Evaluating language models for mathematics through interactions.

There is much excitement about the opportunity to harness the power of large language models (LLMs) when building problem-solving assistants. However, the standard methodology of evaluating LLMs relies on static pairs of inputs and outputs; this is insufficient for making an informed decision about which LLMs are best to use in an interactive setting, and how that varies by setting. Static assessment therefore limits how we understand language model capabilities. We introduce CheckMate, an adaptable prototype platform for humans to interact with and evaluate LLMs. We conduct a study with CheckMate to evaluate three language models (InstructGPT, ChatGPT, and GPT-4) as assistants in proving undergraduate-level mathematics, with a mixed cohort of participants from undergraduate students to professors of mathematics. We release the resulting interaction and rating dataset, MathConverse. By analyzing MathConverse, we derive a taxonomy of human query behaviors and uncover that despite a generally positive correlation, there are notable instances of divergence between correctness and perceived helpfulness in LLM generations, among other findings. Further, we garner a more granular understanding of GPT-4 mathematical problem-solving through a series of case studies, contributed by experienced mathematicians. We conclude with actionable takeaways for ML practitioners and mathematicians: models that communicate uncertainty, respond well to user corrections, and can provide a concise rationale for their recommendations, may constitute better assistants. Humans should inspect LLM output carefully given their current shortcomings and potential for surprising fallibility.

Machine behaviour.

Machines powered by artificial intelligence increasingly mediate our social, cultural, economic and political interactions. Understanding the behaviour of artificial intelligence systems is essential to our ability to control their actions, reap their benefits and minimize their harms. Here we argue that this necessitates a broad scientific research agenda to study machine behaviour that incorporates and expands upon the discipline of computer science and includes insights from across the sciences. We first outline a set of questions that are fundamental to this emerging field and then explore the technical, legal and institutional constraints on the study of machine behaviour.

Bayes and Darwin: How replicator populations implement Bayesian computations.

Bayesian learning theory and evolutionary theory both formalize adaptive competition dynamics in possibly high-dimensional, varying, and noisy environments. What do they have in common and how do they differ? In this paper, we discuss structural and dynamical analogies and their limits, both at a computational and an algorithmic-mechanical level. We point out mathematical equivalences between their basic dynamical equations, generalizing the isomorphism between Bayesian update and replicator dynamics. We discuss how these mechanisms provide analogous answers to the challenge of adapting to stochastically changing environments at multiple timescales. We elucidate an algorithmic equivalence between a sampling approximation, particle filters, and the Wright-Fisher model of population genetics. These equivalences suggest that the frequency distribution of types in replicator populations optimally encodes regularities of a stochastic environment to predict future environments, without invoking the known mechanisms of multilevel selection and evolvability. A unified view of the theories of learning and evolution comes in sight.

The neural architecture of language: Integrative modeling converges on predictive processing.

The neuroscience of perception has recently been revolutionized with an integrative modeling approach in which computation, brain function, and behavior are linked across many datasets and many computational models. By revealing trends across models, this approach yields novel insights into cognitive and neural mechanisms in the target domain. We here present a systematic study taking this approach to higher-level cognition: human language processing, our species' signature cognitive skill. We find that the most powerful "transformer" models predict nearly 100% of explainable variance in neural responses to sentences and generalize across different datasets and imaging modalities (functional MRI and electrocorticography). Models' neural fits ("brain score") and fits to behavioral responses are both strongly correlated with model accuracy on the next-word prediction task (but not other language tasks). Model architecture appears to substantially contribute to neural fit. These results provide computationally explicit evidence that predictive processing fundamentally shapes the language comprehension mechanisms in the human brain.

Emotion prediction as computation over a generative theory of mind.

From sparse descriptions of events, observers can make systematic and nuanced predictions of what emotions the people involved will experience. We propose a formal model of emotion prediction in the context of a public high-stakes social dilemma. This model uses inverse planning to infer a person's beliefs and preferences, including social preferences for equity and for maintaining a good reputation. The model then combines these inferred mental contents with the event to compute 'appraisals': whether the situation conformed to the expectations and fulfilled the preferences. We learn functions mapping computed appraisals to emotion labels, allowing the model to match human observers' quantitative predictions of 20 emotions, including joy, relief, guilt and envy. Model comparison indicates that inferred monetary preferences are not sufficient to explain observers' emotion predictions; inferred social preferences are factored into predictions for nearly every emotion. Human observers and the model both use minimal individualizing information to adjust predictions of how different people will respond to the same event. Thus, our framework integrates inverse planning, event appraisals and emotion concepts in a single computational model to reverse-engineer people's intuitive theory of emotions. This article is part of a discussion meeting issue 'Cognitive artificial intelligence'.

