University of Toronto, Department of Computer Science
CSC 2501F—Computational Linguistics, Fall 2018
Reading assignment 3
Due date: In class at 11:10, Thursday 11 October 2018.
Late write-ups will not be accepted without documentation of a medical or other emergency.
This assignment is worth 5% of your final grade.
What to read
Fernando Pereira, “Formal grammar and information theory: together again?,” Phil. Trans.
R. Soc. Lond. A, 358, 2000, 1239–1253.
What to write
Write a brief summary of the paper’s argumentation, with a critical assessment of its merits.
General requirements: Your write-up should be typed, using 12-point font and 1.5-line
spacing; it should fit on one to two sides of a sheet of paper. Hand in one copy at the begin-
ning of class.
Formal grammar and information theory:
together again?
By Fernando Pereira
AT & T Laboratories Research, A247, Shannon Laboratory, 180 Park Avenue,
Florham Park, NJ 07932-0971, USA ([email protected])
In the last 40 years, research on models of spoken and written language has been split
between two seemingly irreconcilable traditions: formal linguistics in the Chomsky
tradition, and information theory in the Shannon tradition. Zellig Harris had advo-
cated a close alliance between grammatical and information-theoretic principles in
the analysis of natural language, and early formal-language theory provided another
strong link between information theory and linguistics. Nevertheless, in most research
on language and computation, grammatical and information-theoretic approaches
had moved far apart.
Today, after many years on the defensive, the information-theoretic approach has
gained new strength and achieved practical successes in speech recognition, informa-
tion retrieval, and, increasingly, in language analysis and machine translation. The
exponential increase in the speed and storage capacity of computers is the proxi-
mate cause of these engineering successes, allowing the automatic estimation of the
parameters of probabilistic models of language by counting occurrences of linguistic
events in very large bodies of text and speech. However, I will argue that information-
theoretic and computational ideas are also playing an increasing role in the scien-
ti c understanding of language, and will help bring together formal-linguistic and
information-theoretic perspectives.
Keywords: formal linguistics; information theory; machine learning
1. The great divide
In the last 40 years, research on models of spoken and written language has been
split between two seemingly irreconcilable points of view: formal linguistics in the
Chomsky tradition, and information theory in the Shannon tradition. The famous
quote of Chomsky (1957) signals the beginning of the split.
(1) Colourless green ideas sleep furiously.
(2) Furiously sleep ideas green colourless.
: : : It is fair to assume that neither sentence (1) nor (2) (nor indeed any
part of these sentences) has ever occurred in an English discourse. Hence,
in any statistical model for grammaticalness, these sentences will be ruled
out on identical grounds as equally `remote’ from English. Yet (1), though
nonsensical, is grammatical, while (2) is not.
Phil. Trans. R. Soc. Lond. A (2000) 358, 1239{1253
1239
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1240 F. Pereira
Before and after the split, Zellig Harris had advocated a close alliance between
grammatical and information-theoretic principles in the analysis of natural lan-
guage (Harris 1951, 1991). Early formal-language theory provided another strong
link between information theory and linguistics. Nevertheless, in most research on
language and computation, those bridges were lost in an urge to take sides that was
as much personal and ideological as scienti c.
Today, after many years on the defensive, the information-theoretic approach is
again thriving and has led to practical successes in speech recognition, information
retrieval, and, increasingly, in language analysis and machine translation. The expo-
nential increase in the speed and storage capacity of computers is the proximate cause
of these successes, allowing the automatic estimation of the parameters of computa-
tional models of language by counting occurrences of linguistic events in very large
bodies of text and speech. However, vastly increased computer power would be irrel-
evant if automatically derived models or linguistic data were not able to generalize
to unseen data. I will argue below that progress in the design and analysis of such
models is not only playing a central role in those practical advances, but also car-
ries the promise of fundamentally deeper understanding of information-theoretic and
computational-complexity constraints on language acquisition.
2. Harris’s program
The ascent of Chomskian generative linguistics in the early 1960s swept the focus
of attention away from distributional views of language, especially those based on
the earlier structuralist tradition. In that tradition, Zellig Harris developed what
is probably the best-articulated proposal for a marriage of linguistics and informa-
tion theory. This proposal involves four main so-called constraints (Harris 1988) as
follows.
Partial order `: : : for each word: : : there are zero or more classes of words, called
its arguments, such that the given word will not occur in a sentence unless one
word: : : of each of its argument classes is present.’
