Algebraic Circuits Research Articles

SIGACT News Complexity Theory Column 121

This month, Pranjal Dutta and Sumanta Ghosh contributed an in-depth survey on the Polynomial Identity Testing problem, and the efforts made towards obtaining deterministic algorithms for testing identities for various models of algebraic circuits. I wish to thank Pranjal and Sumanta for their thorough work. I hope you enjoy reading their article. On the next issues of SIGACT News, you can expect columns by Dean Doron (probably on probably on binary codes and the GV bound, but stay tuned!), by Stacey Jeffrey (on Quantum Las Vegas Algorithms and Composition), by Igor Carboni Oliveira (on Complexity Theory in Bounded Arithmetic: Recent Results and Current Frontiers) and by Noam Mazor (topic TBA).

The Power Word Problem in Graph Products

The power word problem for a group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document} asks whether an expression u1x1⋯unxn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{u_1^{x_1} \\cdots u_n^{x_n}}$$\\end{document}, where the ui\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{u_i}$$\\end{document} are words over a finite set of generators of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document} and the xi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{x_i}$$\\end{document} binary encoded integers, is equal to the identity of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document}. It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document}). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document} is uNC1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ extsf{uNC}^{1}}$$\\end{document}-many-one reducible to the power word problem for a finite-index subgroup of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{G}$$\\end{document}. For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is AC0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ extsf{AC} ^0}$$\\end{document}-Turing-reducible to the word problem for the free group F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F_2}$$\\end{document} and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\mathcal {C}}$$\\end{document} without order two elements, the uniform power word problem in a graph product can be solved in AC0[C=LUPowWP(C)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ extsf{AC} ^0[\ extsf{C}_=\ extsf{L} ^{{{\\,\ extrm{UPowWP}\\,}}(\\mathcal {C})}]}$$\\end{document}, where UPowWP(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{{{\\,\ extrm{UPowWP}\\,}}(\\mathcal {C})}$$\\end{document} denotes the uniform power word problem for groups from the class C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\mathcal {C}}$$\\end{document}. As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ extsf{NP}}$$\\end{document}-complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated in Stober and Weiß (2022a) is true, our proof relies on this additional assumption.

Logical characterizations of algebraic circuit classes over integral domains

AbstractWe present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the $\mathrm{AC}_{\mathbb{R}}$ and $\mathrm{NC}_{\mathbb{R}}^{}$ classes for this setting. We give a theorem in the style of Immerman’s theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the $\mathrm{AC}_{R}$ and $\mathrm{NC}_R^{}$ hierarchy. Those generalizations apply to the Boolean $\mathrm{AC}$ and $\mathrm{NC}$ hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.

Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits

An Algebraic Circuit for a multivariate polynomial P is a computational model for constructing the polynomial P using only additions and multiplications. It is a syntactic model of computation, as opposed to the Boolean Circuit model, and hence lower bounds for this model are widely expected to be easier to prove than lower bounds for Boolean circuits. Despite this, we do not have superpolynomial lower bounds against general algebraic circuits of depth 3 (except over constant-sized finite fields) and depth 4 (over any field other than F 2 ), while constant-depth Boolean circuit lower bounds have been known since the early 1980s. In this paper, we prove the first superpolynomial lower bounds against algebraic circuits of all constant depths over all fields of characteristic 0. We also observe that our super-polynomial lower bound for constant-depth circuits implies the first deterministic sub-exponential time algorithm for solving the Polynomial Identity Testing (PIT) problem for all small-depth circuits using the known connection between algebraic hardness and randomness.

Secure Genomic String Search with Parallel Homomorphic Encryption

Fully homomorphic encryption (FHE) cryptographic systems enable limitless computations over encrypted data, providing solutions to many of today’s data security problems. While effective FHE platforms can address modern data security concerns in unsecure environments, the extended execution time for these platforms hinders their broader application. This project aims to enhance FHE systems through an efficient parallel framework, specifically building upon the existing torus FHE (TFHE) system chillotti2016faster. The TFHE system was chosen for its superior bootstrapping computations and precise results for countless Boolean gate evaluations, such as AND and XOR. Our first approach was to expand upon the gate operations within the current system, shifting towards algebraic circuits, and using graphics processing units (GPUs) to manage cryptographic operations in parallel. Then, we implemented this GPU-parallel FHE framework into a needed genomic data operation, specifically string search. We utilized popular string distance metrics (hamming distance, edit distance, set maximal matches) to ascertain the disparities between multiple genomic sequences in a secure context with all data and operations occurring under encryption. Our experimental data revealed that our GPU implementation vastly outperforms the former method, providing a 20-fold speedup for any 32-bit Boolean operation and a 14.5-fold increase for multiplications.This paper introduces unique enhancements to existing FHE cryptographic systems using GPUs and additional algorithms to quicken fundamental computations. Looking ahead, the presented framework can be further developed to accommodate more complex, real-world applications.

