Boolean Convolution: Theory & Applications

• Boolean convolution is a noncommutative probability operation defined via the additive K–transform and Boolean cumulants.
• It exhibits unique properties like infinite divisibility and selfdecomposability, which underpin its role in central limit phenomena and stable laws.
• Its applications span analytic, algebraic, combinatorial, and computational frameworks, impacting algorithm design and spectral theory.

Boolean convolution is a fundamental operation in noncommutative probability, encoding an independence notion much simpler than free or classical independence. It arises in several analytic, algebraic, combinatorial, and algorithmic frameworks, affecting the structure of distributions, limit theorems, cumulant theory, and computational complexity. Boolean convolution exists in scalar, operator-valued, and combinatorial settings, is deeply connected to analytic transforms (notably the K–transform), enjoys unique infinite divisibility and selfdecomposability properties, and appears as a modal operation in algebraic logic and convolutional algorithms.

1. Definition, Analytic Transforms, and Algebraic Framework

Boolean additive convolution, denoted by “$\uplus$”, is defined as the law of the sum of two Boolean-independent random variables. If $\mu$ and $\nu$ are probability measures, their Boolean additive convolution $\mu \uplus \nu$ is linearized by the Boolean K–transform: $$K_\mu(z) = z - \frac{1}{G_\mu(z)} \quad\text{where}\quad G_\mu(z) = \int (z-x)^{-1}\, d\mu(x)$$ The defining property is: $K_{\mu \uplus \nu}(z) = K_\mu(z) + K_\nu(z)$ for $z \in \mathbb{C}^+$. The K–transform expansion at infinity yields Boolean cumulants, with $K_\mu(z) = m_1(\mu) + \text{Var}(\mu)/z + \cdots$.

Multiplicative Boolean convolution (for $\mu, \nu$ on $[0,\infty)$) is linearized by the B–transform: $B_{\mu\array\nu}(z) = B_\mu(z) B_\nu(z)$, with $B_\mu(z) = z / \eta_\mu(z)$, $\eta_\mu(z)$ being the composition of auxiliary transforms involving the Cauchy and moment generating functions (Chakraborty et al., 2017).

On complete lattices and relational structures, Boolean convolution corresponds to an operation in the "convolution algebra" $L^\mathfrak{X}$. For $L=2$, this recovers classic complex algebras of modal logic, encapsulating the Boolean convolution as a powerset lifting (Harding et al., 2017, Cranch et al., 2020).

2. Infinite Divisibility, Stable Laws, and Selfdecomposability

Boolean convolution exhibits striking infinite divisibility: every probability measure is infinitely divisible with respect to $\uplus$, reflected in the additive property of the K–transform (Anshelevich et al., 2012). Boolean stable laws, specified by a self-similarity property under $\uplus$, are parameterized by a stability index $\alpha$ and asymmetry coefficient $\rho$. They are characterized by analytic energy/K–function representations, e.g.,

$$K(z) = -e^{i\rho\pi} z^{1-\alpha}, \quad 0 < \alpha < 1$$

(Arizmendi et al., 2012, Arizmend et al., 2014). Free infinite divisibility for Boolean stable laws is completely classified: $0 < \alpha \leq 1$ for all $\rho$, or $1 < \alpha \leq 2$ with $2-1/\alpha < \rho < 1/\alpha-1$.

The class of Boolean selfdecomposable measures $\mathcal{L}(\uplus)$ consists of measures $\mu$ such that for all $0 < c < 1$ there exists $\mu_c$ with $\mu = D_c(\mu)\uplus\mu_c$, where $D_c$ is dilation. Boolean selfdecomposability admits strong regularity: only two atoms are possible, no singular continuous component, and the Lévy measure must exhibit unimodality at zero. Deeply, the standard normal is Boolean selfdecomposable, but shifted normals lose this property for large shift (Hasebe et al., 2022).

3. Limit Theorems and Quantitative Boolean Central Limit Theorem

Boolean probability is marked by a very simple central limit phenomenon: the limit in the Boolean CLT is the symmetric Bernoulli law, not a continuous Gaussian. Let $p$ be a law with zero mean, unit variance, and finite $m_4(p)$; then the normalized Boolean convolution power

$U_n = D_{1/\sqrt{n}}(p^{\uplus n})$

satisfies a Berry–Esseen type rate in Lévy distance to $b = 1/2\,\delta_{-1} + 1/2\,\delta_{1}$: $$L(U_n,b) \leq \frac{2}{\sqrt{m_4(p)-1}}\frac{1}{\sqrt{n}}$$ with sharpness established under bounded support (Arizmendi et al., 2017, Salazar, 2020). Techniques rely on sharp quantitative versions of the Stieltjes–Perron inversion formula, enabling direct control of Lévy distance through the imaginary part of analytic transforms.

For measures with regularly varying tails, Boolean subexponentiality holds: $n$-fold Boolean convolution scales as $n$ times the original tail (Chakraborty et al., 2017). Asymptotics are governed by the K or B-transform expansions, with remainder terms encoding tail decay rates.

Boolean analogs of the Lévy–Khintchine representation express the K–transform in terms analogous to the classical scenario; in some scaling limits (e.g., mixtures of multiplicative and additive convolution), the R–transform of the limit emerges in terms of the principal branch of the Lambert $W$ function (Sakuma et al., 2011).