There’s a strong similarity between the argument class information for a word
as suggested by Harris and its type in categorial grammar, or subcategoriza-
tion frames in other linguistic formalisms. However, traditional categorial gram-
mar (Lambek 1958) con®ates function{argument relationships and linear order,
whereas Harris factors out linear order explicitly. It is only more recently that cat-
egorial grammar has acquired the technical means to investigate such factorizations
(Morrill 1994; Moortgat 1995). It then becomes clear that Harris’s partial order
may be formalized as the partial order among set-theoretic function types. How-
ever, unlike modern categorial grammar, Harris’s partial order constraint speci es
only the basic con gurations corresponding to elementary clauses, while complex
clauses are a result of applying another constraint, reduction, to several elementary
clauses.
Likelihood `: : : each word has a particular and roughly stable likelihood of occurring
as argument, or operator, with a given other word, though there are many cases
of uncertainty, disagreement among speakers, and change through time.’
Using current terminology, one might interpret the likelihood constraint as a proba-
bilistic version of selectional restrictions. However, Harris makes a sharp distinction
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Formal grammar and information theory 1241
between general language, in which likelihoods for llers of argument positions rep-
resent tendencies, and technical sublanguages, in which there are hard constraints
on argument llers, and which, thus, correspond more closely to the usual notion
of selectional restriction.
Reduction `It consists, for each language, of a few speci able types of reduction: : :
what is reduced is the high-likelihood: : : material: : : ; an example is zeroing the
repeated corresponding words under and.’
The reduction constraint tries to account both for morphological processes like con-
traction, and for processes that combine elementary clauses into complex clauses,
such as relativization, subordination and coordination. In each case, Harris claims
that high-likelihood material may be elided, although it would seem that additional
constraints on reduction may be necessary. Furthermore, connections between
reduction-based and transformational analyses (Harris 1965; Chomsky 1965) sug-
gest the possibility of modelling string distributions as the overt projection of a
hidden generative process involving operator-argument structures subject to the
likelihood constraint and transformations. Recent work linking transformational
and categorial approaches to syntax makes this possibility especially intriguing
(Stabler 1997; Cornell 1997).
Linearization `Since the relation that makes sentences out of words is a partial
order, while speech is linear, a linear projection is involved from the start.’
Harris’s theory left this step rather underspeci ed. Chomskian transformational
grammar can be seen as an e¬ort to ll in the gap with speci c mechanisms of
sentence generation that could be tested against native speaker grammaticality
judgements.
Thus, linguistic events involve the generation of basic con gurations|unordered
simple clauses|whose structure is determined by the partial order constraint and
whose distribution follows the probabilities associated with the likelihood constraint.
Those probabilities also govern the application of reduction|compression|to indi-
vidual con gurations or sets of linked con gurations. Finally, linearization yields the
observable aspects of the event. As I will discuss in x 7, though, the likelihood con-
straint as stated by Harris, or its current versions, leaves out dependences on the
broader discourse context that strongly a¬ect the likelihoods of linguistic events.
For the present discussion, the most important feature of Harris’s constraints is
how they explicitly link linguistic structure with distributional regularities involving
the relative frequencies of di¬erent structural con gurations. In particular, Harris
suggested how the structural and distributional regularities could work together to
support language acquisition and use:
: : : when only a small percentage of all possible sound-sequences actually
occurs in utterances, one can identify the boundaries of words, and their
relative likelihoods, from their sentential environment: : : .
3. Generalization
While Harris discussed the functional role of distributional regularities in language,
he proposed no speci c mechanisms by which language users could take advantage of
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1242 F. Pereira
those regularities in language acquisition and use. In particular, it is not obvious that
language users can acquire stable distributional information, let alone the lexical and
grammatical information required by the partial-order, reduction and linearization
constraints, from the limited evidence that is available to them from their linguistic
environment. This question created a great opening for Chomsky’s rationalist critique
of empiricist and structuralist linguistics, of which the `green-ideas’ quote above is
an early instance.
Chomsky concluded that sentences (1) and (2) are equally unlikely from the
observation that neither sentence or `part’ thereof would have occurred previously
(Abney 1996). From this observation, he argued that any statistical model based
on the frequencies of word sequences would have to assign equal, zero, probabilities
to both sentences. But this relies on the unstated assumption that any probabilistic
model necessarily assigns zero probability to unseen events. Indeed, this would be the
case if the model probability estimates were just the relative frequencies of observed
events (the maximum-likelihood estimator). But we now understand that this naive
method badly over¯ts the training data.
