Bootstrapping in a language of thought: a formal model of numerical concept learning

Neural evidence of core foundations and conceptual change in preschool numeracy

Symbolic numeracy first emerges as children learn the meanings of number words and how to use them to precisely count sets of objects. This development starts before children enter school and forms a foundation for lifelong mathematics achievement. Despite its importance, exactly how children acquire this basic knowledge is unclear. Here we test competing theories of early number learning by measuring event-related brain potentials during a novel number word-quantity comparison task in 3-4-year-old preschool children (N = 128). We find several qualitative differences in neural processing of number by conceptual stage of development. Specifically, we find differences in early attention-related parietal electrophysiology (N1), suggesting that less conceptually advanced children process arrays as individual objects and more advanced children distribute attention over the entire set. Subsequently, we find that only more conceptually advanced children show later-going frontal (N2) sensitivity to the numerical-distance relationship between the number word and visual quantity. The nature of this response suggested that exact rather than approximate numerical meanings were being associated with number words over frontal sites. No evidence of numerical distance effects was observed over posterior scalp sites. Together these results suggest that children may engage parallel individuation of objects to learn the meanings of the first few number words, but, ultimately, create new exact cardinal value representations for number words that cannot be defined in terms of core, nonverbal number systems. More broadly, these results document an interaction between attentional and general cognitive mechanisms in cognitive development. RESEARCH HIGHLIGHTS: Conceptual development in numeracy is associated with a shift in attention from objects to sets. Children acquire meanings of the first few number words through associations with parallel attentional individuation of objects. Understanding of cardinality is associated with attentional processing of sets rather than individuals. Brain signatures suggest children attribute exact rather than approximate numerical meanings to the first few number words. Number-quantity relationship processing for the first few number words is evident in frontal but not parietal scalp electrophysiology of young children.

Learning from conditional probabilities

Bayesianism, that is, the formal capturing of belief in terms of probabilities, has had a major impact in cognitive science. Decades of research have examined lay reasoners' learning and reasoning with probabilities. The bulk of that research has concerned the response to new evidence. That response will depend on the conditional probabilities a reasoner assumes, yet little research has addressed the question of how reasoners respond when they are provided with new conditional probabilities. Furthermore, there are not just open empirical questions as to how lay reasoners actually respond, there are also open questions about how they should respond. This is illustrated by philosophical debate about the so-called Judy Benjamin Problem. In this paper, we present experiments on belief revision problems in which the new information is a conditional probability. More specifically, we investigate two versions of these problems. One where basic probability theory (as the core of what it means 'to be Bayesian') provides a single correct answer, and one where that answer is under-constrained. The former provide a new type of evidence on the longstanding question of human probabilistic reasoning skill. The latter informs debate on how to expand the Bayesian toolbox to deal with the issues raised by the Judy Benjamin Problem.

Mapping and modeling the semantic space of math concepts

Mathematics is an underexplored domain of human cognition. While many studies have focused on subsets of math concepts such as numbers, fractions, or geometric shapes, few have ventured beyond these elementary domains. Here, we attempted to map out the full space of math concepts and to answer two specific questions: can distributed semantic models, such a GloVe, provide a satisfactory fit to human semantic judgements in mathematics? And how does this fit vary with education? We first analyzed all of the French and English Wikipedia pages with math contents, and used a semi-automatic procedure to extract the 1000 most frequent math terms in both languages. In a second step, we collected extensive behavioral judgements of familiarity and semantic similarity between them. About half of the variance in human similarity judgements was explained by vector embeddings that attempt to capture latent semantic structures based on cooccurence statistics. Participants' self-reported level of education modulated familiarity and similarity, allowing us to create a partial hierarchy among high-level math concepts. Our results converge onto the proposal of a map of math space, organized as a database of math terms with information about their frequency, familiarity, grade of acquisition, and entanglement with other concepts.

The influence of language on the formation of number concepts: Evidence from preschool children who are bilingual in English and Arabic

We asked whether grammatical number marking has specific influence on the formation of early number concepts. In particular, does comprehension of dual case marking support young children's understanding of cardinality? We assessed number knowledge in 77 3-year-old Arabic-English bilingual children using the Give-a-Number task in both languages. Given recent concerns around the administration and scoring of the Give-a-Number task, we used two complementary approaches: one based on conceptual levels and the other based on overall test scores. We also tested comprehension of dual case marking in Arabic and number sequence knowledge in both languages. Regression analyses showed that dual case comprehension exerts a strong influence on cardinality tested in Arabic independent of age, general language skills, and number sequence knowledge. No such influence was found for cardinality tested in English, indicating a language-specific effect. Further analyses tested for transfer of cardinality knowledge between languages. These revealed, in addition to the findings outlined above, a powerful cross-linguistic transfer effect. Our findings are consistent with a model in which the direct effect of dual case marking is language specific, but concepts, once acquired, may be represented abstractly and transferred between languages.

The relational bottleneck as an inductive bias for efficient abstraction

A central challenge for cognitive science is to explain how abstract concepts are acquired from limited experience. This has often been framed in terms of a dichotomy between connectionist and symbolic cognitive models. Here, we highlight a recently emerging line of work that suggests a novel reconciliation of these approaches, by exploiting an inductive bias that we term the relational bottleneck. In that approach, neural networks are constrained via their architecture to focus on relations between perceptual inputs, rather than the attributes of individual inputs. We review a family of models that employ this approach to induce abstractions in a data-efficient manner, emphasizing their potential as candidate models for the acquisition of abstract concepts in the human mind and brain.

