Iterated lower bound formulas: a diagonalization-based approach to proof complexity

We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally. We use the approach to give an explicit sequence of CNF formulas {φn} such that VNP ≠ VP iff there are no polynomial-size IPS proofs for the formulas φn. This provides a natural equivalence between proof complexity lower bounds and standard algebraic complexity lower bounds. Our proof of this fact uses the implication from IPS lower bounds to algebraic complexity lower bounds due to Grochow and Pitassi together with a diagonalization argument: the formulas φn themselves assert the non-existence of short IPS proofs for formulas encoding VNP ≠ VP at a different input length. Our result also has meta-mathematical implications: it gives evidence for the difficulty of proving strong lower bounds for IPS within IPS. For any strong enough propositional proof system R, we define the *iterated R-lower bound formulas*, which inductively assert the non-existence of short R proofs for formulas encoding the same statement at a different input length, and propose them as explicit hard candidates for the proof system R. We observe that this hypothesis holds for Resolution following recent results of Atserias and Muller and of Garlik, and give evidence in favour of it for other proof systems

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

Iterated lower bound formulas: a diagonalization-based approach to lower bounds in proof complexity

We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally. We use the approach to give an explicit sequence of CNF formulas {φ_n} such that VNP ≠ VP iff there are no polynomial-size IPS proofs for the formulas φ_n. This is the first equivalence between proof complexity lower bounds and standard algebraic complexity lower bounds. Our proof of this fact uses the implication from IPS lower bounds to algebraic complexity lower bounds due to Grochow and Pitassi [GP18] together with a diagonalization argument: the formulas φ_n themselves assert the non-existence of short IPS proofs for formulas encoding VNP ≠ VP at a different input length. Our result also has meta-mathematical implications: it gives evidence for the difficulty of proving strong lower bounds for IPS within IPS. For any strong enough propositional proof system R, we define the iterated R-lower bound formulas, and propose them as explicit hard candidates for the proof system R. We observe that this conjecture holds for Resolution, and give evidence in favour of it for other proof systems

Transplanted neurons integrate into adult retinas and respond to light

Retinal ganglion cells (RGCs) degenerate in diseases like glaucoma and are not replaced in adult mammals. Here we investigate whether transplanted RGCs can integrate into the mature retina. We have transplanted GFP-labelled RGCs into uninjured rat retinas in vivo by intravitreal injection. Transplanted RGCs acquire the general morphology of endogenous RGCs, with axons orienting towards the optic nerve head of the host retina and dendrites growing into the inner plexiform layer. Preliminary data show in some cases GFP(+) axons extending within the host optic nerves and optic tract, reaching usual synaptic targets in the brain, including the lateral geniculate nucleus and superior colliculus. Electrophysiological recordings from transplanted RGCs demonstrate the cells' electrical excitability and light responses similar to host ON, ON–OFF and OFF RGCs, although less rapid and with greater adaptation. These data present a promising approach to develop cell replacement strategies in diseased retinas with degenerating RGCs

Developing Cell-Based Therapies for RPE-Associated Degenerative Eye Diseases

International audienc