Modeling Semantic Cognition as Logical Dimensionality Reduction
Yarden Katz, Noah D. Goodman, Kristian Kersting, Charles Kemp, Joshua B. Tenenbaum{yarden, ndg, kersting, ckemp, jbt}@mit.edu
Department of Brain and Cognitive Sciences andComputer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Abstract
Semantic knowledge is often expressed in the form of intuitive
theories, which organize, predict and explain our observationsof the world. How are these powerful knowledge structures
represented and acquired? We present a framework, logicaldimensionality reduction, that treats theories as compressive
probabilistic models, attempting to express observed data as asample from the logical consequences of the theory’s underlying laws and a small number of core facts. By performing
Bayesian learning and inference on these models we combine
important features of more familiar connectionist and symbolic
approaches to semantic cognition: an ability to handle graded,
uncertain inferences, together with systematicity and compositionality that support appropriate inferences from sparse observations in novel contexts.

Problems of semantic cognition
A person’s store of common-sense knowledge about theworld is vast, but it is more than just a vast collection offacts. In many domains, the mind organizes what it knows
into large-scale systems, with structure at multiple levels of
abstraction. These abstract systems of representation—called
schemas, or theories—are crucial for our ability to make inferences that go beyond the sparse and noisy data of perceptual experience.
How are such powerful knowledge representations acquired, structured, and used? These problems lie at the heartof semantic cognition. Early proposals focused on symbolic
structures, as in the semantic networks of Collins and Quillian(1969) (hereafter CQ) for organizing categories of objects andtheir properties. Categories are placed in a tree-structured
taxonomy, with properties located at nodes of the tree andassumed to inherit down to all categories below them. Each
property needs to be stored only at the highest node of the tree
where it holds generically, leading to a compact encoding ofobjects’ properties and the ability to project known properties
to novel objects.
Later work emphasized the limitations of symbolic representations in handling noisy data or exceptions, and in accounting for graded effects of similarity on people’s inductive
judgments. Symbolic approaches were also criticized for notproviding a working account of how their abstract representations could be learned from experience.
An alternative approach to modeling abstract semantic
knowledge using connectionist networks emerged in the
1980’s. Hinton proposed a connectionist network that could
learn abstract systems of kinship relations (Hinton, 1986),
while RM explored a similar architecture for learning about
categories and properties (Rogers & McClelland, 2004)—
essentially the same problem that CQ treated from a symbolic

perspective. While these models can handle noise and uncertainty, they also have important limitations, complementary to those of symbolic approaches. They do not naturally
display the systematicity and compositionality that characterize people’s intuitive theories and common-sense reasoning(Fodor & Pylyshyn, 1988). This limitation shows up mostclearly when making inferences about novel entities that are
only sparsely observed. Suppose that we encounter two newkinds of organisms, tufas and kibos, and we are told that kibos
have some novel property (e.g., they have omulums). If wethen learn that tufas are kibos, it is likely that tufas also haveomulums. The CQ model makes this prediction, via property
inheritance down through the taxonomy of categories. The
RM network, however, does not automatically yield this inference, if it is trained on the two facts is a(tufa, kibo) andhas a(kibo, omulum). Because the concept ‘kibo’ plays different roles in these two propositions—it is the object of is aand the subject of has a—it is represented in different populations of units, and the effects of training on these two facts
appear in non-overlapping sets of weights. The network does
not equate ‘kibo’ in the first proposition with ‘kibo’ in thesecond proposition, and so fails to draw the obvious inference.
This example does not imply that it would be impossible todesign a connectionist semantic model whose inferences did
respect basic principles of systematicity and compositionality. Our point is only that the connectionist approach does
not naturally capture these aspects of human inference, which
are no less essential than the statistical capacities it does capture well. While it is possible that either the connectionist
approach or the structured symbolic approach could be extended in some way that makes for a satisfactory solution, ouraim here is to explore new alternatives for modeling abstract
semantic knowledge.
We describe an approach that combines valuable capacities of both traditional paradigms: an ability to represent abstract knowledge respecting systematicity and compositionality, and hence to make appropriate inferences from sparse
data in novel situations; and an ability to learn from noisy
data and generalize in graded fashion based on the statistics
of observed data. Our approach is based on a hierarchical
Bayesian model over logical representations. This is similarin spirit to proposals in inductive logic programming (Muggleton & De Raedt, 1994), although these are not typicallyformalized explicitly as hierarchical Bayesian models (for areview of this approach see, Tenenbaum, Griffiths, and Kemp
(2006)). A close relative of our approach is (Conklin & Wit-ten, 1994), where logical theories are learned using a complexity prior. However, that approach does not attempt toconstruct theories with novel unobservable predicates, whichis crucial to understanding many real-world domains and to
our work here.
Our framework can be viewed as a kind of dimensionality reduction for structured logical theories. We observe datain the form of relations and attributes over a set of objects,
and we infer a representation of the abstract structure underlying these data, expressed in terms of a subset of first-order
predicate logic. The observations are high dimensional, andthe inferred underlying abstract structure can be seen as a low
dimensional ‘space’ into which the observed objects are embedded.
While the model instantiates a structured domain theory,
it supports statistical inference via the probabilistic generative process linking the domain theory to the observed data.
Bayesian inversion of this generative model allows us to infer the low-dimensional abstract structure underlying the observed data.
We illustrate our approach on simple versions of the kinship and taxonomic categorization domains where previous
connectionist approaches have been developed. We showthat given an appropriate probabilistic domain theory, we
can make successful inferences from sparse data in novel
contexts—a setting which has proven challenging for connectionist models. The problem of learning an abstract domain
theory remains more difficult for our approach than for connecionist models, but we show that at least in some simple
cases, our hierarchical Bayesian formulation allows the correct abstract domain theory to be inferred from observations.