Exploring Early Number Abilities With Multimodal Transformers

Early number skills represent critical milestones in children's cognitive development and are shaped over years of interacting with quantities and numerals in various contexts. Several connectionist computational models have attempted to emulate how certain number concepts may be learned, represented, and processed in the brain. However, these models mainly used highly simplified inputs and focused on limited tasks. We expand on previous work in two directions: First, we train a model end-to-end on video demonstrations in a synthetic environment with multimodal visual and language inputs. Second, we use a more holistic dataset of 35 tasks, covering enumeration, set comparisons, symbolic digits, and seriation. The order in which the model acquires tasks reflects input length and variability, and the resulting trajectories mostly fit with findings from educational psychology. The trained model also displays symbolic and non-symbolic size and distance effects. Using techniques from interpretability research, we investigate how our attention-based model integrates cross-modal representations and binds them into context-specific associative networks to solve different tasks. We compare models trained with and without symbolic inputs and find that the purely non-symbolic model employs more processing-intensive strategies to determine set size.

Symbolic metaprogram search improves learning efficiency and explains rule learning in humans

Throughout their lives, humans seem to learn a variety of rules for things like applying category labels, following procedures, and explaining causal relationships. These rules are often algorithmically rich but are nonetheless acquired with minimal data and computation. Symbolic models based on program learning successfully explain rule-learning in many domains, but performance degrades quickly as program complexity increases. It remains unclear how to scale symbolic rule-learning methods to model human performance in challenging domains. Here we show that symbolic search over the space of metaprograms-programs that revise programs-dramatically improves learning efficiency. On a behavioral benchmark of 100 algorithmically rich rules, this approach fits human learning more accurately than alternative models while also using orders of magnitude less search. The computation required to match median human performance is consistent with conservative estimates of human thinking time. Our results suggest that metaprogram-like representations may help human learners to efficiently acquire rules.

Early-emerging combinatorial thought: Human infants flexibly combine kind and quantity concepts

Combinatorial thought, or the ability to combine a finite set of concepts into a myriad of complex ideas and knowledge structures, is the key to the productivity of the human mind and underlies communication, science, technology, and art. Despite the importance of combinatorial thought for human cognition and culture, its developmental origins remain unknown. To address this, we tested whether 12-mo-old infants (N = 60), who cannot yet speak and only understand a handful of words, can combine quantity and kind concepts activated by verbal input. We proceeded in two steps: first, we taught infants two novel labels denoting quantity (e.g., "mize" for 1 item; "padu" for 2 items, Experiment 1). Then, we assessed whether they could combine quantity and kind concepts upon hearing complex expressions comprising their labels (e.g., "padu duck", Experiments 2-3). At test, infants viewed four different sets of objects (e.g., 1 duck, 2 ducks, 1 ball, 2 balls) while being presented with the target phrase (e.g., "padu duck") naming one of them (e.g., 2 ducks). They successfully retrieved and combined on-line the labeled concepts, as evidenced by increased looking to the named sets but not to distractor sets. Our results suggest that combinatorial processes for building complex representations are available by the end of the first year of life. The infant mind seems geared to integrate concepts in novel productive ways. This ability may be a precondition for deciphering the ambient language(s) and building abstract models of experience that enable fast and flexible learning.

Recursive Numeral Systems Optimize the Trade-off Between Lexicon Size and Average Morphosyntactic Complexity

Human languages vary in terms of which meanings they lexicalize, but this variation is constrained. It has been argued that languages are under two competing pressures: the pressure to be simple (e.g., to have a small lexicon) and to allow for informative (i.e., precise) communication, and that which meanings get lexicalized may be explained by languages finding a good way to trade off between these two pressures. However, in certain semantic domains, languages can reach very high levels of informativeness even if they lexicalize very few meanings in that domain. This is due to productive morphosyntax and compositional semantics, which may allow for construction of meanings which are not lexicalized. Consider the semantic domain of natural numbers: many languages lexicalize few natural number meanings as monomorphemic expressions, but can precisely convey very many natural number meanings using morphosyntactically complex numerals. In such semantic domains, lexicon size is not in direct competition with informativeness. What explains which meanings are lexicalized in such semantic domains? We will propose that in such cases, languages need to solve a different kind of trade-off problem: the trade-off between the pressure to lexicalize as few meanings as possible (i.e, to minimize lexicon size) and the pressure to produce as morphosyntactically simple utterances as possible (i.e, to minimize average morphosyntactic complexity of utterances). To support this claim, we will present a case study of 128 natural languages' numeral systems, and show computationally that they achieve a near-optimal trade-off between lexicon size and average morphosyntactic complexity of numerals. This study in conjunction with previous work on communicative efficiency suggests that languages' lexicons are shaped by a trade-off between not two but three pressures: be simple, be informative, and minimize average morphosyntactic complexity of utterances.

A model of conceptual bootstrapping in human cognition

To tackle a hard problem, it is often wise to reuse and recombine existing knowledge. Such an ability to bootstrap enables us to grow rich mental concepts despite limited cognitive resources. Here we present a computational model of conceptual bootstrapping. This model uses a dynamic conceptual repertoire that can cache and later reuse elements of earlier insights in principled ways, modelling learning as a series of compositional generalizations. This model predicts systematically different learned concepts when the same evidence is processed in different orders, without any extra assumptions about previous beliefs or background knowledge. Across four behavioural experiments (total n = 570), we demonstrate strong curriculum-order and conceptual garden-pathing effects that closely resemble our model predictions and differ from those of alternative accounts. Taken together, this work offers a computational account of how past experiences shape future conceptual discoveries and showcases the importance of curriculum design in human inductive concept inferences.

The algorithmic origins of counting

The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take finite data and go far beyond what they directly observe or perceive. To illustrate, this paper then outlines a model which acquires a counting procedure using observations of sets and words, extending the proposal of Piantadosi et al. (2012). This new version of the model responds to several critiques of the original work and outlines an approach which is likely appropriate for acquiring further aspects of mathematics.

