---
title: Boolean Rank in 0/1 Matrices & Tensors
url: https://www.emergentmind.com/topics/boolean-rank
type: topic
---

# Boolean Rank in 0/1 Matrices & Tensors

Boolean rank is a rank notion for \(0,1\)-valued matrices and tensors defined over the Boolean semiring, where addition is logical OR and multiplication is logical AND. For matrices, it is the minimum inner dimension in an exact Boolean factorization; equivalently, it is the minimum number of monochromatic all-1 rectangles whose union covers the 1-entries. For bipartite graphs, the same parameter is the biclique cover number of the biadjacency matrix, and for tensors it becomes the minimum number of Boolean rank-one outer products whose Boolean sum equals the array [2601.13900][2509.09570][1112.0298].

## 1. Core definition and equivalent formulations

For a \(0,1\) matrix \(M\), Boolean rank is defined by
\[
\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},
\]
where multiplication is over Boolean arithmetic,
\[
(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).
\]
This formulation makes Boolean rank the Boolean-semiring analogue of ordinary matrix rank. The same literature gives several exact reformulations: it equals \(Cover_1(M)\), the minimum number of monochromatic rectangles needed to cover all 1-entries; it equals the biclique edge cover number of the associated bipartite graph; it equals the intersection number; and it is equivalent to the set basis problem [2601.13900].

The graph-theoretic viewpoint is particularly stable across the literature. If \(A(G)\) is the biadjacency matrix of a bipartite graph \(G\), then
\[
\bc(G)=\brank(A(G)).
\]
In this interpretation, each Boolean rank-one term is a biclique, and a Boolean factorization is precisely a biclique cover. The commutative-algebraic work on monomial ideals adopts the same definition and notation \(\brank(A)\), emphasizing Boolean matrix factorization as exact recovery by latent binary features [2509.09570].

A recurrent source of confusion is the distinction between cover and partition formulations. In the survey literature, Boolean rank is the cover number of the 1s, whereas binary rank is the partition number of the 1s into disjoint rectangles [2601.13900]. Some complexity-theoretic papers, however, use “Boolean rank” or “binary rank” for the partitioning number instead; that terminological shift matters in comparisons with communication complexity and with signed rectangle decompositions [2510.02583].

## 2. Relation to real rank, binary rank, and small-rank regimes

The basic inequalities recorded in the survey are
\[
\text{real rank}(M)\le \text{Boolean rank}(M),\qquad \text{real rank}(M)\le \text{binary rank}(M),
\]
and
\[
\text{Boolean rank}(M)\le \text{binary rank}(M).
\]
Thus Boolean rank is combinatorially more permissive than binary rank, because overlap among covering rectangles is allowed, but it is still constrained by exact \(0/1\) logic rather than arbitrary real coefficients [2601.13900].

For matrices of very small constant real rank, the 2025 classification of small-rank cases gives sharp finite bounds. When the real rank is \(1\) or \(2\), the real, Boolean, and binary ranks coincide. For real rank \(3\) and \(4\), the paper proves the unified bounds
\[
d\le \text{Boolean rank}(M)\le 2d-2,\qquad d-1\le \text{binary rank}(M),\, i(M)\le 2d-2
\]
for \(d=3,4\), and shows that these bounds are tight. The same work identifies the circulant matrices \(C_{2d-2}\) as the extremal examples for \(d=3,4\) [2507.05824].

Large separations are nevertheless possible. For the crown matrix \(C_n\), the exact Boolean rank is
\[
B(C_n)=\sigma(n)=\min\left\{k:\ n\le \binom{k}{\lceil k/2\rceil}\right\},
\]
hence \(B(C_n)=(1+o(1))\log_2 n\), while the survey records that for the same family the binary and real ranks are both \(n\). This exhibits an exponential gap between Boolean rank and real rank [2102.07486][2601.13900].

Boolean rank can also behave very differently under complementation. For infinitely many integers \(k\), there exists a square regular \(0\)-\(1\) matrix \(M\) with binary rank \(k\) such that the Boolean rank of its complement satisfies
\[
\operatorname{BoolRank}(\overline{M})\ge k^{\widetilde{\Omega}(\log k)}.
\]
This settles a conjectural regular-matrix analogue of earlier non-regular separations and shows that complementing a matrix can magnify cover complexity far beyond the original partition complexity [2203.13073].

## 3. Structural results and extremal phenomena

The paper "Ranks of 0-1 arrays of size \(2\times 2\times 2\) and \(2\times 2\times 2\times 2\)" gives a complete Boolean-rank classification for the smallest nontrivial tensors. For \(2\times 2\times 2\) arrays, the possible Boolean ranks are \(0,1,2,3,4\), with counts \(1,27,130,88,10\), so the maximum Boolean rank is \(4\). For \(2\times 2\times 2\times 2\) arrays, the possible Boolean ranks are \(0,1,2,3,4,5,6,7,8\), with maximum \(8\) and full rank distribution explicitly enumerated. In the \(2\times 2\times 2\) case, the Boolean and nonnegative-integer cases coincide, and every rank-\(r\) Boolean array can be decomposed into \(r\) outer products with no overlapping 1-positions; in the \(2\times 2\times 2\times 2\) case, the Boolean and nonnegative-integer classifications no longer coincide [1112.0298].