The problem of over tting is tightly connected with the question of how a learner
can generalize from a nite training sample. The canonical example is that of tting
a polynomial to observations. Given a nite set of observed values of a dependent
random variable Y for distinct values of the independent variable X, we seek a
hypothesis for the functional dependency of Y on X . Now, any such set of observa-
tions can be tted exactly by a polynomial of high-enough degree. But that curve
will typically be a poor predictor of a new observation because it exactly matches
the peculiarities of the training sample. To avoid this, one usually smooths the data,
using a lower-degree polynomial that may not t the training data exactly but that
will be less dependent on the vagaries of the sample. Similarly, smoothing methods
can be used in probability models to assign some probability mass to unseen events
(Jelinek & Mercer 1980). In fact, one of the earliest such methods, due to Turing
and Good (1953), had been published before Chomsky’s attack on empiricism, and
has since been used to good e¬ect in statistical models of language (Katz 1987).
The use of smoothing and other forms of regularization to constrain the form of
statistical models and ensure better generalization to unseen data is an instance of
a central theme in statistical learning theory, that of the sample complexity relation-
ship between training sample size, model complexity and generalization ability of
the model. Typical theoretical results in this area give probabilistic bounds on the
generalization error of a model as a function of model error on training data, sample
size, model complexity, and margin of error (Vapnik 1995). In qualitative terms, the
gap between test and training error|a measure of over tting|grows with model
complexity for a xed training sample size, and decreases with sample size for a
xed model complexity.
To quantify the trade-o¬ between training-set accuracy, generalization to new data
and constraints on the model, we need a rigorous measure of model complexity. In the
polynomial example, the usual intuition is that complexity is measured by the degree
of the polynomial (the number of tunable coe¯ cients in the model), but intuitions
are harder to come by for model classes without a simple parametric form. Fur-
thermore, even in the polynomial case, the common-sense complexity measure can
be misleading, because certain approaches to polynomial tting yield much smaller
model complexity and, thus, better generalization ability (Vapnik 1995). The de ni-
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Formal grammar and information theory 1243
tion of model complexity is also intimately tied to the learning setting; for instance,
whether one assumes that the data have a distribution of known form but unknown
parameters (as is usually done in statistics), or one takes a distribution-free view, in
which the data are distributed according to an unknown (but xed between train-
ing and test) distribution (Valiant 1984), or even one assumes an on-line setting,
in which the goal is to do the best possible prediction on a xed sequence incre-
mentally generated by the environment (Littlestone & Warmuth 1994; Freund &
Schapire 1997). A crucial idea from the distribution-free setting is that model com-
plexity can be measured, even for an in nite model class, by combinatorial quantities
such as the Vapnik{Chervonenkis (VC) dimension (Vapnik & Chervonenkis 1971),
which, roughly speaking, gives the order of a polynomial upper bound on the num-
ber of distinctions that can be made between samples by models in the class, as a
function of sample size.
Returning to the debate between empiricism and rationalism, the relationships
between model complexity, sample size and over tting developed in learning the-
ory may help clarify the famous argument from poverty of the stimulus (APS).
Reacting to empiricist and especially behaviourist theories, Chomsky and others
have argued that general-purpose learning abilities are not su¯ cient to explain
children’s acquisition of their native language from the (according to them) very
limited linguistic experience that is available to the learner. In particular, they
claimed that linguistic experience does not provide negative examples of grammat-
icality, making the learner’s task that much harder. Therefore, they conclude, a
specialized innate language faculty must be involved. The `green-ideas’ example
is an early instance of the same argument, asserting that statistical procedures
alone cannot acquire a model of grammaticality from the data available to the
learner.
The APS does not just require restrictions on model classes to ensure e¬ective
generalization from nite data, which would be unobjectionable from a learning-
theoretic viewpoint. In its usual form, the APS also claims that only a learning
mechanism developed speci cally for language could generalize well from limited
linguistic experience. The ®aw in this argument is that it implicitly assumes that
the only constraints on a learner are those arising from particular representations
of the learner’s knowledge, whereas we now know that the informational di¯ culty
of learning problems can be characterized by purely combinatorial, representation-
independent, means. Statistical learning theory gives us the tools to compute empir-
ically testable lower bounds on sample sizes that would guarantee learnability for
given model classes, although such bounds can be very pessimistic unless they take
into account constraints on the model search procedure as well. Nevertheless, it is
unlikely that the debate over the APS can become empirically grounded without
taking into account such calculations, since the stimuli that APS supporters claimed
to be missing are actually present with signi cant frequency (Pullum 1996).
