Polynomial calculus space and resolution width

We show that if a k-CNF requires width w to refute in resolution, then it requires space square root of √ω to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is the first analogue, in polynomial calculus, of Atserias and Dalmau's result lower-bounding clause space in resolution by resolution width. As a by-product of our new approach to space lower bounds we give a simple proof of Bonacina's recent result that total space in resolution (the total number of variable occurrences that must be kept in memory) is lower-bounded by the width squared. As corollaries of the main result we obtain some new lower bounds on the PCR space needed to refute specific formulas, as well as partial answers to some open problems about relations between space, size, and degree for polynomial calculus

On Vanishing Sums of Roots of Unity in Polynomial Calculus and Sum-Of-Squares

Vanishing sums of roots of unity can be seen as a natural generalization of knapsack from Boolean variables to variables taking values over the roots of unity. We show that these sums are hard to prove for polynomial calculus and for sum-of-squares, both in terms of degree and size

Bounded-depth Frege complexity of Tseitin formulas for all graphs

We prove that there is a constant K such that Tseitin formulas for a connected graph G requires proofs of size 2tw(G)javax.xml.bind.JAXBElement@531a834b in depth-d Frege systems for [Formula presented], where tw(G) is the treewidth of G. This extends Håstad's recent lower bound from grid graphs to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)javax.xml.bind.JAXBElement@25a4b51fpoly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution

Automatizability and Simple Stochastic Games

The complexity of simple stochastic games (SSGs) has been open since they were dened by Condon in 1992. Despite intensive eort, the complexity of this problem is still unresolved. In this paper, building on the results of [4], we establish a connection between the complexity of SSGs and the complexity of an important problem in proof complexity{the proof search problem for low depth Frege systems. We prove that if depth-3 Frege systems are weakly automatizable, then SSGs are solvable in polynomial-time. Moreover we identify a natural combinatorial principle, which is a version of the well-known Graph Ordering Principle (GOP), that we call the integer-valued GOP (IGOP). This principle states that for any graph G with nonnegative integer weights associated with each node, there exists a locally maximal vertex (a vertex whose weight is at least as large as its neighbors). We prove that if depth-2 Frege plus IGOP is weakly automatizable, then SSG is in P. Supported by NSERC.

Identification of Financial Factors in Economic Fluctuations

We estimate demand, supply, monetary, investment and financial shocks in a VAR identified with a minimum set of sign restrictions on US data. We find that financial shocks are major drivers of fluctuations in output, stock prices and investment but have a limited effect on inflation. In a second step we disentangle shocks originating in the housing sector, shocks originating in credit markets and uncertainty shocks. In the extended set-up financial shocks are even more important and a leading role is played by housing shocks that have large and persistent effects on output.publishedVersio

Rare variants in the genetic background modulate cognitive and developmental phenotypes in individuals carrying disease-associated variants.

To assess the contribution of rare variants in the genetic background toward variability of neurodevelopmental phenotypes in individuals with rare copy-number variants (CNVs) and gene-disruptive variants. We analyzed quantitative clinical information, exome sequencing, and microarray data from 757 probands and 233 parents and siblings who carry disease-associated variants. The number of rare likely deleterious variants in functionally intolerant genes ("other hits") correlated with expression of neurodevelopmental phenotypes in probands with 16p12.1 deletion (n=23, p=0.004) and in autism probands carrying gene-disruptive variants (n=184, p=0.03) compared with their carrier family members. Probands with 16p12.1 deletion and a strong family history presented more severe clinical features (p=0.04) and higher burden of other hits compared with those with mild/no family history (p=0.001). The number of other hits also correlated with severity of cognitive impairment in probands carrying pathogenic CNVs (n=53) or de novo pathogenic variants in disease genes (n=290), and negatively correlated with head size among 80 probands with 16p11.2 deletion. These co-occurring hits involved known disease-associated genes such as SETD5, AUTS2, and NRXN1, and were enriched for cellular and developmental processes. Accurate genetic diagnosis of complex disorders will require complete evaluation of the genetic background even after a candidate disease-associated variant is identified