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We extend Newton and Lagrange interpolation to arbitrary dimensions. The core contribution that enables this is a generalized notion of non-tensorial unisolvent nodes, i.e., nodes on which the multivariate polynomial interpolant of a function is unique. By validation, we reach the optimal exponential Trefethen rates for a class of analytic functions, we term Trefethen functions. The number of interpolation nodes required for computing the optimal interpolant depends sub-exponentially on the dimension, hence resisting the curse of dimensionality. Based on these results, we propose an algorithm to efficiently and numerically stably solve arbitrary-dimensional interpolation problems, with at most quadratic runtime and linear memory requirement.

A central component of rational behavior is logical inference: the process of determining which conclusions follow from a set of premises. Psychologists have documented several ways in which humans' inferences deviate from the rules of logic. Do language models, which are trained on text generated by humans, replicate such human biases, or are they able to overcome them? Focusing on the case of syllogisms -- inferences from two simple premises -- we show that, within the PaLM2 family of transformer language models, larger models are more logical than smaller ones, and also more logical than humans. At the same time, even the largest models make systematic errors, some of which mirror human reasoning biases: they show sensitivity to the (irrelevant) ordering of the variables in the syllogism, and draw confident but incorrect inferences from particular syllogisms (syllogistic fallacies). Overall, we find that language models often mimic the human biases included in their training data, but are able to overcome them in some cases.

To make accurate inferences in an interactive setting, an agent must not confuse passive observation of events with having intervened to cause them. The $do$ operator formalises interventions so that we may reason about their effect. Yet there exist pareto optimal mathematical formalisms of general intelligence in an interactive setting which, presupposing no explicit representation of intervention, make maximally accurate inferences. We examine one such formalism. We show that in the absence of a $do$ operator, an intervention can be represented by a variable. We then argue that variables are abstractions, and that need to explicitly represent interventions in advance arises only because we presuppose these sorts of abstractions. The aforementioned formalism avoids this and so, initial conditions permitting, representations of relevant causal interventions will emerge through induction. These emergent abstractions function as representations of one`s self and of any other object, inasmuch as the interventions of those objects impact the satisfaction of goals. We argue that this explains how one might reason about one`s own identity and intent, those of others, of one`s own as perceived by others and so on. In a narrow sense this describes what it is to be aware, and is a mechanistic explanation of aspects of consciousness.

Several techniques have been developed to prove the termination of programs. Finding ranking functions is one of the common approaches to do so. A ranking function must be bounded and must reduce at every iteration for all the reachable program states. Since the set of reachable states is often unknown, invariants serve as an over-approximation. Further, in the case of nested loops, the initial set of program states for the nested loop can be determined by the invariant of the outer loop. So, invariants play an important role in proving the validity of a ranking function in the absence of the exact reachable states. However, in the existing techniques, either the invariants are synthesized independently, or combined with ranking function synthesis into a single query, both of which are inefficient. We observe that a guided search for invariants and ranking functions can have benefits in terms of the number of programs that can be proved to terminate and the time needed to identify a proof of termination. So, in this work, we develop Syndicate, a novel framework that synergistically guides the search for both the ranking function and an invariant that together constitute a proof of termination. Owing to our synergistic approach, Syndicate can not only prove the termination of more benchmarks but also achieves a reduction ranging from 17% to 70% in the average runtime as compared to existing state-of-the-art termination analysis tools. We also prove that Syndicate is relatively complete, i.e., if there exists a ranking function and an invariant in their respective templates that can be used to prove the termination of a program, then Syndicate will always find it if there exist complete procedures for the template-specific functions in our framework.

The power prior is a popular class of informative priors for incorporating information from historical data. It involves raising the likelihood for the historical data to a power, which acts as discounting parameter. When the discounting parameter is modelled as random, the normalized power prior is recommended. In this work, we prove that the marginal posterior for the discounting parameter for generalized linear models converges to a point mass at zero if there is any discrepancy between the historical and current data, and that it does not converge to a point mass at one when they are fully compatible. In addition, we explore the construction of optimal priors for the discounting parameter in a normalized power prior. In particular, we are interested in achieving the dual objectives of encouraging borrowing when the historical and current data are compatible and limiting borrowing when they are in conflict. We propose intuitive procedures for eliciting the shape parameters of a beta prior for the discounting parameter based on two minimization criteria, the Kullback-Leibler divergence and the mean squared error. Based on the proposed criteria, the optimal priors derived are often quite different from commonly used priors such as the uniform prior.

The technique of forgetting in knowledge representation has been shown to be a powerful and useful knowledge engineering tool with widespread application. Yet, very little research has been done on how different policies of forgetting, or use of different forgetting operators, affects the inferential strength of the original theory. The goal of this paper is to define loss functions for measuring changes in inferential strength based on intuitions from model counting and probability theory. Properties of such loss measures are studied and a pragmatic knowledge engineering tool is proposed for computing loss measures using Problog. The paper includes a working methodology for studying and determining the strength of different forgetting policies, in addition to concrete examples showing how to apply the theoretical results using Problog. Although the focus is on forgetting, the results are much more general and should have wider application to other areas.

Various indicators and measures of the real life procedures rise up as functionals of the quantile process of a parent random variable Z. However, Z can be observed only through a response in a linear model whose covariates are not under our control and the probability distribution of error terms is generally unknown. The problem is that of nonparametric estimation or other inference for such functionals. We propose an estimation procedure based on the averaged two-step regression quantile, recently developed by the authors, combined with an R-estimator of slopes of the linear model.

Learning actions that are relevant to decision-making and can be executed effectively is a key problem in autonomous robotics. Current state-of-the-art action representations in robotics lack proper effect-driven learning of the robot's actions. Although successful in solving manipulation tasks, deep learning methods also lack this ability, in addition to their high cost in terms of memory or training data. In this paper, we propose an unsupervised algorithm to discretize a continuous motion space and generate "action prototypes", each producing different effects in the environment. After an exploration phase, the algorithm automatically builds a representation of the effects and groups motions into action prototypes, where motions more likely to produce an effect are represented more than those that lead to negligible changes. We evaluate our method on a simulated stair-climbing reinforcement learning task, and the preliminary results show that our effect driven discretization outperforms uniformly and randomly sampled discretizations in convergence speed and maximum reward.

Recent studies have put into question the belief that emergent abilities in language models are exclusive to large models. This skepticism arises from two observations: 1) smaller models can also exhibit high performance on emergent abilities and 2) there is doubt on the discontinuous metrics used to measure these abilities. In this paper, we propose to study emergent abilities in the lens of pre-training loss, instead of model size or training compute. We demonstrate that the models with the same pre-training loss, but different model and data sizes, generate the same performance on various downstream tasks. We also discover that a model exhibits emergent abilities on certain tasks -- regardless of the continuity of metrics -- when its pre-training loss falls below a specific threshold. Before reaching this threshold, its performance remains at the level of random guessing. This inspires us to redefine emergent abilities as those that manifest in models with lower pre-training losses, highlighting that these abilities cannot be predicted by merely extrapolating the performance trends of models with higher pre-training losses.

As artificial intelligence (AI) models continue to scale up, they are becoming more capable and integrated into various forms of decision-making systems. For models involved in moral decision-making, also known as artificial moral agents (AMA), interpretability provides a way to trust and understand the agent's internal reasoning mechanisms for effective use and error correction. In this paper, we provide an overview of this rapidly-evolving sub-field of AI interpretability, introduce the concept of the Minimum Level of Interpretability (MLI) and recommend an MLI for various types of agents, to aid their safe deployment in real-world settings.