DreamCoder: growing generalizable, interpretable knowledge with wake-sleep Bayesian program learning.

Expert problem-solving is driven by powerful languages for thinking about problems and their solutions. Acquiring expertise means learning these languages-systems of concepts, alongside the skills to use them. We present DreamCoder, a system that learns to solve problems by writing programs. It builds expertise by creating domain-specific programming languages for expressing domain concepts, together with neural networks to guide the search for programs within these languages. A 'wake-sleep' learning algorithm alternately extends the language with new symbolic abstractions and trains the neural network on imagined and replayed problems. DreamCoder solves both classic inductive programming tasks and creative tasks such as drawing pictures and building scenes. It rediscovers the basics of modern functional programming, vector algebra and classical physics, including Newton's and Coulomb's laws. Concepts are built compositionally from those learned earlier, yielding multilayered symbolic representations that are interpretable and transferrable to new tasks, while still growing scalably and flexibly with experience. This article is part of a discussion meeting issue 'Cognitive artificial intelligence'.

Rapid trial-and-error learning with simulation supports flexible tool use and physical reasoning.

Many animals, and an increasing number of artificial agents, display sophisticated capabilities to perceive and manipulate objects. But human beings remain distinctive in their capacity for flexible, creative tool use-using objects in new ways to act on the world, achieve a goal, or solve a problem. To study this type of general physical problem solving, we introduce the Virtual Tools game. In this game, people solve a large range of challenging physical puzzles in just a handful of attempts. We propose that the flexibility of human physical problem solving rests on an ability to imagine the effects of hypothesized actions, while the efficiency of human search arises from rich action priors which are updated via observations of the world. We instantiate these components in the "sample, simulate, update" (SSUP) model and show that it captures human performance across 30 levels of the Virtual Tools game. More broadly, this model provides a mechanism for explaining how people condense general physical knowledge into actionable, task-specific plans to achieve flexible and efficient physical problem solving.

Predicting responsibility judgments from dispositional inferences and causal attributions.

The question of how people hold others responsible has motivated decades of theorizing and empirical work. In this paper, we develop and test a computational model that bridges the gap between broad but qualitative framework theories, and quantitative but narrow models. In our model, responsibility judgments are the result of two cognitive processes: a dispositional inference about a person's character from their action, and a causal attribution about the person's role in bringing about the outcome. We test the model in a group setting in which political committee members vote on whether or not a policy should be passed. We assessed participants' dispositional inferences and causal attributions by asking how surprising and important a committee member's vote was. Participants' answers to these questions in Experiment 1 accurately predicted responsibility judgments in Experiment 2. In Experiments 3 and 4, we show that the model also predicts moral responsibility judgments, and that importance matters more for responsibility, while surprise matters more for judgments of wrongfulness.

Modeling human intuitions about liquid flow with particle-based simulation.

Humans can easily describe, imagine, and, crucially, predict a wide variety of behaviors of liquids-splashing, squirting, gushing, sloshing, soaking, dripping, draining, trickling, pooling, and pouring-despite tremendous variability in their material and dynamical properties. Here we propose and test a computational model of how people perceive and predict these liquid dynamics, based on coarse approximate simulations of fluids as collections of interacting particles. Our model is analogous to a "game engine in the head", drawing on techniques for interactive simulations (as in video games) that optimize for efficiency and natural appearance rather than physical accuracy. In two behavioral experiments, we found that the model accurately captured people's predictions about how liquids flow among complex solid obstacles, and was significantly better than several alternatives based on simple heuristics and deep neural networks. Our model was also able to explain how people's predictions varied as a function of the liquids' properties (e.g., viscosity and stickiness). Together, the model and empirical results extend the recent proposal that human physical scene understanding for the dynamics of rigid, solid objects can be supported by approximate probabilistic simulation, to the more complex and unexplored domain of fluid dynamics.

The Naïve Utility Calculus as a unified, quantitative framework for action understanding.

