A Bayesian Model of the Acquisition of Compositional Semantics
Steven T. Piantadosi, Noah D. Goodman, Benjamin A. Ellis, Joshua B. Tenenbaum{ piantado, ndg, benjelli, jbt } @ mit.edu
MIT Department of Brain and Cognitive Sciences
43 Vassar Street, Building 46, Room 3037, Cambridge, MA 02139
Abstract
We present an unsupervised, cross-situational Bayesian learning model for the acquisition of compositional semantics. We
show that the model acquires the correct grammar for a toy
version of English using a psychologically-plausible amountof data, over a wide range of possible learning environments.
By assuming that speakers typically produce sentences which
are true in the world, the model learns the semantic representation of content and function words, using only positive evidence in the form of sentences and world contexts. We argue that the model can adequately solve both the problem ofreferential uncertainty and the subset problem in this domain,and show that the model makes mistakes analogous to thosemade by children. Keywords: Compositional semantics; language acquisition; Bayesian learning; Combinatory Categorial
GrammarIntroduction
Language is perhaps the only communicative system in nature which compositionally builds structured meanings fromsmaller pieces; this compositionality is the cognitive mechanism that allows for what Humboldt called language’s “infinite use of finite means.” A child acquiring language istherefore faced with learning both which words correspondto which objects and actions in the world (lexical semantics),
and also how words can be pieced together to convey complex
meanings (compositional semantics). This includes learning
the formal semantic characteristics of content words that allow them to be used compositionally, and also inferring how
function words transform abstract representations of meaning. For example, the word “every” in the sentence “Every
person laughed” has an abstract meaning which quantifies the
proposition LAUGHED(x) over all people x. Because function words do not generally refer to entities in the world, butrather represent abstract syntactic and semantic operations,they provide an especially interesting learning problem.
Though computational models exist for many problems inlanguage acquisition, the area of compositional semantics has
been relatively unstudied. We present a novel, unsupervised,cross-situational learning model which is capable of learning compositional semantics in a relatively unconstrained hypothesis space. The model uses positive evidence in the formof sentences and contextual information from world environments to learn mappings from words to pairs of Combinatory
Categorial Grammar (CCG) syntactic classes and lambda expressions. Together, the syntactic and semantic forms allow
the model to compositionally build the meaning of sentences.
The model contains an inductive bias to prefer grammars withshorter description lengths, and posits that the utterances ithears are typically true in its current context. We observe thatthe model faces two key problems in this domain: the prob-

lem of referential uncertainty and the subset problem. The
problem of referential uncertainty concerns the fact that any
sentence can potentially refer to an infinity of possible actions, objects, and logical relations. The subset problem isthat learners may consider an under-restrictive grammar to becorrect. If this happens, they will receive no direct evidence
that contradicts their grammar, and may not acquire the correct one. We show that the model can solve these problems
using a Bayesian cross-situational learning algorithm.
Language acquisition is a complex and multi-faceted problem, and for reasons of computational tractability, we focus
only on compositional semantics. Thus, the model assumes
that lexical semantics has already been acquired—it is given
a mapping from words to objects or actions in the world.
The model learns the semantic structures that govern how
these words can combine in a toy version of English which
includes nouns, quantifiers, transitive verbs, and intransitive
verbs. The model also learns the semantic representations ofwords such as “every” which do not correspond to objects
in the world, but provide consistent abstract operations onmeanings. Focusing on only this dimension of the language
acquisition problem allows significant simplification conceptually; however, this model could in principle be combinedwith similar models for learning lexical semantics from context (Frank, Goodman and Tenenbaum 2007). More generally, it could be combined with similar models for tasks suchas concept learning (Goodman et al. 2008), word segmentation (Goldwater, Griffiths and Johnson 2006) and syntactic
learning (Klein and Manning 2005; Perfors, Tenenbaum andRegier 2006) to provide a comprehensive statistical model oflanguage acquisition.
In addition, the model accounts for quantifier spreading,a well-studied phenomenon in the quantifier acquisition. Inhelder and Piaget (1964) observed that children up to ages 4
and 5 often answer “no” to the question “Is every man wearing a hat?” in a situation in which every man is wearing ahat, but one hat is not being worn by anyone. A variety of explanations have been offered for this effect, children’s misunderstanding of part-whole relationships (Inhelder and Piaget
1964), non-adult linguistic representations (Philip 1995), andpragmatic difficulties (Crain 2000). We show that our learning model makes analogous mistakes in learning, providing astatistical learning account of the effect.