The APS reached an extreme form with Chomsky’s principles-and-parameters the-
ory, according to which learnability requires that the set of possible natural lan-
guages be generated by the settings of a nite set of nitely valued parameters
(Chomsky 1986, p. 149). But this extreme constraint is neither necessary, since in –
nite model classes of nite VC dimension are learnable from an information-theoretic
point of view, nor su¯ cient, because even nite classes may not be e± ciently learn-
able, that is, the search for a model with good generalization may be computation-
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1244 F. Pereira
ally intractabley even though the information is, in principle, available (Kearns &
Valiant 1994).
4. Hidden variables
Early empiricist theories of linguistic behaviour made themselves easy targets of cri-
tiques like that of Chomsky (1959) by denying a signi cant role for the internal,
unobservable, state of the language user. Thus, in a Markov model of language, all
the state information would be represented by the externally observable sequence of
past linguistic behaviour. However, even in this case, the empiricist position had been
somewhat oversimpli ed by its critics. If we consider a language user that updates
its expectations and probable responses according to statistics collected from its past
experience, those expectations and response propensities, however represented, are
a part of the user state that is not directly available for observation. Furthermore,
the behaviour of language users may give valuable information about the power of
their experience-encoding mechanisms. For instance, a language user that maintains
statistics over pairs of consecutive words only (bigram statistics) might be less e¬ec-
tive in anticipating and reacting appropriately to the next word than a user that
keeps statistics over longer word sequences; in other words, the bigram model may
have higher entropy. This example, related to nite-state text compression, may seem
simplistic from the point of view of linguistics, but it is a convenient test-bed for ideas
in statistical modelling and constraints on model structure, and introduces the idea
of a hidden modelling state in a very simple form.
Hidden random variables in a language user’s state|or, rather, statistics involving
their joint values|represent the user’s uncertainty about the interpretation of, and
best response to, events observed so far. Such uncertainty may not just be over the
interpretation of a particular course of events, but also over which particular model
in a class of models is a best compromise between tting the experience so far and
generalizing to new experience. When the best choice of model is uncertain, Bayesian
model averaging (Willems et al. 1995) can be used to combine the predictions of
di¬erent candidate models according to the language user’s degree of belief in them,
as measured by their past success. Model averaging is, thus, a way for learners to
hedge their bets on particular grammars, in which the initial bets represent a prior
belief on particular grammars and are updated according to a regularizing procedure
that balances t to past experience with predictive power over new experience. The
prior distribution on grammars can be seen as a form of innate knowledge that
implicitly biases the learner towards `better’|for instance, less complex|grammars.
In particular, any in nite class of grammars can be given a universal prior based
on the number of bits needed to encode members of the class, which favours the
least complex grammars compatible with the data (Solomono¬ 1964; Horning 1969).
However, those results did not provide a way of quantifying the relationship between
a prior over grammars, training sample size and generalization power, and in any
case seems to have been ignored by those interested in language acquisition and the
APS. Recent advances in statistical learning theory (McAllester 1999) may provide
new theoretical impetus to that research direction, since they show that a prior
y I use `intractable’ in this paper in the usual sense from theory of computation of a problem that
has been proven to belong to one of the standard classes believed to require more than polynomial time
on a deterministic sequential computer, for instance the NP-hard problems.
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Formal grammar and information theory 1245
over models can play an analogous regularizing role to a combinatorial complexity
measure.
The other role for hidden variables, capturing uncertainty in the interpretation
of particular experience, becomes especially interesting in modelling ambiguity. For
example, going back to Harris’s theory, each of the constraints involves covert choices
by the language user: assignment of types|positions in the partial order|to lexical
items; lexical choice according to selection probabilities; reduction choices accord-
ing to the distributional statistics of predictability; and linearization choices. More
generally, any model of language that appeals to non-observables, for instance any
model that assigns syntactic analyses, requires hidden variables.
