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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.

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.

Temporal connectives are function words that relate events intime. Despite their ubiquity and utility, children acquire themeanings of temporal connective words late in development.Experimental work has uncovered patterns in the acquisitionof temporal connectives that clarify the learning challenge thatthese words pose to children. In particular, developmentalstudies have identified differing acquisition trajectories acrossconnective types, asymmetries in learning within pairs of re-lated connectives, and monotonic increases in comprehensionwith age. Expanding on prior theoretical accounts, we formal-ize temporal connective acquisition in a computational wordlearning framework. We demonstrate that each of the empir-ically determined acquisition patterns emerges in the learningbehavior of our computational model. Finally, we discuss ourfindings in relation to earlier theories and to general learnabil-ity concerns in language acquisition.

From early in life, people implicitly associate time, number,and other abstract conceptual domains with space. Accord-ing to the Generalized Magnitude System proposal, these men-tal mappings reflect a common neural system for represent-ing various magnitudes, and share a common spatial organiza-tion. In a test of this proposal, here we measured mappings ofsize, time, and number in the Tsimane’, an indigenous Ama-zonian group with few of the cultural practices (like readingand math) that spatialize size, time, and number in the expe-rience of industrialized adults. On three spatial axes, the Tsi-mane’ systematically arranged imagistic stimuli according totheir magnitudes, but they showed no directional preferencesoverall and individuals often mapped different domains in op-posite directions. The results are inconsistent with predictionsof the Generalized Magnitude System proposal but can be ex-plained by Hierarchical Mental Metaphor Theory, accordingto which mental mappings initially reflect a set of correlationsobservable in the natural world.

Probabilities can be described by a numerator and a de-nominator and students and decision-makers are not in-different to numerical values of the components. For in-stance, when people compare two equal ratios their choicesgravitate to the option with larger number, even if theyknow both ratios are equal. To the date, however, it is un-clear if whole number biases are present in other cultures.We tested a farming-foraging group living in the Bolivianrain forest in a simple 2AFC ratio comparison task. Af-ter appropriate training, the Tsimane were highly accu-rate in this task, confirming that visual proportional rea-soning is present across cultures. Importantly, they hada strong tendency to favor large numbers in equal ratiocomparisons, similar to what is found in educated popu-lations. Even though our sample size is moderate (n=76),the whole number bias we found occurred under good pro-portional reasoning. The bias may be a general feature ofcognition, rather than a cultural or education artifact, thatmay help humans solve ambiguous situations.

BackgroundSeasons affect many social, economic, and biological outcomes, particularly in low-resource settings, and some studies suggest that birth season affects child growth.AimTo study a predictor of stunting that has received limited attention: birth season.Subjects and methodsThis study uses cross-sectional data collected during 2008 in a low-resource society of horticulturists-foragers in the Bolivian Amazon, Tsimane'. It estimates the associations between birth months and height-for-age Z-scores (HAZ) for 562 girls and 546 boys separately, from birth until age 11 years or pre-puberty, which in this society occurs ∼13-14 years.ResultsChildren born during the rainy season (February-May) were shorter, while children born during the end of the dry season and the start of the rainy season (August-November) were taller, both compared with their age-sex peers born during the rest of the year. The correlations of birth season with HAZ were stronger for boys than for girls. Controlling for birth season, there is some evidence of eventual partial catch-up growth, with the HAZ of girls or boys worsening until ∼ age 4-5 years, but improving thereafter. By age 6 years, many girls and boys had ceased to be stunted, irrespective of birth season.ConclusionThe results suggest that redressing stunting will require attention to conditions in utero, infancy and late childhood.

When learning about the world, we develop mental represen-tations or concepts for things we have never seen. At the sametime, we also develop representations for things that are similarto what we have experienced. Traditionally, similarity-basedand rule-based systems have been used as distinct models tocapture conceptual representation. However, it seems implau-sible that we do not flexibly deploy both systems. Whetherboth systems can be used simultaneously to represent compo-nents of a single concept is an open empirical question. Oneexample suggesting that the use of both systems is possible isthe concept of a ZEBRA , which looks like a horse but striped.Using an artificial concept learning task, we test whether peo-ple can combine similarity and rules compositionally in orderto represent concepts. Our results suggest that people are ableto compose similarity and rules when mentally representing asingle concept.

