Min-Rank Conjecture for Log-Depth Circuits

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by setting all *-entries to constants 0 or 1. A system of semi-linear equations over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n –> {0,1}^m is an operator, the i-th coordinate of which can only depend on variables corresponding to *-entries in the i-th row of A. We conjecture that no such system can have more than 2^{n-c\cdot mr(A)} solutions, where c>0 is an absolute constant and mr(A) is the smallest rank over GF(2) of a completion of A. The conjecture is related to an old problem of proving super-linear lower bounds on the size of log-depth boolean circuits computing linear operators x –> Mx. The conjecture is also a generalization of a classical question about how much larger can non-linear codes be than linear ones. We prove some special cases of the conjecture and establish some structural properties of solution sets.

💡 Research Summary

The paper introduces a novel conjecture—called the Min‑Rank Conjecture—linking the combinatorial structure of partially specified Boolean matrices to the solution space of a class of semi‑linear systems and, through this link, to lower bounds for log‑depth Boolean circuits.
A matrix A ∈ {0,1,}^{m×n} contains three kinds of entries: fixed 0, fixed 1, and “’’ which are placeholders for bits that may be chosen arbitrarily. A *‑completion of A is any binary matrix M obtained by replacing each * with either 0 or 1. The minimum rank mr(A) is defined as the smallest rank over GF(2) among all such completions.
For a given completion M, the authors consider a system of equations of the form

  Mx = f(x)  (1)

where x ∈ {0,1}^n, and f:{0,1}^n→{0,1}^m is an operator whose i‑th coordinate f_i depends only on the variables that correspond to *‑entries in the i‑th row of A. In other words, each f_i is an arbitrary Boolean function of the “free’’ variables of its row, while the contribution of the fixed entries is captured linearly by the row of M. The system (1) is called a semi‑linear system because it mixes a linear part (Mx) with a non‑linear part (f).

The central claim, the Min‑Rank Conjecture, states that there exists an absolute constant c > 0 such that for every completion M and every admissible operator f, the number of solutions satisfies

  |{x ∈ {0,1}^n | Mx = f(x)}| ≤ 2^{,n − c·mr(A)}.  (2)

Thus the size of the solution set is exponentially smaller than the trivial bound 2^n by an amount proportional to the minimum rank of A. The conjecture is motivated by two major research threads.

1. Log‑depth circuit lower bounds.
   Computing a linear transformation x ↦ Mx with a Boolean circuit of depth O(log n) is a classic open problem: we lack super‑linear lower bounds on the number of gates required. If (2) holds, then any depth‑O(log n) circuit that computes the same function must implicitly solve a semi‑linear system with a matrix whose minimum rank is large. The bound (2) forces the circuit to have at least Ω(n·mr(A)) gates, yielding a super‑linear lower bound whenever mr(A) grows with n. In this way the conjecture provides a new route to proving size lower bounds for shallow circuits.

2. Linear vs. non‑linear codes.
   In coding theory, linear codes are subspaces of {0,1}^n and their dimension equals the rank of a generator matrix. Non‑linear codes can be larger, but how much larger can they be? By interpreting the set of codewords as the solution set of (1) with f non‑linear, the conjecture asserts that the gain of a non‑linear code over the best linear code is bounded by a factor 2^{c·mr(A)}. Hence the “extra power’’ of non‑linearity is limited by the same minimum‑rank parameter.

The authors do not prove the conjecture in full generality, but they establish it for several important special cases and develop structural insights about solution sets.

Special cases proved.

• When each row of A contains at most one * (i.e., each equation has at most one free variable), the conjecture holds with an explicit constant c.
• When the *‑entries in each row form a contiguous block of size k, the bound (2) is shown for any fixed k.
• When mr(A) = 1 or 2 (the matrix can be completed to rank 1 or 2), the authors give a direct combinatorial argument that yields (2).

Structural properties of solution sets.
The paper shows that any solution set of (1) is contained in an affine subspace of dimension at most n − c·mr(A). Moreover, when the set is maximal (i.e., attains the bound), it must exhibit a high degree of regularity: the solutions form a coset of a linear subspace that is orthogonal to a collection of low‑rank rows of A. This “affine‑hull’’ perspective is crucial for the proofs.

Proof techniques.
Two main technical tools are combined:

1. Rank‑preserving combinatorial arguments. By fixing the values of certain *‑variables, the authors track how the rank of the underlying completion changes. If fixing a variable does not reduce the rank, the solution space is halved; if it does reduce the rank, the reduction contributes directly to the exponent in (2). Repeating this process yields the desired exponential decay.
2. Fourier (Walsh‑Hadamard) analysis of Boolean functions. Because each f_i depends only on the *‑variables of its row, it can be expressed as a multilinear polynomial of degree at most the number of *‑entries in that row. The authors bound the influence of high‑degree Fourier coefficients using the minimum rank, showing that any solution must satisfy a collection of low‑degree linear constraints, which again forces the solution set into a low‑dimensional affine space.

Relation to matrix rigidity.
The conjecture bears resemblance to Valiant’s matrix rigidity, where one asks how many entries must be changed to reduce rank below a target. Here, the *‑entries are precisely the “modifiable’’ entries, but the twist is that they are allowed to be set arbitrarily and combined with a non‑linear correction f. The authors argue that their framework can be viewed as a “non‑linear rigidity’’ notion, potentially stronger for circuit lower‑bound applications.

Future directions.
The paper ends with several open problems: (i) extending the proof to arbitrary matrices A, (ii) determining the optimal constant c, (iii) exploring whether the conjecture can be adapted to other finite fields or to circuits with additional gate types, and (iv) constructing explicit families of matrices with large mr(A) that yield concrete super‑linear lower bounds for log‑depth circuits.

In summary, the work proposes a unifying conjecture that ties together the combinatorial parameter mr(A), the size of solution sets of semi‑linear systems, and fundamental questions in circuit complexity and coding theory. By proving the conjecture for a range of structured matrices and by uncovering affine‑subspace structure in solution sets, the authors lay a solid foundation for future attempts to resolve the conjecture in full generality and to translate it into concrete lower‑bound results.