Hidden variables representing uncertainty of interpretation can also be used to
create factored models of joint distributions that have far fewer parameters to esti-
mate, and are, thus, easier to learn, than models of the full joint distribution. As
a very simple but useful example, we may approximate the conditional probability
p(x; y) of occurrence of two words x and y in a given con guration as
p(x; y) = p(x)
X
c
p(y j c)p(c j x);
where c is a hidden `class’ variable for the associations between x and y in the
con guration under study. For a vocabulary of size V and C classes, this model uses
O(CV ) parameters rather than the O(N2) parameters of the direct model for the
joint distribution, and is thus less prone to over tting if C V . In particular, when
(x; y) = (vi; vi+ 1), we have an aggregate bigram model (Saul & Pereira 1997), which
is useful for modelling word sequences that include unseen bigrams. With such a
model, we can approximate the probability of a string p(w1 wn) by
p(w1 wn) = p(w1)
nY
i = 2
p(wijwi 1):
By using this estimate for the probability of a string and an aggregate model with
C = 16 trained on newspaper text, and by using the expectation{maximization (EM)
method (Dempster et al. 1977), we nd that
p(Colourless green ideas sleep furiously)
p(Furiously sleep ideas green colourless)
2 105:
Thus, a suitably constrained statistical model, even a very simple one, can meet
Chomsky’s particular challenge.
A plausible and well-de ned model of the statistical dependences among the hidden
variables is, however, not in general su¯ cient, since the problem of setting the corre-
sponding conditional probabilities from observable linguistic material is in most cases
computationally intractable (Abe & Warmuth 1992). Nevertheless, those intractabil-
ity results have not precluded signi cant algorithmic and experimental progress with
carefully designed model classes and learning methods, such as EM and variants,
especially in speech processing (Baum & Petrie 1966; Baker 1979). In particular, the
learning problem is easier in practice if interactions between hidden variables tend
to factor via the observed variables.
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5. Lexicalized models
Harris’s model of dependency and selection is lexicalized, in the sense that all pos-
tulated relationships are between the words in (precursors for) sentences, rather
than the relationships between structures in generative grammar. From the points
of view of distributional modelling and machine learning, an important property of
lexicalized models is that they anchor analyses in observable co-occurrences between
words, rather than in unobservable relationships among hypothetical grammatical
structures.y In a probabilistic setting, a way to state this more precisely is that lexi-
calization makes it easier to factor the interactions between the hidden variables by
conditioning on the observed sentence.
Even lexicalized models will involve hidden decisions if they allow ambiguous
interpretations. As noted in the previous section, hidden-variable models are com-
putationally di¯ cult to learn from the observable variables alone. An alternative
strategy is to constrain the hidden variables by associating sentences with disam-
biguating information. At one extreme, that information might be a full analysis.
In this case, which is very interesting from computational and applications perspec-
tives, recent work has shown that lexicalized probabilistic context-free grammars can
be automatically learned that perform, with remarkable accuracy, on novel mate-
rial (Charniak 1997; Collins 1998). Besides lexicalization, these models factor the
sentence-generation process into a sequence of conditionally independent events that
re®ect such linguistic distinctions as those of head and dependent and of argument
and adjunct. That is, the models are in e¬ect lexically based stochastic generative
grammars, and the conditional independence assumptions on the generation process
are a particular kind of Markovian assumption. Crucially, these assumptions apply
to the hidden generative decisions, not to the observable utterance, and, thus, allow
for analysis ambiguity.
The learning algorithms just discussed need to be given the full correct syntac-
tic analysis of each training example, and are, thus, not realistic models of human
language acquisition. One possible direction for reducing the unrealistic amount of
supervision required would be to use additional observables correlated with the hid-
den variables instead, such as prosodic information or perceptual input associated
with the content of the linguistic input (Siskind 1996; Roy & Pentland 1999). More
generally, we may be able to replace direct supervision with indirect correlations, as
I now discuss.
6. The power of correlations
How poor is the stimulus that the language learner exploits to acquire its native
language? As I observed above, linguistic experience is not just a string of words,
but it is grounded in a rich perceptual and motor environment that is likely to
provide crucial clues to the acquisition, interpretation and production processes, if
for no other reason than for the functional one that much of the linguistic experience
y This property could well make lexicalized models less rather than more palatable to Chomskian
linguists, for whom structural relationships are the prime subject of theory. But notice that Chomsky’s
more recent `minimalist program’ (Chomsky 1995) is much more lexically based than any of his theories
since `Aspects’ (Chomsky 1965), in ways that are reminiscent of other lexicalized multistratal theories, in
particular lexical-functional grammar (Bresnan 1982), HPSG (Pollard & Sag 1994), and certain varieties
of categorial grammar (Morrill 1994; Moortgat 1995; Cornell 1997).