Formalizing theories of child development: Introduction to the special section

Here we introduce a Special Section of Child Development entitled "Formalizing Theories of Child Development." This Special Section features five papers that use mathematical models to advance our understanding of central questions in the study of child development. This landmark collection is timely: it signifies growing awareness that rigorous empirical bricks are not enough; we need solid theory to build the house. By stating theory in mathematical terms, formal models make concepts, assumptions, and reasoning more explicit than verbal theory does. This increases falsifiability, promotes cumulative science, and enables integration with mathematical theory in allied disciplines. The Special Section contributions cover a range of topics: the developmental origins of counting, interactions between mathematics and language development, visual exploration and word learning in infancy, referent identification by toddlers, and the emergence of typical and atypical development. All are written in an accessible manner and for a broad audience.

Questions About Quantifiers: Symbolic and Nonsymbolic Quantity Processing by the Brain

One approach to understanding how the human cognitive system stores and operates with quantifiers such as "some," "many," and "all" is to investigate their interaction with the cognitive mechanisms for estimating and comparing quantities from perceptual input (i.e., nonsymbolic quantities). While a potential link between quantifier processing and nonsymbolic quantity processing has been considered in the past, it has never been discussed extensively. Simultaneously, there is a long line of research within the field of numerical cognition on the relationship between processing exact number symbols (such as "3" or "three") and nonsymbolic quantity. This accumulated knowledge can potentially be harvested for research on quantifiers since quantifiers and number symbols are two different ways of referring to quantity information symbolically. The goal of the present review is to survey the research on the relationship between quantifiers and nonsymbolic quantity processing mechanisms and provide a set of research directions and specific questions for the investigation of quantifier processing.

Recognition of small numbers in subset knowers Cardinal knowledge in early childhood

Previous research suggests that subset-knowers have an approximate understanding of small numbers. However, it is still unclear exactly what subset-knowers understand about small numbers. To investigate this further, we tested 133 participants, ages 2.6-4 years, on a newly developed eye-tracking task targeting cardinal recognition. Participants were presented with two sets differing in cardinality (1-4 items) and asked to find a specific cardinality. Our main finding showed that on a group level, subset-knowers could identify all presented targets at rates above chance, further supporting that subset-knowers understand several of the basic principles of small numbers. Exploratory analyses tentatively suggest that 1-knowers could identify the targets 1 and 2, but struggled when the target was 3 and 4, whereas 2-knowers and above could identify all targets at rates above chance. This might tentatively suggest that subset-knowers have an approximate understanding of numbers that is just (i.e. +1) above their current knower level. We discuss the implications of these results at length.

Probabilistic Learning and Psychological Similarity

The notions of psychological similarity and probabilistic learning are key posits in cognitive, computational, and developmental psychology and in machine learning. However, their explanatory relationship is rarely made explicit within and across these research fields. This opinionated review critically evaluates how these notions can mutually inform each other within computational cognitive science. Using probabilistic models of concept learning as a case study, I argue that two notions of psychological similarity offer important normative constraints to guide modelers' interpretations of representational primitives. In particular, the two notions furnish probabilistic models of cognition with meaningful interpretations of what the associated subjective probabilities in the model represent and how they attach to experiences from which the agent learns. Similarity representations thereby provide probabilistic models with cognitive, as opposed to purely mathematical, content.

Concept learning in a probabilistic language-of-thought. How is it possible and what does it presuppose?

Where does a probabilistic language-of-thought (PLoT) come from? How can we learn new concepts based on probabilistic inferences operating on a PLoT? Here, I explore these questions, sketching a traditional circularity objection to LoT and canvassing various approaches to addressing it. I conclude that PLoT-based cognitive architectures can support genuine concept learning; but, currently, it is unclear that they enjoy more explanatory breadth in relation to concept learning than alternative architectures that do not posit any LoT.

Active inductive inference in children and adults: A constructivist perspective

A defining aspect of being human is an ability to reason about the world by generating and adapting ideas and hypotheses. Here we explore how this ability develops by comparing children's and adults' active search and explicit hypothesis generation patterns in a task that mimics the open-ended process of scientific induction. In our experiment, 54 children (aged 8.97±1.11) and 50 adults performed inductive inferences about a series of causal rules through active testing. Children were more elaborate in their testing behavior and generated substantially more complex guesses about the hidden rules. We take a 'computational constructivist' perspective to explaining these patterns, arguing that these inferences are driven by a combination of thinking (generating and modifying symbolic concepts) and exploring (discovering and investigating patterns in the physical world). We show how this framework and rich new dataset speak to questions about developmental differences in hypothesis generation, active learning and inductive generalization. In particular, we find children's learning is driven by less fine-tuned construction mechanisms than adults', resulting in a greater diversity of ideas but less reliable discovery of simple explanations.

Socially intelligent machines that learn from humans and help humans learn

A hallmark of human intelligence is the ability to understand and influence other minds. Humans engage in inferential social learning (ISL) by using commonsense psychology to learn from others and help others learn. Recent advances in artificial intelligence (AI) are raising new questions about the feasibility of human-machine interactions that support such powerful modes of social learning. Here, we envision what it means to develop socially intelligent machines that can learn, teach, and communicate in ways that are characteristic of ISL. Rather than machines that simply predict human behaviours or recapitulate superficial aspects of human sociality (e.g. smiling, imitating), we should aim to build machines that can learn from human inputs and generate outputs for humans by proactively considering human values, intentions and beliefs. While such machines can inspire next-generation AI systems that learn more effectively from humans (as learners) and even help humans acquire new knowledge (as teachers), achieving these goals will also require scientific studies of its counterpart: how humans reason about machine minds and behaviours. We close by discussing the need for closer collaborations between the AI/ML and cognitive science communities to advance a science of both natural and artificial intelligence. This article is part of a discussion meeting issue 'Cognitive artificial intelligence'.

