Determinantal Point Processes for Memory and Structured InferenceSteven M. Frankland

steven.frankland@princeton.eduPrinceton Neuroscience Institute

Princeton, NJ

Jonathan D. CohenPrinceton Neuroscience Institute

Princeton, NJ

AbstractDeterminantal Point Processes (DPPs) are probabilisticmodels of repulsion, capturing negative dependenciesbetween states. Here, we show that a DPP inrepresentation-space predicts inferential biases towardmutual exclusivity commonly observed in word learning(mutual exclusivity bias) and reasoning (disjunctivesyllogism) tasks. It does so without requiring explicitrule representations, without supervision, and withoutexplicit knowledge transfer. The DPP attempts tomaximize the total ”volume” spanned by the set ofinferred code-vectors. In a representational system inwhich combinatorial codes are constructed by re-usingcomponents, a DPP will naturally favor the combinationof previously un-used components. We suggest thatthis bias toward the selection of volume-maximizingcombinations may exist to promote the efficient retrievalof individuals from memory. In support of this, we showthe same algorithm implements efficient ”hashing”,minimizing collisions between key/value pairs withoutexpanding the required storage space. We suggestthat the mechanisms that promote efficient memorysearch may also underlie cognitive biases in structuredinference.Keywords: mutual exclusivity; determinantal point process;memory; binding; compositionality; probabilistic models

Background and MotivationImagine that Tim and Tom are playing an Argentinian boardgame called El Estanciero. Tim won. What happened toTom? Although you may not be certain, you don’t needany more information to infer that it’s likely that Tom lost.This is true despite the fact that you have no familiarity withthe particulars of the game or the people involved. Thisreflects an inferential bias toward mutual exclusivity (ME):the tendency to map individuals to 1/n possible relationalpositions, and not to multiple positions in the same instance.Although, in the El Estanciero example, you came equippedwith rich knowledge about the relational structure of gamesthat you could import, ME inferences have been observedsurprisingly early in human development (Halberda 2003,Cesana-Arlotti et al. 2018), and in non-human species(Pepperberg et al. 2019), and aren’t constrained to binaryrelations.

For example, a classic finding in developmentalpsychology is that, all else being equal, young childrenprefer to map a novel word (”zurp”) to a novel referent(cathode-ray tube), rather than mapping many words to thesame object, or many objects to the same word (Markman& Wachtel, 1988; Halberda, 2003), a phenomenon knownas the mutual exclusivity bias (ME) in word-learning. Atthe same time, work in a different cognitive domain hasfound that pre-verbal infants assume that an individual objectcan not be in two places at the same time, and thus makeinferences that resemble a formal disjunctive syllogism (A orB (but not both). Not A. Therefore, B) (Cesana-Arlotti et al.2018; See also Mody & Carey, 2016)).

Here, we argue that the class of ME inferences canbe fruitfully modeled using a Determinantal Point Process(DPP) operating over a representational space. DPPs areprobabilistic models of repulsion between states: the moresimilar two states are, the less likely they are to co-occur (SeeFigure 1). DPPs originated in statistical physics to modelthe location of fermions at thermal equilibrium (Macchi,1975), but have since been extended to other branches ofmathematics and machine learning (Kulesza & Taskar, 2012).In machine learning, they have recently gained traction inthe generation of sets of samples when sample diversity isdesirable, such as recommender systems looking to presenta broad sample of item-types to users ((Kulesza & Taskar,2012, Gillenwater et al. 2012).

Here, we consider the inferential biases afforded byDPPs over a representational space. Specifically, we show(1) that when representations of possible combinations(e.g., word/object or location/object combinations) arecombinations of re-usable codes, DPPs naturally predictthe mutual exclusivity bias in word learning and reasoningby disjunctive syllogism. We then (2) suggest that theseinferential biases may owe to basic desiderata placed on thedata structures employed by an efficient memory system, andprovide evidence that DPPs effectively navigate a space/timetradeoff regarding storage space and access time in encodingand retrieval.

General MethodsWe focus on two well-studied cases of mutual exclusivityin cognitive science: The ”mutual exclusivity bias” in wordlearning (Markman Wachtel, 1988; Merriman Bowman,

3302©2020 The Author(s). This work is licensed under a CreativeCommons Attribution 4.0 International License (CC BY).