FIXP-Membership via Convex Optimization: Games, Cakes, and Markets

We introduce a new technique for proving membership of problems in FIXP - the class capturing the complexity of computing a fixed-point of an algebraic circuit. Our technique constructs a "pseudogate" which can be used as a black box when building FIXP circuits. This pseudogate, which we term the "OPT-gate", can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets. In particular, we prove complexity results for two classic problems: computing a market equilibrium in the Arrow-Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with very general valuations is FIXP-complete. We further showcase the wide applicability of our technique, by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games, as well as the pseudomarket mechanism of Hylland and Zeckhauser.

A Polynomial Degree Bound on Equations for Non-rigid Matrices and Small Linear Circuits

We show that there is an equation of degree at most poly( n ) for the (Zariski closure of the) set of the non-rigid matrices: That is, we show that for every large enough field 𝔽, there is a non-zero n 2 -variate polynomial P ε 𝔽[ x 1, 1 , ..., x n, n ] of degree at most poly( n ) such that every matrix M that can be written as a sum of a matrix of rank at most n /100 and a matrix of sparsity at most n 2 /100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer, and Landsberg [ 9 ] and improves the best upper bound known for this problem down from exp ( n 2 ) [ 9 , 12 ] to poly( n ). We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n 2 /200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [ 21 ] to construct low-degree “universal” maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low-degree annihilating polynomial completes the proof. As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [ 11 ].

Discovering the Roots: Uniform Closure Results for Algebraic Classes Under Factoring

Newton iteration is an almost 350-year-old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all roots simultaneously. In this form, the process yields better circuit complexity in the case when the number of rootsris small but the multiplicities are exponentially large. Our method sets up a linear system inrunknowns and iteratively builds the roots as formal power series. For an algebraic circuit\( f(x_1,\ldots ,x_n) \)of sizes, we prove that each factor has size at most a polynomial insand the degree of the squarefree part off. Consequently, if\( f_1 \)is a\( 2^{\Omega (n)} \)-hard polynomial, then any nonzero multiple\( \prod _{i} f_i^{e_i} \)is equally hard for arbitrary positive\( e_i \)’s, assuming that\( \sum _i\deg (f_i) \)is at most\( 2^{O(n)} \).It is an old open question whether the class of poly(n) size formulas (respectively, algebraic branching programs) is closed under factoring. We show that given a polynomialfof degree\( n^{O(1)} \)and formula (respectively, algebraic branching program) size\( n^{O(\log n)} \), we can find a similar-size formula (respectively, algebraic branching program) factor in randomized poly(\( n^{\log n} \)) time. Consequently, if the determinant requires an\( n^{\Omega (\log n)} \)size formula, then the same can be said about any of its nonzero multiples.In all of our proofs, we exploit the following property of multivariate polynomial factorization. Under a random linear transformation\( \tau \), the polynomial\( f(\tau \overline{x}) \)completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. Therefore, with the help of the strong mathematical characterizations and the ‘allRootsNI’ technique, we make significant progress towards the old open problems; supplementing the vast body of classical results and concepts in algebraic circuit factorization (e.g., [17,51,54,111]).

Guest Column

Given a polynomial fpx1; : : : ; xNq, we can wonder how we can evaluate it with a minimal number of arithmetic operations. Indeed, the task of the algorithm designer is to nd evolved processes which decrease this number. In the other direction, it could be interesting to be able to prove that there is no algorithm which does the computation in less than some number of operations. But how one could prove something like that? We will present here some methods which allow us to tackle the problem in the case of constant-depth algebraic circuits.

SIGACT News Complexity Theory Column 113

My deepest thanks to Nutan Limaye, Srikanth Srinivasan, and Sebastien Tavenas for their fascinating article, Lower bounds Against Constant-Depth Algebraic Circuits.

Derandomization from Algebraic Hardness

A hitting-set generator (HSG) is a polynomial map $G:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $C$ of small enough circuit size and degree, if $C$ is nonzero, then $C\circ G$ is nonzero. In this paper, we give a new construction of such an HSG assuming that we have an explicit polynomial of sufficient hardness. Formally, we prove the following over any field of characteristic zero: Let $k\in \mathbb{N}$ and $\delta > 0$ be arbitrary constants. Suppose $\{P_d\}_{d\in \mathbb{N}}$ is an explicit family of $k$-variate polynomials such that $\operatorname{deg} P_d = d$ and $P_d$ requires algebraic circuits of size $d^\delta$. Then, there are explicit hitting sets of polynomial size for $\mathsf{VP}$. This is the first HSG in the algebraic setting that yields a complete derandomization of polynomial identity testing (PIT) for general circuits from a suitable algebraic hardness assumption. As a direct consequence, we show that even saving a single point from the trivial explicit, exponential sized hitting sets for constant-variate polynomials of low individual degree which are computable by small circuits, implies a deterministic polynomial time algorithm for PIT. More precisely, we show the following: Let $k\in \mathbb{N}$ and $\delta > 0$ be arbitrary constants. Suppose for every $s$ large enough, there is an explicit hitting set of size at most $((s+1)^k - 1)$ for the class of $k$-variate polynomials of individual degree $s$ that are computable by size $s^\delta$ circuits. Then there is an explicit hitting set of size $\operatorname{poly}(s)$ for the class of $s$-variate polynomials, of degree $s$, that are computable by size $s$ circuits. As a consequence, we give a deterministic polynomial time construction of hitting sets for algebraic circuits, if a strengthening of the $\tau$-Conjecture of Shub and Smale is true.

