A mathematical theory of relational generalization in transitive inference.

Humans and animals routinely infer relations between different items or events and generalize these relations to novel combinations of items. This allows them to respond appropriately to radically novel circumstances and is fundamental to advanced cognition. However, how learning systems (including the brain) can implement the necessary inductive biases has been unclear. We investigated transitive inference (TI), a classic relational task paradigm in which subjects must learn a relation ([Formula: see text] and [Formula: see text]) and generalize it to new combinations of items ([Formula: see text]). Through mathematical analysis, we found that a broad range of biologically relevant learning models (e.g. gradient flow or ridge regression) perform TI successfully and recapitulate signature behavioral patterns long observed in living subjects. First, we found that models with item-wise additive representations automatically encode transitive relations. Second, for more general representations, a single scalar "conjunctivity factor" determines model behavior on TI and, further, the principle of norm minimization (a standard statistical inductive bias) enables models with fixed, partly conjunctive representations to generalize transitively. Finally, neural networks in the "rich regime," which enables representation learning and improves generalization on many tasks, unexpectedly show poor generalization and anomalous behavior on TI. We find that such networks implement a form of norm minimization (over hidden weights) that yields a local encoding mechanism lacking transitivity. Our findings show how minimal statistical learning principles give rise to a classical relational inductive bias (transitivity), explain empirically observed behaviors, and establish a formal approach to understanding the neural basis of relational abstraction.

Inferior parietal cortex represents relational structures for explicit transitive inference.

The human brain is distinguished by its ability to perform explicit logical reasoning like transitive inference. This study investigated the functional role of the inferior parietal cortex in transitive inference with functional MRI. Participants viewed premises describing abstract relations among items. They accurately recalled the relationship between old pairs of items, effectively inferred the relationship between new pairs of items, and discriminated between true and false relationships for new pairs. First, the inferior parietal cortex, but not the hippocampus or lateral prefrontal cortex, was associated with transitive inference. The inferior parietal activity and functional connectivity were modulated by inference (new versus old pairs) and discrimination (true versus false pairs). Moreover, the new/old and true/false pairs were decodable from the inferior parietal representation. Second, the inferior parietal cortex represented an integrated relational structure (ordered and directed series). The inferior parietal activity was modulated by serial position (larger end versus center pairs). The inferior parietal representation was modulated by symbolic distance (adjacent versus distant pairs) and direction (preceding versus following pairs). It suggests that the inferior parietal cortex may flexibly integrate observed relations into a relational structure and use the relational structure to infer unobserved relations and discriminate between true and false relations.

Cross-frequency and inter-regional phase synchronization in explicit transitive inference.

Explicit logical reasoning, like transitive inference, is a hallmark of human intelligence. This study investigated cortical oscillations and their interactions in transitive inference with EEG. Participants viewed premises describing abstract relations among items. They accurately recalled the relationship between old pairs of items, effectively inferred the relationship between new pairs of items, and discriminated between true and false relationships for new pairs. First, theta (4-7 Hz) and alpha oscillations (8-15 Hz) had distinct functional roles. Frontal theta oscillations distinguished between new and old pairs, reflecting the inference of new information. Parietal alpha oscillations changed with serial position and symbolic distance of the pairs, representing the underlying relational structure. Frontal alpha oscillations distinguished between true and false pairs, linking the new information with the underlying relational structure. Second, theta and alpha oscillations interacted through cross-frequency and inter-regional phase synchronization. Frontal theta-alpha 1:2 phase locking appeared to coordinate spectrally diverse neural activity, enhanced for new versus old pairs and true versus false pairs. Alpha-band frontal-parietal phase coherence appeared to coordinate anatomically distributed neural activity, enhanced for new versus old pairs and false versus true pairs. It suggests that cross-frequency and inter-regional phase synchronization among theta and alpha oscillations supports human transitive inference.

Superstitious learning of abstract order from random reinforcement.