Humans master complex systems of interrelated concepts likemathematics and natural language. Previous work suggestslearning these systems relies on iteratively and directly re-vising a language-like conceptual representation. We intro-duce and assess a novel concept learning paradigm calledMartha’s Magical Machines that captures complex relation-ships between concepts. We model human concept learning inthis paradigm as a search in the space of term rewriting sys-tems, previously developed as an abstract model of compu-tation. Our model accurately predicts that participants learnsome transformations more easily than others and that theylearn harder concepts more easily using a bootstrapping cur-riculum focused on their compositional parts. Our results sug-gest that term rewriting systems may be a useful model of hu-man conceptual representations.

In many domains, people are able to transfer abstract knowl-edge about objects, events, or contexts that are superficiallydissimilar, enabling striking new insights and inferences. Weprovide evidence that this ability is naturally explained as theaddition of new primitive elements to a compositional mentalrepresentation, such as that in the probabilistic Language OfThought (LOT). We conducted a transfer-learning experimentin which participants learned about two sequences, one afterthe other. We show that participants’ ability to learn the secondsequence is affected the first sequence they saw. We test twoprobabilistic models to evaluate alternative theories of how al-gorithmic knowledge is transferred from the first to second se-quence: one model rationally updates the prior probability ofthe primitive operations in the LOT based on what was used inthe first sequence; the other stores previously likely hypothesesas new primitives. Both models perform better than baselinesin explaining behavior, with the human subjects appearing totransfer entire hypotheses when they can, and otherwise updat-ing the prior on primitives.

A widely observed phenomenon in children’s word-extensionsand generalizations is the characteristic-to-defining shift,whereby young children initially generalize words based ontypical properties and gradually transition into generalizingwords using abstract, logical information. In this paper, wepropose a statistically principled model of conceptual devel-opment grounded in the trade-off between simplicity and fit tothe data. We run our model based on informant-provided fam-ily trees and the real-life characteristic features of people onthose trees. We demonstrate that the characteristic-to-definingshift does not necessarily depend on discrete change in rep-resentation or processes. Instead, the shift could fall out nat-urally from statistical inference over conceptual hypotheses.Our model finds that the shift occurs even when abstract logicalrelations are present from the outset of learning as long as char-acteristic features are informative but imperfect in their abilityto capture the underlying concept to be learned—a property ofour elicited features.

People have the capability to process text three times fasterthan they would naturally read it, yet many current theories ofsentence processing rely on natural reading times as a proxyfor processing difficulty. How can people read material soquickly in spite of information processing limitations sug-gested by sentence processing theories? One possibility is thatsurprisal effects on reading time, the hallmark of processingdifficulty under sentence processing theories, might arise fromperceptual processing, implying no relation between surprisaland sentence processing difficulty. In this paper, we conducteda novel self-paced rapid serial visual presentation (RSVP) ex-periment, which controlled perceptual processes to probe forsentence processing related surprisal effects. We further testedhow readers might compensate for information processing lim-its during RSVP. We find support for sentence processing re-lated surprisal effects, the pattern of which is consistent with aFirst-In, First-Out (FIFO) buffer model.

Zipf famously stated that, if natural language lexicons are structured for efficient communication, the words that are used the most frequently should require the least effort. This observation explains the famous finding that the most frequent words in a language tend to be short. A related prediction is that, even within words of the same length, the most frequent word forms should be the ones that are easiest to produce and understand. Using orthographics as a proxy for phonetics, we test this hypothesis using corpora of 96 languages from Wikipedia. We find that, across a variety of languages and language families and controlling for length, the most frequent forms in a language tend to be more orthographically well-formed and have more orthographic neighbors than less frequent forms. We interpret this result as evidence that lexicons are structured by language usage pressures to facilitate efficient communication.

