Beyond Boolean logic: exploring representation
languages for learning complex concepts
Steven T. Piantadosi, Joshua B. Tenenbaum, Noah D. Goodman{ piantado, jbt, ndg } @ mit.edu
MIT Department of Brain and Cognitive Sciences
43 Vassar Street, Building 46, Room 3037G, Cambridge, MA 02139
Abstract
We study concept learning for semantically-motivated, settheoretic concepts. We first present an experiment in which weshow that subjects learn concepts which cannot be represented
by a simple Boolean logic. We then present a computationalmodel which is similarly capable of learning these concepts,
and show that it provides a good fit to human learning curves.
Additionally, we compare the performance of several potential
representation languages which are richer than Boolean logic
in predicting human response distributions.
Keywords: Rule-based concept learning; probabilistic model;
semantics.

Introduction
Every cognitive theory requires a hypothesis about mental
representation—what structures and operations form the basis for complex ideas. High-level, symbolic theories often
characterize this representation using a representation language (RL) (Fodor 1975) that specifies primitive elements andcomposition laws which can be used to form complex cognitive structures. This approach has been extensively studied
within two traditions in cognitive science: concept learningand linguistic semantics. In models of rule-based concept
learning (Bruner, Goodnow, & Austin 1956, Shepard, Hovland, & Jenkins 1961, Feldman 2000) the representation language has typically been simple Boolean logic, which represents concepts—stable mental representations—using conjunctions, disjunctions, and negation of simple perceptual
primitives. Goodman et al. (2008) presented a model of probabilistic learning for rule-based concepts that represents concepts in a simple propositional language and achieves stateof-the-art fits to experimentally-measured difficulty of learning Boolean concepts. The logical complexity of concepts
appears to play a crucial role in determining how these concepts are learned: learners are biased to preferentially learn
concepts with simpler representations.
However, simple Boolean concepts can capture only a very
limited range of the human conceptual repertoire. People
readily conceptualize context-dependent meanings such as“happiest,” and can form more complex and abstract relational concepts like “everyone with two or more siblings.”
Semantic theories capture such meanings using primitive operations which manipulate and quantify over sets of objects,
rather than simply features and propositional connectives.
The denotation of a quantifier like “some,” for instance, is afunction which takes two sets, A and B, and is true only when
the intersection of A ∩ B is nonempty1 (Montague 1974).
1 In a sentence like “Some boy smiled,” the set of boys would

In this paper we extend the probabilistic approach to concept induction to representation languages which manipulate sets of objects. We first describe an experiment that
explores the difficulty of learning concepts that involve setmanipulation and quantification. Second, we compare humandifficulty to the predictions of models with varying RLs. Ourmodeling work has two goals: the first is to test different RLs
to see which provide the best account of people’s learning
behavior. Each possible RL differs in representational power
and the way in which it assigns probability to potential concepts. This means that different RLs make different predictions about people’s learning trajectories and we can therefore
compare RLs by determining how well they match subjects’
empirical response distributions. The second goal of the modeling work is to provide an explicit learning theory for theseconcepts. Work on boolean concept learning has provided aprobabilistic model which accounts for subjects’ behavior in
acquiring boolean concepts, but there is no comparable formal theory for concepts which require a richer representation
language. Such a theory would importantly extend rule-based
concept learning in cognitive science to richer, linguisticallyinteresting semantic representations.

Behavioral experiment
The experiment we present aims to extend the rule-based concept learning paradigm to concepts which refer to sets andproperties of sets of objects. To do this, we used a learning
paradigm where subjects see a set of objects, guess at a labeling of the objects according to the unknown target concept,
and receive feedback on their responses. Subjects used thisfeedback to infer the target concept.

Procedure
Amazon’s Mechanical Turk was used to run 381 subjects.
Each subject was told that they had to learn the meaning of
a novel word, wudsy, from an alien language. Subjects weretold that aliens use wudsy, to refer to some objects in a collection of objects, and that they have to figure out what makes
an object wudsy. Subjects were informed that what makes anobject wudsy may depend on which other objects are present.
During the experiment subjects were shown a set of four
objects which varied in size, color, shape, and backgroundcolor. An example set of items is shown below:
be A and the set of things which smiled would be set B. “Someboy smiled” is true if and only if the intersection of boys and things
which smiled is nonempty.

largest. As such, they require answering NA when there is nota unique largest element3 .