the theory.) Laws in our theories take the form of typed Hornclauses, commonly used in the logic and inductive logic programming literature. A proposal for Horn clauses as a psychologically plausible representation of theories is found in(Kemp, Goodman, & Tenenbaum, 2007).
A feature of our framework is that the values of derivative
relations (such as mother, in kinship) can be compressed intoa particular assignment of values for the core relations (suchas parent), via the theory’s laws. The probabilistic generative model we define favors theories that adopt as few core
relations as possible, seeking the optimal compression of thegiven set of observations.
=(a)spouseparentChristopher = Penelope Andrew = Christine
Margaret = Arthur

Colin
(b)
animalhas_wings

Jennifer = Charles

Victoria = JamesCharlotte
has_skineats
breathes

has_feathers

bird

can_fly

canaryis_a
has_a
has_fins
can_swim

ﬁsh
eagle

shark

has_gills

salmoncan_sing is_yellow has_claws is_strong can_bite is_dangerous swims_upstream is_pink

Figure 1: (a) A segment of a family tree for the kinship theory
used in Hinton (1986). (b) A Collins & Quillian-like taxonomy. Categories in bold red, properties in italics blue.

Theories: structure and form
We use the term theory formally as a specification of a set of
relations and an associated set of laws and types that govern
them. Theories are instantiated by models, or possible worlds,
corresponding to ways in which a theory may hold for a particular set of objects. For example, we can apply the theoryof kinship to reason about a set of individuals in a family, andlater apply it to an entirely different family. We will refer eachof these collections of objects that a theory can apply to, andtheir associated abstract structures, as contexts. Relations include ‘father’, ‘child’, or ‘spouse’, while a law might be that‘the child of an individual’s spouse is also their child.’ Everyfamily forms a context, and two models might be one where
Alice is the spouse of Bob, and another in which Alice is the
spouse of Carroll.
The general framework we follow is shown in Figure 2(a).
In a generative fashion, our theories produce models, each ofwhich in turn generates observations. Each theory specifies
a set of core relations, whose values are not directly observable. These are analogous to the lower-dimensional space innumerical dimensionality reduction. The laws of the theory
then relate the core relations to an observable set of derivative or observable relations (the ‘high-dimensional space’ of

(a)

(b)

Core rels: femaleC : Person × Person
spouseCF : Person × PersonchildCF : Person × Person
Types: Person
Laws:

Core rels: is aCF : Cat × Cat
has aC : Cat × Prop
Types: Cat, Prop
Laws:

female(X) ← femaleC (X)

is a(X, Y) ← is aCF (X, Y)

spouse(X, Y) ← spouseCF (X, Y)

has a(X, Y) ← has aC (X, Y)

spouse(X, Y) ← spouse(Y, X)

is a(X, Y) ← is a(X, Z) ∧ is a(Z, Y)

child(X, Y) ← childCF (X, Y)

has a(X, Y) ← is a(X, Z) ∧ has a(Z, Y)

child(X, Y) ← child(X, Z) ∧ spouse(Z, Y)
mother(X, Y) ← female(X) ∧ child(Y, X)
father(X,Y) ← ¬female(X) ∧ child(Y, X)
daughter(X, Y) ← female(X) ∧ child(X, Y)son(X, Y) ← ¬female(X) ∧ child(X,Y)
wife(X,Y) ← female(X) ∧ spouse(X,Y)
husband(X,Y) ← ¬female(X) ∧ spouse(X,Y)

Table 1: Logical representations of (a) portion of the kinshiptheory, and (b) taxonomy theory. Core relations and functions
are denoted with C and F subscripts, respectively.

A generative model for theories
In this section we expand the basic generative model (Figure2(a)), describing in more detail how theories are represented,

(a)Theory

λ

α

(b) T

θ

(d) Sample a set of positive observations, Obsk , from M :P (Obsk ) =YP (oi ∈ Obsk )

i

C

(

Obj

=

Model

i

M
Observations

Obs
Contexts

Y

K

Figure 2: (a) The generic structure of a theory learning framework based on dimensionality reduction. (b) An instantiationof the generic framework in K contexts, for theories where
the core relations are generated independently. Shaded nodes
denote observed variables for inference given an existing theory.

how a theory generates a model of the relations over a particular set of objects, and how observations are generated froma model.
Formally, a theory T is a triple hCore, Laws, τ i of core
relations1 , laws, and types, respectively. The core relations
specify an unobservable, compressed representation of thedomain, while the laws are a set of rules for recovering theobservable properties of the domain from this core representation. Because the laws determine the observable relations given the core relations, fixing the extension of all core
relations uniquely determines a model. The core relations
themselves are generated independently according to extension weights θi (each core relation Ri has its own extension
weight determining the fraction of “core facts” that are expected to be true). As a further compression, we allow thatsome core relations are functions: if R is a functional relation, then for each i, R(i, j) holds for at most one j (notethat this reduces the number of independent core facts, sincecolumns of R are no longer independent). More formally, thecore relations are generated as follows:
1. For each Ri ∈ Core, draw a θi ∼ Beta(α, β).
2. For every context k ≤ K,(a) Choose a group of objects Ot ⊆ Obj k that belong toeach type t ∈ τ .
(b) Generate the core extension Cki for Ri : for every a, b ∈Obj k , Ri holds with probability P (Ri (a, b)) = θi when
the types of a, b match the type signature of Ri (andprobability 0 otherwise).
(c) Complete the model M for Ck , by iterative applicationof every L ∈ Laws to Ck , until no additional inferences
are made.
1
For simplicity, we describe the generative process only for thecase of binary relations; similar processes describe unary or higherarity relations.

1
size



oi ∈ M,oi 6∈ M.

where size is the number of true facts in M . A small
non-zero value of  allows the theory to tolerate noise in
the observed data.
Note that while we focus here on the problem of learningfrom positive observations only, the model can easily be extended to learn from negative observations as well. Learning from positive data is often considerably more difficult
than learning from both positive and negative data, and weshow that promising generalization is possible even in thisless richly observed setting.

Model inference
Given this generative process, we can compute the probability that a query q—a logical atom, such as female(mary) or
spouse(mary, jon)—is true given a set of contexts, a theory,
and a setting of the hyperparameters α=(α, β). Summingover models of the theory (and restricting to K = 1 for clarity), we see that:
P (q | Obs, Obj, T, α) =X
P (q | M )P (M | C)P (C | Obs, Obj, T ).
C,MThe laws of T uniquely determine M given C, so the sum
over M and the term P (M | C) can be dropped. The remaining sum can be expressed via Bayes’ rule as follows:
P (q | Obs, Obj, T ) ∝X
P (q | C)P (Obs | C, T )P (C | Obj, T ).

(1)

CWhen there are few objects, the sum over the extensions of
the core relations can be computed exactly. In practice, wemust often approximate the sum using Gibbs sampling orother Monte Carlo methods; these methods can also be used
to search for single most probable model.