These tensor classifications also show that Boolean rank does not admit the same kind of orbit-canonical-form theory that appears over fields. In the Boolean case, the paper explicitly states that canonical forms for a group action do not exist, and the resulting classifications are organized instead by rank and by the number of entries equal to \(1\) [1112.0298].

Another structural theme concerns stability under augmentation. The augmentation-property paper defines a matrix \(A\) to have the Augmentation property for a rank function \(R\) if every collection of columns that can be added individually without increasing rank can also be added simultaneously without increasing rank. For Boolean rank, the paper proves the exact criterion
\[
A \text{ has the Augmentation property} \iff A \text{ has a unique base that spans all other bases.}
\]
It also shows that failure can be severe: for any \(k\), there exists a matrix \(A_k\) and vectors \(x_1,\dots,x_k\) such that each \(R_{\text{bool}}(A_k\mid x_i)=R_{\text{bool}}(A_k)=4\), while
\[
R_{\text{bool}}(A_k\mid x_1,\dots,x_k)\ge k.
\]
This behavior has no analogue for ordinary linear rank [1706.06900].

Kronecker products supply a different extremal mechanism. Boolean rank is always submultiplicative:
\[
B(A\otimes B)\le B(A)\,B(B).
\]
For the crown matrices, the 2021 paper proves that this inequality is strict for all \(n,m\ge 7\):
\[
B(C_n\otimes C_m)<B(C_n)\,B(C_m),
\]
hence in particular
\[
B(C_n\otimes C_n)<B(C_n)^2 \qquad (n\ge 7).
\]
For \(C_n\otimes C_n\), the same paper constructs explicit covers of size \(O(B(C_n)\log B(C_n))\), which is near the linear lower-bound regime rather than the naive quadratic regime [2102.07486].

## 4. Communication complexity and algebraic lower bounds

Boolean rank is directly tied to nondeterministic communication complexity. The survey states
\[
N_1(M)=\left\lceil \log_2(\text{Boolean rank}(M))\right\rceil,
\]
while binary rank gives the analogous unambiguous nondeterministic quantity
\[
U(M)=\left\lceil \log_2(\text{binary rank}(M))\right\rceil.
\]
This connection explains why rectangle covers, biclique covers, and factorization ranks recur in communication-complexity lower bounds and in log-rank-type questions [2601.13900].

A related but non-identical parameter appears in recent work on the log-rank conjecture. That paper uses the partitioning number of a Boolean matrix—also called Boolean rank or binary rank in its terminology—and inserts between real rank and partition number a new parameter, signed rectangle rank: the minimum number of primitive all-1 rectangles in a \(\pm 1\)-sum representation,
\[
M=\sum_{i=1}^t \varepsilon_i R_i,\qquad \varepsilon_i\in\{1,-1\}.
\]
Its basic inequality is
\[
\text{rank}(M)\le \text{signed rectangle rank}(M)\le \text{partition number}(M),
\]
and its main theorem shows
\[
\text{signed rectangle rank}(M)\le O(\operatorname{rank}(M)\log \operatorname{rank}(M)).
\]
This reframes the log-rank conjecture as the problem of converting signed rectangle decompositions into positive rectangle partitions with only quasipolynomial blowup [2510.02583].

Lower bounds for Boolean rank also admit commutative-algebraic formulations. Given the bipartite graph \(G\) of a binary matrix \(A\), one lower-bound framework uses the edge ideal \(I_G\) and proves
\[
\reg(R/I_G)\le \brank(A(G)).
\]
Combined with induced matching bounds, this yields
\[
\indmatch(G)\le \reg(R/I_G)\le \brank(A).
\]
A second framework introduces the isolation ideal \(J_A\) attached to isolated sets of 1-entries and proves
\[
\reg(k[A]/J_A)=\iota(A)=\dim(\Delta_A)+1\le \brank(A).
\]
These identities do not turn Boolean rank into a homological invariant, but they place classical lower-bound parameters inside a standard regularity calculus [2509.09570].

## 5. Computational complexity, testing, and model selection

Exact computation of Boolean rank is NP-complete. The survey attributes this both to reductions through biclique edge cover and to the equivalence with the set basis problem, and it further states that Boolean rank is hard to approximate within a factor of
\[
n^{1-\varepsilon}
\]
for any \(\varepsilon>0\) on an \(n\times n\) matrix [2601.13900].

Despite this hardness, Boolean rank is fixed-parameter tractable with respect to the target rank \(d\). A basic kernelization principle is that if Boolean rank is at most \(d\), then after removing all-zero rows and columns and merging duplicates, the matrix has at most \(2^d\) distinct rows and at most \(2^d\) distinct columns. This reduction underlies parameterized exact algorithms and also clarifies why low-rank Boolean structure can be computationally accessible on compressed kernels even when the original matrix is large [2601.13900].

