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 $\mathsf{PV}$

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 $\mathsf{PV}$

Peripapillary microcirculation in Leber hereditary optic neuropathy

Purpose In this prospective observational comparative case series, we aimed to study the peripapillary capillary network with spectral‐domain optical coherence tomography angiography (OCT‐A) in Leber hereditary optic neuropathy (LHON). Methods Twelve eyes of six individuals, of these three males (five eyes) after clinical onset of visual impairment were imaged by OCT‐A with scans centred on optic discs. Control group consisted of 6 eyes with no visual impairment. Results The three affected individuals lost vision 6 years (at age 22 years), 2 years and 3 months (at age 26 years) and 1 year and 2 months (at age 30 years) prior to OCT‐A examination. All five affected eyes had alterations in density of the radial peripapillary microvascular network at the level of retinal nerve fibre layer, including an eye of a patient treated with idebenone that underwent almost full recovery (best corrected visual acuity 0.87). Interestingly, the other eye showed normal ocular findings 14 months after onset. Results of OCT‐A examination in this eye were unfortunately inconclusive due to a delineation error. At the level of the ganglion cell layer differences could be also noted, but only in two severely affected individuals. There were no differences between unaffected mutation carriers and control eyes. Conclusion Optical coherence tomography angiography scans confirmed that the peripapillary microvascular network is highly abnormal in eyes manifesting visual impairment due to LHON. These findings support the hypothesis that microangiopathy contributes to the development of vision loss in this mitochondrial disorder

Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that requireboolean circuits of super-linear size is a major frontier in complexity theory.While such lower bounds are known for larger complexity classes, existingresults only show that the corresponding problems are hard on infinitely manyinput lengths. For instance, proving almost-everywhere circuit lower bounds isopen even for problems in $\mathsf{MAEXP}$. Giving the notorious difficulty ofproving lower bounds that hold for all large input lengths, we ask thefollowing question: Can we show that a large set of techniques cannot provethat $\mathsf{NP}$ is easy infinitely often? Motivated by this and relatedquestions 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 isconsistent with theory $T$ that computational class ${\mathcal C} \not\subseteq \textit{i.o.}\mathrm{SIZE}(n^k)$, where $(T, \mathcal{C})$ is one ofthe 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 establishinfinitely often circuit upper bounds for the corresponding problems. This isof interest because the weaker theory $\mathsf{PV}$ already formalizessophisticated arguments, such as a proof of the PCP Theorem. These consistencystatements are unconditional and improve on earlier theorems of [KO17] and[BM18] on the consistency of lower bounds with $\mathsf{PV}$

Magnetic Non-destructive Testing of Plastically Deformed Mild Steel

The Barkhausen noise analysis and coercive field measurement have been used as magnetic non-destructive testing methods for plastically deformed high quality carbon steel specimens. The strain dependence of root mean square value and power spectrum of the Barkhausen noise and the coercive field are explained in terms of the dislocation density. The specimens have been subjected to different magnetizing frequencies to show the overlapping nature of the Barkhausen noise. The results are discussed in the context of usage of magnetic non-destructive testing to evaluate the plastic deformation of high quality carbon steel products

Adaptive testing of Materials using Preisach Model Parameters Variations - Introductory Tests

A new diagnostic method (MAT - Magnetic Adaptive Testing) for non-destructive testing of ferromagnetic construction materials (i.e. iron based) under mechanical stress is under development, [1]. The method is based on the investigation of the correlation between the mechanical load and the parameters of Preisach-like model describing magnetic properties of such materials as the differential permeability matrix. To get the set of model parameters needed, a number of minor hysteresis loops under defined exciting magnetic field strength waveform shape H(t), especially with constant field change rate dH(t)/dt required (which implies the inducted voltage to be proportional to the differential permeability), is to be measured. The influence of initial magnetic state of the investigated material, algorithm of demagnetisation process, the slope of time dependence of exciting magnetic field on the signal-to-noise ratio and stability of the measured signal is discussed