Inference in sparse contexts
Once a learner acquires a systematic theory, such as kinship or taxonomy (as shown in Table 1), a number of inferences about novel, sparsely observed objects can be made.
When we place probabilities over these logical representations, three general classes of inferences are possible: deductive, inductive, and deductive consequences of inductive inferences. We show examples of all three for the theories oftaxonomy and kinship.

(a) a b c
abc

motherfatherdaughter

son

wife

husband

(b)

e= f
g=h

a=b
c

d i

j

Figure 3: (a) (a . . . j × a . . . j) matrices of observable relations in the kinship domain. Black and grey entries indicate
observed and inferred relations, respectively. White entries
indicate relations inferred to be false. (b) Inferred family tree,compactly representing all inferences in (a). Females shownin bold italics, males in ordinary font (c’s gender is unknown,
indicated by a circle.)
Kinship Consider the family relations shown in Figure 3.
In light of the logical theory of kinship, many inferences
could be made. For example, the gender of the family members can be deduced with certainty, since mothers are always
female, sons are always males, and so forth.
In addition to deductive inferences, the observations invite
several plausible inductive inferences. Since a and b are bothobserved to be parents of c, it is plausible that a is the spouseof b. In light of this inductive inference, one can infer—byapplication the logical laws of kinship—that d is the daughterof b, since d was already observed to be the daughter of a.
This is an example of a deductive consequence of an inductive
inference. Note how our initial inductive leap, that a and b are
married, led to a deductive inference that allowed us to makeefficient use of our observations. The best model found forthis context via greedy stochastic search contains all of these
inferences, deductive and inductive. The core relations forthis best-scoring model correspond to the family tree shownin Figure 3(b); the observable relations that it predicts will betrue (or false) are indicated by the squares colored gray (orwhite, respectively) in Figure 3(a).
Essential to these inferences is the fact that the identity ofobjects remains the same even if they play different ‘roles’
in distinct scenarios. In our observations, we see two roles
for a: one in which it is the mother of c (as first argument to
the mother predicate), and another in which it is the parentof d (as the the second argument to daughter.) The effective
integration of information from both the deductive and inductive inferences just shown relies heavily on the identity of ain these two distinct roles.
Taxonomy Consider the full taxonomy given in Figure1(b). Suppose that we observe only the is a links correspond-

ing to direct edges in the hierarchy, along with the properties
true of the leaf-node categories. For instance, we observethat canaries (a leaf-node category) can sing, are yellow, havewings, and have skin, that eagles also have wings and skin,and that canaries and eagles are both birds and are animals(along with many other facts). We make no direct observations about the properties of birds, fish or animals, though.
If we then search for the best-scoring model, we recover theconfiguration of core relations shown in Figure 1(b): the extension of is aCF and has aC includes only the minimal setof is a and has a links needed to capture all observations under the theory’s laws. Each property is attached to only one
category in the is a hierarchy, the lowest superordinate of all
categories with that property. We compress out the correlations in the observed properties of leaf-node categories, by
positing properties true of abstract superordinates which are
not themselves ever directly observed.
Now suppose that we learn of two new objects, a and b,
by making the following minimal observations: is a(a, fish),is a(b, animal). The classic CQ approach would infer many
common-sense inferences about a, b, e.g. that both breathe,that a can swim, and so on. Our best-scoring model does
as well. Conditioned on the inferences described in the previous paragraph, that familiar properties like swimming andbreathing are probably true of the unobserved superordinate
categories fish and animal, we now infer that these properties
also hold for the new species a and b, for which we have observed only a single is a relation each. This example shows
how we capture a general feature of common-sense reasoningby combining the power of induction and deduction: an inductive leap to the likely properties of unobserved superordinate categories, with deductive inference of the consequencesthat follow.
Property induction We now show how intuitive patterns ofgraded, uncertain inference, usually thought to weigh against
symbolic representations of human semantic knowledge, canalso be captured in our probabilistic logical framework. We
consider a simple case of property induction. For tractability
and ease of presentation, we explore these phenomena using a pared-down version of the Collins & Quillian example,
shown in Figure 4(a). We call this structure a balanced taxonomy, where both properties and nodes are distributed evenly
along all branches of the tree. In this case, each node hasa single unique property, in addition to properties it inherits
from its parent nodes. For instance, a’s unique property is p3,
while f’s unique property is p5, and both inherit p1 from theirparent node g.
Given observations of direct is a links and properties ofleaf node category, we can then query for the probability thateach property is true of each category in the hierarchy. Figure 4(c) shows the probabilities of all queries for the property
p3, which is observed only at one leaf node category, a. As
expected, the probability that a has p3 is near certain. However, other less certain generalizations are found. It is plausible but not likely that e has p3, and very unlikely that g or f

