相关论文: Cheryl's Birthday
Black holes are extreme spacetime deformations where even light is imprisoned. There is an extensive astrophysical evidence for the real and abundant existence of these prisons of matter and light in the Universe. Mathematically, black…
When exploring time series datasets, analysts often pose "which and when" questions. For example, with world life expectancy data over one hundred years, they may inquire about the top 10 countries in life expectancy and the time period…
Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive…
English translation of Russian book compiled to honor the memory of Ilya Mikhailovich Kapchinsky - To the 90th Birthday Collection of Memories. The idea for this publication belongs to Nikolai Vladimirovich Lazarev, a close collaborator of…
Recently long range correlations were detected in nucleotide sequences and in human writings by several authors. We undertake here a systematic investigation of two books, Moby Dick by H. Melville and Grimm's tales, with respect to the…
In 1964, Paul Erd\H{o}s published a paper settling a question about function spaces that he had seen in a problem book. Erd\H{o}s proved that the answer was yes if and only if the continuum hypothesis was false: an innocent-looking question…
Chen et al. recently established bijections for $(d+1)$-noncrossing/ nonnesting matchings, oscillating tableaux of bounded height $d$, and oscillating lattice walks in the $d$-dimensional Weyl chamber. Stanley asked what is the total number…
This paper is a survey of a large number of informal definitions of ``intelligence'' that the authors have collected over the years. Naturally, compiling a complete list would be impossible as many definitions of intelligence are buried…
This is a structured compilation of some of my favourite open problems.
We give a unified geometric approach to some theorems about primitive elements and palindromes in free groups of rank 2. The geometric treatment gives new proofs of the theorems. Dedicated to Bill Harvey on his 65th birthday.
Traditional symbolic reasoning engines, while attractive for their precision and explicability, have a few major drawbacks: the use of brittle inference procedures that rely on exact matching (unification) of logical terms, an inability to…
Consider the math problem: "Lily received 3 cookies from her best friend yesterday and ate 5 for breakfast. Today, her friend gave her 3 more cookies. How many cookies does Lily have now?" Many large language models (LLMs) in previous…
This expository paper is written in celebration of Boris Zilber's 75th birthday. We discuss Ax-Schanuel type statements focusing on the case of positive characteristic.
Define a building blocks set to be a collection of n cubes (each with six sides) where each side is assigned one letter and one color from a palette of m colors. We propose a novel problem of assigning letters and colors to each face so as…
Cylindric algebras, or concept algebras in another name, form an interface between algebra, geometry and logic; they were invented by Alfred Tarski around 1947. We prove that there are 2 to the alpha many varieties of geometric (i.e.,…
This is a very basic introduction to some notions related to logic and complexity.
In his celebrated book "On Number and Games" (Academic Press, New-York, 1976), J.H. Conway introduced twelve versions of compound games. We analyze these twelve versions for the Node-Kayles game on paths. For usual disjunctive compound,…
Adversarial evaluation stress tests a model's understanding of natural language. While past approaches expose superficial patterns, the resulting adversarial examples are limited in complexity and diversity. We propose human-in-the-loop…
We help Alice play a certain "convergence game" against Bob and win the prize, which is a constructive solution to a problem by Erd\H{o}s and Graham, posed in their 1980 book on open questions in combinatorial number theory. Namely, after…
In Probabilistic Logic Nilsson uses the device of a probability distribution over a set of possible worlds to assign probabilities to the sentences of a logical language. In his paper Nilsson concentrated on inference and associated…