Balancing Straight-line Programs

We show that a context-free grammar of size that produces a single string of length (such a grammar is also called a string straight-line program) can be transformed in linear time into a context-free grammar for of size , whose unique derivation tree has depth . This solves an open problem in the area of grammar-based compression, improves many results in this area, and greatly simplifies many existing constructions. Similar results are shown for two formalisms for grammar-based tree compression: top dags and forest straight-line programs. These balancing results can be all deduced from a single meta-theorem stating that the depth of an algebraic circuit over an algebra with a certain finite base property can be reduced to with the cost of a constant multiplicative size increase. Here, refers to the size of the unfolding (or unravelling) of the circuit. In particular, this results applies to standard arithmetic circuits over (noncommutative) semirings.

Strongly Exponential Separation between Monotone VP and Monotone VNP

We show that there is a sequence of explicit multilinear polynomials P n (x 1 , … ,x n ) ϵ R [x 1 , … ,x n ] with non-negative coefficients that lies in monotone VNP such that any monotone algebraic circuit for P n must have size exp (Ω ( n )) This builds on (and strengthens) a result of Yehudayoff (STOC 2019) who showed a lower bound of exp (Ω(√n)).

Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity over Any Field.

$ \newcommand{\cclass}[1]{{\textsf{#1}}} \newcommand{\Pclass}{\cclass{P}} \newcommand{\NP}{\cclass{NP}} \newcommand{\AM}{\cclass{AM}} \newcommand{\coAM}{\cclass{coAM}} \newcommand{\AMcapcoAM}{\AM\,\cap\,\coAM} \newcommand{\PSPACE}{\cclass{PSPACE}} \newcommand{\EXPSPACE}{\cclass{EXPSPACE}} \newcommand{\VP}{\cclass{VP}} \newcommand{\VPbar}{\overline{\VP}} $ Testing whether a set $\mathbf{f}$ of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. The complexity of algebraic independence testing is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). Previously, the best complexity bound known was $\NP^{\#\Pclass}$ (Mittmann, Saxena, Scheiblechner, Trans. AMS 2014). In this article we put the problem in $\AMcapcoAM$. In particular, dependence testing is unlikely to be $\NP$-hard. Our proof uses methods of algebraic geometry. We estimate the size of the image and the sizes of the preimages of the polynomial map $\mathbf{f}$ over the finite field. A gap between the corresponding sizes for independent and for dependent sets of polynomials is utilized in the $\AM$ protocols. Next, we study the open question of testing whether every annihilator of $\mathbf{f}$ has zero constant term (Kayal, CCC'09). We introduce a new problem called approximate polynomial satisfiability (APS), which is equivalent to the preceding question by a classical characterization in terms of the Zariski closure of the image of $\mathbf{f}$. We show that APS is $\NP$-hard and, using ideas from algebraic geometry, we put APS in $\PSPACE$. (The best previous bound was $\EXPSPACE$ via Gröbner basis computation.) As an unexpected application of this to approximative complexity theory we obtain that, over any field, hitting sets for $\VPbar$ can be constructed in $\PSPACE$. This solves an open problem posed in (Mulmuley, FOCS'12, J. AMS 2017), greatly mitigating the GCT Chasm (exponentially in terms of space complexity).

Circuit Complexity, Proof Complexity, and Polynomial Identity Testing

We introduce a new and natural algebraic proof system, whose complexity measure is essentially the algebraic circuit size of Nullstellensatz certificates. This enables us to exhibit close connections between effective Nullstellensatzë, proof complexity, and (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits (VNP ≠ VP). We also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs imply the Permanent versus Determinant Conjecture. Note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems. In doing so, we highlight the importance of polynomial identity testing (PIT) in proof complexity. In particular, we use PIT to illuminate AC 0 [ p ]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. Finally, we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity. Using the algebraic structure of our proof system, we propose a novel route to such lower bounds. Although such lower bounds remain elusive, this proposal should be contrasted with the difficulty of extending AC 0 [ p ] circuit lower bounds to AC 0 [ p ]-Frege lower bounds.

