An average-case depth hierarchy theorem for Boolean circuits

We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of $\mathsf{AND}$, $\mathsf{OR}$, and $\mathsf{NOT}$ gates. Our hierarchy theorem says that for every $d \geq 2$, there is an explicit $n$-variable Boolean function $f$, computed by a linear-size depth-$d$ formula, which is such that any depth-$(d-1)$ circuit that agrees with $f$ on $(1/2 + o_n(1))$ fraction of all inputs must have size $\exp({n^{\Omega(1/d)}}).$ This answers an open question posed by H{\aa}stad in his Ph.D. thesis. Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus answering a question posed by O'Donnell, Kalai, and Hatami. A key ingredient in our proof is a notion of \emph{random projections} which generalize random restrictions

Lower End of the Linial-Post Spectrum

We show that recognizing axiomatizations of the Hilbert-style calculus containing only the axiom a -> (b -> a) is undecidable (a reduction from the Post correspondence problem is formalized in the Lean theorem prover). Interestingly, the problem remains undecidable considering only axioms which, when seen as simple types, are principal for some lambda-terms in beta-normal form. This problem is closely related to type-based composition synthesis, i.e. finding a composition of given building blocks (typed terms) satisfying a desired specification (goal type). Contrary to the above result, axiomatizations of the Hilbert-style calculus containing only the axiom a -> (b -> b) are recognizable in linear time

Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time $\widetilde{O}\left(m\log \kappa \log^2 (1/\epsilon)\right)$ where $\epsilon$ is the amount of error we are willing to tolerate. Here, $\kappa$ represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever $\kappa$ is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time $\widetilde{O}(m^{3/2} \log (1/\epsilon))$. In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201

Mixin Composition Synthesis based on Intersection Types

We present a method for synthesizing compositions of mixins using type inhabitation in intersection types. First, recursively defined classes and mixins, which are functions over classes, are expressed as terms in a lambda calculus with records. Intersection types with records and record-merge are used to assign meaningful types to these terms without resorting to recursive types. Second, typed terms are translated to a repository of typed combinators. We show a relation between record types with record-merge and intersection types with constructors. This relation is used to prove soundness and partial completeness of the translation with respect to mixin composition synthesis. Furthermore, we demonstrate how a translated repository and goal type can be used as input to an existing framework for composition synthesis in bounded combinatory logic via type inhabitation. The computed result is a class typed by the goal type and generated by a mixin composition applied to an existing class

On the Round Complexity of Randomized Byzantine Agreement

We prove lower bounds on the round complexity of randomized Byzantine agreement (BA) protocols, bounding the halting probability of such protocols after one and two rounds. In particular, we prove that: 1) BA protocols resilient against n/3 [resp., n/4] corruptions terminate (under attack) at the end of the first round with probability at most o(1) [resp., 1/2+ o(1)]. 2) BA protocols resilient against n/4 corruptions terminate at the end of the second round with probability at most 1-Theta(1). 3) For a large class of protocols (including all BA protocols used in practice) and under a plausible combinatorial conjecture, BA protocols resilient against n/3 [resp., n/4] corruptions terminate at the end of the second round with probability at most o(1) [resp., 1/2 + o(1)]. The above bounds hold even when the parties use a trusted setup phase, e.g., a public-key infrastructure (PKI). The third bound essentially matches the recent protocol of Micali (ITCS\u2717) that tolerates up to n/3 corruptions and terminates at the end of the third round with constant probability