(a)

is_a
has_a

p1
g

p2

p5
f

e

# Errors(b)

Probability

(c)

a

b

c

d

p3

p4

p6

p7

10
5
0

100%

80%

70%

60%

1

0.5
0

3)
3)
3)
3) 3)
3)
3)
a, p b, p c, p d, p e, p (f, p g, p_a( s_a( as_a( s_a( as_a( as_a s_a(
s
a
a
a
a
h
h
h
h
h
h
h

Figure 4: (a) Balanced taxonomy. (b) Simulations of CQ contexts with different sparsity levels. (c) The probabilities ofvarious queries under full observations of leaf node properties. Results obtained by Gibbs sampling core extensions of
the theory (using 6000 samples.)1
has it. This is explained by the size
factor in our likelihood.
If g had p3, all nodes in the hierarchy would inherit p3, making the fact of our first and only observation of the property
at one specific node a relatively less likely. The same size
principle weighs against generalization to e, but less so, since
e is not so far up in the taxonomy. We also see a pattern ofsimilarity-based generalization along the leaf node categoriesof the is a hierarchy. The closer a category is in the tree to a,
the more likely it is to have p3.

Systematic study of generalization
We now study more systematically how well our approach
generalizes beyond the observed data, as a function of thesparsity of the data it receives. Inspired by the simulations ofHinton (1986) for the theory of kinship, we ran simulations inwhich we observed randomly sampled sets of the observablefacts in the balanced taxonomy domain and searched for thebest scoring model under the taxonomic theory. In this domain there are a total of 98 observable propositions (7 × 7 =49 is a propositions, and the same number of has a propositions), of which 27 are true and 71 are false. For instance,
has a(a, p2) is true, while has a(b, p5) is false. In keepingwith our focus on learning from positive examples, we observed a fraction of the true observable propositions—100%,80%, 70%, or 60% of the 27 possible positive examples—andsearched for the best scoring model under the taxonomic theory. Rather than merely memorizing the given true facts, weseek to compress the data via inferring the underlying taxonomy and then predict the unobserved propositions this modelentails.
We evaluated performance by computing the number ofincorrect observable propositions entailed by the recovered
model. This number includes two types of errors: “false
alarms” (false propositions predicted to be true) and “misses”(true propositions predicted to be false). Figure 4(b) shows
the results, averaged across five trials at every level of sparsity. Generalization is perfect when all 27 true propositions
are observed. This means that none of the 71 false propositions were incorrectly predicted to be true. Generalization
decreases gradually as the data become sparser, with errors
distributed among both misses and false alarms. Even at thehighest levels of sparsity, generalization is quite good: weobserve only 16 (≈ 60% of 27) of the 98 observable propositions and we make about 7 errors on average, inferring thecorrect truth values for (on average) 91/98 observables.

Learning a logical theory
We now address the problem of learning the theories we’ve
described. We do so by defining a prior distribution P (T )over theories. Following Kemp et. al., we take a representation length (RL) approach . Intuitively, given the choice
between two theories, a RL prior will favor the one that isless complicated to write. The precise definition of RL is tied
to a choice of language, which in our case is the language ofHorn clauses.
As before, the distribution is described as a generative process. Given an assignment of values to the hyperparametersα, λ, theories are generated as follows:
1. Generate a number γ of core relations in T : γ ∼Poisson(λ).
(a) For every generated core relation R, choose its arity,
where P (R is binary) ∼ Bern(θa ) and whether it isfunctional, P (R is functional) ∼ Bern(θf ). We assume
θa , θf are given.
2. Draw a set of laws, scored according to their RL. The RL
in our case is a count of the number of total predicates,
variables and clauses that appear in the laws:
P (Laws) ∝ 2−rl(Laws)
where rl(Laws) = #cp + #vars + #clauses . We assume
every L ∈ Laws is syntactically well-formed (otherwise,
P (Laws) = 0.)
We now consider a case of competing theories, by evaluating the true taxonomy theory discussed earlier against
seven variants, shown in Table 2. These were generated
by considering all theories that can be constructed by inclusion/omission of the following features: (1) the is a transitivity law (L1), (2) the property has a inheritance law (L2),and (3) having the is a relation be a function. L1 and L2
are the third and fourth laws in Table 1(b), respectively. To
see which theory is favored on a given data set, we firstlook for the best scoring model of each theory using greedy