Humans and other animals often infer spurious associations among unrelated events. However, such superstitious learning is usually accounted for by conditioned associations, raising the question of whether an animal could develop more complex cognitive structures independent of reinforcement. Here, we tasked monkeys with discovering the serial order of two pictorial sets: a "learnable" set in which the stimuli were implicitly ordered and monkeys were rewarded for choosing the higher-rank stimulus and an "unlearnable" set in which stimuli were unordered and feedback was random regardless of the choice. We replicated prior results that monkeys reliably learned the implicit order of the learnable set. Surprisingly, the monkeys behaved as though some ordering also existed in the unlearnable set, showing consistent choice preference that transferred to novel untrained pairs in this set, even under a preference-discouraging reward schedule that gave rewards more frequently to the stimulus that was selected less often. In simulations, a model-free reinforcement learning algorithm (Q-learning) displayed a degree of consistent ordering among the unlearnable set but, unlike the monkeys, failed to do so under the preference-discouraging reward schedule. Our results suggest that monkeys infer abstract structures from objectively random events using heuristics that extend beyond stimulus-outcome conditional learning to more cognitive model-based learning mechanisms.

Role of Sleep in Formation of Relational Associative Memory.

Relational memory, the ability to make and remember associations between objects, is an essential component of mammalian reasoning. In relational memory tasks, it has been shown that periods of offline processing, such as sleep, are critical to making indirect associations. To understand biophysical mechanisms behind the role of sleep in improving relational memory, we developed a model of the thalamocortical network to test how slow-wave sleep affects performance on an unordered relational memory task. First, the model was trained in the awake state on a paired associate inference task, in which the model learned to recall direct associations. After a period of subsequent slow-wave sleep, the model developed the ability to recall indirect associations. We found that replay, during sleep, of memory patterns learned in awake increased synaptic connectivity between neurons representing the item that was overlapping between tasks and neurons representing the unlinked items of the different tasks; this forms an attractor that enables indirect memory recall. Our study predicts that overlapping items between indirectly associated tasks are essential for relational memory, and sleep can reactivate pathways to and from overlapping items to the unlinked objects to strengthen these pathways and form new relational memories.SIGNIFICANCE STATEMENT Experimental studies have shown that some types of associative memory, such as transitive inference and relational memory, can improve after sleep. Still, it remains unknown what specific mechanisms are responsible for these sleep-related changes. In this new work, we addressed this problem by building a thalamocortical network model that can learn relational memory tasks and that can be simulated in awake or sleep states. We found that memory traces learned in awake were replayed during slow waves of NREM sleep and revealed that replay increased connections to and from overlapping memory items to form new relational memories. Our work discovered specific mechanisms behind the role of sleep in associative memory and made testable predictions about how sleep augments associative learning.

Representation of the inferred relationships in a map-like space.

A cognitive map is an internal representation of the external world that guides flexible behavior in a complex environment. Cognitive map theory assumes that relationships between entities can be organized using Euclidean-based coordinates. Previous studies revealed that cognitive map theory can also be generalized to inferences about abstract spaces, such as social spaces. However, it is still unclear whether humans can construct a cognitive map by combining relational knowledge between discrete entities with multiple abstract dimensions in nonsocial spaces. Here we asked subjects to learn to navigate a novel object space defined by two feature dimensions, price and abstraction. The subjects first learned the rank relationships between objects in each feature dimension and then completed a transitive inferences task. We recorded brain activity using functional magnetic resonance imaging (fMRI) while they performed the transitive inference task. By analyzing the behavioral data, we found that the Euclidean distance between objects had a significant effect on response time (RT). The longer the one-dimensional rank distance and two-dimensional (2D) Euclidean distance between objects the shorter the RT. The task-fMRI data were analyzed using both univariate analysis and representational similarity analysis. We found that the hippocampus, entorhinal cortex, and medial orbitofrontal cortex were able to represent the Euclidean distance between objects in 2D space. Our findings suggest that relationship inferences between discrete objects can be made in a 2D nonsocial space and that the neural basis of this inference is related to cognitive maps.

Thinking about order: a review of common processing of magnitude and learned orders in animals.

