Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve

Variety Membership Testing in Algebraic Complexity Theory

In this thesis, we study some of the central problems in algebraic complexity theory through the lens of the variety membership testing problem. In the first part, we investigate whether separations between algebraic complexity classes can be phrased as instances of the variety membership testing problem. For this, we compare some complexity classes with their closures. We show that monotone commutative single-(source, sink) ABPs are closed. Further, we prove that multi-(source, sink) ABPs are not closed in both the monotone commutative and the noncommutative settings. However, the corresponding complexity classes are closed in all these settings. Next, we observe a separation between the complexity class VQP and the closure of VNP. In the second part, we cover the blackbox polynomial identity testing (PIT) problem, and the rank computation problem of symbolic matrices, both phrasable as instances of the variety membership testing problem. For the blackbox PIT, we give a randomized polynomial time algorithm that uses the number of random bits that matches the information-theoretic lower bound, differing from it only in the lower order terms. For the rank computation problem, we give a deterministic polynomial time approximation scheme (PTAS) when the degrees of the entries of the matrices are bounded by a constant. Finally, we show NP-hardness of two problems on 3-tensors, both of which are instances of the variety membership testing problem. The first problem is the orbit closure containment problem for the action of GLk x GLm x GLn on 3-tensors, while the second problem is to decide whether the slice rank of a given 3-tensor is at most r

Pseudorandomness via the discrete Fourier transform

We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs. We give an explicit construction of a pseudorandom generator that fools the discrete Fourier transforms of linear functions with seed-length that is nearly logarithmic (up to polyloglog factors) in the input size and the desired error parameter. Our result gives a single pseudorandom generator that fools several important classes of tests computable in logspace that have been considered in the literature, including halfspaces (over general domains), modular tests and combinatorial shapes. For all these classes, our generator is the first that achieves near logarithmic seed-length in both the input length and the error parameter. Getting such a seed-length is a natural challenge in its own right, which needs to be overcome in order to derandomize RL - a central question in complexity theory. Our construction combines ideas from a large body of prior work, ranging from a classical construction of [NN93] to the recent gradually increasing independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some novel analytic machinery which might find other applications

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)

Truth Table Minimization of Computational Models

Complexity theory offers a variety of concise computational models for computing boolean functions - branching programs, circuits, decision trees and ordered binary decision diagrams to name a few. A natural question that arises in this context with respect to any such model is this: Given a function f:{0,1}^n \to {0,1}, can we compute the optimal complexity of computing f in the computational model in question? (according to some desirable measure). A critical issue regarding this question is how exactly is f given, since a more elaborate description of f allows the algorithm to use more computational resources. Among the possible representations are black-box access to f (such as in computational learning theory), a representation of f in the desired computational model or a representation of f in some other model. One might conjecture that if f is given as its complete truth table (i.e., a list of f's values on each of its 2^n possible inputs), the most elaborate description conceivable, then any computational model can be efficiently computed, since the algorithm computing it can run poly(2^n) time. Several recent studies show that this is far from the truth - some models have efficient and simple algorithms that yield the desired result, others are believed to be hard, and for some models this problem remains open. In this thesis we will discuss the computational complexity of this question regarding several common types of computational models. We shall present several new hardness results and efficient algorithms, as well as new proofs and extensions for known theorems, for variants of decision trees, formulas and branching programs