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Children can take many paths to become scientists. But they undoubtedly include the following steps Play, play and play; Observe; Ask (themselves). In this paper I will talk about some of my experiences teaching girls and boys, teenagers…
This work contains the mathematical exploration of a few prototypical games in which central concepts from statistics and probability theory naturally emerge. The first two kinds of games are termed Fisher and Bayesian games, which are…
Influence diagrams provide a compact graphical representation of decision problems. Several algorithms for the quick computation of their associated expected utilities are available in the literature. However, often they rely on a full…
Dyck paths are one of the most important objects in enumerative combinatorics, and there are many papers devoted to counting selected families of Dyck paths. Here we present two approaches for the automatic counting of many such families,…
In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using…
We analyze the computational complexity of the popular computer games Threes!, 1024!, 2048 and many of their variants. For most known versions expanded to an m x n board, we show that it is NP-hard to decide whether a given starting…
If the state space of a homogeneous continuous-time Markov chain is too large, making inferences - here limited to determining marginal or limit expectations - becomes computationally infeasible. Fortunately, the state space of such a chain…
Lample and Charton (2019) describe a system that uses deep learning technology to compute symbolic, indefinite integrals, and to find symbolic solutions to first- and second-order ordinary differential equations, when the solutions are…
Neural networks are capable of superposition -- representing more features than there are dimensions. Recent work considers the analogous concept for computation instead of storage, proposing theoretical constructions. But there has been…
Large language models (LLMs) have achieved striking successes on many benchmarks, yet recent studies continue to expose fundamental weaknesses. In this paper, we introduce Concept, a simple word-guessing board game, as a benchmark for…
As children enter elementary school, their understanding of the ordinal structure of numbers transitions from a memorized count list of the first 50-100 numbers to knowing the successor function and understanding the countably infinite. We…
Simulating sampling algorithms with people has proven a useful method for efficiently probing and understanding their mental representations. We propose that the same methods can be used to study the representations of Large Language Models…
The strength of chess engines together with the availability of numerous chess games have attracted the attention of chess players, data scientists, and researchers during the last decades. State-of-the-art engines now provide an…
Parrondo's paradox was introduced by Juan Parrondo in 1996. In game theory, this paradox is described as: A combination of losing strategies becomes a winning strategy. At first glance, this paradox is quite surprising, but we can easily…
This paper presents an intervention study on the effects of the combined methods of (1) the Socratic method, (2) Chain of Thought (CoT) reasoning, (3) simplified gamification and (4) formative feedback on university students' Maths learning…
The ability of intelligent agents to play games in human-like fashion is popularly considered a benchmark of progress in Artificial Intelligence. Similarly, performance on multi-disciplinary tasks such as Visual Question Answering (VQA) is…
Markov Logic Networks (MLNs), which elegantly combine logic rules and probabilistic graphical models, can be used to address many knowledge graph problems. However, inference in MLN is computationally intensive, making the industrial-scale…
Chain-of-thought prompting (e.g., "Let's think step-by-step") primes large language models to verbalize rationalization for their predictions. While chain-of-thought can lead to dramatic performance gains, benefits appear to emerge only for…
A new object of the probability theory, two-sided chain of events (symbols), is introduced. A theory of multi-steps Markov chains with long-range memory, proposed earlier in Phys. Rev. E 68, 06117 (2003), is developed and used to establish…
We consider the problem of inferring the directed, causal graph from observational data, assuming no hidden confounders. We take an information theoretic approach, and make three main contributions. First, we show how through algorithmic…