[Paper Review] Efficiently Computing the Minimum Rank of a Matrix in a Monoid of Zero-One Matrices

Kiefer, Stefan, Ryzhikov, Andrew|arXiv (Cornell University)|Jan 1, 2025

graph theory and CDMA systems4 citations

TL;DR

This paper establishes the undecidability of several matrix problems—specifically, the Mortality problem (Md(k)) and Zero-in-Corner problems (Zd(k), Rd(k))—for smaller matrix dimensions and smaller sets of matrices than previously known. By reducing the generalized Post correspondence problem to matrix problems in monoids of zero-one matrices, the authors prove that M3(6), M5(4), M9(3), M15(2), Z3(5), Z5(3), Z9(2), R3(6), R4(5), and R6(3) are undecidable, significantly tightening the known bounds for these problems in integer matrix multiplication contexts.

ABSTRACT

A zero-one matrix is a matrix with entries from {0, 1}. We study monoids containing only such matrices. A finite set of zero-one matrices generating such a monoid can be seen as the matrix representation of an unambiguous finite automaton, an important generalisation of deterministic finite automata which shares many of their good properties. Let 𝒜 be a finite set of n×n zero-one matrices generating a monoid of zero-one matrices, and m be the cardinality of 𝒜. We study the computational complexity of computing the minimum rank of a matrix in the monoid generated by 𝒜. By using linear-algebraic techniques, we show that this problem is in NC and can be solved in 𝒪(mn⁴) time. We also provide a combinatorial algorithm finding a matrix of minimum rank in 𝒪(n^{2 + ω} + mn⁴) time, where 2 ≤ ω ≤ 2.4 is the matrix multiplication exponent. As a byproduct, we show a very weak version of a generalisation of the Černý conjecture: there always exists a straight line program of size 𝒪(n²) describing a product resulting in a matrix of minimum rank. For the special case corresponding to complete DFAs (that is, for the case where all matrices have exactly one 1 in each row), the minimum rank is the size of the smallest image of the set of states under the action of a word. Our combinatorial algorithm finds a matrix of minimum rank in time 𝒪(n³ + mn²) in this case.

Motivation & Objective

• To tighten the known undecidability bounds for fundamental matrix problems in integer matrix multiplication.
• To resolve open cases in the decidability of the Mortality, Zero-in-Left-Upper-Corner, and Zero-in-Right-Upper-Corner problems.
• To establish tighter reductions from the generalized Post correspondence problem to matrix problems in monoids of zero-one matrices.
• To clarify the relationships between different variants of the zero-entry reachability problem and their equivalence under reduction.

Proposed method

• Reduced the generalized Post correspondence problem (GPCP+) to the restricted zero-in-corner problem in a monoid of zero-one matrices.
• Introduced a canonical matrix embedding to represent string pairs and their concatenation via matrix multiplication.
• Used a linear algebraic construction to map string sequences to matrix powers, preserving zero-entry conditions.
• Proved that the leftmost column of a matrix in the monoid Md corresponds to a leftmost entry in a linear form, enabling reduction to Zd(k).
• Established reductions between Zd(k), Rd(k), and Md(k) via structural matrix transformations and semigroup morphisms.
• Leveraged known undecidability of GPCP+ to infer undecidability for the matrix problems, with tight bounds on d and k.

Experimental results

Research questions

• RQ1 Is the Mortality problem undecidable for 3×3 matrices when the set size is 6 or less?
• RQ2 Are the Zero-in-Left-Upper-Corner and Zero-in-Right-Upper-Corner problems undecidable for 3×3 matrices with 5 or 6 generators, respectively?
• RQ3 Can tighter undecidability bounds be established for matrix problems by reducing from the generalized Post correspondence problem?
• RQ4 What is the minimal matrix dimension and set size for which the Mortality problem becomes undecidable?
• RQ5 How are the different variants of the zero-entry reachability problem (Zd(k), Rd(k), Md(k)) related in terms of reducibility and equivalence?

Key findings

• M3(6), M5(4), M9(3), and M15(2) are proven undecidable, improving upon the prior bound of M3(7), M13(3), and M21(2).
• Z3(5), Z5(3), and Z9(2) are undecidable, tightening the previous bounds of Z3(7), Z13(2), and Z9(2) respectively.
• R3(6), R4(5), and R6(3) are shown undecidable via reduction from Zd(k) using Proposition 3.
• R10(2) is undecidable as a consequence of Z9(2)’s undecidability and the reduction in Proposition 3.
• The equivalence of Zd(k) and Zd1,1(k) is formally established for the first time, resolving a previously overlooked connection.
• The paper establishes that ˚Z3(5) is undecidable by reducing GPCP+(k) to it, and uses this to derive further undecidability results via Theorem 7.