Rich behavioral and neurobiological evidence suggests cognitive and neural overlap in how quantitatively comparable dimensions such as quantity, time, and space are processed in humans and animals. While magnitude domains such as physical magnitude, time, and space represent information that can be quantitatively compared (4 "is half of" 8), they also represent information that can be organized ordinally (1→2→3→4). Recent evidence suggests that the common representations seen across physical magnitude, time, and space domains in humans may be due to their common ordinal features rather than their common quantitative features, as these common representations appear to extend beyond magnitude domains to include learned orders. In this review, we bring together separate lines of research on multiple ordinal domains including magnitude-based and learned orders in animals to explore the extent to which there is support for a common cognitive process underlying ordinal processing. Animals show similarities in performance patterns across natural quantitatively comparable ordered domains (physical magnitude, time, space, dominance) and learned orders (acquired through transitive inference or simultaneous chaining). Additionally, they show transfer and interference across tasks within and between ordinal domains that support the theory of a common ordinal representation across domains. This review provides some support for the development of a unified theory of ordinality and suggests areas for future research to better characterize the extent to which there are commonalities in cognitive processing of ordinal information generally.

Heuristics Facilitates the Evolution of Transitive Inference and Social Hierarchy in a Large Group.

Transitive inference (TI) refers to social cognition that facilitates the discernment of unknown relationships between individuals using known relationships. It is extensively reported that TI evolves in animals living in a large group because TI could assess relative rank without deducing all dyadic relationships, which averts costly fights. The relationships in a large group become so complex that social cognition may not be developed adequately to handle such complexity. If members apply TI to all possible members in the group, TI requires extremely highly developed cognitive abilities especially in a large group. Instead of developing cognitive abilities significantly, animals may apply simplified TI we call reference TI in this study as heuristic approaches. The reference TI allows members to recognize and remember social interactions only among a set of reference members rather than all potential members. Our study assumes that information processes in the reference TI comprises (1) the number of reference members based on which individuals infer transitively, (2) the number of reference members shared by the same strategists, and (3) memory capacity. We examined how information processes evolve in a large group using evolutionary simulations in the hawk-dove game. Information processes with almost any numbers of reference members could evolve in a large group as long as the numbers of shared reference member are high because information from the others' experiences is shared. TI dominates immediate inference, which assesses relative rank on direct interactions, because TI could establish social hierarchy more rapidly applying information from others' experiences.

Neural correlates of transitive inference: An SDM meta-analysis on 32 fMRI studies.

Transitive inference (TI) is a critical capacity involving the integration of relevant information into prior knowledge structure for drawing novel inferences on unobserved relationships. To date, the neural correlates of TI remain unclear due to the small sample size and heterogeneity of various experimental tasks from individual studies. Here, the meta-analysis on 32 fMRI studies was performed to detect brain activation patterns of TI and its three paradigms (spatial inference, hierarchical inference, and associative inference). We found the hippocampus, prefrontal cortex (PFC), putamen, posterior parietal cortex (PPC), retrosplenial cortex (RSC), supplementary motor area (SMA), precentral gyrus (PreCG), and median cingulate cortex (MCC) were engaged in TI. Specifically, the RSC was implicated in the associative inference, whereas PPC, SMA, PreCG, and MCC were implicated in the hierarchical inference. In addition, the hierarchical inference and associative inference both evoked activation in the hippocampus, medial PFC, and PCC. Although the meta-analysis on spatial inference did not generate a reliable result due to insufficient amount of investigations, the present work still offers a new insight for better understanding the neural basis underlying TI.

Positional inference in rhesus macaques.

