A feasible interpolation for random resolution

Random resolution, defined by Buss, Kolodziejczyk and Thapen (JSL, 2014), is a sound propositional proof system that extends the resolution proof system by the possibility to augment any set of initial clauses by a set of randomly chosen clauses (modulo a technical condition). We show how to apply the general feasible interpolation theorem for semantic derivations of Krajicek (JSL, 1997) to random resolution. As a consequence we get a lower bound for random resolution refutations of the clique-coloring formulas.Comment: Preprint April 2016, revised September and October 201

Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere circuit lower bounds is open even for problems in $\mathsf{MAEXP}$. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question: Can we show that a large set of techniques cannot prove that $\mathsf{NP}$ is easy infinitely often? Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k \geq 1$ it is consistent with theory $T$ that computational class ${\mathcal C} \not \subseteq \textit{i.o.}\mathrm{SIZE}(n^k)$, where $(T, \mathcal{C})$ is one of the pairs: $T = \mathsf{T}^1_2$ and ${\mathcal C} = \mathsf{P}^\mathsf{NP}$, $T = \mathsf{S}^1_2$ and ${\mathcal C} = \mathsf{NP}$, $T = \mathsf{PV}$ and ${\mathcal C} = \mathsf{P}$. In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory $\mathsf{PV}$ already formalizes sophisticated arguments, such as a proof of the PCP Theorem. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with $\mathsf{PV}$

On monotone circuits with local oracles and clique lower bounds

We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs $y_i = y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $\mu \in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n,k}, V_{n,k} \subseteq \{0,1\}^m$ be the set of $k$-cliques and the set of complete $(k-1)$-partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-$2$ monotone circuits with local oracles, we show that the size of the smallest circuits separating $U_{n,3}$ (triangles) and $V_{n,3}$ (complete bipartite graphs) undergoes two phase transitions according to $\mu$. 2. For $5 \leq k(n) \leq n^{1/4}$, arbitrary depth, and $\mu \leq 1/50$, we prove that the monotone circuit size complexity of separating the sets $U_{n,k}$ and $V_{n,k}$ is $n^{\Theta(\sqrt{k})}$, under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of $k$-clique obtained by Alon and Boppana (1987).Comment: Updated acknowledgements and funding informatio

Using WEPP to Model Better Management Practices to Reduce Erosion of an Agricultural Field in Marcellus, New

Using the Water Erosion Prediction Project model, this study analyzes soil erosion and deposition values of an agricultural field located in Marcellus, New York that is currently suffering from erosion problems. Sensitivity and calibration analyses are performed on the model using observed data from the 1940s. Appropriate erosion control management practices including no-till crop management, terraces, and grass filter strips are simulated. Recommendations are given comparing the simulated erosion values and a cost-benefit analysis of each better management practice in an attempt to reduce erosion rates and maximize the field’s profitability

Unprovability of circuit upper bounds in Cook's theory PV

We establish unconditionally that for every integer $k \geq 1$ there is a language L \in \mbox{P} such that it is consistent with Cook's theory PV that $L \notin Size(n^k)$. Our argument is non-constructive and does not provide an explicit description of this language

Ischemic Cholangiopathy 11 Years after Liver Transplantation from Asymptomatic Chronic Hepatic Artery Thrombosis

Hepatic artery thrombosis is a concerning complication of orthotopic liver transplantation, and it most often occurs early in the posttransplant period. However, on rare occasions it can occur at a time remote from transplant. We present a case of ischemic cholangiopathy complicated by stricture and anastomotic bile leak from chronic hepatic artery thrombosis that occurred 11 years after the transplant. The initial biliary stenting helped with the resolution of the leak but she was found to have stones, sludge and copious pus at the time of stent exchange. Hepatic arteriography demonstrated complete occlusion of the transplant hepatic artery with periportal collaterals reconstituting intrahepatic hepatic arterial branches. The patient was subsequently referred for repeat liver transplantation

The Cook-Reckhow definition

The Cook-Reckhow 1979 paper defined the area of research we now call Proof Complexity. There were earlier papers which contributed to the subject as we understand it today, the most significant being Tseitin's 1968 paper, but none of them introduced general notions that would allow to make an explicit and universal link between lengths-of-proofs problems and computational complexity theory. In this note we shall highlight three particular definitions from the paper: of proof systems, p-simulations and the pigeonhole principle formula, and discuss their role in defining the field. We will also mention some related developments and open problems