[Paper Review] Generating Matrix Identities and Proof Complexity

Fu Li, Iddo Tzameret|arXiv (Cornell University)|Dec 21, 2013

Complexity and Algorithms in Graphs25 references3 citations

TL;DR

This paper introduces a hierarchy of arithmetic proof systems for proving matrix identities over finite fields, establishing an unconditional $Ω(n^{2d})$ lower bound on proof size for $d \times d$ matrix identities with $n$ variables. It leverages polynomial identity theory and non-commutative algebra to construct hard instances for strong proof systems, offering a new route to understanding proof complexity lower bounds.

ABSTRACT

Motivated by the fundamental lower bounds questions in proof complexity, we initiate the study of matrix identities as hard instances for strong proof systems. A matrix identity of $d imes d$ matrices over a field $\mathbb{F}$, is a non-commutative polynomial $f(x_1,\ldots,x_n)$ over $\mathbb{F}$ such that $f$ vanishes on every $d imes d$ matrix assignment to its variables. We focus on arithmetic proofs, which are proofs of polynomial identities operating with arithmetic circuits and whose axioms are the polynomial-ring axioms (these proofs serve as an algebraic analogue of the Extended Frege propositional proof system; and over $GF(2)$ they constitute formally a sub-system of Extended Frege [HT12]). We introduce a decreasing in strength hierarchy of proof systems within arithmetic proofs, in which the $d$th level is a sound and complete proof system for proving $d imes d$ matrix identities (over a given field). For each level $d>2$ in the hierarchy, we establish a proof-size lower bound in terms of the number of variables in the matrix identity proved: we show the existence of a family of matrix identities $f_n$ with $n$ variables, such that any proof of $f_n=0$ requires $Ω(n^{2d})$ number of lines. The lower bound argument uses fundamental results from the theory of algebras with polynomial identities together with a generalization of the arguments in [Hru11]. We then set out to study matrix identities as hard instances for (full) arithmetic proofs. We present two conjectures, one about non-commutative arithmetic circuit complexity and the other about proof complexity, under which up to exponential-size lower bounds on arithmetic proofs (in terms of the arithmetic circuit size of the identities proved) hold. Finally, we discuss the applicability of our approach to strong propositional proof systems such as Extended Frege.

Motivation & Objective

• To develop a hierarchy of sound and complete proof systems for $d \times d$ matrix identities over a field $\mathbb{F}$, with increasing strength as $d$ grows.
• To establish unconditional proof-size lower bounds for arithmetic proofs of matrix identities, particularly in terms of the number of variables $n$.
• To explore whether matrix identities can serve as hard instances for strong proof systems like Extended Frege, especially in the context of circuit complexity and proof complexity.
• To investigate the connection between non-commutative arithmetic circuit complexity and proof complexity via matrix identities, aiming for exponential-size lower bounds.

Proposed method

• Introduces a decreasing-in-strength hierarchy of arithmetic proof systems, where the $d$-th level is complete for proving $d \times d$ matrix identities.
• Uses fundamental results from the theory of algebras with polynomial identities to analyze the minimal number of generators needed to generate a matrix identity.
• Applies a generalized version of arguments from [7] to derive lower bounds on proof size based on the structure of matrix identities.
• Leverages the fact that matrix identities exhibit significant cancellation properties, resembling commutative polynomials, to analyze proof complexity.
• Constructs a family of identities $f_n$ with $n$ variables such that any proof of $f_n = 0$ requires $\Omega(n^{2d})$ lines.
• Relies on the notion of $Q$-measure and non-substitution-invariance to rule out small proofs for certain hard families of identities.

Experimental results

Research questions

• RQ1 Can matrix identities serve as hard instances for strong proof systems such as Extended Frege, particularly in the context of arithmetic proofs?
• RQ2 What is the minimal number of lines required to prove a $d \times d$ matrix identity with $n$ variables, and can this be bounded below by $\Omega(n^{2d})$?
• RQ3 Is there a connection between the $Q$-measure of a polynomial and its circuit complexity, especially when the polynomial is a matrix identity?
• RQ4 Can the proof complexity of matrix identities be used to derive exponential lower bounds on arithmetic proofs, assuming plausible conjectures in non-commutative circuit complexity?
• RQ5 Are there families of matrix identities that are small in size but have high $Q$-measure, indicating inherent proof complexity?

Key findings

• An unconditional proof-size lower bound of $\Omega(n^{2d})$ is established for proving $d \times d$ matrix identities with $n$ variables, using non-commutative algebraic techniques.
• The paper proves that the minimal number of generators needed to generate a matrix identity is bounded below by a function that grows with $d$, a result of independent interest in polynomial identity theory.
• It is shown that matrix identities over ${\rm Mat}_d(\mathbb{F})$ are highly structured and exhibit extensive cancellation, making them behave more like commutative polynomials than general non-commutative ones.
• The authors demonstrate that even with a black-box PIT for non-commutative circuits, recognizing matrix identities remains difficult, as identities like $s_4$ do not vanish as non-commutative polynomials.
• For any polynomial $f$ of degree $d$, the $Q$-measure satisfies $Q(f) = O(n^{d-1})$, but this bound does not preclude high $Q$-measure for specific hard families.
• The paper provides evidence that families of identities not being substitution instances of a finite basis are essential for proving strong $Q$-measure lower bounds, as such families avoid trivial reductions.