Understanding how organisms make transitive inferences is critical to understanding their general ability to learn serial relationships. In this context, transitive inference (TI) can be understood as a specific heuristic that applies broadly to many different serial learning tasks, which have been the focus of hundreds of studies involving dozens of species. In the present study, monkeys learned the order of 7-item lists of photographic stimuli by trial and error, and were then tested on "derived" lists. These derived test lists combined stimuli from multiple training lists in ambiguous ways, sometimes changing their order relative to training. We found that subjects displayed strong preferences when presented with novel test pairs, even when those pairs were drawn from different training lists. These preferences were helpful when test pairs had an ordering congruent with their ranks during training, but yielded consistently below-chance performance when pairs had an incongruent order relative to training. This behavior can be explained by the joint contributions of transitive inference and another heuristic that we refer to as "positional inference." Positional inferences play a complementary role to transitive inferences in facilitating choices between novel pairs of stimuli. The theoretical framework that best explains both transitive and positional inferences is a spatial model that represents both the position of each stimulus and its uncertainty. A computational implementation of this framework yields accurate predictions about both correct responses and errors on derived lists.

Fish can infer relations between colour cues in a non-social learning task.

Transitive inference (TI) describes the ability to infer relationships between stimuli that have never been seen together before. Social cichlids can use TI in a social setting where observers assess dominance status after witnessing contests between different dyads of conspecifics. If cognitive processes are domain-general, animals should use abilities evolved in a social context also in a non-social context. Therefore, if TI is domain-general in fish, social fish should also be able to use TI in non-social tasks. Here we tested whether the cooperatively breeding cichlid Neolamprologus pulcher can infer transitive relationships between artificial stimuli in a non-social context. We used an associative learning paradigm where the fish received a food reward when correctly solving a colour discrimination task. Eleven of 12 subjects chose the predicted outcome for TI in the first test trial and five subjects performed with 100% accuracy in six successive test trials. We found no evidence that the fish solved the TI task by value transfer. Our findings show that fish also use TI in non-social tasks with artificial stimuli, thus generalizing past results reported in a social context and hinting toward a domain-general cognitive mechanism.

Atypical Reinforcement Learning in Developmental Dyslexia.

OBJECTIVES: According to the Procedural Deficit Hypothesis, abnormalities in corticostriatal pathways could account for the language-related deficits observed in developmental dyslexia. The same neural network has also been implicated in the ability to learn contingencies based on trial and error (i.e., reinforcement learning [RL]). On this basis, the present study tested the assumption that dyslexic individuals would be impaired in RL compared with neurotypicals in two different tasks. METHODS: In a probabilistic selection task, participants were required to learn reinforcement contingencies based on probabilistic feedback. In an implicit transitive inference task, participants were also required to base their decisions on reinforcement histories, but feedback was deterministic and stimulus pairs were partially overlapping, such that participants were required to learn hierarchical relations. RESULTS: Across tasks, results revealed that although the ability to learn from positive/negative feedback did not differ between the two groups, the learning of reinforcement contingencies was poorer in the dyslexia group compared with the neurotypicals group. Furthermore, in novel test pairs where previously learned information was presented in new combinations, dyslexic individuals performed similarly to neurotypicals. CONCLUSIONS: Taken together, these results suggest that learning of reinforcement contingencies occurs less robustly in individuals with developmental dyslexia. Inferences for the neuro-cognitive mechanisms of developmental dyslexia are discussed.

Chained study and the discovery of relational structure.

Prior knowledge of relational structure allows people to quickly make sense of and respond to new experiences. When awareness of such structure is not necessary to support learning, however, it is unclear when and why individuals "spontaneously discover" an underlying relational schema. The present study examines the determinants of such discovery in discrimination-based transitive inference (TI), whereby people learn about a hierarchy of interrelated premises and are tested on their ability to draw inferences that bridge studied relations. Experiencing "chained" sequences of overlapping premises during training was predicted to facilitate the discovery of relational structure. Among individuals without prior knowledge of the hierarchy, chaining improved relational learning and was most likely to result in explicit awareness of the underlying relations between items. Observation of chained training sequences was also more effective than the self-generation of training sequences. These findings add to growing evidence that the temporal dynamics of training, including successive presentation of overlapping associations, are key to understanding spontaneous relational discovery during learning.

Towards a resolution of some outstanding issues in transitive research: An empirical test on middle childhood.