Previous research in compositional semantics
learning
Models of compositional semantics acquisition have previously been proposed in natural language processing (Zettle-

moyer and Collins 2005, Wong and Moony 2007), robotics(Sugita and Tani 2003), and the description of visual scenes(Roy 2002). In general, these models would provide unsatisfactory psychological models due to their reliance on supervised learning, or implausible linguistic representations.
Siskind (1996) proposed a cross-situational word learning algorithm as a psychological model, which employed an elementary compositional semantics. He showed how crosssituational inference can solve the problems of referential uncertainty, homonomy and noisy input.
We try to address several shortcomings of Siskind’s model.
First, Siskind’s model treats sentences as unordered sets, and
it is not clear how to extend his results to fit with contemporary syntactic theories. In contrast, this paper studies
mappings from lexical items to lambda calculus expressions,which interface with many syntactic theories, including generative grammar and CCG (Steedman 2001). Siskind does
not explicitly address the subset problem, and it is not clearthat if his algorithm were extended to a learning domain forwhich the subset problem was a serious issue—such as thelearning of quantifiers—it would perform adequately. In addition, Siskind’s algorithm follows a set of seemingly ad hoc
rules, which makes it difficult to extract specific learning principles which can be used to extend the work and combine itwith other learning theories. In contrast, this paper does not
attempt to provide an algorithmic theory, but rather shows
how the compositional semantics can be acquired by any algorithm which solves the statistical problem we formalize below.

Preliminaries
We choose to formalize syntax with CCG and semantics withthe lambda calculus. These are used because CCG is a lexicalized syntactic formalism which easily interfaces with thelambda calculus. A lexicalized grammar is desireable because it simplifies the search problem: to find the best grammar, we must search over mappings of words to CCG syntactic types, and not separately learn rules of the grammar.
While CCG generally makes use of type-raising rules, weuse a simplified CCG grammar which does not require typeraising. Before introducing the model, we present the simplified versions of these formalisms used in the model.
A wide range of linguistic phenomena have been modeledusing CCG, including coordination, WH-questions, prosody,and crossed-serial dependencies (Steedman 2001). In CCG,
the syntactic category of each word specifies what syntactic
types it requires to the left or to the right, and which syntactic
type results from combining a word with its neighbor. For example, the word “loved” may be of type (S\NP)/NP, where
the outermost “/” denotes that “loved” requires an NP to theright and returns an element of type S\NP. The type S\NP requires an NP to the left and returns a type S, for “sentence.” Inthese expressions, “/” denotes a requirement to the right, and“\” denotes a requirement to the left. Thus, if “Mark” and“Jenn” are of type NP, the sentence “Mark loved Jenn” canbe represented syntactically by NP (S\NP)/NP NP. First,(S\NP)/NP and the final NP combine, giving NP S\NP.
Then, the first NP combines with S\NP, giving a syntactic
type S.
Many semantic formalisms—including CCG—represent
word meanings as functions and constants. Semantic representations are well-integrated with the CCG syntax: anytime two syntactic elements combine, their respective semantic functions are combined with functional composition.
Thus, the syntax determines the order of composition1 . The
lambda calculus provides a convenient formalism for representing functions, and was originally studied in the context
of computability and logic. In extremely simplified systems,
for example, a verb such as “loved” may be mapped to thelambda expression
λxλy.LOV ED(y, x).