The human ability to reason about the causes behind other people' behavior is critical for navigating the social world. Recent empirical research with both children and adults suggests that this ability is structured around an assumption that other agents act to maximize some notion of subjective utility. In this paper, we present a formal theory of this Naïve Utility Calculus as a probabilistic generative model, which highlights the role of cost and reward tradeoffs in a Bayesian framework for action-understanding. Our model predicts with quantitative accuracy how people infer agents' subjective costs and rewards based on their observable actions. By distinguishing between desires, goals, and intentions, the model extends to complex action scenarios unfolding over space and time in scenes with multiple objects and multiple action episodes. We contrast our account with simpler model variants and a set of special-case heuristics across a wide range of action-understanding tasks: inferring costs and rewards, making confidence judgments about relative costs and rewards, combining inferences from multiple events, predicting future behavior, inferring knowledge or ignorance, and reasoning about social goals. Our work sheds light on the basic representations and computations that structure our everyday ability to make sense of and navigate the social world.

Social Pragmatics: Preschoolers Rely on Commonsense Psychology to Resolve Referential Underspecification.

Four experiments show that 4- and 5-year-olds (total N = 112) can identify the referent of underdetermined utterances through their Naïve Utility Calculus-an intuitive theory of people's behavior structured around an assumption that agents maximize utilities. In Experiments 1-2, a puppet asked for help without specifying to whom she was talking ("Can you help me?"). In Experiments 3-4, a puppet asked the child to pass an object without specifying what she wanted ("Can you pass me that one?"). Children's responses suggest that they considered cost trade-offs between the members in the interaction. These findings add to a body of work showing that reference resolution is informed by commonsense psychology from early in childhood.

Functional neuroanatomy of intuitive physical inference.

To engage with the world-to understand the scene in front of us, plan actions, and predict what will happen next-we must have an intuitive grasp of the world's physical structure and dynamics. How do the objects in front of us rest on and support each other, how much force would be required to move them, and how will they behave when they fall, roll, or collide? Despite the centrality of physical inferences in daily life, little is known about the brain mechanisms recruited to interpret the physical structure of a scene and predict how physical events will unfold. Here, in a series of fMRI experiments, we identified a set of cortical regions that are selectively engaged when people watch and predict the unfolding of physical events-a "physics engine" in the brain. These brain regions are selective to physical inferences relative to nonphysical but otherwise highly similar scenes and tasks. However, these regions are not exclusively engaged in physical inferences per se or, indeed, even in scene understanding; they overlap with the domain-general "multiple demand" system, especially the parts of that system involved in action planning and tool use, pointing to a close relationship between the cognitive and neural mechanisms involved in parsing the physical content of a scene and preparing an appropriate action.

Subjective randomness as statistical inference.

Some events seem more random than others. For example, when tossing a coin, a sequence of eight heads in a row does not seem very random. Where do these intuitions about randomness come from? We argue that subjective randomness can be understood as the result of a statistical inference assessing the evidence that an event provides for having been produced by a random generating process. We show how this account provides a link to previous work relating randomness to algorithmic complexity, in which random events are those that cannot be described by short computer programs. Algorithmic complexity is both incomputable and too general to capture the regularities that people can recognize, but viewing randomness as statistical inference provides two paths to addressing these problems: considering regularities generated by simpler computing machines, and restricting the set of probability distributions that characterize regularity. Building on previous work exploring these different routes to a more restricted notion of randomness, we define strong quantitative models of human randomness judgments that apply not just to binary sequences - which have been the focus of much of the previous work on subjective randomness - but also to binary matrices and spatial clustering.

Intuitive experimentation in the physical world.

Many aspects of our physical environment are hidden. For example, it is hard to estimate how heavy an object is from visual observation alone. In this paper we examine how people actively "experiment" within the physical world to discover such latent properties. In the first part of the paper, we develop a novel framework for the quantitative analysis of the information produced by physical interactions. We then describe two experiments that present participants with moving objects in "microworlds" that operate according to continuous spatiotemporal dynamics similar to everyday physics (i.e., forces of gravity, friction, etc.). Participants were asked to interact with objects in the microworlds in order to identify their masses, or the forces of attraction/repulsion that governed their movement. Using our modeling framework, we find that learners who freely interacted with the physical system selectively produced evidence that revealed the physical property consistent with their inquiry goal. As a result, their inferences were more accurate than for passive observers and, in some contexts, for yoked participants who watched video replays of an active learner's interactions. We characterize active learners' actions into a range of micro-experiment strategies and discuss how these might be learned or generalized from past experience. The technical contribution of this work is the development of a novel analytic framework and methodology for the study of interactively learning about the physical world. Its empirical contribution is the demonstration of sophisticated goal directed human active learning in a naturalistic context.

