Abstract
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).
Highlights
Computational complexity theory studies the limits of efficient computation, and a particular goal is to quantify the power of different computational resources such as time, space, non-determinism, and randomness
The Forbes-Shpilka generator given in Lemma 7.1 is a width w2-read-once oblivious algebraic branching programs (roABPs) succinct generator for degree d roABPs that read the variables in order X1, X2, . . . , XN
The generator given by Forbes and Shpilka in Lemma 7.1 hits roABPs that read the variables in the order X1, X2, . . . , XN and not necessarily in any variable order
Summary
Computational complexity theory studies the limits of efficient computation, and a particular goal is to quantify the power of different computational resources such as time, space, non-determinism, and randomness. Such questions can be instantiated as asking to prove equalities or separations between complexity classes, such as resolving P versus NP. The setting of Razborov and Rudich [86] is that of non-uniform complexity, where instead of considering a Turing machine solving a problem on all input sizes, one considers a model such as Boolean circuits where the computational device can change with the size of the input. N≥1 with the following properties, where we denote N := 2n to be the input size to the property.
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