Proof Complexity Lower Bounds from Algebraic Circuit Complexity

We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi (J. ACM, 2018), where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the Boolean setting for subsystems of Extended Frege proofs, where proof-lines are circuits from restricted Boolean circuit classes. Except one, all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity). We give two general methods of converting certain algebraic circuit lower bounds into proof complexity ones. However, we need to strengthen existing lower bounds to hold for either the functional model or for multiplicities (see below). Our techniques are reminiscent of existing methods for converting Boolean circuit lower bounds into related proof complexity results, such as feasible interpolation. We obtain the relevant types of lower bounds for a variety of classes (sparse polynomials, depth-3 powering formulas, read-once oblivious algebraic branching programs, and multilinear formulas), and infer the relevant proof complexity results. We complement our lower bounds by giving short refutations of the previously studied subset-sum axiom using IPS subsystems, allowing us to conclude strict separations between some of these subsystems. Our first method is a functional lower bound, a notion due to Grigoriev and Razborov (Appl. Algebra Eng. Commun. Comput., 2000), which says that not only does a polynomial f require large algebraic circuits, but that any polynomial g agreeing with f on the Boolean cube also requires large algebraic circuits. For our classes of interest, we develop functional lower bounds where g(x¯¯¯) equals 1/p(x¯¯¯) where p is a constant-degree polynomial, which in turn yield corresponding IPS lower bounds for proving that p is nonzero over the Boolean cube. In particular, we show superpolynomial lower bounds for refuting variants of the subset-sum axiom in various IPS subsystems. Our second method is to give lower bounds for multiples, that is, to give explicit polynomials whose all (nonzero) multiples require large algebraic circuit complexity. By extending known techniques, we are able to obtain such lower bounds for our classes of interest, which we then use to derive corresponding IPS lower bounds. Such lower bounds for multiples are of independent interest, as they have tight connections with the algebraic hardness versus randomness paradigm

Stem cell treatment of degenerative eye disease

Stem cell therapies are being explored extensively as treatments for degenerative eye disease, either for replacing lost neurons, restoring neural circuits or, based on more recent evidence, as paracrine-mediated therapies in which stem cell-derived trophic factors protect compromised endogenous retinal neurons from death and induce the growth of new connections. Retinal progenitor phenotypes induced from embryonic stem cells/induced pluripotent stem cells (ESCs/iPSCs) and endogenous retinal stem cells may replace lost photoreceptors and retinal pigment epithelial (RPE) cells and restore vision in the diseased eye, whereas treatment of injured retinal ganglion cells (RGCs) has so far been reliant on mesenchymal stem cells (MSC). Here, we review the properties of non-retinal-derived adult stem cells, in particular neural stem cells (NSCs), MSC derived from bone marrow (BMSC), adipose tissues (ADSC) and dental pulp (DPSC), together with ESC/iPSC and discuss and compare their potential advantages as therapies designed to provide trophic support, repair and replacement of retinal neurons, RPE and glia in degenerative retinal diseases. We conclude that ESCs/iPSCs have the potential to replace lost retinal cells, whereas MSC may be a useful source of paracrine factors that protect RGC and stimulate regeneration of their axons in the optic nerve in degenerate eye disease. NSC may have potential as both a source of replacement cells and also as mediators of paracrine treatment

Comparing solar water heater popularization policies in China, Israel and Australia : the roles of governments in adopting green innovations

Studying the roles of governments in adopting green innovations is significant for analysing the transition to a more sustainable energy system. This article presents a comparative study of policies for popularizing domestic solar water heaters in three countries: China, Israel and Australia. Expanding the analysis beyond the economics of innovation, it demonstrates the institutional dimension of green technology deployment in these three countries. By examining the diverging roles of governments in facilitating green technology adoption in existing social routines and practices, it finds that governments' motivations, support and implementation mechanisms are remarkably different in these three countries. In particular, the paper argues that solar water heater popularization has been distinguished as a business opportunity in China, energy security in Israel and environmental responsibility in Australia. In addition, the institutional settings have a real impact on governments' roles in adopting green innovations, in terms of the policy instruments chosen and implementation mechanisms.11 page(s