Carving joints into nature: reengineering scientific concepts in light of concept-laden evidence

A new wave of proposals suggests that scientists must reassess scientific concepts in light of accumulated evidence. However, reengineering scientific concepts in light of data is challenging because scientific concepts affect the evidence itself in multiple ways. Among other possible influences, concepts (i) prime scientists to overemphasize within-concept similarities and between-concept differences; (ii) lead scientists to measure conceptually relevant dimensions more accurately; (iii) serve as units of scientific experimentation, communication, and theory-building; and (iv) affect the phenomena themselves. When looking for improved ways to carve nature at its joints, scholars must take the concept-laden nature of evidence into account to avoid entering a vicious circle of concept-evidence mutual substantiation.

Précis of What Babies Know

Where does human knowledge begin? Research on human infants, children, adults, and nonhuman animals, using diverse methods from the cognitive, brain, and computational sciences, provides evidence for six early emerging, domain-specific systems of core knowledge. These automatic, unconscious systems are situated between perceptual systems and systems of explicit concepts and beliefs. They emerge early in infancy, guide children's learning, and function throughout life.

When Naïve Pedagogy Breaks Down: Adults Rationally Decide How to Teach, but Misrepresent Learners' Beliefs

From early in childhood, humans exhibit sophisticated intuitions about how to share knowledge efficiently in simple controlled studies. Yet, untrained adults often fail to teach effectively in real-world situations. Here, we explored what causes adults to struggle in informal pedagogical exchanges. In Experiment 1, we first showed evidence of this effect, finding that adult participants failed to communicate their knowledge to naïve learners in a simple teaching task, despite reporting high confidence that they taught effectively. Using a computational model of rational teaching, we found that adults assigned to our teaching condition provided highly informative examples but failed to teach effectively because their examples were tailored to learners who were only considering a small set of possible explanations. In Experiment 2, we then found experimental evidence for this possibility, showing that knowledgeable participants systematically misunderstand the beliefs of naïve participants. Specifically, knowledgeable participants assumed naïve agents would primarily consider hypotheses close to the correct one. Finally, in Experiment 3, we aligned learners' beliefs to knowledgeable agents' expectations and showed learners the same examples selected by participants assigned to teach in Experiment 1. We found that these same examples were significantly more informative once learners' hypothesis spaces were constrained to match teachers' expectations. Our findings show that, in informal settings, adult pedagogical failures result from an inaccurate representation of what naïve learners believe is plausible and not an inability to select informative data in a rational way.

The best game in town: The reemergence of the language-of-thought hypothesis across the cognitive sciences

Mental representations remain the central posits of psychology after many decades of scrutiny. However, there is no consensus about the representational format(s) of biological cognition. This paper provides a survey of evidence from computational cognitive psychology, perceptual psychology, developmental psychology, comparative psychology, and social psychology, and concludes that one type of format that routinely crops up is the language-of-thought (LoT). We outline six core properties of LoTs: (i) discrete constituents; (ii) role-filler independence; (iii) predicate-argument structure; (iv) logical operators; (v) inferential promiscuity; and (vi) abstract content. These properties cluster together throughout cognitive science. Bayesian computational modeling, compositional features of object perception, complex infant and animal reasoning, and automatic, intuitive cognition in adults all implicate LoT-like structures. Instead of regarding LoT as a relic of the previous century, researchers in cognitive science and philosophy-of-mind must take seriously the explanatory breadth of LoT-based architectures. We grant that the mind may harbor many formats and architectures, including iconic and associative structures as well as deep-neural-network-like architectures. However, as computational/representational approaches to the mind continue to advance, classical compositional symbolic structures - that is, LoTs - only prove more flexible and well-supported over time.

Communicative Efficiency or Iconic Learning: Do Acquisition and Communicative Pressures Interact to Shape Colour- Naming Systems?

Language evolution is driven by pressures for simplicity and informativity; however, the timescale on which these pressures operate is debated. Over several generations, learners' biases for simple and informative systems can guide language evolution. Over repeated instances of dyadic communication, the principle of least effort dictates that speakers should bias systems towards simplicity and listeners towards informativity, similarly guiding language evolution. At the same time, it has been argued that learners only provide a bias for simplicity and, thus, language users must provide a bias for informativity. To what extent do languages evolve during acquisition versus use? We address this question by formally defining and investigating the communicative efficiency of acquisition trajectories. We illustrate our approach using colour-naming systems, replicating the communicative efficiency model of Zaslavsky, Kemp, Regier & Tishby (2018, PNAS) and the acquisition model of Beekhuizen & Stevenson (2018, Cogn. Sci.). We find that to the extent that language is iconic, learning alone is sufficient to shape language evolution. Regarding colour-naming systems specifically, we find that incorporating learning biases into communicative efficiency accounts might explain how speakers and listeners trade off communicative effort.

Adaptive search space pruning in complex strategic problems

People have limited computational resources, yet they make complex strategic decisions over enormous spaces of possibilities. How do people efficiently search spaces with combinatorially branching paths? Here, we study players’ search strategies for a winning move in a “k-in-a-row” game. We find that players use scoring strategies to prune the search space and augment this pruning by a “shutter” heuristic that focuses the search on the paths emanating from their previous move. This strong pruning has its costs—both computational simulations and behavioral data indicate that the shutter size is correlated with players’ blindness to their opponent’s winning moves. However, simulations of the search while varying the shutter size, complexity levels, noise levels, branching factor, and computational limitations indicate that despite its costs, a narrow shutter strategy is the dominant strategy for most of the parameter space. Finally, we show that in the presence of computational limitations, the shutter heuristic enhances the performance of deep learning networks in these end-game scenarios. Together, our findings suggest a novel adaptive heuristic that benefits search in a vast space of possibilities of a strategic game.

