Boolean Rank in 0/1 Matrices & Tensors

• Boolean rank is a measure that quantifies the minimal number of Boolean rank-one components required to exactly decompose 0/1 matrices and tensors using logical OR and AND operations.
• Its methodology involves covering all 1-entries with monochromatic rectangles, equating to biclique covers in bipartite graphs and precise outer product decompositions in tensors.
• The concept influences various fields by linking combinatorial covers with computational complexity, NP-completeness, fixed-parameter tractability, and communication complexity lower bounds.

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 (Parnas, 20 Jan 2026, Geraci et al., 11 Sep 2025, Bremner et al., 2011).

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 (Parnas, 20 Jan 2026).

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 (Geraci et al., 11 Sep 2025).

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 (Parnas, 20 Jan 2026). 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 (Hambardzumyan et al., 2 Oct 2025).

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

The basic inequalities recorded in the survey are

$0,1$0

and

$0,1$1

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$2 logic rather than arbitrary real coefficients (Parnas, 20 Jan 2026).

For matrices of very small constant real rank, the 2025 classification of small-rank cases gives sharp finite bounds. When the real rank is $0,1$3 or $0,1$4, the real, Boolean, and binary ranks coincide. For real rank $0,1$5 and $0,1$6, the paper proves the unified bounds

$0,1$7

for $0,1$8, and shows that these bounds are tight. The same work identifies the circulant matrices $0,1$9 as the extremal examples for $M$0 (Parnas et al., 8 Jul 2025).

Large separations are nevertheless possible. For the crown matrix $M$1, the exact Boolean rank is

$M$2

hence $M$3, while the survey records that for the same family the binary and real ranks are both $M$4. This exhibits an exponential gap between Boolean rank and real rank (Haviv et al., 2021, Parnas, 20 Jan 2026).

Boolean rank can also behave very differently under complementation. For infinitely many integers $M$5, there exists a square regular $M$6-$M$7 matrix $M$8 with binary rank $M$9 such that the Boolean rank of its complement satisfies

$$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$0

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 (Haviv et al., 2022).

3. Structural results and extremal phenomena

The paper "Ranks of 0-1 arrays of size $$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$1 and $$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$2" gives a complete Boolean-rank classification for the smallest nontrivial tensors. For $$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$3 arrays, the possible Boolean ranks are $$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$4, with counts $$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$5, so the maximum Boolean rank is $$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$6. For $$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$7 arrays, the possible Boolean ranks are $$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$8, with maximum $$\operatorname{brank}(M)=\min\{d:\; M=A\cdot B,\ A\in\{0,1\}^{n\times d},\ B\in\{0,1\}^{d\times m}\},$$9 and full rank distribution explicitly enumerated. In the $$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$0 case, the Boolean and nonnegative-integer cases coincide, and every rank-$$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$1 Boolean array can be decomposed into $$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$2 outer products with no overlapping 1-positions; in the $$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$3 case, the Boolean and nonnegative-integer classifications no longer coincide (Bremner et al., 2011).

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 $$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$4 (Bremner et al., 2011).

Another structural theme concerns stability under augmentation. The augmentation-property paper defines a matrix $$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$5 to have the Augmentation property for a rank function $$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$6 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

$$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$7

It also shows that failure can be severe: for any $$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$8, there exists a matrix $$(A\cdot B)_{ij}=\bigvee_{t=1}^d (A_{it}\wedge B_{tj}).$$9 and vectors $Cover_1(M)$0 such that each $Cover_1(M)$1, while

$Cover_1(M)$2

This behavior has no analogue for ordinary linear rank (Parnas et al., 2017).

Kronecker products supply a different extremal mechanism. Boolean rank is always submultiplicative: $Cover_1(M)$3 For the crown matrices, the 2021 paper proves that this inequality is strict for all $Cover_1(M)$4: $Cover_1(M)$5 hence in particular

$Cover_1(M)$6

For $Cover_1(M)$7, the same paper constructs explicit covers of size $Cover_1(M)$8, which is near the linear lower-bound regime rather than the naive quadratic regime (Haviv et al., 2021).