We develop a mathematical approach to formally proving that certain neural computations and representations exist based on patterns observed in an organism's behaviour. To illustrate, we provide a simple set of conditions under which an ant's ability to determine how far it is from its nest would logically imply neural structures isomorphic to the natural numbers ℕ$$ \mathrm{\mathbb{N}} $$. We generalise these results to arbitrary behaviours and representations and show what mathematical characterisation of neural computation and representation is simplest while being maximally predictive of behaviour. We develop this framework in detail using a path integration example, where an organism's ability to search for its nest in the correct location implies representational structures isomorphic to two‐dimensional coordinates under addition. We also study a system for processing anbn$$ {a}^n{b}^n $$ strings common in comparative work. Our approach provides an objective way to determine what theory of a physical system is best, addressing a fundamental challenge in neuroscientific inference. These results motivate considering which neurobiological structures have the requisite formal structure and are otherwise physically plausible given relevant physical considerations such as generalisability, information density, thermodynamic stability and energetic cost. [ABSTRACT FROM AUTHOR]

How do people rapidly learn rich, structured concepts fromsparse input? Recent approaches to concept learning havefound success by integrating rules and statistics. We describe ahierarchical model in this spirit in which the rules are stochas-tic, generative processes, and the rules themselves arise froma higher-level stochastic, generative process. We evaluate thisprobabilistic language-of-thought model with data from an ab-stract rule learning experiment carried out with adults. In thisexperiment, we find novel generalization effects, and we showthat the model gives a qualitatively good account of the exper-imental data. We then discuss the role of this kind of model inthe larger context of concept learning.

We apply Bayesian data analysis to a structured cognitivemodel in order to determine the priors that support humangeneralizations in a simple concept learning task. We mod-eled 250,000 ratings in a “number game” experiment wheresubjects took examples of a numbers produced by a program(e.g. 4, 16, 32) and rated how likely other numbers (e.g. 8vs. 9) would be to be generated. This paper develops a dataanalysis technique for a family of compositional “Language ofThought” (LOT) models which permits discovery of subjects’prior probability of mental operations (e.g. addition, multi-plication, etc.) in this domain. Our results reveal high cor-relations between model mean predictions and subject gener-alizations, but with some qualitative mismatch for a stronglycompositional prior.

We study the projection of cognitive representations intocontinuous motor (reaching) responses with a computationalmodel that unifies three influential approaches: accumulationof evidence, statistical inference, and optimalfeedback control. We modeled a number comparison taskthat asked participants to respond with a reaching gesturewhich of two side had more dots. The model successfullyreproduced subjects’ pattern of reach and performanceacross varying difficulties of numerical comparison.Our model parameterized several potentially relevantcognitive variables, including a threshold, memory decay,and mental sampling rate. Remarkably, a threshold formovement was not needed for modeling human behaviorwhen statistical inference is combined with optimal motorplanning. Overall, the model indicates that the motorsystempositions the effectors optimally, both biomechanicallythrough an optimal feedback controller, and cognitivelyby means of continuous statistical inference on theavailable evidence.

Context plays a ubiquitous role in language processing. Forthe most part, work in language processing investigates theeffects of context without investigating questions about whatdetermines a context. For example, interpretation of any referentialexpression must take into account the notion of a referentialdomain. Here we investigate the influence of perceptualcues in establishing a referential domain, or linguistic context.We demonstrate that people use perceptual cues to establish alinguistic context; the influence of perceptual cues is gradientwith respect to cue magnitude; and the contribution of a perceptualcue in constructing a linguistic context is not an effectof attention or salience. We provide these results as a first steptoward developing a formal model for the establishment of linguisticcontext.

Current models of word learning focus on the mapping betweenwords and their referents and remain mute with regardto conceptual representation. We develop a cross-situationalmodel of word learning that captures word-concept mappingby jointly inferring the referents and underlying concepts foreach word. We also develop a variant of our model that incorporatesrecursion, which entertains the idea that children canuse learned words to aid future learning. We demonstrate bothmodels’ ability to learn kinship terms and show that addingrecursion into the model speeds acquisition

Although the mapping between form and meaning is often regarded as arbitrary, there are in fact well-known constraints on words which are the result of functional pressures associated with language use and its acquisition. In particular, languages have been shown to encode meaning distinctions in their sound properties, which may be important for language learning. Here, we investigate the relationship between semantic distance and phonological distance in the large-scale structure of the lexicon. We show evidence in 100 languages from a diverse array of language families that more semantically similar word pairs are also more phonologically similar. This suggests that there is an important statistical trend for lexicons to have semantically similar words be phonologically similar as well, possibly for functional reasons associated with language learning.