Transitive Inference (deduce B > D from B > C and C > D) can help us to understand other areas of sociocognitive development. Across three experiments, learning, memory, and the validity of two transitive paradigms were investigated. In Experiment 1 (N = 121), 7-year-olds completed a three-term nontraining task or a five-term task requiring extensive-training. Performance was superior on the three-term task. Experiment 2 presented 5-10-year-olds with a new five-term task, increasing learning opportunities without lengthening training (N = 71). Inferences improved, suggesting children can learn five-term series rapidly. Regarding memory, the minor (CD) premise was the best predictor of BD-inferential performance in both task-types. However, tasks exhibited different profiles according to associations between the major (BC) premise and BD inference, correlations between the premises, and the role of age. Experiment 3 (N = 227) helped rule out the possible objection that the above findings simply stemmed from three-term tasks with real objects being easier to solve than computer-tasks. It also confirmed that, unlike for five-term task (Experiments 1 & 2), inferences on three-term tasks improve with age, whether the age range is wide (Experiment 3) or narrow (Experiment 2). I conclude that the tasks indexed different routes within a dual-process conception of transitive reasoning: The five-term tasks indexes Type 1 (associative) processing, and the three-term task indexes Type 2 (analytic) processing. As well as demonstrating that both tasks are perfectly valid, these findings open up opportunities to use transitive tasks for educability, to investigate the role of transitivity in other domains of reasoning, and potentially to benefit the lived experiences of persons with developmental issues.

Category learning in a transitive inference paradigm.

The implied order of a ranked set of visual images can be learned without reliance on information that explicitly signals their order. Such learning is difficult to explain by associative mechanisms, but can be accounted for by cognitive representations and processes such as transitive inference. Our study sought to determine if those processes also apply to learning categories of images. We asked whether participants can (a) infer that stimulus images belonged to familiar categories, even when the images for each trial were unique, and (b) sort those categories into an ordering that obeys transitivity. Participants received minimal verbal instruction and a single session of training. Despite this, they learned the implied order of lists of fixed stimuli and lists of ordered categories, using trial-unique exemplars. We trained two groups, one for which stimuli were constant throughout training and testing (n = 60), and one for which exemplars of each category were trial-unique (n = 50). Our findings suggest that differing cognitive processes may underpin serial learning when learning about specific stimuli as opposed to stimulus categories.

Associative models fail to characterize transitive inference performance in rhesus monkeys (Macaca mulatta).

It has been suggested that non-verbal transitive inference (if A > B and B > C, then A > C) can be accounted for by associative models. However, little is known about the applicability of such models to primate data. In Experiment 1, we tested the fit of two associative models to primate data from both sequential training, in which the training pairs were presented in a backward order, and simultaneous training, in which all training pairs are presented intermixed from the beginning. We found that the models provided an equally poor fit for both sequential and simultaneous training presentations, contrary to the case with data from pigeons. The models were also unable to predict the robust symbolic distance effects characteristic of primate transitive choices. In Experiment 2, we used the models to fit a list-linking design in which two seven-item transitive lists were first trained independently (A > B…. > F > G and H > I …. > M > N) then combined via a linking pair (G+ H-) into a single, 14-item list. The model produced accurate predictions for between-list pairs, but did not predict transitive responses for within-list pairs from list 2. Overall, our results support research indicating that associative strength does not adequately account for the behavior of primates in transitive inference tasks. The results also suggest that transitive choices may result from different processes, or different weighting of multiple processes, across species.

Exploring the interaction between approach-avoidance conflict and memory processing.