Property testing gives a sublinear alternative to exact computation. For Boolean rank, the 2019 testing paper proves a one-sided-error non-adaptive tester with query complexity
\[
\tilde{O}\!\left(\frac{d^4}{\epsilon^6}\right),
\]
which always accepts matrices of Boolean rank at most \(d\) and rejects with probability at least \(2/3\) when the matrix is \(\epsilon\)-far from rank at most \(d\). The proof is built around combinatorial objects called skeletons, beneficial entries, and influential entries, and it exploits the fact that every low-rank Boolean submatrix must admit a skeleton with no beneficial entries [1908.11632].

In applied Boolean matrix factorization, Boolean rank also functions as a model-order parameter. The 2025 bfact paper defines a Boolean matrix factorization of \(\mathbf{X}\in\{0,1\}^{M\times N}\) into binary factors \(\mathbf{L}\in\{0,1\}^{M\times K}\) and \(\mathbf{R}\in\{0,1\}^{K\times N}\) via
\[
X_{ij}=\bigvee_{k=1}^{K} L_{ik}\land R_{kj},
\]
and treats the rank \(K\) as the number of factors used to explain the matrix. Because exact BMF is NP-complete, the paper approaches rank estimation through a hybrid restricted master problem with cluster-derived candidate factors, refinement, pruning, and comparison across candidate ranks \(K_c\). In its reported experiments, this pipeline performs particularly well at estimating the true rank in simulated settings and achieves strong signal recovery with much lower rank on Human Lung Cell Atlas benchmarks [2509.06192].

## 6. Tensors and other meanings of “Boolean rank”

For tensors, Boolean rank is defined by replacing matrix products with Boolean outer products. In the finite-array classification paper, a rank-one \(n\)-dimensional \(2\times\cdots\times 2\) Boolean array has the form
\[
X=V_1\cdots V_n,\qquad V_i\in\{[1,0],[0,1],[1,1]\},
\]
and the rank of \(X\) is the minimum \(R\) such that
\[
X=\sum_{r=1}^R V_1^{(r)}\cdots V_n^{(r)},
\]
where the sum is taken entrywise with Boolean addition \(1+1=1\). The essential distinction from arithmetic rank is that overlap of rank-one summands is absorbed rather than counted or cancelled [1112.0298].

Algorithmic tensor decomposition adopts the same semantics. The GETF paper writes a \(k\)-order Boolean tensor as
\[
\mathscr{X}^{m_1\times \cdots \times m_k}
=
\vee_{j=1}^l
\left(
A_{:j}^{m_1\times l,1}\otimes
A_{:j}^{m_2\times l,2}\cdots
A_{:j}^{m_k\times l,k}
\right),
\]
so the effective Boolean rank is the number \(l\) of Boolean rank-one components. Its main algorithm extracts such components sequentially from a residual tensor using left-triangular-like reordering and geometric seeding, and the paper derives \(O(n)\) complexity when a \(k\)-order tensor has \(n=m^k\) entries [2007.15821].

Recent complexity-theoretic work extends related decomposition ideas to higher order. For an order-\(\ell\) Boolean tensor \(T\) of tensor rank \(r\), the signed-rectangle-rank paper proves that \(T\) can be written as a \(\pm 1\)-linear combination of at most
\[
(cr\log r)^{\ell-1}
\]
primitive tensors, and notes that the same bound holds if tensor rank is replaced by flattening rank. This result concerns signed decompositions rather than Boolean covers, but it places tensor Boolean structures into the same rank-versus-rectangle-decomposition landscape as the matrix case [2510.02583].

The phrase “Boolean rank” is also overloaded outside matrix and tensor factorization. In finite dynamical systems, rank means image size: for a Boolean network \(f:[2]^n\to[2]^n\),
\[
\operatorname{rank}(f)=|\operatorname{Im}(f)|,
\]
and for interaction graphs contained in a digraph \(D\), the maximum Boolean-network rank is \(2^{\alpha_1(D)}\) [1512.01448]. In decision-tree complexity, rank means the Horton–Strahler number of a decision tree, with
\[
Rank(f)=\min\{Rank(T):T\text{ computes }f\},
\]
a parameter satisfying
\[
Rank(f)\le \log DTSize(f)\le Rank(f)\log\!\left(\frac{en}{Rank(f)}\right)
\]
for non-constant \(f\) on \(n\) variables [2209.12877]. These notions are structurally unrelated to Boolean matrix rank, and the shared terminology should not be conflated.

Boolean rank therefore names a family of exact-cover-type invariants whose central matrix meaning is stable—Boolean factorization, rectangle cover number, biclique cover number, intersection number, and set basis size—but whose surrounding theory spans sharp finite classifications, strong separations from real and binary rank, communication-complexity reformulations, algebraic lower bounds, parameterized and sublinear algorithms, and tensor generalizations [2601.13900].

Source: https://www.emergentmind.com/topics/boolean-rank