Search problems usually have a common trade-off between accuracy and computational resources; Finding the best solution usually requires an exhaustive search, while limiting computations usually decreases the quality of solutions. Yet, humans provide high-quality solutions for complex problems despite having limited computational resources. How do they do that? Here, we analyze people’s behavior in a strategic game of “k-in-a-row” that has an enormous space of possibilities. We find that people strongly prune the search space by using scoring strategies to evaluate each possibility and augment this pruning with a shutter heuristic that limits their search to the possible winning paths from their last move. Similar to other adaptive heuristics, the shutter heuristic provides a strong reduction in the computations the searcher needs to carry out while maintaining on par accuracy rates. Finally, this adaptive heuristic generalizes to the performance of deep learning networks when playing with limited computational resources.

Symbols and mental programs: a hypothesis about human singularity

Natural language is often seen as the single factor that explains the cognitive singularity of the human species. Instead, we propose that humans possess multiple internal languages of thought, akin to computer languages, which encode and compress structures in various domains (mathematics, music, shape…). These languages rely on cortical circuits distinct from classical language areas. Each is characterized by: (i) the discretization of a domain using a small set of symbols, and (ii) their recursive composition into mental programs that encode nested repetitions with variations. In various tasks of elementary shape or sequence perception, minimum description length in the proposed languages captures human behavior and brain activity, whereas non-human primate data are captured by simpler nonsymbolic models. Our research argues in favor of discrete symbolic models of human thought.

Verbal counting and the timing of number acquisition in an indigenous Amazonian group

Children in industrialized cultures typically succeed on Give-N, a test of counting ability, by age 4. On the other hand, counting appears to be learned much later in the Tsimane’, an indigenous group in the Bolivian Amazon. This study tests three hypotheses for what may cause this difference in timing: (a) Tsimane’ children may be shy in providing behavioral responses to number tasks, (b) Tsimane’ children may not memorize the verbal list of number words early in acquisition, and/or (c) home environments may not support mathematical learning in the same way as in US samples, leading Tsimane’ children to primarily acquire mathematics through formalized schooling. Our results suggest that most of our subjects are not inhibited by shyness in responding to experimental tasks. We also find that Tsimane’ children (N = 100, ages 4-11) learn the verbal list later than US children, but even upon acquiring this list, still take time to pass Give-N tasks. We find that performance in counting varies across tasks and is related to formal schooling. These results highlight the importance of formal education, including instruction in the count list, in learning the meanings of the number words.

Neural Networks Track the Logical Complexity of Boolean Concepts

The language of thought hypothesis and connectionism provide two main accounts of category acquisition in the cognitive sciences. However, it is unclear to what extent their predictions agree. In this article, we tackle this problem by comparing the two accounts with respect to a common set of predictions about the effort required to acquire categories. We find that the two accounts produce similar predictions in the domain of Boolean categorization, however, with substantial variation depending on the operators in the language of thought.

Algorithms of adaptation in inductive inference

We investigate the idea that human concept inference utilizes local adaptive search within a compositional mental theory space. To explore this, we study human judgments in a challenging task that involves actively gathering evidence about a symbolic rule governing the behavior of a simulated environment. Participants learn by performing mini-experiments before making generalizations and explicit guesses about a hidden rule. They then collect additional evidence themselves (Experiment 1) or observe evidence gathered by someone else (Experiment 2) before revising their own generalizations and guesses. In each case, we focus on the relationship between participants' initial and revised guesses about the hidden rule concept. We find an order effect whereby revised guesses are anchored to idiosyncratic elements of the earlier guess. To explain this pattern, we develop a family of process accounts that combine program induction ideas with local (MCMC-like) adaptation mechanisms. A particularly local variant of this adaptive account captures participants' hypothesis revisions better than a range of alternative explanations. We take this as suggestive that people deal with the inherent complexity of concept inference partly through use of local adaptive search in a latent compositional theory space.

Exact Number Concepts Are Limited to the Verbal Count Range

Previous findings suggest that mentally representing exact numbers larger than four depends on a verbal count routine (e.g., "one, two, three . . ."). However, these findings are controversial because they rely on comparisons across radically different languages and cultures. We tested the role of language in number concepts within a single population-the Tsimane' of Bolivia-in which knowledge of number words varies across individual adults. We used a novel data-analysis model to quantify the point at which participants (N = 30) switched from exact to approximate number representations during a simple numerical matching task. The results show that these behavioral switch points were bounded by participants' verbal count ranges; their representations of exact cardinalities were limited to the number words they knew. Beyond that range, they resorted to numerical approximation. These results resolve competing accounts of previous findings and provide unambiguous evidence that large exact number concepts are enabled by language.

Assessing the knower-level framework: How reliable is the Give-a-Number task?

The Give-a-Number task has become a gold standard of children's number word comprehension in developmental psychology. Recently, researchers have begun to use the task as a predictor of other developmental milestones. This raises the question of how reliable the task is, since test-retest reliability of any measure places an upper bound on the size of reliable correlations that can be found between it and other measures. In Experiment 1, we presented 81 2- to 5-year-old children with Wynn (1992) titrated version of the Give-a-Number task twice within a single session. We found that the reliability of this version of the task was high overall, but varied importantly across different assigned knower levels, and was very low for some knower levels. In Experiment 2, we assessed the test-retest reliability of the non-titrated version of the Give-a-Number task with another group of 81 children and found a similar pattern of results. Finally, in Experiment 3, we asked whether the two versions of Give-a-Number generated different knower levels within-subjects, by testing 75 children with both tasks. Also, we asked how both tasks relate to another commonly used test of number knowledge, the "What's-On-This-Card" task. We found that overall, the titrated and non-titrated versions of Give-a-Number yielded similar knower levels, though the non-titrated version was slightly more conservative than the titrated version, which produced modestly higher knower levels. Neither was more closely related to "What's-On-This-Card" than the other. We discuss the theoretical and practical implications of these results.