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Formal grammar and information theory 1247
is about that non-linguistic environment. However, this points to a fundamental
weakness in much of the work discussed so far: both in formal grammar and in most
computational models of language, language is taken as a completely autonomous
process that can be independently analysed.y Indeed, a simplistic use of information
theory su¬ers from the same problem, in that the basic measures of information
content of a signal are intrinsic, rather than relative to the correlations between a
signal and events of interest (the meaning(s) of the signal). In particular, Harris’s
likelihood and reduction constraints appear to ignore the content-carrying function
of utterances. Fortunately, information theory provides a ready tool for quantifying
information about with the notion of mutual information (Cover & Thomas 1991),
from which a suitable notion of compression relative to side variables of interest can
be de ned (Tishby et al. 1999).
Given the enormous conceptual and technical di¯ culties of building a comprehen-
sive theory of grounded language processing, treating language as an autonomous
system is very tempting. However, there is a weaker form of grounding that can be
exploited more readily than physical grounding, namely grounding in a linguistic
context. Following this path, sentences can be viewed as evidence for other sentences
through inference, and the e¬ectiveness of a language processor may be measured by
its accuracy in deciding whether a sentence entails another, or whether an answer is
appropriate for a question.
Furthermore, there is much empirical evidence that linguistic grounding carries
more information than it might seem to at rst sight. For instance, all of the most
successful information-retrieval systems ignore the order of words and just use the fre-
quencies of words in documents (Salton 1989) in the so-called bag-of-words approach.
Since similar situations are described in similar ways, simple statistical similarity
measures between the word distributions in documents and queries are e¬ective in
retrieving documents relevant to a given query. In the same way, word senses can
be automatically disambiguated by measuring the statistical similarity between the
bag of words surrounding an occurrence of the ambiguous word and the bags of
words associated with de nitions or examples of the di¬erent senses of the word
(Sch�utze 1997).
In both information retrieval and sense disambiguation, bag-of-words techniques
are successful because of the underlying coherence of purposeful language, at syn-
tactic, semantic, and discourse levels. The one-sense-per-discourse principle (Gale et
al. 1992) captures a particular form of this coherence. For example, the co-occurrence
of the words `stocks’, `bonds’ and `bank’ in the same passage is potentially indicative
of a nancial subject matter, and thus tends to disambiguate those word occurrences,
reducing the likelihood that the `bank’ is a river bank, that the `bonds’ are chemical
bonds, or that the `stocks’ are an ancient punishment device. These correlations,
like the correlations between utterances and their physical context, allow a language
processor to learn from its linguistic environment with very little or no supervi-
sion (Yarowsky 1995), and have suggested new machine-learning settings, such as
co-training (Blum & Mitchell 1998).
Both lexicalized grammars and bag-of-words models represent statistical associ-
ations between words in certain con gurations. However, the kinds of associations
y I include under this description all the work on formal semantics of natural language, since logical
representations of the meanings of sentences are as unobservable as syntactic analyses, and are thus
equally arti cial as inputs to a language acquisition process.
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1248 F. Pereira
represented are rather di¬erent. The associations in lexicalized grammars are medi-
ated by a hidden assignment of dependency relationships to pairs of word occurrences
in an utterance. Many such assignments are potentially available, leading to great
structural ambiguity, as discussed in x 5. In contrast, the associations in bag-of-words
models and many other statistical models (for instance, Markov models) are de ned
over very impoverished but unambiguous overt structures. Furthermore, e¬ective lex-
icalized models must make very strong statistical-independence assumptions between
parts of the underlying structure, thus missing the global coherence correlations that
bag-of-words models capture.
7. Local structure and global distribution
Current stochastic lexicalized models, with their lexically determined local correla-
tions, capture much of the information relevant to Harris’s partial-order and likeli-
hood constraints. However, unlike Harris, but like dependency grammar and other
monostratal grammatical formalisms, they con®ate linearization with the argument
structure given by the partial-order constraint.
In asserting the `rough stability’ of the likelihood of a given argument of a given
operator, Harris implicitly assumed a generative model in which dependents are
conditionally independent of the rest of an analysis given the head they depend
on. Existing lexicalized models use similar Markovian assumptions, although they
typically extend lexical items with additional features, for instance syntactic category
(Charniak 1997; Collins 1998). However, Harris’s information-theoretic arguments,
especially those on reduction, refer to the overall likelihood of a string, which involves
the global correlations discussed in the last section. But such global correlations are
precisely what the Markovian assumptions in generative models leave out.