Figure 1: Sampling in a plane for three types of pointprocesses. PPP’s contain no relational information, andtherefore, although likely to be spread across the space, canbe susceptible to ”clumping”. MPPs, by contrast, carryrelational information about the similarity between states,directly promoting such clumping. DPPs use the samesimilarity representation as MPPs, but, critically, repel similarstates, ensuring that samples are well distributed across thespace.

1991; Halberda, 2003) and the ability to complete disjunctivesyllogisms (Mody & Carey, 2016; Cesana-Arlotti et al.2018). We first describe the modeling approach in abstractterms, explaining the features that are common to bothuse-cases.

DPP framework. Our model assumes a point process Poperating over a finite ground set Y of discrete items. Here,the items are possible representations (i.e., encodings) andmore specifically, representations of particular combinations(e.g., word/object or location/object combinations). We willoften use the generic terms ”keys” and ”values” when talkingabout the representational components, motivated largely bythe connections we wish to make to memory.

The process defines a probability of selecting particularsubsets (S) of these combinations drawn from the ground set.

P(S⊆ Y ) = det(KS)

where K is an NxN positive semi-definite kernel matrixwhose entries encode pairwise similarities between Npossible discrete states, where N = m keys X n values. Forall analyses reported here, we use a linear kernel, computedas a normalized inner product of each combination of themXn key/value vectors under consideration (See Figure 2 forillustration).

DPPs select a particular configuration of items so asto maximize the determinant (det) of the correspondingsub-matrix of K, indexed by S. (Macchi, 1975; KuleszaTaskar, 2012). Thus, the central computation in the currentcase is arg maxS det(KS).

Geometrically, we can think of this determinant as thevolume spanned by the parallelpiped of the code vectors. Themore similar a set of vectors (small determinant), the lesslikely they are to co-occur in a set. The less similar the vectors(larger determinant), the more likely they are to co-occur inS. This enables the modeling of repulsion between possible

states; it is central to the current work and its ability to modelaspects of higher-level cognition.

Here, the relevant representational states areconcatenations of two types of code vectors correspondingto pre-factorized representations. As we noted above,these factors may be thought of as ”keys” and ”values”, oralternatively ”relations” and ”content”. Concretely, however,in the situations we explore, they are representations ofwords and their referents in the ME-bias case, and spatiallocations and objects in the disjunctive syllogism case. Togenerate the code for a possible combination, we simplyconcatenate vectors for the key/value components, keepingthe ordering and codes consistent across uses, consistentwith compositionality.1 Although we report simple key/valueconcatenations here, we obtain the same results both bysumming key/value representations.

Although finding the maximum a posteriori (MAP) subsetin a DPP is NP-hard (Kulesza & Taskar, 2012), thereexist greedy methods that can effectively approximate it(Gillenwater et al. 2012, Han et al. 2017). However, here,we make the simplifying assumption that Y is itself a subsetof the vastly larger set of possible items that could have beenunder consideration. For present purposes, we assume thatthis context-dependent restriction of the possibility space canbe carried out by standard attentional mechanisms. Withinthis small-cardinality space, we are able to exhaustivelysearch for the MAP subset (the sub-matrix that maximizes thedeterminant). However, generating more plausible heuristicmethods that can scale to larger spaces remains a focus forfuture work.

Throughout, we compare the DPP to two alternativepoint process models in order to emphasize the conceptualcontribution of the DPP. First, a Poisson Point Process (PPP),which assumes no similarity kernel K, treating each item asindependent. Here, we use P(S⊆ Y ) = ∏i∈y pi ∏i/∈y (1− pi),where p is a flat prior across states. This is random uniformselection. Second, a generic Markov Point Process (MPP),which selects items based on the kernel K used for theDPP, but selecting directly on the similarity scores of thesub-matrix, rather than its determinant. In this, the MPP overK can be considered in opposition to the DPP, favoring itemsthat are nearby in the code-space rather than far apart. Takentogether, one can think of these three models as capturing thepossibility of (a) random inference (PPP), (b) inference bysimilarity (MPP) and (c) inference by repulsion (DPP). SeeFigure 1. We note that we are not choosing to compare the

1We note that, although this encoding framework is simple,it is motivated by empirical evidence concerning the nature andorganization of the projections from the mammalian entorhinalcortex (EC) to the hippocampal sub-fields: a key circuit bothfor simple forms of reasoning and memory (Zeithamova et al.,2012). Specifically, a medial region of EC contains low-dimensionalrepresentations of the spatial structure of the environment, while alateral region encodes sensory content (Behrens et al. 2018). Theseseparate representations are believed to then be bound togetherin the hippocampus in order to encode different structure/contentcombinations (Whittington et al. 2018).