(1)

Informally, (1) can be interpreted as a function which takes
two arguments, x and y, and returns the logical expression
LOV ED(y, x). One can tell that x and y are arguments since
they are each preceded by a λ.
The words “Mark” and “Sallie” may be mapped to the logical atoms MARK and SALLIE respectively. To arrive at thesemantics of a sentence such as “Mark loved Sallie” we firstcompose the lambda expressions for “loved” and “Sallie,”giving λy.LOV ED(y, SALLIE). The result is then composedwith the lambda expression for “Mark,” giving the logical expression LOV ED(MARK, SALLIE), which is intended to be
the logical representation of the meaning of “Mark loved Sallie.”
In more complex linguistic constructions, the arguments toa lambda expression may be other lambda expressions. For
example, in the sentence “Every man loved Sallie,” a simplified analysis would map “man” to λy.MAN(y), and “every”
toλx.λy.∀z.x(z) → y(z).
(2)
Thus, “every” is mapped to a lambda expression that takes
two arguments, x and y, both of which are other lambda
expressions. It returns the expression ∀z.x(z) → y(z),where x(z) represents the lambda expression x applied toz, and y(z) represents the lambda expression y applied toz. Thus, in the sentence “Every man loved Sallie,” shownin Figure 1, we first compose “Every” and “man” to get
λy.∀z.MAN(s) → y(z). We also compose “loved” and “Sallie” to get λy.LOV ED(y, SALLIE). These expressions arethen composed to give∀z.MAN(z) → LOV ED(z, SALLIE).

(3)This expression represents the meaning of the sentence in that(3) is true if and only if the sentence “Every man laughed.” istrue. In this way, CCG and the lambda calculus can be used todefine a first-order logical expression for any sentence, whichthen can be evaluated as true or false in a world context.
1 Many semantic formalisms also use a type-system for this purpose.

Figure 1: An example derivation showing syntactic CCG types and lambda calculus expressions combining to form a logicalrepresentation of the sentence “Every man loved Sallie.”

The model

Problems in compositional semantics learning
As the amount of debate concerning the semantic structure ofcommon words such as the definite article shows, the compositional semantic structure of language is neither obvious nor
trivial. In addition to the sheer complexity of semantic representations children apparently acquire, learners also face impoverished input and a potentially infinite hypothesis space.
Two problems in particular pose a substantial challenge tolearners.
First, the problem of referential uncertainty—sometimes
called the gavagai problem—concerns the fact that there isa large or potentially infinite set of meanings for each utterance a child hears (Quine 1960, Siskind 1996). Not only caneach word potentially pick out an infinite set of objects inthe world, each word may represent one a large set of possible logical meanings, or lambda expressions. For example,
even if the child knows lexical semantics in that they know
the word “loved” is some expression involving the logical orconceptual propositional LOV ED, there are still many potential alternatives to the correct semantics (1) above. For example, the child might potentially consider expressions such as
λxλy.LOV ED(x, y) or λxλy.∀zLOV ED(x, z) ↔ LOV ED(y, z).
A well-formulated learning model must show how children
could use observed sentences and contextual information tosolve this problem.
Another key problem faced in compositional semantics
learning is a variant of the subset problem. Like some models of grammar learning (Berwick 1985, Manzini and Wexler
1987) and phonological learning (Hayes 2004, Prince andTesar 2004), it is possible that a learner would incorrectly infer an under-constrained hypothesis to be correct. For example, a possible logical representation of the sentence “Everyman loved Sallie” is∃z.MAN(z) → LOV ED(z, SALLIE).

(4)

Note that the correct semantics (3) is logically stronger than
(4). Therefore, a naive model which assumes the learnerchooses a compositional semantics which maps sentences tomeanings which are most often true would incorrectly choose
a grammar which gives the sentence meaning (4). The learnermust therefore have some mechanism to prevent them fromlearning grammars which are logically too weak.