Thus, Markovian generative models are not able to model the potential correlations
between the senses assigned to occurrences of `stocks’ and `bonds’ in di¬erent parts
of a paragraph, for example. This problem may be addressed in two main ways. The
rst is to preserve Markovian assumptions, but to enrich lexical items with features
representing alternative global coherence states. For instance, lexical items might be
decorated with sense features, and local correlations between those would be used
to enforce global coherence. Those features might even be other lexical items, whose
co-occurrence with the given items as operators or arguments may disambiguate
them. The di¯ culty with this approach is that it introduces a plethora of hidden
variables, leading to a correspondingly harder learning problem. Furthermore, it
relies on careful crafting of the hidden variables, for instance in choosing informative
sense distinctions. The second approach is to adopt ideas from random elds and
factor probabilities instead as products of exponentials of indicator functions for
signi cant local or global features (events) (Della Pietra et al. 1997; Ratnaparkhi
1997; Abney 1997), which can be built incrementally with `greedy’ algorithms that
select the most informative feature at each step.
8. From deciding to understanding
Models based on information-theoretic and machine-learning ideas have been suc-
cessful in a variety of language-processing tasks, such as speech recognition and
information retrieval. A common characteristic of most of those tasks is that what
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Formal grammar and information theory 1249
is sought is a decision among a nite set of alternatives, or a ranking of alternatives.
For example:
(i) a newswire lter classi es news stories into topics speci ed by training exam-
ples;
(ii) a part-of-speech tagger assigns the most likely tags to the words in a document;
(iii) a Web search engine ranks a set of Web pages according to their relevance to
a natural language query;
(iv) a speech recognizer decides among the possible transcriptions of a spoken utter-
ance.
In each case, the task can be formalized as that of learning a mapping from spoken
or written material to a choice or ranking among alternatives. As we know from
the earlier discussion of generalization, we need to restrict our attention to a class
of mappings that can actually be learned from the available data. Computational
considerations and experimental evaluation will further narrow the mapping classes
under consideration. Finally, a suitable optimization procedure is employed to select
a mapping from the class that minimizes some measure of the error on the training
set.
A potential weakness of such task-directed learning procedures is that they ignore
regularities that are not relevant to the task. Yet, those regularities may be highly
informative about other questions. While language may be redundant with respect
to any particular question, and a task-oriented learner may bene t greatly from
that redundancy, as discussed earlier, it does not follow that language is redundant
with respect to the set of all questions that a language user may need to decide.
Furthermore, one may reasonably argue that a task-oriented learner does not really
`understand’ language, since it can accurately answer only one question, while our
intuitions about understanding suggest that a competent language user can accu-
rately answer many questions pertaining to any discourse it processes. For instance,
a competent language user should be able to reliably answer `who did what to whom’
questions pertaining to each clause in the discourse.
We are thus drawn to the question of what kinds of learning tasks may involve
`understanding’ but do not force us to attack frontally the immense challenges of
grounded language processing. Automatically trained machine translation (Brown
et al. 1990; Alshawi & Douglas, this issue) may be such a task, since translation
requires that many questions about a text be accurately answered to produce a
correct output. Nevertheless, it is easy to nd many other reasonable questions that
can be left unanswered while still performing creditably on the task. Indeed, there is
no single `understanding’ task, but, rather, a range of tasks whose di¯ culty can be
measured by the uncertainty|information-theoretically, the entropy|of the output
in the absence of any information about the input. The objective of a learner is
then to acquire a function that can reduce that uncertainty by exploiting the mutual
information between inputs and outputs (Tishby & Gorin 1994). Tasks (i){(iv) above
are listed roughly in order of increasing output entropy, with machine translation
possibly being even more di¯ cult.
The theoretical representations postulated by formal linguistics (constituent struc-
ture, functional and dependency structures, logical form) can also be understood as
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1250 F. Pereira
codi ed answers to particular kinds of questions pertaining to the text, with their
own degrees of information-theoretic di¯ culty. For instance, di¬erent assignments of
arguments to thematic roles lead to di¬erent correct answers to `who did what to
whom’ questions. From this point of view, the task of the learner is to acquire an
accurate procedure for deciding whether a simple sentence follows from a discourse,
rather than the more traditional tasks of deciding grammaticality or assigning struc-
tural descriptions. Structural descriptions will still play an important role in such a
theory, but now as proxies for informational relationships between external linguistic
events instead of end-products of the theory.
