al., 2020).

3. Specialization to the Boolean Lattice

In the Boolean lattice $2^X$3, denote $2^X$4. The focal points become: $2^X$5 Let $2^X$6. Restricted transforms are: $2^X$7 When $2^X$8, these sums exclude all sets outside $2^X$9, resulting in significant computational savings.

4. Complexity: Naive, Fast, and Focal-Point-Reduced Approaches

The computational cost of evaluating $f: 2^X \to \mathbb{R}$0 from $f: 2^X \to \mathbb{R}$1 (and vice versa) depends on the chosen method:

Method	Addition Count	Space Complexity
Naive (full lattice)	$f: 2^X \to \mathbb{R}$2	$f: 2^X \to \mathbb{R}$3
Fast Möbius Transform (FMT)	$f: 2^X \to \mathbb{R}$4	$f: 2^X \to \mathbb{R}$5
Focal-Point-Reduced (this work)	$f: 2^X \to \mathbb{R}$6	$f: 2^X \to \mathbb{R}$7

• Naive: For each $f: 2^X \to \mathbb{R}$8, sum over all $f: 2^X \to \mathbb{R}$9. The total number of additions is $g: 2^X \to \mathbb{R}$0.
• FMT: Bitwise dynamic programming achieves $g: 2^X \to \mathbb{R}$1 additions.
• Focal-point-reduced: For each focal point $g: 2^X \to \mathbb{R}$2, add $g: 2^X \to \mathbb{R}$3 to all $g: 2^X \to \mathbb{R}$4. The total complexity is $g: 2^X \to \mathbb{R}$5, which can be exponentially smaller when $g: 2^X \to \mathbb{R}$6.

This focal-point reduction delivers exact results with potentially dramatic savings in both time and space, especially for sparse or concentrated support. In the worst case ($g: 2^X \to \mathbb{R}$7) the costs revert to $g: 2^X \to \mathbb{R}$8, but for $g: 2^X \to \mathbb{R}$9 the computational improvements can be exponential (Chaveroche et al., 2020).

5. Detailed Example: $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$0

Let $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$1 with $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$2, $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$3, others zero. Then,

$$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$4

• Naive: $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$5 sums over all eight subsets. Up to $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$6 additions.
• Focal-reduced: Only $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$7 with $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$8, i.e.: $$\forall\,B \subseteq X,\quad g(B) = \sum_{A \subseteq B} f(A).$$9 Only $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$0 additions are required.

This demonstrates the strict reduction of computational workload for sparse-support functions.

6. Focal-Point Algorithms and Minimality

Focal points constitute the smallest domain over which exact zeta transforms and Möbius inversions need to be computed. In settings such as Dempster–Shafer theory, the effective support (number of nonzero masses) and its join-closure are frequently much less than $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$1, resulting in substantially reduced time and space requirements compared to FMT.

Pseudocode for the focal-point zeta transform, given $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$2: $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$9 Cost: $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$3 additions.

Pseudocode for Möbius inversion restricted to focal points: $g = f * \zeta, \qquad f = g * \mu,$0 Cost: $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$4 (# focal supersets of $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$5), which is at most $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$6.

The focal-point-reduced method is provably the minimal exact restriction: no further reduction in the computational domain is possible without compromising correctness (Chaveroche et al., 2020). Hence, for all $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$7 with focal-point closure $$\forall\,A \subseteq X,\quad f(A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|}\,g(B).$$8, focal-point techniques provide the fastest exact computation, especially in domains with structured or sparse supports.