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Figure 2: (A). Example of the word learning problem. The model’s task is to select a word/object mapping, conditioned on3 existing associations. Humans tend to map the held-out word to the held-out object. (B) shows the representations used bythe models to guide inference. We assume separate, but combinable, codes for words (keys) and objects (values). The squarematrix in (C) represents pairwise similarities between possible word/object combinations. Brighter colors reflect more similarcombinatorial codes, darker colors less similar codes. Black bars across rows and columns reflect a hypothetical subset ofword/object mappings, as in A. (D) We evaluate the probability that the held-out word is mapped to the held-out object (mutualexclusivity bias) across 1000 simulations with different word/object codes. A DPP naturally selects the novel unused word andun-used object, exhibiting the mutual exclusivity bias.

DPP to these other point process models because we believethat MPPs and PPPs are a priori particularly plausible modelsof the relevant types of structured inference. Instead, webelieve they are useful to highlight fundamental properties ofthe DPP (e.g., repulsion). We believe that first comparing thepredictions of a DPP against these clearly situates the DPP ina consistent conceptual framework for expository purposes.

Mutual Exclusivity Bias in Word Learning. The ”mutualexclusivity (ME) bias” in word learning refers to theempirical observation that, all else being equal, youngchildren and adults prefer to map a novel word to a novelreferent. They prefer this both to mapping many words tothe same object, or many objects to the same word (Markman& Wachtel, (1988); Merriman & Bowman, (1989); Halberda,(2003); Lake, Linzen, Baroni (2019)). Here, we suggest thatthis inferential bias follows directly from the considerationof an associative encoding system that selects which codesto bind under a Determinantal Point Process. A learner thatperforms inference using a DPP will prefer combinationsthat maximize the total volume of the representational space.Under the assumption that components re-use codes acrosspossible uses (consistent with compositionality), a DPPnaturally favors combinations of previously un-used wordsand objects.

We model a case involving 4 words and 4 objects, in whichthe learner has 3 extant word/object associations (See Figure2a). Here, we are agnostic as to whether those associationswere acquired in this particular episode, or whether theywere brought to the episode. Words and objects are randomcode vectors, sampled from a multivariate Gaussian N (0,1).These codes are concatenated to form possible word/objectcombinations (See Figure 2c). We compute a linear kernel

over the representations of these combinations, reflectingthe covariance structure amongst the codes for differentword/object pairs. The different point process models thenselect from the 13 remaining word/object combinations,conditioned on the 3 previous word/object combinations.

We ran 1000 simulations involving different random wordand object vectors, and found that the DPP exhibits themutual exclusivity bias 99.3 % of the time. See Figure 2.As would be expected, the PPP model randomly selects fromthe 13 remaining possible conjunctions. The MPP prioritizesre-use of codes across instances (re-using a word to referto multiple objects), given that its desideratum is to selectcombinations similar to those already encountered. It has aninferential bias of ”many-to-one”.

Thus, when codes for combinatorial states arecompositions of words and objects (keys and values), asimple algorithm that maximizes the volume spanned by thecode vectors naturally produces a mutual exclusivity bias inthe word learning process. Re-using either words or objectsacross different mappings in the same context works againstmaximizing the volume spanned by the vectors, as the samevector will contribute to multiple combinations.Disjunctive Syllogism. Our second example involves theability to make inferences like those in a classical DisjunctiveSyllogism (DS) (Mody & Carey, 2016; Cesana-Arlottiet al., 2018; Pepperberg et al. 2019). Formally, adisjunctive syllogism starts with the representation of adisjunction (premise 1: A or B), where ’or’ is XOR (oneor the other, but not both). Next, one acquires somepiece of information (premise 2: not A). Finally, a ruleis applied to derive the conclusion, conditioned on thepremises. (conclusion: Therefore, B). Notably, young