Results
Figure 1: An example set.
After seeing a set of objects, subjects were told to guess
which objects were the wudsy ones. For each object in theset, they were required to response “Yes,” “No,” or “NA,” andwere told to respond “NA” when it is unspecified whether an
object is wudsy. For this example, subjects might entertain
the concept is “red objects,” in which case they should respond that the second and fourth objects are in the concept
and the first and third are not. However, subjects might also
entertain that the meaning of wudsy is context-dependent, as
in, for instance “unique smallest.” Similarly, the concept may
also be complex, such as “same shape as the object with thedarkest background.” The shape with the darkest backgroundis a circle, so subjects should say all the circles are in theconcept; if all backgrounds are the same color, subjects may
respond “NA.” After responding, subjects were told what thecorrect answer was according to the target concept, but nevergiven explicit instruction on the target concept. Subjects who
responded incorrectly to any element of the set were penalized with a 5 second delay, during which they saw the set andthe correct responses for each object.

Materials & Concepts
The meanings subjects were required to learn consisted of theconcepts shown in Figure 2. These concepts include simple
boolean rule-based concepts (e.g. “circles” and “circles or
blue objects”), as well as more complex concepts which cannot be expressed in boolean logic (“larger than all the otherobjects”), and concepts which require several bound variables
to express (“Same shape as the largest blue object”).
Several of the concepts we studied focus on size predicates.
This is because size predicates, such as “largest” and “smallest,” are salient properties of objects in sets. They are perhaps the simplest words whose meaning is context-sensitive,
and therefore not expressible with only conjunctions, disjunctions, and negation of object features. We included three simple size relations, “there exists a smaller object,” “larger than
all other objects,” and “one of the largest objects.” Note thatthe latter two differ with respect to uniqueness: if there are
two objects of the maximal size, then neither is larger than allother objects, but both are one of the largest2 .
Because we included these simple size predicates, it is natural to include complex concepts which are also based onsize, such as “same shape as the largest object,” “same shapeas the largest blue object,” and “unique largest blue object.”
All three of these concepts require finding the largest object
and selecting other elements based on the properties of the
2 These concepts are interesting in part because it is unclear

which of these meanings corresponds to the denotation of “largest”
in English, and also what role pragmatics plays in understanding“largest” in normal conversation.

The plots in Figure 2 show subjects’ accuracy at labelingwhich objects are wudsy (y-axis), as a function of the amountof labeled data they received (x-axis). Subjects who weremore than 3 standard deviations below the mean accuracy
for each concept were removed in order to exclude subjects
who were not performing the task. The vertical error bars
show binomial 95% confidence intervals, and the red lines
show the best fitting model, which is discussed in the next
section. These results reveal several interesting qualitative
trends. First, subjects accuracies increase for almost all of
the concepts. Importantly, even though the subjects receive
labeled data, they are never explicitly instructed on the concept. This means that high accuracy can only be achieved bygeneralizing from the observed data, which requires inferring
abstract rules for these concepts.
Two interesting exceptions to subjects’ general ability tolearn these concepts are “Everything iff there is a triangle”
and “Everything iff there is a single blue object.” Subject performance on these concepts does not substantially improve,and these are intuitively somewhat unnatural concepts which
require all elements of a set to be selected based on what
the set contains. Words do exist with similar denotations inEnglish—for instance, a set is contaminated if one elementof the set is bad—but subjects find these types of concepts
unusually hard to learn.
Figure 2 reveals a number of places where subject performance drops temporarily for a single set–for instance at item
32 of “there exists a smaller object.” Post-hoc analysis revealed that many of these dips are caused when subjects see
one of the first exceptions to a plausible alternative concept:
item 32 is the first time that all objects in the set are the same
size. Subjects responded true to objects in this item, consistent with a concept such as “not smaller than the rest,” butincorrect according to the target concept.

