Related papers: Anna Karenina and The Two Envelopes Problem
Frankl's conjecture, also known as the union-closed sets conjecture, can be equivalently expressed in terms of intersection-closed set families by considering the complements of sets. It posits that any family of sets closed under…
Counterfactual explanations indicate the smallest change in input that can translate to a different outcome for a machine learning model. Counterfactuals have generated immense interest in high-stakes applications such as finance,…
Sentences with gapping, such as Paul likes coffee and Mary tea, lack an overt predicate to indicate the relation between two or more arguments. Surface syntax representations of such sentences are often produced poorly by parsers, and even…
Entropy serves as a central observable which indicates uncertainty in many chemical, thermodynamical, biological and ecological systems, and the principle of the maximum entropy (MaxEnt) is widely supported in natural science. Recently,…
Zipf's, Heaps' and Taylor's laws are ubiquitous in many different systems where innovation processes are at play. Together, they represent a compelling set of stylized facts regarding the overall statistics, the innovation rate and the…
Established frameworks to understand problems with reproducibility in science begin with the relationship between our understanding of the prior probability of a claim and the statistical certainty that should be demanded of it, and explore…
We construct an oracle relative to which $\mathrm{P} = \mathrm{NP} \cap \mathrm{coNP}$, but there are no many-one complete sets in $\mathrm{UP}$, no many-one complete disjoint $\mathrm{NP}$-pairs, and no many-one complete disjoint…
We provide a natural answer to Lewis Carroll's pillow problem of what is the probability that a triangle is obtuse, Prob(Obtuse). This arises by straightforward combination of a) Kendall's Theorem - that the space of all triangles is a…
The study of entanglement and magic properties in topologically frustrated systems suggests that, in the thermodynamic limit, these quantities decompose into two distinct contributions. One is determined by the specific nature of the model…
Set theoretical paradoxes have a common root -- lack of understanding of why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such…
NP complete problem is one of the most challenging issues. The question of whether all problems in NP are also in P is generally considered one of the most important open questions in mathematics and theoretical computer science as it has…
What is the relationship between plausibility logic and the principle of maximum entropy? When does the principle give unreasonable or wrong results? When is it appropriate to use the rule `expectation = average'? Can plausibility logic…
Individual probabilities refer to the probabilities of outcomes that are realized only once: the probability that it will rain tomorrow, the probability that Alice will die within the next 12 months, the probability that Bob will be…
Characterization of complexity within the sociological interpretation has resulted in a large number of notions, which are relevant in different situations. From the statistical mechanics point of view, these notions resemble entropy. In a…
The objective of this article is to formalize the definition of NP problems. We construct a mathematical model of discrete problems as independence systems with weighted elements. We introduce two auxiliary sets that characterize the…
The 2015 Nobel Prize in Economic Sciences was awarded to Eugene Fama, Lars Peter Hansen and Robert Shiller for their contributions to the empirical analysis of asset prices. Eugene Fama [1] is an advocate of the efficient market hypothesis.…
We investigate several technical and conceptual questions. Our main subject is the investigation of independence as a ternary relation in the context of non-monotonic logic. In the context of probability, this investigation was started by…
Defects in topologically ordered models have interesting properties that are reminiscent of the anyonic excitations of the models themselves. For example, dislocations in the toric code model are known as twists and possess properties that…
We fulfill the rough topological analysis of the problem of the motion of the Kovalevskaya top in a double field. This problem is described by a completely integrable system with three degrees of freedom not reducible to a family of systems…
The original Wigner's friend paradox is a gedankenexperiment involving an observer described by an external agent. The paradox highlights the tension between unitary evolution and collapse in quantum theory, and is sometimes taken as…