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Figure 3: (A). Rendering of Cesana-Arlotti (2018)’s experimental paradigm, based on their Figure 1. (B). To model this, weassume a factored space of object and location codes under consideration and (C) populate a square matrix with the pairwisesimilarities between possible object/location combinations (K). We highlight in red the items selected by arg max S ∈Y det(KS),conditioned separately on each combination that could be observed (the diagonal). The DPP reliably favors the combinationthat is most dissimilar (dark blue) to the observed object/location combination (bright yellow). (D). Cesana-Arlotti et al. (2018)found that infants exhibit increased looking time to DS-inconsistent cases (results schematically depicted here). (E) The DPPmodel naturally selects a combination of the un-used location and un-used object. If these inferences are used to generatepredictions and compared against the DS-consistent and DS-inconsistent cases, the DPP exhibits greater prediction errors whenthe revealed object/location combination is DS-inconsistent, like pre-verbal infants.

children (Mody & Carey, 2016) including infants as youngas 12 months (Cesana-Arlotti et al. 2018) and non-humananimals (Pepperberg et al. 2019) all show aspects of thisinferential ability. Here, we show that a DPP defined overthe space of combinatorial representations predicts the keyempirical pattern.

For expository purposes, we focus on Cesana-Arlotti et al.(2018)’s paradigm with pre-verbal infants. See Figure 3a fora schematic of a trial. A trial begins with two objects ona screen. Both are temporarily hidden behind an occluder,obstructing the objects from the infant’s view. One objectis then seen to be scooped out from behind the occluder,though the infant is unable to determine which of the twoobjects it was. The occluder is then removed revealing (e.g.,)object A. The inference by disjunctive syllogism, of course,is that object B must therefore be the object in the bucket.Infants’ expectations are assessed by measuring their lookingtime. If it is then revealed that the bucket contains objectA, rather than object B (the ”DS-inconsistent” condition),infants as young as 12 months old are surprised, evidenced byincreased looking time relative to the alternative outcome inwhich object B is in the bucket (”DS-consistent” condition).

To model this, we assume 1x100 random code vectorsdrawn from a multivariate Gaussian N (0,1) for each of twoobjects (values) and two locations (keys). We concatenatethese to form a 4x200 matrix, in which the rows arecompositions of possible object/location combinations, andthe columns are the random features. As above, we computethe covariance between each of the mXn combinations, hereobtaining a 4x4 kernel K encoding the similarities betweenthe codes for possible combinatorial states. For our analyses,we simulated 1000 different possible instances of randomvectors, while also randomly selecting different superficialtrial structures (e.g., that the DS-consistent combination wasobject A/location 1, object A/location2, object B/location1,objectB/location2). As expected, given one conjunction(e.g., object A in location 1), a DPP reliably selects theun-used object and the un-used location (here, object Bin location 2), as it maximizes the volume spanned byvectors encoding the combinations. See Figure 3. Tomore directly relate the models’ inferences to the infantlooking time data, we next computed the MSE between theobject/location combination selected by the model and thecode for the stimulus in the DS-consistent (low-surprise)

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and DS-inconsistent (high-surprise) conditions. As expected,the prediction error is high for the DPP model in theDS-inconsistent, and at zero for the DS-consistent condition.The MPP (similarity-based), and PPP (random) models donot predict this direction of the ”surprisal” effect.

DPP Hashing for Collision-Free Encoding.The empirical findings that we model here demonstrate anumber of notable features of ME biases, chief amongstthem: (a) they appear to be present remarkably early indevelopment (Cesana-Arlotti et al. 2018; Lewis et al.,2020), (b) they have been observed, in different forms,across representational domains (word learning and logicalreasoning about physical states), and (c) related worksuggests that at least one type (DS-like inferences) maybe present in some non-human species (Pepperberg et al.2019). This raises a family of interesting theoretical questionsregarding their acquisition and nature. For example, are thereseparate domain-specific biases, the existence of each owingto its utility in a particular domain, or do these emerge froma shared system? What are the relevant representational andinferential systems, and how are they implemented? And, ofcourse, familiar questions regarding whether such inferentialbiases are acquired over phylogenetic or ontogenetic time. 2

Although we certainly do not intend to definitively answerthese questions here, we add one theoretical suggestion to theliterature: we propose that ME may be a consequence of amore general strategy for efficiently encoding and retrievingunique tokens of key/value associations in memory. On suchan account, it would not be surprising that ME biases arepresent across disparate representational domains, as longas the domain requires binding of component parts andstorage for later retrieval. Moreover, it seems possible thatgeneral mechanisms for encoding and retrieving of tokens(individuals) might be be present early in development andwould be employed by different species.