Analysis
We first used a regression to analyze how subjects’ learning
rate varied across the 12 concepts studied4 . In each logistic
regression the outcome was whether the subject’s response toeach object in a set was correct, and the independent variable was the number of items each subject had seen so far(0 . . . 70). The key prediction we tested is whether slopes(regression coefficients)—which quantify the effect of additional data on accuracy—differed between concepts.
3 This makes it difficult to compare these concepts with the simple size-predicate concepts since the latter never require NA, which
may be a difficult response for subjects to learn, independent of theconcept.
4 Because subjects typically were only run on one concept, subject effects are confounded with concept. We therefore performed
a separate mixed-effect logistic regression (Gelman & Hill 2007)
within each concept including slopes and intercepts by subjects. Regression coefficients across concepts were compared using t-tests.

Figure 2: Subject accuracy (y-axis) in labeling the wudsy objects as a function of trial number (x-axis). Black lines show subject
mean percent correct. Error bars are 95% confidence intervals. The red line shows the best-fitting model, although note that themodel is fit based on agreement with the full distribution of responses, not the accuracies shown here. Numbers in the lower
right show the correlation between the model and human accuracies.
Our results replicate basic effects in Boolean concept learning (Bruner, Goodnow, & Austin 1956, Shepard, Hovland, &Jenkins 1961). As is clear from Figure 2, simple concepts(“blue objects”) are easier to learn than complex concepts(“circles and blue objects”) (t = 2.70, p < 0.01). In addition,
our results replicate that conjunctions (“circles and blue objects”) are easier to learn than disjunctions (“circles or blue
objects”) (t = 3.10, p < 0.01). These replications provide validation for our experimental paradigm.
These effects of complexity also generalized to more complex functions than those expressible in Boolean logic. For
instance, “The unique largest blue object” was easier to learn
than “The same shape as the unique largest blue object” (t =2.71, p < 0.01). This effect is interesting because it shows
the additional difficulty associated with more complex settheoretic concepts. The latter concept requires an additionalbound variable to express in first-order logic, or a lambda abstraction to express in lambda calculus, and the effect of thiscomplexity is reflected in subjects’ learning rates. The regression revealed no difference between uniqueness presuppositions for concepts involving the largest element of a set:“larger than all other objects” was no more difficult than “oneof the largest objects” in either slopes (t < 1.26, p > 0.20) orintercepts (t < 1.83, p > 0.05).
Importantly, subjects may infer a different concept fromthe one that was used to generate the data—high accuracy
on some concepts can be achieved by inferring related concepts. To address this, we compared how well closely-related
concepts predicted subjects’ responses in the last half of theexperiment. For each target concept in Table 1, we looked
at data points for which the target concept made a different prediction from the specified alternative hypothesis. For
instance, we looked at sets for which “Largest blue object”
and “blue objects” made different predictions—that is, when
there are multiple blue objects, so not all of them are thelargest. We then computed the percent of subjects who responded more than half the time in agreement with the target
concept, as well as the overall proportion of time subjects
responded with the target concept. These results show thatfor most of the concepts, subjects typically responded in accord with the target concept and not a close alternative. The
only exception to this is the comparison between “One of thelargest objects” and “size = 5” (In the items, “5” is the maximal size for objects), which showed that subjects may havebeen learning to identify objects based on comparing themto absolute size, rather than a context-sensitive measure of“largest”. In general, though, these results show that subject’s
pattern of learning cannot be explained by simpler theories
which make reference only to only individual objects’ properties. This is especially striking given that many of the alter-

TargetAlternative

Subject pct.

Response Pct.

One of the largest
Larger than all other objects objects
Size = 5Size ≥ 4
Size ≥ 3

1.00

0.94

0.81
1.00
1.00

0.67
0.82
0.88

This expression represents a function which checks whether
its argument, x, is red and a square, and returns true or false
for any object (since red and square are assumed to return
true or false). We might also have concepts which take twoarguments, a contextually-relevant set S and an element x:

One of the largest
objects
Blue Objects
Larger than all otherobjects

0.79

0.66

λS λx.(equal x (unique-smallest S))

0.79
0.79

0.74
0.63

Same shape as the uniquelargest blue object

Same shape as thelargest object

0.52

0.58

One of the largest objects

Size = 5Size ≥ 4
Size ≥ 3

0.36
1.00
1.00

0.46
0.67
0.67

Size = 5
There exists a smaller object Size ≥ 4
Size ≥ 3

1.00
1.00
1.00

0.67
0.73
0.91

Unique largest blue object

Table 1: Comparison of subject agreement with target concepts
compared to alternative concepts. Subject pct. shows the proportion of subjects agreeing more than 50% with the target concept,
Response Pct. shows the overall percent agreement with the target.

native hypotheses are much simpler than the target concepts,
and provides strong evidence that subjects are attending tomore than simple object properties.