Learning physical parameters from dynamic scenes.

Humans acquire their most basic physical concepts early in development, and continue to enrich and expand their intuitive physics throughout life as they are exposed to more and varied dynamical environments. We introduce a hierarchical Bayesian framework to explain how people can learn physical parameters at multiple levels. In contrast to previous Bayesian models of theory acquisition (Tenenbaum, Kemp, Griffiths, & Goodman, 2011), we work with more expressive probabilistic program representations suitable for learning the forces and properties that govern how objects interact in dynamic scenes unfolding over time. We compare our model to human learners on a challenging task of estimating multiple physical parameters in novel microworlds given short movies. This task requires people to reason simultaneously about multiple interacting physical laws and properties. People are generally able to learn in this setting and are consistent in their judgments. Yet they also make systematic errors indicative of the approximations people might make in solving this computationally demanding problem with limited computational resources. We propose two approximations that complement the top-down Bayesian approach. One approximation model relies on a more bottom-up feature-based inference scheme. The second approximation combines the strengths of the bottom-up and top-down approaches, by taking the feature-based inference as its point of departure for a search in physical-parameter space.

Changing minds: Children's inferences about third party belief revision.

By the age of 5, children explicitly represent that agents can have both true and false beliefs based on epistemic access to information (e.g., Wellman, Cross, & Watson, 2001). Children also begin to understand that agents can view identical evidence and draw different inferences from it (e.g., Carpendale & Chandler, 1996). However, much less is known about when, and under what conditions, children expect other agents to change their minds. Here, inspired by formal ideal observer models of learning, we investigate children's expectations of the dynamics that underlie third parties' belief revision. We introduce an agent who has prior beliefs about the location of a population of toys and then observes evidence that, from an ideal observer perspective, either does, or does not justify revising those beliefs. We show that children's inferences on behalf of third parties are consistent with the ideal observer perspective, but not with a number of alternative possibilities, including that children expect other agents to be influenced only by their prior beliefs, only by the sampling process, or only by the observed data. Rather, children integrate all three factors in determining how and when agents will update their beliefs from evidence.

Eye-Tracking Causality.

How do people make causal judgments? What role, if any, does counterfactual simulation play? Counterfactual theories of causal judgments predict that people compare what actually happened with what would have happened if the candidate cause had been absent. Process theories predict that people focus only on what actually happened, to assess the mechanism linking candidate cause and outcome. We tracked participants' eye movements while they judged whether one billiard ball caused another one to go through a gate or prevented it from going through. Both participants' looking patterns and their judgments demonstrated that counterfactual simulation played a critical role. Participants simulated where the target ball would have gone if the candidate cause had been removed from the scene. The more certain participants were that the outcome would have been different, the stronger the causal judgments. These results provide the first direct evidence for spontaneous counterfactual simulation in an important domain of high-level cognition.

Compositional inductive biases in function learning.

How do people recognize and learn about complex functional structure? Taking inspiration from other areas of cognitive science, we propose that this is achieved by harnessing compositionality: complex structure is decomposed into simpler building blocks. We formalize this idea within the framework of Bayesian regression using a grammar over Gaussian process kernels, and compare this approach with other structure learning approaches. Participants consistently chose compositional (over non-compositional) extrapolations and interpolations of functions. Experiments designed to elicit priors over functional patterns revealed an inductive bias for compositional structure. Compositional functions were perceived as subjectively more predictable than non-compositional functions, and exhibited other signatures of predictability, such as enhanced memorability and reduced numerosity. Taken together, these results support the view that the human intuitive theory of functions is inherently compositional.

Ingredients of intelligence: From classic debates to an engineering roadmap.

We were encouraged by the broad enthusiasm for building machines that learn and think in more human-like ways. Many commentators saw our set of key ingredients as helpful, but there was disagreement regarding the origin and structure of those ingredients. Our response covers three main dimensions of this disagreement: nature versus nurture, coherent theories versus theory fragments, and symbolic versus sub-symbolic representations. These dimensions align with classic debates in artificial intelligence and cognitive science, although, rather than embracing these debates, we emphasize ways of moving beyond them. Several commentators saw our set of key ingredients as incomplete and offered a wide range of additions. We agree that these additional ingredients are important in the long run and discuss prospects for incorporating them. Finally, we consider some of the ethical questions raised regarding the research program as a whole.