4. Operator-Valued, Infinitesimal, and Multi-Faced Extensions

Operator-valued Boolean convolution generalizes the scalar theory by allowing convolution powers with respect to completely positive linear maps on a base $C^*$-algebra $B$. The Boolean convolution power $p^{\uplus\eta}$ is defined via $B_{p^{\uplus\eta}} = \eta\circ B_p$, and the resulting structure forms a commutative evolution semigroup (Anshelevich et al., 2011, Anshelevich et al., 2014). This semigroup encodes flows that connect to the Bercovici–Pata bijection and the geometry of free Brownian motion (the operator-valued Burgers equation).

Infinitesimal operator-valued independence theory (OVI) further extends Boolean convolution to encode small perturbations ("first-order corrections") of expectations and cumulants, with explicit convolution formulas obtained by lifting the algebra to a $2\times2$ upper-triangular matrix algebra (Perales et al., 2020). This provides a framework for subtle fluctuation and first-order correction phenomena in random matrix limits and noncommutative probability.

Boolean convolution also admits a bi-faced extension in bi-Boolean independence for pairs of algebras. The bi-Boolean additive convolution is linearized by a two-variable "partial self-energy" function, generalizing Boolean cumulants to $B$-$(\ell,r)$-cumulants and paralleling the bi-free Lévy–Khintchine theory, but with even more restrictive possibilities for infinite divisibility (Gu et al., 2017).

A further variant, cyclic–Boolean independence, has been formulated to describe joint independence for families of subalgebras with respect to both a state and a trace, motivated by problems in spectral graph theory and noncommutative Markov chains (Arizmendi et al., 2022). Its convolution is governed by an explicit transform additivity, and the associated class of infinitely divisible distributions is exceedingly rigid.

5. Boolean Convolution in Computational Complexity and Algorithms

Boolean convolution as a combinatorial operation is central in algorithms and circuit complexity. The Boolean convolution of $n$-dimensional Boolean vectors (with wraparound) is given by: $$(A \star B)[k] = \bigvee_{i \in [n]} (A[i] \wedge B[(k-i)\bmod n])$$ This operation is equivalent to computing sumsets in finite cyclic groups, a fundamental primitive in combinatorial optimization—e.g., in Modular Subset Sum and dynamic programming recurrences (Bringmann et al., 2021).

Significant advances in algorithm design have produced deterministic and randomized (Las Vegas) algorithms to compute $n$-fold Boolean convolution in time nearly linear in the combined input and output size, leveraging Kneser's theorem from additive combinatorics to detect and exploit periodicity in sumsets.

Circuit complexity analyses demonstrate that computing Boolean convolution is intrinsically costly: the monotone circuit complexity (number of AND/OR gates) is essentially $\Theta(n^2)$ up to polylogarithmic factors in $n$ (Grinchuk et al., 2017, Paterson, 2017). Lower bounds are established using properties of thin circulant matrices and sumset geometry.

6. Connections to Modal, Algebraic, and Categorical Structures

General Boolean convolution operations abstract to modal, lattice-theoretic, and algebraic frameworks. In the categorical language, convolution algebras $L^\mathfrak{X}$ (with $L$ a complete lattice or even a quantale) can be formed from relational structures $(X, (R_i))$, endowing the function space $L^X$ with modal/convolution operators $f_i$ that aggregate according to $R_i$ (Harding et al., 2017, Cranch et al., 2020). When $L=\{0,1\}$, convolutions of predicates induced by $R_i$ retrieve classical complex algebras for modal and algebraic logic, providing an expansive generalization of Boolean convolution in algebraic settings.

The convolution framework is bifunctorial, captures products and morphisms in relational and algebraic categories, and separates additive (join-based) and multiplicative (meet-based) convolution flavors, unifying algebraic, combinatorial, and logical perspectives.

7. Applications and Open Problems

Boolean convolution and its derivatives appear in random matrix theory (limiting spectral distributions), combinatorial optimization (subset sum, Boolean sum-systems), spectral theory (graph products), analysis of limit theorems, and modal algebra.

A recent result (Chakraborty, 23 Aug 2025) establishes a deep connection between Boolean convolution and symmetry resistance for Bernoulli random variables: any symmetrizer (Boolean-independent $Y$ so that $X \uplus Y$ is symmetric about 0) must have variance at least $pq$ (for $X\sim$ Bernoulli($p$)). This lower bound, mirroring the classical case, is forced by the expansion and parity (oddness) of the K–transform, which encodes moments and symmetry constraints. In the free convolution context, the analogous minimality of variance remains an open problem, attesting to nuanced differences between free and Boolean independence (and convolution).

In combinatorial probability, Boolean subexponentiality parallels classical and free subexponential laws under regular variation, with detailed behavior of heavy-tailed convolution governed by the asymptotics of analytic transforms and their Taylor expansions.

Further open questions concern the fine structure of operator-valued Boolean convolution powers, connections to monotone convolution, multidimensional extensions (bi-Boolean, cyclic–Boolean), and the optimization of fast convolution algorithms for sparse and structured input.

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Boolean convolution thus occupies a central role bridging analytic, algebraic, combinatorial, and algorithmic dimensions in noncommutative probability and related mathematical disciplines, with broad implications for limit theorems, distributional structure, spectral theory, modal logic, and computational complexity.