We assume the task of the learner is to find the “best” way tomap sentences that they hear into first-order logical forms.
This mapping is formalized using a grammar G, which isa function from words to pairs of CCG syntactic types andlambda expressions. Once the learner has a CCG syntactic
type and lambda expression for each word, we assume thatthey follow the syntactic and semantic composition rules toarrive at the representations of sentences in first-order logic.
In addition, the model assumes that the sentences are utteredin world contexts and that the truth value of each logical expression can be evaluated in the world context.
Formally, let A1 , A2 , . . . be sets of sentences that are uttered in their respective world contexts, C1 ,C2 , . . .. Here, Ci
is taken to be a logical model of the world. We construct
the Ci by defining truth values for all atomic propositions
such as LOV ED(BEN, JENN) and CRIED(SALLIE), whichthen recursively defines truth values for combinations of these
propositions with logical connectives and quantifiers. Like
Siskind (1996), we assume that Quine (1960)’s problem is atleast mitigated by some perceptual or conceptual system thatnarrows down the range of possible meanings to a salient set.
The task of the learner is then to find a mapping G fromwords to pairs of lambda expressions and syntactic types thatmaximizes P(G | C1 ,C2 , . . . , A1 , A2 , . . .). We assume that eachsentence is generated independently given a context and that
contexts are independent of the grammar G. By Bayes rules,
we therefore have,P(G | C1 ,C2 , . . . , A1 , A2 , . . .) ∝ P(A1 , . . . | C1 , . . . , G)P(G)= P(G) ∏ P(Ai | Ci , G)i

= P(G) ∏ ∏ P(a | Ci , G)i a∈Ai(5)
We choose simple forms for both the likelihood P(a | Ci , G)
and the prior P(G). Namely, we assume that with probability
α, the speaker chooses a true sentence uniformly at random,
and with probability (1 − α) the speaker chooses a false sentence uniformly at random. Thus we have(α
if a is truei(6)p(a | Ci , G) = T1−αFi if a is false
where Ti is the number of sentences which are true in context
Ci according to G and Fi is the number that are false. For theTable 1: Grammar for the target language
Wordlaughed, cried
loved, hated
Mark, Ben, Sallie, Jennperson, man, woman
every
some

Syntactic type

Semantics

S\NP

λx.LAUGHED(x)(S\NP)/NP

λxλy.LOV ED(y,x)

NP

MARK

NP

λx.PERSON(x)(S/(S\NP))/NP

λx.λy∀z.x(z)→y(z)(S/(S\NP))/NP

λx.λy∃z.x(z)∧y(z)simulations here, we fix α = 0.9.
The prior P(G) is defined to penalize complex grammars,
which we take to be grammars which map words to lambda
expressions containing many logical elements such as quantifiers, lambdas, atomic propositions, and logical connectives.
In this way, the prior can be seen as a Minimum Description Length prior which penalizes grammars whose semantics require long expressions in the lambda calculus. Thus, if
Cplx(l) is the number of logical elements in the expressionsl, and GL is a mapping from words to lambda expressions, wedefineP(G) ∝ exp(− ∑ Cplx(GL (w))),w∈W

where W is the set of words in the language.
In addition, the model assumes that words of the same classare mapped to “similar” lambda expressions. That is, if thelambda expression for “loved” is λxλy.LOV ED(y, x), then thelambda expression for “hated” must be λxλy.HAT ED(y, x),
where the only change is that LOV ED has been replaced by
HAT ED. This significantly reduces the size of the hypothesis space that needs to be considered and is generally true insemantic theories (Steedman 2001). Moreover, these wordclasses are plausibly learned by other means, such as distributional analysis (Cartwright and Brent 1996; Redington,
Chater and Finch 1998; Mintz, Newport and Bever 1998).
Thus, in order to maximize (5), the learner must find a“simple” grammar which assigns high likelihood to the sentences heard. To solve this computational problem, we employ a backtracking search that finds the exact maximum forthe hypothesis space defined above2 . We prune the search by
not considering grammars that give malformed syntactic orsemantic expressions, thereby assuming that the learner only
considers grammars which can perfectly parse all of the sentences heard. This assumption is not realistic, but simplifies
the search problem considerably.

The target grammar and model implementation
We work with a toy version of English which, though simple, still faces the subset problem and problem of referential
uncertainty discussed above. The target grammar is shownin Table 1 and possible sentences are of the following forms:
intransitives (“Mark smiled”), transitives (“Mark loved Sallie”), subject-quantified intransitives (“Every boy smiled”),
and subject-quantified transitives (“Every boy loved Sallie”).
2 We choose at random when grammars tie.