Computational model
The behavioral experiment shows that generally subjects areable to induce these types of set-theoretic concepts from thelabeled data. Although it is important that subjects can eventually learn most of these concepts, we are also interested inwhether their learning trajectory—their guesses and hypothesized concepts at each point in time—follow sensibly fromthe observed data. It may be rational to initially learn simpler
related concepts which give approximately correct answers.
There is no guarantee, for instance, that 70 items are enoughto justify learning the correct form of the target concepts. We
next present a computational model which can learn thesetypes of set-theoretic concepts.
Our computational model aims to extend the rational rules
model of Goodman et al. (2008) to a richer hypothesis
space—one which is capable of representing these types of
set-theoretic concepts. The probabilistic structure of themodel and inference algorithm we use are neutral with respect
to the RL, meaning that any potential RL can be incorporated
and tested to see what distribution of responses it predicts foreach object in each concept.
Each potential RL L defines a hypothesis space of potential
concepts corresponding to the set of all ways to compose the
RL’s primitive functions in order to create functions which
map objects to labels (true, false, NA). For instance one concept might be5
λx.(red x) ∧ (square x)
5 We write functions as lambda expressions, meaning that thename for the argument is preceded by λ. We also use prefix notation: a function f applied to an argument x is written (f x).

This function checks if x is the object which is the unique,smallest element of S.
We use a probabilistic context-free grammar (PCFG)
which assigns probability to every possible composition ofprimitive elements. This PCFG functions as a prior over concepts and for simplicity, we assume that all PCFG expansions
are equally probable6 . In general, the PCFG assigns highprobability to short or “simple” compositions of L ’s primitives, and lower probability to complex rules. For instance,a function λx.(red x) will be higher probability a-priori thanλx.(red x) ∧ ((square x) ∨ (circle x)). This captures the notion that people should be biased to prefer simple explanations of the labeled data they observe.
The second part of the model is a likelihood function which
provides the probability of labels according to a hypothesized
RL expression. Specifically, for any composition E of primitives in L , the correct label is generated with probability α,
and a label is chosen uniformly at random with probability
1 − α. However, it is also likely that memory factors comeinto play in remembering past labeled examples. We include
this in the model by weighting the log likelihood for the n’th
data point back in time by n−β , where β > 0. As β → 0, themodel has perfect memory, and as β → ∞ the model quickly
forgets past data points. This leaves us with two unknown
free parameters: α, which controls how reliably set elements
are labeled, and β which controls how much more recent set
elements matter than past ones.
Together, the prior and likelihood specify a complete probabilistic model for any RL. Formally, we can score the probability of a hypothesized concept expression E conditioned
on a collection of example sets S with corresponding labels L
according to Bayes rule:P(E | S, L, L ) ∝ P(L | S, E)P(E | L ).

(1)

Here, P(E | L ) is the probability of E according to the PCFG
for L and P(L | S, E) scores the likelihood of the labels L
under the observed sets of objects S and hypothesized expression E. While Equation 1 scores the probability of any given
expression E, it is a complex inference problem to actually
determine what expressions are likely given the data. This
problem is difficult because the space of possible expressions
E is in principle infinite and difficult to search. We solved thisproblem using a Markov-Chain Monte-Carlo (MCMC) similar to Goodman et al (2008)’s method, which takes samples
6 Unlike the rational-rules model, we do not integrate out the

PCFG production probabilities. This is because primitives which
introduce new bound variables, such as quantifiers, make this integration difficult and potentially not analytically tractable in general.

