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)

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)

The Exact Complexity of Pseudorandom Functions and Tight Barriers to Lower Bound Proofs

How much computational resource do we need for cryptography? This is an important question of both theoretical and practical interests. In this paper, we study the problem on pseudorandom functions (PRFs) in the context of circuit complexity. Perhaps surprisingly, we prove extremely tight upper and lower bounds in various circuit models. * In general $B_2$ circuits, assuming the existence of PRFs, PRFs can be constructed in $2n + o(n)$ size, simplifying and improving the $O(n)$ bound by Ishai et al. (STOC 2008). We show that such construction is almost optimal by giving an unconditional $2n-O(1)$ lower bound. * In logarithmic depth circuits, assuming the existence of $NC^1$ PRFs, PRFs can be constructed in $2n + o(n)$ size and $(1+\epsilon) \log n$ depth simultaneously. * In constant depth linear threshold circuits, assuming the existence of $TC^0$ PRFs, PRFs can be constructed with wire complexity $n^{1+O(1.61^{-d})}$. We also give an $n^{1+\Omega(c^{-d})}$ wire complexity lower bound for some constant $c$. The upper bounds are proved with generalized Levin\u27s trick and novel constructions of almost universal hash functions; the lower bound for general circuits is proved via a tricky but elementary wire-counting argument; and the lower bound for $TC^0$ circuits is proved by extracting a black-box property of $TC^0$ circuits from the white-box restriction lemma of Chen, Santhanam, and Srinivasan (Theory Comput. 2018). As a byproduct, we prove unconditional tight upper and lower bounds for almost universal hashing, which we believe to have independent interests. Following Natural Proofs by Razborov and Rudich (J. Comput. Syst. Sci. 1997), our results make progress in realizing the difficulty to improve known circuit lower bounds which recently becomes significant due to the discovery of several bootstrapping results . In $TC^0$, this reveals the limitation of the current restriction-based methods; in particular, it brings new insights in understanding the strange phenomenon of sharp threshold results such as the one presented by Chen and Tell (STOC 2019)

Toward Better Formula Lower Bounds: An Information Complexity Approach to the KRW Composition Conjecture

One of the major open problems in complexity theory is proving super-polynomial lower bounds for circuits with logarithmic depth (i.e., P ̸ ⊆ NC1). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving superpolynomial circuit lower bounds. Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean functions f and g, the depth complexity of the composed function g ◦ f is roughly the sum of the depth complexities of f and g. They showed that the validity of this conjecture would imply that P ̸ ⊆ NC1. As a starting point for studying the composition of functions, they introduced a relation called “the universal relation”, and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by H˚astad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open. In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation. We also suggest a candidate for the next step and provide initial results toward it. Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations – communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.

Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere circuit lower bounds is open even for problems in $\mathsf{MAEXP}$. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question: Can we show that a large set of techniques cannot prove that $\mathsf{NP}$ is easy infinitely often? Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k \geq 1$ it is consistent with theory $T$ that computational class ${\mathcal C} \not \subseteq \textit{i.o.}\mathrm{SIZE}(n^k)$, where $(T, \mathcal{C})$ is one of the pairs: $T = \mathsf{T}^1_2$ and ${\mathcal C} = \mathsf{P}^\mathsf{NP}$, $T = \mathsf{S}^1_2$ and ${\mathcal C} = \mathsf{NP}$, $T = \mathsf{PV}$ and ${\mathcal C} = \mathsf{P}$. In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory $\mathsf{PV}$ already formalizes sophisticated arguments, such as a proof of the PCP Theorem. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with $\mathsf{PV}$

Circuit Complexity Meets Ontology-Based Data Access

Ontology-based data access is an approach to organizing access to a database augmented with a logical theory. In this approach query answering proceeds through a reformulation of a given query into a new one which can be answered without any use of theory. Thus the problem reduces to the standard database setting. However, the size of the query may increase substantially during the reformulation. In this survey we review a recently developed framework on proving lower and upper bounds on the size of this reformulation by employing methods and results from Boolean circuit complexity.Comment: To appear in proceedings of CSR 2015, LNCS 9139, Springe