L1
L2
is a func.
log score

T1
√
√
√

T2
√
√

−150

−180

×

T3
×
√
√

T4
×
√

−163

−185

×

T5
√

T6
√

×
√

×
×

−171

−200

T7
×
×
√

T8
×
×
×

−164

−206

Table 2: Log scores for true taxonomy theory (shown in bold)
and its variants.
stochastic search. We then score each theory together withits best model according to the joint posterior probability
P (T, M, C | Obs, Obj, λ, α). Table 2 shows the log scoresof the eight theories, taking as data all true observable propositions in the balanced taxonomy domain.
Note that all the variants of the true theory are favored byour prior, as they are less complex. We can also see that thescore ordering for these theories is sometimes structured andmonotonically decreasing as we go from more to less complex theories. While the prior favors simplicity, the posterior
ought to favor the more complex of the variants we consider,since these have greater predictive power. For example, T2
is penalized for lacking is a as a function, and T3 for lacking
L1, but T4 is penalized more heavily than either for lacking
both.
Similarly, theories having is a as a function are preferredto those that do not. A learner who believes is a is only arelation and not a function might believe that a shark is botha fish and a bird, leading to a penalty in the likelihood of ourmodel. Fixing is a as a function rules out these cases and sogives a more compact encoding of the observations.
This monotonicity property does not hold of all theories(compare T3, T5, and T7.) This can be explained by the factthat L1 and L2 are not independent. A learner who does notknow about the inheritance of properties (T7) would gain little by adopting L1, in the current data set, and vice versa,
depending on the statistics of the observations.
Our purpose was to demonstrate that our generative model
can be used as a evaluation metric for theories. We leave
open the orthogonal question of how actual learners selecthypotheses to evaluate from this vast space. Though we useda stochastic search for hypothesis selection here, more work
is needed to see if any search-based account could provide
a solid foundation for theory learning. For any search-based
proposal to work, the probability landscape of theories musthave tractable properties, such as the monotonicity property
we considered above. An area of future work is determining
when these conditions hold for theories in general.

Conclusions and future work
In a classic paper on distributed representations, Hinton outlined three questions to ask of all systems of knowledge representation: (1) Does the representation create “sensible internal representations” for the entities it processes, in a waythat is sensitive to the (potentially latent) regularities in its input? (2) Does it generalize to unobserved truths of a domain?
And finally, (3) Does it compress and generalize informationfrom distinct but isomorphic context?
We believe that our proposed framework fares well on all
three. First, the use of a logical framework makes our representation transparent, allowing for direct and interpretablecomparisons of the internal structure of two learned representations. It is not obvious how to make such comparisonsin a connectionist setting. At the same time, Bayesian inference over these structured representations allows us to makeinductive leaps from sparse and noisy data, overcoming a major flaw of traditional symbolic approaches. An advantage ofour framework is that the engine of deduction (logical representations), and the engine of induction (Bayesian inference),
cooperate and supplement each other to reason in sparse contexts. We have shown the benefits of this approach for thetheories of kinship and taxonomy.
In future work, we would like to develop more scalable inference algorithms and a more expressive language, allowing
unobserved entities and probabilistic generative mechanisms.
This should let us explore more realistic theories, such as the
dynamic theory of Mendelian genetics. A second direction is
to study how well our approach captures people’s reasoning
in the laboratory, following (Kemp, Goodman, & Tenenbaum,2008).
Acknowledgements We thank Vikash Mansinghka, Brian
Milch, and Daniel Roy for helpful discussions. This work was
funded in part by AFOSR grant FA9550-07-1-0075 and the James
S. McDonnell Foundation Causal Learning Research Collaborative.

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