from the posterior distribution P(E | S, L, L ). This method
takes a biased random walk around the space of hypotheses
by making local changes to hypothesized expressions E, andcan be shown to, in the limit, draw samples from the posterior distribution. We ran the MCMC algorithm for a rangeof α and β values for each amount of data, in each sequence,
conditioning on the correct, observed labels for all previous
sets in the sequence. This gives a distribution P(E | S, L, L )on expressions E in the RL L at each point during learning.
These expressions can be evaluated on the next item in order to provide a model prediction of subject’s distribution of
responses, conditioned on the observed labeled data. Thus,
the model was run conditioned on the same labeled data human subjects were given, and—just like human subjects—
was asked to make predictions about the correct labels for the
next data point. Ideally, subjects’ distribution of responses at
each point in time during learning should correspond to thepredictions of the model, conditioned on the exact same sequence of training data.
One goal of the model is to test different representational
languages to see which provide the best theory of people’s
inductive biases in learning these concepts. We computed theposterior predictive distribution of responses for each representation language L and saw which assigned the human responses highest likelihood7 . We compared four different RLs
with differing primitives and representational power:
Language
R ESPONSE - BIASEDS IMPLE - BOOLEAN
S ET- FUNCTIONSQ UANTIFIERS

Primitive Operations
true, false, undefined
and, or, not, shape, size, color,background-color, equal
contains, filter, only, uniquelargest, unique-smallest, set-oflargest, set-of-smallest, sameobject
exists, forall

Each RL is a superset of the preceding languages, except that none other than RESPONSE - BIASED contain true,
false, and na as primitives. Here, shape, color, andbackground-color are functions which extract the corresponding properties of objects. equal tests if two properties
are equal. contains returns true if a set contains an element,
filter removes all elements in a set not satisfying a predicate,and only return the only element of a set and NA if the set
has more than one element. The primitives unique-largest
and set-of-largest return the unique largest element in a set(and NA otherwise), and the set of elements for which none
are larger, respectively. same-object tests if two objects are
identical on all dimensions. exists and forall are first-order
existential and universal quantifiers.
Intuitively, the R ESPONSE - BIASED language allows learners only to infer a distribution on responses, but not give responses which depend on the current objects. This serves
7 Model predictive distributions were smoothed to give each re-sponse a minimum possible probability of 0.01, to prevent divergence.

as a baseline, and way to test if subjects are really performing the task. The SIMPLE - BOOLEAN language is one
which include basic logical operations and object properties, and implements the representational system studied mostin previous rule-based concept learning experiments. The
SET- FUNCTIONS language extends the SIMPLE - BOOLEAN
language by including primitive operation for testing if setscontain elements, extracting sets or elements with maximalor minimal properties along the size-dimension and filtering
sets by elements. The QUANTIFIERS language extends the
SET- FUNCTIONS language by incorporating quantification.

Results & Discussion
Table 2 shows the performance of these models in predicting
the human distribution of responses across the 12 concepts
studied. This shows the average log-likelihood of the human
responses for the best-fitting values of α and β within eachconcept8 . This table illustrates several key properties of the
RLs. First, the RESPONSE - BIASED model is overall the worst
predictor of human responses. This is important because itshows that subjects are performing the task, and performing
nontrivial inferences about the target concepts.
In addition, this figure shows that while SIMPLE - BOOLEAN
is a good predictor for the simple Boolean concepts, SETFUNCTIONS and QUANTIFIERS provide a better account forthe set-theoretic concepts that subjects are able to learn.
SIMPLE - BOOLEAN provides the worst account for “sameshape as the largest object” and “same shape as the uniquelargest blue object.” While subjects do not learn these concepts especially well, these results show that the SIMPLE BOOLEAN does not account well for subject responses.
Overall, the best RL is QUANTIFIERS; however, the differences between QUANTIFIERS and SET- FUNCTIONS is small.
Richer representation languages not only have the formal
power to represent the types of set-theoretic and logical concepts required by human conceptual systems, but also provide
a better account of human inductive leaning than the other
RLs considered here.
As discussed above, the black line in Figure 2 shows learning curves showing percent accuracy over time for humansubjects. This figure also shows a red line, corresponding to
the performance of the RL QUANTIFIERS for the best-fitting
α and β within each concept. We chose the best-fitting model
parameters based on which parameter values assigned highest
likelihood to the observed distribution of human responses,“true,” “false,” and “NA.” Doing this does not necessarily provide the best fit to the human learning curves in Figure 2 since
the model is not fit to human accuracy (correct/incorrect).
This means that Figure 2 shows a conservative view of theagreement between human accuracies and model accuracies.
For the concepts “circles or blue objects” and “unique largest
blue object” the model’s learning trajectory would increase
8 That is, these numbers are the total log likelihood assigned tohuman responses, divided by the number of responses. This wasnecessary for cross-concept comparison since concepts may havediffering numbers of subject responses.