The medial temporal lobe (MTL) has been implicated in approach-avoidance (AA) conflict processing, which arises when a stimulus is imbued with both positive and negative valences. Notably, since the MTL has been traditionally viewed as a mnemonic brain region, a pertinent question is how AA conflict and memory processing interact with each other behaviourally. We conducted two behavioural experiments to examine whether increased AA conflict processing has a significant impact on incidental mnemonic encoding and inferential reasoning. In Experiment 1, participants first completed a reward and punishment AA task and were subsequently administered a surprise recognition memory test for stimuli that were presented during high and no AA conflict trials. In Experiment 2, participants completed a reward and punishment task in which they learned the valences of objects presented in pairs (AB, BC pairs). Next, we assessed their ability to integrate information across these pairs (infer A-C relationships) and examined whether inferential reasoning was more challenging across objects with conflicting compared to non-conflicting incentive values. We observed that increased motivational conflict did not significantly impact encoding or inferential reasoning. Potential explanations for these findings are considered, including the possibility that AA conflict and memory processing are not necessarily intertwined behaviourally.

Transitive inference in pigeons may result from differential tendencies to reject the test stimuli acquired during training.

In the five-term, transitive inference task used with animals, pigeons are trained on four simultaneous discrimination premise pairs: A + B -, B + C -, C + D -, D + E -. Typically, when tested with the BD pair, most pigeons show a transitive inference effect, choosing B over D. Two non-inferential hypotheses have been proposed to account for this effect but neither has been reliably supported by research. Here we test a third non-inferential hypothesis that the preference for B arises because the animals have not had as much experience with B - in the A + B - discrimination as they have had with the D - in the C + D - discrimination. To test this hypothesis we trained the Experimental Group with the A + B - discrimination in which, over trials, there were four possible A + stimuli that could appear. This was done to encourage the pigeons to learn to reject the B - stimulus. For the Control Group there was only one A + stimulus over trials, as is typically the case. We also varied the nature of the stimuli between groups, such that colors served as the stimuli for half of the pigeons, whereas flags of different counties served as stimuli for the remaining pigeons. In both stimulus conditions, for the Experiment Group, we found little preference for stimulus B over stimulus D, whereas for the Control Group we found the typical preference for stimulus B. Thus, we propose that it is not necessary to attribute the transitive inference effect to an inferential process.

Transitive inference in Polistes paper wasps.

Transitive inference (TI) is a form of logical reasoning that involves using known relationships to infer unknown relationships (A > B; B > C; then A > C). TI has been found in a wide range of vertebrates but not in insects. Here, we test whether Polistes dominula and Polistes metricus paper wasps can solve a TI problem. Wasps were trained to discriminate between five elements in series (A0B-, B0C-, C0D-, D0E-), then tested on novel, untrained pairs (B versus D). Consistent with TI, wasps chose B more frequently than D. Wasps organized the trained stimuli into an implicit hierarchy and used TI to choose between untrained pairs. Species that form social hierarchies like Polistes may be predisposed to spontaneously organize information along a common underlying dimension. This work contributes to a growing body of evidence that the miniature nervous system of insects does not limit sophisticated behaviours.

Inferential Learning of Serial Order of Perceptual Categories by Rhesus Monkeys (Macaca mulatta).

Category learning in animals is typically trained explicitly, in most instances by varying the exemplars of a single category in a matching-to-sample task. Here, we show that male rhesus macaques can learn categories by a transitive inference paradigm in which novel exemplars of five categories were presented throughout training. Instead of requiring decisions about a constant set of repetitively presented stimuli, we studied the macaque's ability to determine the relative order of multiple exemplars of particular stimuli that were rarely repeated. Ordinal decisions generalized both to novel stimuli and, as a consequence, to novel pairings. Thus, we showed that rhesus monkeys could learn to categorize on the basis of implied ordinal position, without prior matching-to-sample training, and that they could then make inferences about category order. Our results challenge the plausibility of association models of category learning and broaden the scope of the transitive inference paradigm.SIGNIFICANCE STATEMENT The cognitive abilities of nonhuman animals are of enduring interest to scientists and the general public because they blur the dividing line between human and nonhuman intelligence. Categorization and sequence learning are highly abstract cognitive abilities each in their own right. This study is the first to provide evidence that visual categories can be ordered serially by macaque monkeys using a behavioral paradigm that provides no explicit feedback about category or serial order. These results strongly challenge accounts of learning based on stimulus-response associations.