Figure 2: Probability of learning a grammar which gives thecorrect meaning for each sentence type, plotted against the
number of sentences of that type seen in the input (K = 0.15).
As above, we assume that words in each row of Table 1 aremapped to “similar” lambda expressions. In general the number of possible grammars grows exponentially in the numberof words; therefore for computational tractability we makeseveral simplifying assumptions in the model implementation:
The lexical semantics is already known by the model. We
take this to mean that the model knows the lambda expression
for a word such as “loved” may or may not involve the logical element LOV ED, but definitely will not involve the logical elements HAT ED or CRIED. Again, we believe this is areasonable assumption because other methods such as Frank,
Goodman, and Tenenbaum (2007) have shown that lexical semantics can be learned by a similar model. Though this mitigates the problem of referential uncertainty for the lexicon,
there is still considerable uncertainty about what logical relations in the world sentences refer to.
Sentences without quantifiers are learned before sentenceswith quantifiers. This considerably shrinks the hypothesis
space: we first search over lambda expressions and CCG
syntactic types for proper nouns (“Mark,” “Ben,” “Sallie,”“Jenn”), intransitives (“laughed” and “cried”) and transitives(“loved” and “hated”). We then fix these and search overlambda expressions for lambda expressions and CCG syntactic types for common nouns (“Person,” “man,” “woman”),“every,” and “some.” This assumption is empirically justified
in that children do not typically learn quantifiers until long after they have learned simple verbs (Inhelder and Piaget 1964).
The space of lambda expressions considered by the model
is finite but large. This spaces includes all expressions withless than or equal to two function applications, one logicalconnective (→, ↔, ∨, ∧, ¬), one quantifier, and two boundlambda variables. In addition, we removed lambda expressions which contained unused variables bound by quantifiersor lambdas, resulting in a total of 2657 possible lambda expressions for each word.
However, the computational task faced by this model is still
formidable. For both sentences with quantifiers and sentences
without quantifiers, the model considers over a trillion poten-

Figure 3: Probability of learning
the correct mapping fromsentences to logical forms for
non-quantified sentences andquantified sentences for various
K given 10 sentences percontext. Each point represents
the an average of 50 runs of themodel.
tial grammars, many of which are eliminated by backtracking. As far as we know, such as large hypothesis space has
not been considered previously in this domain, and we showthat even an unsupervised model is capable of learning thecorrect grammar. We expect that the model could be scaled
up to larger grammars by using appropriate algorithms.

Results and discussion
The model we present learns compositional semantics in theform of lambda expressions and CCG syntactic types, which
are used to map sentences to first-order logical expressions.
The model assumes that the sentences are typically true inthe world context, and attempts to find a simple grammar
which assigns the observed data high likelihood. However,
because we know of no data set which gives both the contexts and sentences that occur in them, we created such adata set ourselves. One free parameter, K, governs what proportion of atomic propositions such as SMILED(BEN) andLOV ED(JENN, MARK) are true on average in each context.
Intuitively, acquisition will be difficult or impossible when K
is close to 0 or 1, since in these instances the world context
will not be informative about the meanings of sentences.
Figure 2 shows how often the model acquires the correct
semantic mapping for sentences with transitive verbs, intransitive verbs, “every” and “some” as a function of how many
sentences it is given containing each of these words. Note thattransitive and intransitive verbs are acquired very quickly—
often after only a few sentences containing them. This is because their semantics are short and therefore favored by theprior. The correct semantics for sentences containing “every”
and “some” is also learned quickly, typically after about 20
sentences. Interestingly, though, the semantics of “every” islearned less quickly than “some.”
Though it is not clear what an ecologically valid value for
K is, we show that the model can acquire the correct semantics over a wide range of values for K. Figure 3 shows theprobability that the model acquires a mapping which gives thecorrect meaning, as a function of K and the number of contexts the model has seen. Note that the model quickly learns
to map sentences to the correct logical forms for a wide rangeof K values—often after 1 context for transitive and intransitive verbs, or 50 contexts for quantifiers. It has also been suggested that rarity of true propositions (corresponding to smallK) is an important property in reasoning and learning (Oaksford and Chater 1994). Indeed, Figure 3 shows an asymmetry,learning more quickly when K is small than when K is large.
We also analyzed the mistakes that the model makes late inacquisition. Table 2 shows probability with which the model
assigned the sentence “Every woman laughed” various semantic values, for K = 0.1, 0.8, 0.9 after 50 contexts3 . For
other values of K the model always learned the correct semantics. We note that for K = 0.1, the only mistake the models makes is to interpret the sentence “Every woman laughed”
as being true if and only if all women laughed, and the only
people who laugh are women. This is analogous to the mistake children make in quantifier spreading. From a learning perspective, this mistake is reasonable because the biconditional and implication operators are logically very similar, and therefore require specific and potentially rare kindsof data to distinguish. For K = 0.8 and 0.9, the model oftenchooses ∀z.LAUGHED(z), ignoring the restriction to women.
Similar mistakes have been reported in children (Kang 1999),
but it is likely that typical K values are much lower (Oaksford
and Chater 1994)“Every woman laughed.”∀z.WOMAN(z)→LAUGHED(z)∀z.WOMAN(z)↔LAUGHED(z)∀z.LAUGHED(z)∀z.LAUGHED(z)∨WOMAN(z)