Concept
Blue objects
CirclesCircles or blue objects
Circles and blue objects
Everything iff there is a triangle
There exists a smaller object
Larger than all other objects
One of the largest objects
Everything iff there is a single blue object
Same shape as the largest object
Unique largest blue object
Same shape as the unique largest blue object
Mean

R ESPONSE - BIASED
-0.66
-0.73
-0.81
-0.30
-0.80
-0.81
-0.58
-0.80
-0.85
-1.10
-1.05
-1.08
-0.797

S IMPLE - BOOLEAN
-0.18
-0.17
-0.73
-0.27
-0.73
-0.51
-0.48
-0.63
-0.78
-1.75
-1.54
-1.34
-0.760S ET- FUNCTIONS
-0.19
-0.17
-0.74
-0.27
-0.73
-0.40
-0.46
-0.28
-0.78
-0.99
-1.06
-1.06
-0.594Q UANTIFIERS
-0.18
-0.17
-0.74
-0.27
-0.73
-0.41
-0.36
-0.28
-0.78
-0.99
-1.06
-1.04
-0.584

Table 2: Model log likelihoods per response for each concept. These represent the model log likelihood assigned to human
responses, divided by the total number of responses in each concept to allow comparisons across concepts.
more for other values of α and β, and thus look more like
subject’s accuracies, but provide a less-good fit to subjects’
overall response distribution.
This figure shows good fit between the probabilistic modeland human learning. This fit appears especially remarkablefor concepts which subjects have a difficult time learning,such as “Everything iff there is a triangle.” Because subjects
do not learn this concept well, the best-fitting α is low and
β is highly negative, meaning that the model is not penalized much for incorrect answers and down-weights old data.
The model therefore responds with in simple ways, such as
always responding true, or responding true to only the triangles; subjects appear to use similar strategies, and thus bothshow similar patterns of response accuracies9
The model also shows more subtle agreement patterns withhuman subjects. First, it is capable of learning simple boolean
concepts in a way similar to humans, quickly arriving at thecorrect meaning given the training data. This is also true forconcepts like “there exists a smaller object” and the othersize-predicates. The model also matches local dips and peaks
in reasonably well. This is because the model, like people,
may temporarily be led to a concept which is not the target
concept, just as subjects (e.g. at item 32 of “there exists a
smaller object”). This provides evidence that people makethe same rational, statistical inferences given the same data.

Conclusion
While the SIMPLE - BOOLEAN RL provided a good fit to human response data in some cases, it is insufficient to represent
some of the complex concepts that subjects learned. Subjects’ability to learn these concepts was demonstrated by theirlearning curves for several context-dependent concepts. The
comparison of different RLs suggests a potentially fruitful approach to discovering the precise form of semantic representations. Recently, Pietroski et al. (2009) and Hackl (2009)
have used psychophysical measures to make inferences aboutplausible representations and computations that underlie se9 The best fitting β also shows a modest negative correlation(R = −0.55, p = 0.06, N = 12) with response accuracies over the
12 concepts, suggesting an interaction between the target concept
and the attentional or memory resources people allocate.

mantic meaning for words like “most.” Our work provides
a complementary approach to the same problem—instead of
measuring response times, we studied what RLs provide agood account of human inductive biases during learning. Thismethod may be broadly applicable to discovering the form of
semantic representations in natural language.
Of course, the RLs we study here are still incomplete withrespect to the full richness of human conceptual systems;
however, this work suggests that rule-based concept-learning
can be extended to complex concepts which can begin to approach the complexity and context-dependence observed inhuman linguistic systems. Furthermore, the model provides
one potential acquisition theory for semantic concepts. Children may learn semantic meanings like adults in our experiment did—by inducing concepts in a sufficiently-powerfulcompositional RL.

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