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 $\mathsf{PV}$

Axiomatic method and the law

Whether an axiomatic approach to law is possible and useful today has to be perceived as unanswered. Perception of the axiomatic method among lawyers, however, is clouded by misunderstanding. Clarifying them may generate new discussion about the axiomatization of legal theories

Free will, robots, and the axiom of choice

There are quite a number of similarities between the moral concept of choice and the mathematical axiom of choice. These similarities shed light on how to adapt law to solve cases that arise with the increasing “autonomy” of robots

Subsidized Jobs for Unemployed Workers in Slovakia

This paper uses an administrative dataset from the Slovak Republic on durations of individual unemployment spells. The focus of the analysis is on the effect of the duration of temporary subsidized jobs on the job finding rate of unemployed workers. It appears that the duration of the temporary jobs is an important determinant of the speed by which unemployed workers find regular jobs. In this sense shorter temporary jobs are more effective than long temporary jobs. The main reason for this is probably that temporary jobs with a long duration induce workers in the first period on the temporary job to search less intensive for a regular job than temporary jobs with a short duration do.http://deepblue.lib.umich.edu/bitstream/2027.42/39695/3/wp311.pd

Why is there a Spike in the Job Finding Rate at Benefit Exhaustion?

Putting a limit on the duration of unemployment benefits tends to introduce a “spike” in the job finding rate shortly before benefits are exhausted. Current theories explain this spike from workers’ behavior. We present a theoretical model in which also the nature of the job matters. End-of-benefit spikes in job finding rates are related to optimizing behavior of unemployed workers who rationally assume that employers will accept delays in the starting date of a new job, especially if these jobs are permanent. We use a dataset on Slovenian unemployment spells to test this prediction and find supporting evidence. We conclude that the spike in the job finding rate suggests that workers exploit unemployment insurance benefits for subsidized leisure.unemployment benefits, spikes

Why Is There a Spike in the Job Finding Rate at Benefit Exhaustion?

Putting a limit on the duration of unemployment benefits tends to introduce a "spike" in the job finding rate shortly before benefits are exhausted. Current theories explain this spike from workers’ behavior. We present a theoretical model in which also the nature of the job matters. End-of-benefit spikes in job finding rates are related to optimizing behavior of unemployed workers who rationally assume that employers will accept delays in the starting date of a new job, especially if these jobs are permanent. We use a dataset on Slovenian unemployment spells to test this prediction and find supporting evidence. We conclude that the spike in the job finding rate suggests that workers exploit unemployment insurance benefits for subsidized leisure.unemployment benefits, spikes