4. Communication complexity and algebraic lower bounds

Boolean rank is directly tied to nondeterministic communication complexity. The survey states

$Cover_1(M)$9

while binary rank gives the analogous unambiguous nondeterministic quantity

$A(G)$0

This connection explains why rectangle covers, biclique covers, and factorization ranks recur in communication-complexity lower bounds and in log-rank-type questions (Parnas, 20 Jan 2026).

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 $A(G)$1-sum representation,

$A(G)$2

Its basic inequality is

$A(G)$3

and its main theorem shows

$A(G)$4

This reframes the log-rank conjecture as the problem of converting signed rectangle decompositions into positive rectangle partitions with only quasipolynomial blowup (Hambardzumyan et al., 2 Oct 2025).

Lower bounds for Boolean rank also admit commutative-algebraic formulations. Given the bipartite graph $A(G)$5 of a binary matrix $A(G)$6, one lower-bound framework uses the edge ideal $A(G)$7 and proves

$A(G)$8

Combined with induced matching bounds, this yields

$A(G)$9

A second framework introduces the isolation ideal $G$0 attached to isolated sets of 1-entries and proves

$G$1

These identities do not turn Boolean rank into a homological invariant, but they place classical lower-bound parameters inside a standard regularity calculus (Geraci et al., 11 Sep 2025).

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

$G$2

for any $G$3 on an $G$4 matrix (Parnas, 20 Jan 2026).

Despite this hardness, Boolean rank is fixed-parameter tractable with respect to the target rank $G$5. A basic kernelization principle is that if Boolean rank is at most $G$6, then after removing all-zero rows and columns and merging duplicates, the matrix has at most $G$7 distinct rows and at most $G$8 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 (Parnas, 20 Jan 2026).

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

$G$9

which always accepts matrices of Boolean rank at most $\bc(G)=\brank(A(G)).$0 and rejects with probability at least $\bc(G)=\brank(A(G)).$1 when the matrix is $\bc(G)=\brank(A(G)).$2-far from rank at most $\bc(G)=\brank(A(G)).$3. 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 (Parnas et al., 2019).

In applied Boolean matrix factorization, Boolean rank also functions as a model-order parameter. The 2025 bfact paper defines a Boolean matrix factorization of $\bc(G)=\brank(A(G)).$4 into binary factors $\bc(G)=\brank(A(G)).$5 and $\bc(G)=\brank(A(G)).$6 via

$\bc(G)=\brank(A(G)).$7

and treats the rank $\bc(G)=\brank(A(G)).$8 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 $\bc(G)=\brank(A(G)).$9. 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 (Visscher et al., 7 Sep 2025).

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 $\brank(A)$0-dimensional $\brank(A)$1 Boolean array has the form

$\brank(A)$2

and the rank of $\brank(A)$3 is the minimum $\brank(A)$4 such that

$\brank(A)$5

where the sum is taken entrywise with Boolean addition $\brank(A)$6. The essential distinction from arithmetic rank is that overlap of rank-one summands is absorbed rather than counted or cancelled (Bremner et al., 2011).

Algorithmic tensor decomposition adopts the same semantics. The GETF paper writes a $\brank(A)$7-order Boolean tensor as

$\brank(A)$8

so the effective Boolean rank is the number $\brank(A)$9 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 $0,1$00 complexity when a $0,1$01-order tensor has $0,1$02 entries (Wan et al., 2020).

Recent complexity-theoretic work extends related decomposition ideas to higher order. For an order-$0,1$03 Boolean tensor $0,1$04 of tensor rank $0,1$05, the signed-rectangle-rank paper proves that $0,1$06 can be written as a $0,1$07-linear combination of at most

$0,1$08

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 (Hambardzumyan et al., 2 Oct 2025).

The phrase “Boolean rank” is also overloaded outside matrix and tensor factorization. In finite dynamical systems, rank means image size: for a Boolean network $0,1$09,

$0,1$10

and for interaction graphs contained in a digraph $0,1$11, the maximum Boolean-network rank is $0,1$12 (Gadouleau, 2015). In decision-tree complexity, rank means the Horton–Strahler number of a decision tree, with

$0,1$13

a parameter satisfying

$0,1$14

for non-constant $0,1$15 on $0,1$16 variables (Dahiya et al., 2022). 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 (Parnas, 20 Jan 2026).