Why would one think that encoding and retrievingunique tokens of key/value associations in memory hasanything to do with mutual exclusivity? Recall, first, that weassume that the bindings under consideration (word/objector object/location combinations) are fundamentallycompositional: re-using codes to promote generalization tonovel combinations. However, a compositional encodingscheme also creates the possibility of collisions betweendistinct instances of similar states (imagine the classicexample of where you parked your car yesterday vs.two days ago). One might therefore wish to index therepresentations of individual instances in such a way asto avoid mapping similar states to the same address. One

2The early-onset of the DS results of Cesana-Arlotti et al (2018)at least raise the possibility that pieces of the relevant machinerycould be innate. However, recent computational modeling workfrom Lake (2019) provides an existence proof that an ME bias canitself be induced from domain-experience. Lake (2019) shows thata neural network equipped with an external memory and trained in ameta learning sequence-to-sequence paradigm can learn to apply anME bias to novel instances of word-object pairings.

way to achieve this is to spread the keys broadly across therepresentational space (maximizing volume, as in a DPP).That is, we suggest that ME biases may exist across domainsbecause those domains all draw on a shared memory systemfor associative binding, and an important feature of thissystem is its ability to avoid collisions by maximizing therepresentational ”volume” of the memory keys. On this view,ME biases would arise in any representational domain thatrequires inference regarding novel combinations of familiarcomponents (word/object, object/location).

To better illustrate the potential benefit of dispersing keysfor efficient memory retrieval, it is instructive to considerdata structures for ”hashing” in computer science. Ahash-function is a way of mapping from a datum to aunique index, such as a position in an array. Effectivehashing seeks to avoid the time demands that are produced bysequential search techniques, which have a time-complexityof (O(n)) or binary search (O(log n)). Instead, a goodhash-function enables (O(1)) access times, in which readouttime is invariant to the number of items in the memory. Thisis a classic example of a space/time tradeoff (Sedgewick &Wayne, 2011): If one is willing to expend the resourcesnecessary to construct a vast associative array, data wouldalmost never be mapped to the same position, and collisionswould be minimized. However, this is costly in terms ofspace. By contrast, restricting the size of the array reduces theamount of space consumed, but risks dramatically increasingretrieval time, as one would have to search all the items in theparticular location currently indexed (i.e, linear probing andchaining methods). Hashing seeks to effectively navigate thistrade-off, constraining both the size of the array that is needed(space), while also minimizing the amount of computationspent resolving collisions (time). DPPs in representationalspace effectively avoid this tradeoff.

To see this, consider a toy case in which locations inmemory are indexed by a finite set of keys, and we are ableto select a key for each datum. We compare hypotheticalkey-selection algorithms that hash based on PPPs, MPPs,and DPPs where the latter two cases are defined over thesimilarity kernel for the space of key/value combinations inthe dataset, as above. DPP-based key selection begins byrandomly sampling a key/value pair. Then, each subsequentvalue in the dataset is tagged with the particular key thatmaximizes the total volume of the dataset of key/value pairsthat have been hashed to that point (when concatenatedwith the value). DPP-based key selection (unlike PPP andMPP) thus implicitly discourages the re-use of keys, asthis would reduce the volume of the parallelpided spannedby the code vectors for the association. Figure 4 showsthe results of 1000 simulations comparing the performanceof these different models. The probability of a collisionin such an idealized memory system is near 0 (Figure4a).3 Notably, this ”collision-free” property is accomplished

3We find that these infrequent collisions in the DPP can becompletely eliminated by use of Pearson correlation to compute the

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Figure 4: DPP-driven codes enable efficient retrieval of unique items. We allow the keys of key/value pairs to be selectedas a MPP, PPP, or DPP. DPPs (A) minimize collisions between items. They do so in virtue of (B) selecting un-used keys tomaximize the volume spanned by the key/value code vectors, across the dataset. They thus efficiently manage the resourcetradeoff between search time and storage space.

without dramatically expanding the size of the array, as thearray size is no larger than the number of individual states wewish to encode (Figure 4b). A DPP-based hash algorithm,unlike random selection in a PPP, thus exploits the repulsiveproperty to distribute codes evenly across the representationalspace.

