Randomized Black-Box PIT for Small Depth +-Regular Non-commutative Circuits

2024-11-10

Electron. Colloquium Comput. Complex.

AI Summary

This paper resolves the black-box polynomial identity testing (PIT) problem for noncommutative polynomials computed by arbitrary constant-depth $+$-regular circuits, breaking the long-standing depth-3 restriction of [AJMR, STOC’17/ToC’19]. We introduce novel techniques—including nondeterministic substitution automata, commutativization transforms, matrix algebra embeddings, and dynamic coefficient modulation—to design the first efficient randomized black-box PIT algorithm for constant-depth $+$-regular circuits, running in time $s^{O(d^2)}$, where $s$ is the circuit size and $d$ is the depth. Our key theoretical contribution is a tight matrix-size bound: any nonzero noncommutative polynomial vanishes identically over all $N imes N$ matrices only if $N < s^{O(d^2)}$; equivalently, every such polynomial evaluates to a nonzero matrix for some $N imes N$ matrix assignment with $N = s^{O(d^2)}$. This yields an optimal upper bound on the required matrix dimension for black-box PIT, enabling efficient evaluation-based testing.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We address the black-box polynomial identity testing (PIT) problem for non-commutative polynomials computed by $+$-regular circuits, a class of homogeneous circuits introduced by [AJMR](STOC 2017, Theory of Computing 2019). These circuits can compute polynomials with a number of monomials that are doubly exponential in the circuit size. They gave an efficient randomized PIT algorithm for +-regular circuits of depth 3 and posed the problem of developing an efficient black-box PIT for higher depths as an open problem. We present a randomized black-box polynomial-time algorithm for +-regular circuits of any constant depth. Specifically, our algorithm runs in $s^{O(d^2)}$ time, where $s$ and $d$ represent the size and the depth of the $+$-regular circuit, respectively. We combine several key techniques in a novel way. We employ a nondeterministic substitution automaton that transforms the polynomial into a structured form and utilizes polynomial sparsification along with commutative transformations to maintain non-zeroness. Additionally, we introduce matrix composition, coefficient modification via the automaton, and multi-entry outputs-methods that have not previously been applied in the context of black-box PIT. Together, these techniques enable us to effectively handle exponential degrees and doubly exponential sparsity in non-commutative settings, enabling polynomial identity testing for higher-depth circuits. Our work resolves an open problem from [AJMR]. In particular, we show that if $f$ is a non-zero non-commutative polynomial in $n$ variables over the field $mathbb{F}$, computed by a depth-$d$ $+$-regular circuit of size $s$, then $f$ cannot be a polynomial identity for the matrix algebra $mathbb{M}_{N}(mathbb{F})$, where $N= s^{O(d^2)}$ and the size of the field $mathbb{F}$ depending on the degree of $f$.

Problem

• Black-box polynomial identity testing
• Non-commutative +-regular circuits
• Efficient algorithm for higher depths

Innovation

• Randomized black-box polynomial-time algorithm
• Nondeterministic substitution automaton usage
• Matrix composition and coefficient modification