One model for the learning of language

A major goal of linguistics and cognitive science is to understand what class of learning systems can acquire natural language. Until recently, the computational requirements of language have been used to argue that learning is impossible without a highly constrained hypothesis space. Here, we describe a learning system that is maximally unconstrained, operating over the space of all computations, and is able to acquire many of the key structures present in natural language from positive evidence alone. We demonstrate this by providing the same learning model with data from 74 distinct formal languages which have been argued to capture key features of language, have been studied in experimental work, or come from an interesting complexity class. The model is able to successfully induce the latent system generating the observed strings from small amounts of evidence in almost all cases, including for regular (e.g., an , [Formula: see text], and [Formula: see text]), context-free (e.g., [Formula: see text], and [Formula: see text]), and context-sensitive (e.g., [Formula: see text], and xx) languages, as well as for many languages studied in learning experiments. These results show that relatively small amounts of positive evidence can support learning of rich classes of generative computations over structures. The model provides an idealized learning setup upon which additional cognitive constraints and biases can be formalized.

Logical word learning: The case of kinship

We examine the conceptual development of kinship through the lens of program induction. We present a computational model for the acquisition of kinship term concepts, resulting in the first computational model of kinship learning that is closely tied to developmental phenomena. We demonstrate that our model can learn several kinship systems of varying complexity using cross-linguistic data from English, Pukapuka, Turkish, and Yanomamö. More importantly, the behavioral patterns observed in children learning kinship terms, under-extension and over-generalization, fall out naturally from our learning model. We then conducted interviews to simulate realistic learning environments and demonstrate that the characteristic-to-defining shift is a consequence of our learning model in naturalistic contexts containing abstract and concrete features. We use model simulations to understand the influence of logical simplicity and children’s learning environment on the order of acquisition of kinship terms, providing novel predictions for the learning trajectories of these words. We conclude with a discussion of how this model framework generalizes beyond kinship terms, as well as a discussion of its limitations.

Testing the role of symbols in preschool numeracy: An experimental computer-based intervention study

Numeracy is of critical importance for scholastic success and modern-day living, but the precise mechanisms that drive its development are poorly understood. Here we used novel experimental training methods to begin to investigate the role of symbols in the development of numeracy in preschool-aged children. We assigned pre-school children in the U.S. and Italy (N = 215; Mean age = 49.15 months) to play one of five versions of a computer-based numerical comparison game for two weeks. The different versions of the game were equated on basic features of gameplay and demands but systematically varied in numerical content. Critically, some versions included non-symbolic numerical comparisons only, while others combined non-symbolic numerical comparison with symbolic aids of various types. Before and after training we assessed four components of early numeracy: counting proficiency, non-symbolic numerical comparison, one-to-one correspondence, and arithmetic set transformation. We found that overall children showed improvement in most of these components after completing these short trainings. However, children trained on numerical comparisons with symbolic aids made larger gains on assessments of one-to-one correspondence and arithmetic transformation compared to children whose training involved non-symbolic numerical comparison only. Further exploratory analyses suggested that, although there were no major differences between children trained with verbal symbols (e.g., verbal counting) and non-verbal visuo-spatial symbols (i.e., abacus counting), the gains in one-to-one correspondence may have been driven by abacus training, while the gains in non-verbal arithmetic transformations may have been driven by verbal training. These results provide initial evidence that the introduction of symbols may contribute to the emergence of numeracy by enhancing the capacity for thinking about exact equality and the numerical effects of set transformations. More broadly, this study provides an empirical basis to motivate further focused study of the processes by which children’s mastery of symbols influences children’s developing mastery of numeracy.

Theory and predictions for the development of morphology and syntax: A Universal Grammar + statistics approach

The key aim of this special issue is to make developmental theory proposals concrete enough to evaluate with empirical data. With this in mind, I discuss proposals from the "Universal Grammar + statistics" (UG+stats) perspective for learning several morphology and syntax phenomena. I briefly review why UG has traditionally been part of many developmental theories of language, as well as common statistical learning approaches that are part of UG+stats proposals. I then discuss each morphology or syntax phenomenon in turn, giving an overview of relevant UG+stats proposals for that phenomenon, specific predictions made by each proposal, and what we currently know about how those predictions hold up. I conclude by briefly discussing where we seem to be when it comes to how well UG+stats proposals help us understand the development of morphology and syntax knowledge.

Children's understanding of the abstract logic of counting

When children learn to count, do they understand its logic independent of the number list that they learned to count with? Here we tested CP-knowers' (ages three to five) understanding of how counting reveals a set's cardinality, even when non-numerical lists are used to count. Participants watched an agent count unobservable objects in two boxes and were asked to identify the larger set. Participants successfully identified the box with more objects when the agent counted using their familiar number list (Experiment 1) and when the agent counted using a non-numeric ordered list, as long as the items in the list were not linguistically used as number words (Experiments 2-3). Additionally, children's performance was strongly influenced by visual cues that helped them link the list's order to representations of magnitude (Experiment 4). Our findings suggest that three- to six-year-olds who can count also understand how counting reveals a set's cardinality, but they require additional time to understand how symbols on any arbitrary ordered list can be used as numerals.

The computational origin of representation

Each of our theories of mental representation provides some insight into how the mind works. However, these insights often seem incompatible, as the debates between symbolic, dynamical, emergentist, sub-symbolic, and grounded approaches to cognition attest. Mental representations-whatever they are-must share many features with each of our theories of representation, and yet there are few hypotheses about how a synthesis could be possible. Here, I develop a theory of the underpinnings of symbolic cognition that shows how sub-symbolic dynamics may give rise to higher-level cognitive representations of structures, systems of knowledge, and algorithmic processes. This theory implements a version of conceptual role semantics by positing an internal universal representation language in which learners may create mental models to capture dynamics they observe in the world. The theory formalizes one account of how truly novel conceptual content may arise, allowing us to explain how even elementary logical and computational operations may be learned from a more primitive basis. I provide an implementation that learns to represent a variety of structures, including logic, number, kinship trees, regular languages, context-free languages, domains of theories like magnetism, dominance hierarchies, list structures, quantification, and computational primitives like repetition, reversal, and recursion. This account is based on simple discrete dynamical processes that could be implemented in a variety of different physical or biological systems. In particular, I describe how the required dynamics can be directly implemented in a connectionist framework. The resulting theory provides an "assembly language" for cognition, where high-level theories of symbolic computation can be implemented in simple dynamics that themselves could be encoded in biologically plausible systems.