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. [ABSTRACT FROM AUTHOR]

We investigate number and arithmetic learning among a Bolivian indigenous people, the Tsimane', for whom formal schooling is comparatively recent in history and variable in both extent and consistency. We first present a large-scale meta-analysis on child number development involving over 800 Tsimane' children. The results emphasize the impact of formal schooling: Children are only found to be full counters when they have attended school, suggesting the importance of cultural support for early mathematics. We then test especially remote Tsimane' communities and document the development of specialized arithmetical knowledge in the absence of direct formal education. Specifically, we describe individuals who succeed on arithmetic problems involving the number five--which has a distinct role in the local economy--even though they do not succeed on some lower numbers. Some of these participants can perform multiplication with fives at greater accuracy than addition by one. These results highlight the importance of cultural factors in early mathematics and suggest that psychological theories of number where quantities are derived from lower numbers via repeated addition (e.g., a successor function) are unlikely to explain the diversity of human mathematical ability. [ABSTRACT FROM AUTHOR]

We show that children in the Tsimane', a farming-foraging group in the Bolivian rain-forest, learn number words along a similar developmental trajectory to children from industrialized countries. Tsimane' children successively acquire the first three or four number words before fully learning how counting works. However, their learning is substantially delayed relative to children from the United States, Russia, and Japan. The presence of a similar developmental trajectory likely indicates that the incremental stages of numerical knowledge--but not their timing--reflect a fundamental property of number concept acquisition which is relatively independent of language, culture, age, and early education.

Infants must learn about many cognitive domains (e.g., language, music) from auditory statistics, yet capacity limits on their cognitive resources restrict the quantity that they can encode. Previous research has established that infants can attend to only a subset of available acoustic input. Yet few previous studies have directly examined infant auditory attention, and none have directly tested theorized mechanisms of attentional selection based on stimulus complexity. This work utilizes model-based behavioral methods that were recently developed to examine visual attention in infants (e.g., Kidd, Piantadosi, & Aslin, 2012). The present results demonstrate that 7- to 8-month-old infants selectively attend to nonsocial auditory stimuli that are intermediately predictable/complex with respect to their current implicit beliefs and expectations. These findings provide evidence of a broad principle of infant attention across modalities and suggest that sound-to-sound transitional statistics heavily influence the allocation of auditory attention in human infants.

A major open question in natural language research is the role of communicative efficiency in the origin and on-line processing of language structures. Here, we use word pairs like "chimp/chimpanzee", which differ in length but have nearly identical meanings, to investigate the communicative properties of lexical systems and the communicative pressures on language users. If language is designed to be information-theoretically optimal, then shorter words should convey less information than their longer counterparts, when controlling for meaning. Consistent with this prediction, a corpus analysis revealed that the short form of our meaning-matched pairs occurs in more predictive contexts than the longer form. Second, a behavioral study showed that language users choose the short form more often in predictive contexts, suggesting that tendencies to be information-theoretically efficient manifest in explicit behavioral choices. Our findings, which demonstrate the prominent role of communicative efficiency in the structure of the lexicon, complement and extend the results of Piantadosi, Tily, and Gibson (2011), who showed that word length is better correlated with Shannon information content than with frequency. Crucially, we show that this effect arises at least in part from active speaker choice. (Contains 2 figures.)

The capacity to generate recursive sequences is a marker of rich, algorithmic cognition, and perhaps unique to humans. Yet, the precise processes driving recursive sequence generation remain mysterious. We investigated three potential cognitive mechanisms underlying recursive pattern processing: hierarchical reasoning, ordinal reasoning, and associative chaining. We developed a Bayesian mixture model to quantify the extent to which these three cognitive mechanisms contribute to adult humans' performance in a sequence generation task. We further tested whether recursive rule discovery depends upon relational information, either perceptual or semantic. We found that the presence of relational information facilitates hierarchical reasoning and drives the generation of recursive sequences across novel depths of center embedding. In the absence of relational information, the use of ordinal reasoning predominates. Our results suggest that hierarchical reasoning is an important cognitive mechanism underlying recursive pattern processing and can be deployed across embedding depths and relational domains. [ABSTRACT FROM AUTHOR]