DiscussionWe have shown that Determinantal Point Processes(DPPs)–probabilistic models of the negative interactionsbetween states– predict a class of commonly observedbiases in structured inference: specifically, inferential biasestoward (a) mutual exclusivity in word learning (Markman& Wachtel, 1988; Halberda, 2003) and (b) completion ofdisjunctive syllogisms (Mody & Carey, 2016; Cesana-Arlottiet al., 2018). These inferences arise naturally from a DPPbecause a DPP selects subsets so as to maximize the volumespanned by the vectors (here, a subset of the possiblecombinations). When the similarity is defined over re-usablekeys and values, the DPP prefers combinations of previouslyunused components.

This framework does not require that the cognitivesystem have explicit representations of rules, receive directsupervision, or have mechanisms for transferring knowledgebetween domains. This puts ME biases well-within thecognitive reach of pre-verbal infants, language-learningchildren, and non-human species that may lack the relevantexperience, cortical machinery, or both necessary to representand operate over abstract logical rules (See Mody & Carey,(2016) for related discussion regarding disjunctive syllogismin young children).

Instead, we suggest that the central driver of this biasmay be the promotion of an efficient memory system.Maximizing the volume spanned by the vectors in codespace promotes memory retrieval by minimizing interference

similarity matrix.

(”collisions”). This is closely related to classic ideasregarding pattern separation in an episodic memory system(Marr, 1971; Treves & Rolls, 1994; O’Reilly & McClelland,1994): a reduction in the similarity between two statesin a function’s output relative to their similarity in theinput. Pattern separation is canonically implemented byprojecting codes into a high-dimensional space where theprobability of collisions is low. The DPP has a similarmotivation here. However, a DPP precludes the needto project into a higher-dimensionality in order to avoidcollisions, as it is able to uniquely map items to locationswith an array size equal to the number of keys (See Figure4). Thus, a DPP may better navigate the time/spacetradeoff (Sedgewick & Wayne, 2011) than the strategyof interference-reduction through dimensionality-expansionstandard in pattern separation. However, this savings instorage may come at a computational cost, as computingdeterminants has a time complexity of either O(n3) orO(!) (depending on the algorithm)4. These quantitiestherefore likely need to be approximated in order tobe implemented in a neural substrate. Approximatingthem in a biologically plausible algorithm remains atopic of ongoing work. Although pattern separation isconventionally studied in the episodic memory literature, thetheoretical points that we make throughout regarding DPPsin representation-space apply to working memory as well. Insome ways, the considerations regarding structured inferenceare more closely to tied to what is conventionally thought

4We note, however, that although such exponential (or factorial)scaling is detrimental in applied use-cases of hashing, it has anintriguing connection to some empirically observed set-size effectsin uniform domains, characterized by rapid, non-linear decrease inperformance as n grows) (Miller, 1956; Luck & Vogel, 1997). Onepossibility is that capacity limits that appear to stem from a fixednumber of ”slots” may instead owe to the computational complexityof computing (or approximating) the determinants necessary toencode unique (non-colliding) conjunctions. At the moment, thisremains speculative, however.

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of as ”working memory”, as we assume that attentionalmechanisms have already windowed into a smaller regionof the possibility space so that we can easily computethe MAP over a sub-set of the broader set of possibleitems. Better understanding how DPPs may relate tothe particular factorizations of memory systems standardin cognitive science (e.g., episodic / working / semantic),as well as specific aspects of the entorhinal/hippocampalsystem5 also remain important topics of ongoing work. Forpresent purposes, however, the central distinction in memorysystems is simply that between a stable set of re-usablerepresentations and combinations of those representations inparticular instances (key/value pairs). While re-using codespromotes generalization, it increases the risk of collisionsin the memory system. Here, we have suggested thatan algorithm that seeks to maximize the total volume ofthe constructed combinations in a representational-space(exhibiting repulsion) not only promotes efficient memoryencoding and retrieval, but may also underlie inferentialbiases toward mutual exclusivity.

AcknowledgmentsThis project / publication was made possible through thesupport of grants from the John Templeton Foundation andNIH grant T32MH065214. The opinions expressed in thispublication are those of the authors and do not necessarilyreflect the views of the John Templeton Foundation.