A theory of memory for binary sequences: Evidence for a mental compression algorithm in humans

Working memory capacity can be improved by recoding the memorized information in a condensed form. Here, we tested the theory that human adults encode binary sequences of stimuli in memory using an abstract internal language and a recursive compression algorithm. The theory predicts that the psychological complexity of a given sequence should be proportional to the length of its shortest description in the proposed language, which can capture any nested pattern of repetitions and alternations using a limited number of instructions. Five experiments examine the capacity of the theory to predict human adults' memory for a variety of auditory and visual sequences. We probed memory using a sequence violation paradigm in which participants attempted to detect occasional violations in an otherwise fixed sequence. Both subjective complexity ratings and objective violation detection performance were well predicted by our theoretical measure of complexity, which simply reflects a weighted sum of the number of elementary instructions and digits in the shortest formula that captures the sequence in our language. While a simpler transition probability model, when tested as a single predictor in the statistical analyses, accounted for significant variance in the data, the goodness-of-fit with the data significantly improved when the language-based complexity measure was included in the statistical model, while the variance explained by the transition probability model largely decreased. Model comparison also showed that shortest description length in a recursive language provides a better fit than six alternative previously proposed models of sequence encoding. The data support the hypothesis that, beyond the extraction of statistical knowledge, human sequence coding relies on an internal compression using language-like nested structures.

Numeral Systems Across Languages Support Efficient Communication: From Approximate Numerosity to Recursion

Languages differ qualitatively in their numeral systems. At one extreme, some languages have a small set of number terms, which denote approximate or inexact numerosities; at the other extreme, many languages have forms for exact numerosities over a very large range, through a recursively defined counting system. Why do numeral systems vary as they do? Here, we use computational analyses to explore the numeral systems of 30 languages that span this spectrum. We find that these numeral systems all reflect a functional need for efficient communication, mirroring existing arguments in other semantic domains such as color, kinship, and space. Our findings suggest that cross-language variation in numeral systems may be understood in terms of a shared functional need to communicate precisely while using minimal cognitive resources.

The Child as Hacker

The scope of human learning and development poses a radical challenge for cognitive science. We propose that developmental theories can address this challenge by adopting perspectives from computer science. Many of our best models treat learning as analogous to computer programming because symbolic programs provide the most compelling account of sophisticated mental representations. We specifically propose that children's learning is analogous to a particular style of programming called hacking, making code better along many dimensions through an open-ended set of goals and activities. By contrast to existing theories, which depend primarily on local search and simple metrics, this view highlights the many features of good mental representations and the multiple complementary processes children use to create them.

Brain computation by assemblies of neurons

Our expanding understanding of the brain at the level of neurons and synapses, and the level of cognitive phenomena such as language, leaves a formidable gap between these two scales. Here we introduce a computational system which promises to bridge this gap: the Assembly Calculus. It encompasses operations on assemblies of neurons, such as project, associate, and merge, which appear to be implicated in cognitive phenomena, and can be shown, analytically as well as through simulations, to be plausibly realizable at the level of neurons and synapses. We demonstrate the reach of this system by proposing a brain architecture for syntactic processing in the production of language, compatible with recent experimental results.

Assemblies are large populations of neurons believed to imprint memories, concepts, words, and other cognitive information. We identify a repertoire of operations on assemblies. These operations correspond to properties of assemblies observed in experiments, and can be shown, analytically and through simulations, to be realizable by generic, randomly connected populations of neurons with Hebbian plasticity and inhibition. Assemblies and their operations constitute a computational model of the brain which we call the Assembly Calculus, occupying a level of detail intermediate between the level of spiking neurons and synapses and that of the whole brain. The resulting computational system can be shown, under assumptions, to be, in principle, capable of carrying out arbitrary computations. We hypothesize that something like it may underlie higher human cognitive functions such as reasoning, planning, and language. In particular, we propose a plausible brain architecture based on assemblies for implementing the syntactic processing of language in cortex, which is consistent with recent experimental results.

Recursive sequence generation in monkeys, children, U.S. adults, and native Amazonians

The question of what computational capacities, if any, differ between humans and nonhuman animals has been at the core of foundational debates in cognitive psychology, anthropology, linguistics, and animal behavior. The capacity to form nested hierarchical representations is hypothesized to be essential to uniquely human thought, but its origins in evolution, development, and culture are controversial. We used a nonlinguistic sequence generation task to test whether subjects generalize sequential groupings of items to a center-embedded, recursive structure. Children (3 to 5 years old), U.S. adults, and adults from a Bolivian indigenous group spontaneously induced recursive structures from ambiguous training data. In contrast, monkeys did so only with additional exposure. We quantify these patterns using a Bayesian mixture model over logically possible strategies. Our results show that recursive hierarchical strategies are robust in human thought, both early in development and across cultures, but the capacity itself is not unique to humans.