0.1
0.74
0.26
0
0

K
0.8
0.98
0
0.02
0

0.9
0.62
0
0.36
0.02

Table 2: Example mistakes made in learning
As discussed above, unsupervised learners of compositional semantics face two primary problems: the problem ofreferential uncertainty, and the subset problem. In this model,
the cross-situational aspect of the learning allows the problem
of referential uncertainty to be solved. The optimal grammar is the one which performs best across a wide range ofcontexts, so even though individual sentences could refer to arange of possible logical relations, the model chooses a grammar which consistently produces highly-probable meanings
across all contexts it sees.
The subset problem is solved in this model by the size principle (Tenenbaum 1999). Because the model assumes that
3 These probabilities are the average of 50 model runs.

(a)
(b)
(c)

Rank
1
2
731

Log Posterior
-1716
-1723
-2035

LL
-1698
-1705
-2017

Log Prior
-18
-18
-18

“man”
λx.MAN(x)λx.MAN(x)λx.MAN(x)“every”
λxλy∀z.x(z) → y(z)λxλy∀z.y(z) ↔ x(z)λxλy∃z.x(z) → y(z)“some”
λxλy∃z.y(z) ∧ x(z)λxλy∃z.y(z) ∧ x(z)λxλy∃z.y(z) ∧ x(z)Table 3: Key semantic values for K = 0.15 with 50 contexts. These grammars all have the correct CCG syntactic types.

true sentences are chosen from a probability distribution, it
will be penalized if the grammar maps sentences to logicalforms which are true too often4 . That is, the likelihood of
all sentences will be lowered if the model posits incorrectlythat some sentences are true. Table 3 illustrates this withthe results of a typical run. The table shows the rank of thegrammar’s overall posterior relative to all other grammars,
as well as the posterior, prior, and likelihood. Note that (a),the highest scoring grammar, is the correct one on this runof the model. The second highest scoring grammar is (b),which uses the biconditional rather than implication in thelambda expression for “every.” The fact that these are closeand highly-ranked is consistent with Table 2, which showsthat the model often chooses the wrong one, making the same
mistake as children. In addition, (c) is ranked relatively low.
This is interesting because (c) is the logically weaker versionof the correct grammar that, and thus would be preferred by alearner which chooses grammars which are most often true.

Conclusion
We have shown that an unsupervised, Bayesian model cancorrectly learn compositional semantics and syntax in a toy
version of English, using only positive evidence and contextual information. Moreover, this model considers a relatively
unconstrained hypothesis space, and makes mistakes in acquiring quantifiers analogous to those children make. We
have argued that the toy language presents some of the core
problems that children face in acquiring a full compositionalsemantics. The model succeeds due to very general principles in Bayesian learning and uses linguistically-realistic representations of semantics. This suggests that the model could
be scaled up to include more realistic models for sentence
production, or a more elaborate syntax and semantics.

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