ReferencesBehrens, T. E., Muller, T. H., Whittington, J. C., Mark, S.,

Baram, A. B., Stachenfeld, K. L., & Kurth-Nelson, Z.(2018). What is a cognitive map? organizing knowledgefor flexible behavior. Neuron, 100(2), 490–509.

Cesana-Arlotti, N., Martı́n, A., Téglás, E., Vorobyova, L.,Cetnarski, R., & Bonatti, L. L. (2018). Precursors of logicalreasoning in preverbal human infants. Science, 359(6381),1263–1266.

Chanales, A. J., Oza, A., Favila, S. E., & Kuhl, B. A.(2017). Overlap among spatial memories triggers repulsionof hippocampal representations. Current Biology, 27(15),2307–2317.

Gillenwater, J., Kulesza, A., & Taskar, B. (2012).Near-optimal map inference for determinantal pointprocesses. In Advances in neural information processingsystems (pp. 2735–2743).

Halberda, J. (2003). The development of a word-learningstrategy. Cognition, 87(1), B23–B34.

Kulesza, A., Taskar, B., et al. (2012). Determinantal pointprocesses for machine learning. Foundations and Trends R©in Machine Learning, 5(2–3), 123–286.

Lake, B. M. (2019). Compositional generalization throughmeta sequence-to-sequence learning. In Advances in neuralinformation processing systems (pp. 9788–9798).5See Chanales et al. (2017) for intriguing fMRI evidence of

”repulsion” in hippocampal codes, in which similar states havedissimilar representations.

Lake, B. M., Linzen, T., & Baroni, M. (2019). Humanfew-shot learning of compositional instructions. arXivpreprint arXiv:1901.04587.

Lewis, M., Cristiano, V., Lake, B. M., Kwan, T., & Frank,M. C. (2020). The role of developmental changeand linguistic experience in the mutual exclusivity effect.Cognition, 198, 104191.

Luck, S. J., & Vogel, E. K. (1997). The capacity of visualworking memory for features and conjunctions. Nature,390(6657), 279–281.

Markman, E. M., & Wachtel, G. F. (1988). Children’s useof mutual exclusivity to constrain the meanings of words.Cognitive psychology, 20(2), 121–157.

Marr, D. (1971). Simple memory: A theory for archicortex.Philosophical Transactions of the Royal Society of London.Series B, Biological Sciences, 262(841), 23–81. Retrievedfrom http://www.jstor.org/stable/2417171

McClelland, J. L., McNaughton, B. L., & O’Reilly, R. C.(1995). Why there are complementary learning systemsin the hippocampus and neocortex: insights from thesuccesses and failures of connectionist models of learningand memory. Psychological review, 102(3), 419.

Merriman, W. E., Bowman, L. L., & MacWhinney, B.(1989). The mutual exclusivity bias in children’s wordlearning. Monographs of the society for research in childdevelopment, i–129.

Miller, G. A. (1956). The magical number seven, plus orminus two: Some limits on our capacity for processinginformation. Psychological review, 63(2), 81.

Mody, S., & Carey, S. (2016). The emergence of reasoningby the disjunctive syllogism in early childhood. Cognition,154, 40–48.

O’reilly, R. C., & McClelland, J. L. (1994). Hippocampalconjunctive encoding, storage, and recall: Avoiding atrade-off. Hippocampus, 4(6), 661–682.

Pepperberg, I. M., Gray, S. L., Mody, S., Cornero, F. M.,& Carey, S. (2019). Logical reasoning by a grey parrot?a case study of the disjunctive syllogism. Behaviour,156(5-8), 409–445.

Sedgewick, R., & Wayne, K. (2011). Algorithms.Addison-wesley professional.

Treves, A., & Rolls, E. T. (1994). Computational analysisof the role of the hippocampus in memory. Hippocampus,4(3), 374–391.

Whittington, J., Muller, T., Mark, S., Barry, C., & Behrens,T. (2018). Generalisation of structural knowledge in thehippocampal-entorhinal system. In Advances in neuralinformation processing systems (pp. 8484–8495).

Zeithamova, D., Schlichting, M. L., & Preston, A. R. (2012).The hippocampus and inferential reasoning: buildingmemories to navigate future decisions. Frontiers in humanneuroscience, 6, 70.

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