Abstract representations of events arise from mental errors in learning and memory

Humans are adept at uncovering abstract associations in the world around them, yet the underlying mechanisms remain poorly understood. Intuitively, learning the higher-order structure of statistical relationships should involve complex mental processes. Here we propose an alternative perspective: that higher-order associations instead arise from natural errors in learning and memory. Using the free energy principle, which bridges information theory and Bayesian inference, we derive a maximum entropy model of people's internal representations of the transitions between stimuli. Importantly, our model (i) affords a concise analytic form, (ii) qualitatively explains the effects of transition network structure on human expectations, and (iii) quantitatively predicts human reaction times in probabilistic sequential motor tasks. Together, these results suggest that mental errors influence our abstract representations of the world in significant and predictable ways, with direct implications for the study and design of optimally learnable information sources.

Baboons (Papio papio) Process a Context-Free but Not a Context-Sensitive Grammar

Language processing involves the ability to master supra-regular grammars, that go beyond the level of complexity of regular grammars. This ability has been hypothesized to be a uniquely human capacity. Our study probed baboons' capacity to learn two supra-regular grammars of different levels of complexity: a context-free grammar generating sequences following a mirror structure (e.g., AB | BA, ABC | CBA) and a context-sensitive grammar generating sequences following a repeat structure (e.g., AB | AB, ABC | ABC), the latter requiring greater computational power to be processed. Fourteen baboons were tested in a prediction task, requiring them to track a moving target on a touchscreen. In distinct experiments, sequences of target locations followed one of the above two grammars, with rare violations. Baboons showed slower response times when violations occurred in mirror sequences, but did not react to violations in repeat sequences, suggesting that they learned the context-free (mirror) but not the context-sensitive (repeat) grammar. By contrast, humans tested with the same task learned both grammars. These data suggest a difference in sensitivity in baboons between a context-free and a context-sensitive grammar.

The Challenge of Modeling the Acquisition of Mathematical Concepts

As a full-blown research topic, numerical cognition is investigated by a variety of disciplines including cognitive science, developmental and educational psychology, linguistics, anthropology and, more recently, biology and neuroscience. However, despite the great progress achieved by such a broad and diversified scientific inquiry, we are still lacking a comprehensive theory that could explain how numerical concepts are learned by the human brain. In this perspective, I argue that computer simulation should have a primary role in filling this gap because it allows identifying the finer-grained computational mechanisms underlying complex behavior and cognition. Modeling efforts will be most effective if carried out at cross-disciplinary intersections, as attested by the recent success in simulating human cognition using techniques developed in the fields of artificial intelligence and machine learning. In this respect, deep learning models have provided valuable insights into our most basic quantification abilities, showing how numerosity perception could emerge in multi-layered neural networks that learn the statistical structure of their visual environment. Nevertheless, this modeling approach has not yet scaled to more sophisticated cognitive skills that are foundational to higher-level mathematical thinking, such as those involving the use of symbolic numbers and arithmetic principles. I will discuss promising directions to push deep learning into this uncharted territory. If successful, such endeavor would allow simulating the acquisition of numerical concepts in its full complexity, guiding empirical investigation on the richest soil and possibly offering far-reaching implications for educational practice.

Beyond Reward Prediction Errors: Human Striatum Updates Rule Values During Learning

Humans naturally group the world into coherent categories defined by membership rules. Rules can be learned implicitly by building stimulus-response associations using reinforcement learning or by using explicit reasoning. We tested if the striatum, in which activation reliably scales with reward prediction error, would track prediction errors in a task that required explicit rule generation. Using functional magnetic resonance imaging during a categorization task, we show that striatal responses to feedback scale with a "surprise" signal derived from a Bayesian rule-learning model and are inconsistent with RL prediction error. We also find that striatum and caudal inferior frontal sulcus (cIFS) are involved in updating the likelihood of discriminative rules. We conclude that the striatum, in cooperation with the cIFS, is involved in updating the values assigned to categorization rules when people learn using explicit reasoning.

One-to-one correspondence without language

A logical rule important in counting and representing exact number is one-to-one correspondence, the understanding that two sets are equal if each item in one set corresponds to exactly one item in the second set. The role of this rule in children's development of counting remains unclear, possibly due to individual differences in the development of language. We report that non-human primates, which do not have language, have at least a partial understanding of this principle. Baboons were given a quantity discrimination task where two caches were baited with different quantities of food. When the quantities were baited in a manner that highlighted the one-to-one relation between those quantities, baboons performed significantly better than when one-to-one correspondence cues were not provided. The implication is that one-to-one correspondence, which requires intuitions about equality and is a possible building block of counting, has a pre-linguistic origin.

Emotion words, emotion concepts, and emotional development in children: A constructionist hypothesis

In this article, we integrate two constructionist approaches-the theory of constructed emotion and rational constructivism-to introduce several novel hypotheses for understanding emotional development. We first discuss the hypothesis that emotion categories are abstract and conceptual, whose instances share a goal-based function in a particular context but are highly variable in their affective, physical, and perceptual features. Next, we discuss the possibility that emotional development is the process of developing emotion concepts, and that emotion words may be a critical part of this process. We hypothesize that infants and children learn emotion categories the way they learn other abstract conceptual categories-by observing others use the same emotion word to label highly variable events. Finally, we hypothesize that emotional development can be understood as a concept construction problem: a child becomes capable of experiencing and perceiving emotion only when her brain develops the capacity to assemble ad hoc, situated emotion concepts for the purposes of guiding behavior and giving meaning to sensory inputs. Specifically, we offer a predictive processing account of emotional development. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Ontogenetic Origins of Human Integer Representations

Do children learn number words by associating them with perceptual magnitudes? Recent studies argue that approximate numerical magnitudes play a foundational role in the development of integer concepts. Against this, we argue that approximate number representations fail both empirically and in principle to provide the content required of integer concepts. Instead, we suggest that children's understanding of integer concepts proceeds in two phases. In the first phase, children learn small exact number word meanings by associating words with small sets. In the second phase, children learn the meanings of larger number words by mastering the logic of exact counting algorithms, which implement the successor function and Hume's principle (that one-to-one correspondence guarantees exact equality). In neither phase do approximate number representations play a foundational role.