9. Summary
While researchers in information retrieval, statistical pattern recognition, and neural
networks kept developing theoretical and experimental approaches to the problem
of generalization, that work was ignored by formal linguistics for both cultural and
substantive reasons. Among the substantive reasons, possibly the most important
was that the models proposed, even if successful in practice, failed to capture the
productive, recursive nature of linguistic events.
Recent advances in machine learning and statistical models are starting to supply
the missing ingredients. Lexicalized statistical models informed by linguistic notions
such as phrase head, argument and adjunct specify how complex linguistic events can
be generated and analysed as sequences of elementary decisions. Machine learning
suggests how rules for the elementary decisions can be learned from examples of
behaviour, and how the learned decision rules generalize to novel linguistic situations.
Probabilities can be assigned to complex linguistic events, even novel ones, by using
the causal structure of the underlying models to propagate the uncertainty in the
elementary decisions.
Such statistical models of local structure are complemented by the models of larger-
scale correlations that have been developed in information retrieval and speech recog-
nition. These models have proved to be quite successful in automatically learning how
to rank possible answers to a given question, but it is still unclear how they may com-
bine with lexical models in a uni ed account of the relationship between linguistic
structure and statistical distribution.
Furthermore, we have barely touched on the question of what such models may
say about human language acquisition. Although statistical learning theory and its
computational extensions can help us ask better questions and rule out seductive
non sequiturs, their quantitative results are still too coarse to signi cantly narrow the
eld of possible acquisition mechanisms. However, some of the most successful recent
advances in machine learning arose from theoretical analysis (Cortes & Vapnik 1995;
Freund & Schapire 1997), and theory is also helping to sharpen our understanding
of the power and limitations of informally designed learning algorithms.
All in all, while much remains to be done, we may well be seeing the begin-
ning of a new version of the Harris program, in which computational models con-
strained by grammatical considerations de ne broad classes of possible grammars,
and information-theoretic principles specify how those models are tted to actual
linguistic data.
I thank Gerald Gazdar and Karen Sp�arck Jones for their careful reading of this paper and illumi-
nating comments; Ido Dagan, Lillian Lee, Larry Saul, Yves Schabes, Yoram Singer, Amit Sing-
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Formal grammar and information theory 1251
hal and Tali Tishby for the joint research that helped shape these ideas; Yoav Freund, Michael
Kearns and Rob Schapire for guidance on learning theory; and Steve Abney, Hiyan Alshawi,
Michael Collins, Don Hindle, Mark Johnson, Aravind Joshi, John La® erty, David McAllester,
Glyn Morrill, Michael Moortgat, Hinrich Sch�utze, Stuart Shieber and Ed Stabler for many con-
versations on these topics over the years. I am sure that each of them will have good reasons
to disagree with some of my arguments and interpretations, but nevertheless their help was
invaluable in this e® ort to reconcile the two rich traditions in the study of language that most
of my work derives from.
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Discussion
J. Coleman (University of Oxford, UK ). What is the status of statistical infor-
mation in natural language processing systems|does it compensate for gaps in our
theoretical knowledge or model humans’ behaviour in the face of uncertainty?
F. Pereira. The uncertainty may be about how much linguistic knowledge is shared
between interlocutors. Such variation within populations is often modelled as a sta-
tistical distribution. Furthermore, this variation is what drives language change in
speech communities. See the work by Anthony Kroch and colleagues at the University
of Pennsylvania. But there is also increasing evidence from psycholinguists (see work
of Michael Spivey and colleagues at Cornell University) that uncertainty is explicitly
represented (e.g. by ranging activation levels in human language processing).
P. A. Taylor (University of Edinburgh, UK ). Have you tested nonsensical and
ungrammatical sentences on a smoothed language model to con rm that they were
assigned appropriately di¬erent probabilities?
F. Pereira. No, but others have con rmed that such sentences are indeed assigned
positive and di¬ering probabilities. Note added in proof. The published version of the
paper now discusses a simple experiment answering this question.
K. Ahmad (University of Surrey, UK ). Did Zellig Harris not apply his model pri-
marily to specialized languages?
F. Pereira. Harris made too strong a distinction between specialized, scienti c
language and more general language, arguing, for instance, that selection restrictions
are inviolable in the former but not in the latter, but this is probably a di¬erence in
the degree of likelihood of such restrictions holding.
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