Abstract

We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on $AC^0[p]$, multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen), the connected components technique (Ben-Or), depth 3 arithmetic circuit lower bounds over finite fields (Grigoriev-Karpinski), lower bounds on permanent versus determinant (Mignon-Ressayre, Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (B\"{u}rgisser-Ikenmeyer) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi; Agrawal-Vinay; Koiran), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification and broad generalization of known results. It also shows that the framework of GCT is at least as powerful as known methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT; we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT naturally provides new properties to consider in the search for new lower bounds.

Highlights

  • Geometric complexity theory (GCT) is a program toward lower bounds—such as P = NP—using algebraic geometry and representation theory (see Mulmuley (2011b) for an overview, and references therein)

  • We show that most algebraic circuit lower bounds naturally fit into the representation-theoretic framework used in GCT

  • By showing that previous lower bounds and GCT share a common representation-theoretic viewpoint, we reveal many new contexts in which it might hopefully be easier to develop the tools and techniques of algebraic geometry and representation theory needed for the GCT approach to bigger problems such as permanent versus determinant or P versus NP

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Summary

Introduction

GCT provides a unifying framework for many known lower bounds, vastly generalizing known lower bound techniques This representation-theoretic viewpoint opens the door for new potentially fruitful two-way interactions between previous results and new progress in (geometric) complexity theory (see Sections 1.2 and 4.2 for details). We can already give one such argument: Most algebraic circuit lower bounds already use separating modules This new viewpoint makes new tools available and suggests new conjectures and directions to better understand complexity classes and lower bounds. This is one of the ways in which GCT suggests how we might understand previous lower bounds better, even ones that are essentially tight. This reduces an amorphous search for new useful properties to a comparatively feasible search for separating modules, which can even be made computational (see Appendix B.3 and Section 4 for more)

Definitions and a motivating example
On the necessity and utility of separating modules and border complexity
Most algebraic circuit lower bounds yield separating modules
Relations between lower bounds yield relations between separating modules
Proof of the correspondence between invariant properties and test modules
More on the necessity and utility of separating modules
Discussion of terminology
Standard notation in the literature
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