Flux-Charge Description of Circuits With Non-Volatile Switching Memristor Devices

This brief aims at defining a class ${\mathcal{ D}}$ of extended memristors by using a combination of fundamental algebraic circuit elements corresponding to ideal memristors and nonlinear resistors. By massaging the characteristics of the constitutive elements, such class ${\mathcal{ D}}$ of extended memristors is able to approximate rectifying effects and asymmetric pinched hysteresis loops of real non-volatile switching memristor devices. Finally, it is shown that if the nonlinear resistors are described in terms of piecewise linear characteristics, then the flux-charge analysis method permits to analyze the dynamics of nonlinear circuits with extended memristors in ${\mathcal{ D}}$ .

Witnessing matrix identities and proof complexity

We use results from the theory of algebras with polynomial identities (PI-algebras) to study the witness complexity of matrix identities. A matrix identity of [Formula: see text] matrices over a field [Formula: see text]is a non-commutative polynomial (f(x1, …, xn)) over [Formula: see text], such that [Formula: see text] vanishes on every [Formula: see text] matrix assignment to its variables. For every field [Formula: see text]of characteristic 0, every [Formula: see text] and every finite basis of [Formula: see text] matrix identities over [Formula: see text], we show there exists a family of matrix identities [Formula: see text], such that each [Formula: see text] has [Formula: see text] variables and requires at least [Formula: see text] many generators to generate, where the generators are substitution instances of elements from the basis. The lower bound argument uses fundamental results from PI-algebras together with a generalization of the arguments in [P. Hrubeš, How much commutativity is needed to prove polynomial identities? Electronic colloquium on computational complexity, ECCC, Report No.: TR11-088, June 2011].We apply this result in algebraic proof complexity, focusing on proof systems for polynomial identities (PI proofs) which operate with algebraic circuits and whose axioms are the polynomial-ring axioms [P. Hrubeš and I. Tzameret, The proof complexity of polynomial identities, in Proc. 24th Annual IEEE Conf. Computational Complexity, CCC 2009, 15–18 July 2009, Paris, France (2009), pp. 41–51; Short proofs for the determinant identities, SIAM J. Comput. 44(2) (2015) 340–383], and their subsystems. We identify a decrease in strength hierarchy of subsystems of PI proofs, in which the [Formula: see text]th level is a sound and complete proof system for proving [Formula: see text] matrix identities (over a given field). For each level [Formula: see text] in the hierarchy, we establish an [Formula: see text] lower bound on the number of proof-steps needed to prove certain identities.Finally, we present several concrete open problems about non-commutative algebraic circuits and speed-ups in proof complexity, whose solution would establish stronger size lower bounds on PI proofs of matrix identities, and beyond.

Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich (1997) for Boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike in the Boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support “cryptography” secure against algebraic circuits. Following a similar result of Williams (2016) in the Boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the polynomial identity testing problem, that is, to the existence of a hitting set for the class of poly(N)-degree poly(N)-size circuits which consists of coefficient vectors of polynomials of polylog(N) degree with polylog(N)-size circuits. Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices. Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that none of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier. Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni (2001), except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits. A conference version of this paper appeared in the Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC 2017).

Algebraic proof complexity

We survey recent progress in the proof complexity of strong proof systems and its connection to algebraic circuit complexity, showing how the synergy between the two gives rise to new approaches to fundamental open questions, solutions to old problems, and new directions of research. In particular, we focus on tight connections between proof complexity lower bounds (namely, lower bounds on the size of proofs of certain tautologies), algebraic circuit lower bounds, and the Polynomial Identity Testing problem from derandomization theory.

Complexity Theory Column 88

Algebraic complexity theory studies the complexity of computing (multivariate) polynomials efficiently using algebraic circuits. This succinct representation leads to fundamental algorithmic challenges such as the polynomial identity testing (PIT) problem (decide nonzeroness of the computed polynomial) and the polynomial factorization problem (compute succinct representations of the factors of the circuit). While the Schwartz-Zippel-DeMillo-Lipton Lemma [Sch80,Zip79,DL78] gives an easy randomized algorithm for PIT, randomized algorithms for factorization require more ideas as given by Kaltofen [Kal89]. However, even derandomizing PIT remains a fundamental problem in understanding the power of randomness. In this column, we survey the factorization problem, discussing the algorithmic ideas as well as the applications to other problems. We then discuss the challenges ahead, in particular focusing on the goal of obtaining deterministic factoring algorithms. While deterministic PIT algorithms have been developed for various restricted circuit classes, there are very few corresponding factoring algorithms. We discuss some recent progress on the divisibility testing problem (test if a given polynomial divides another given polynomial) which captures some of the difficulty of factoring. Along the way we attempt to highlight key challenges whose solutions we hope will drive progress in the area.