Grammar, Geometry, and Brain

J. E. Fenstad has a provocative little book called Grammar, Geometry, and Brain (CSLI 2010). It is speculative and is sort of like an annotated bibliography, but as such it is very valuable as a discussion of work in cognitive science and brain modeling at the interface with linguistics. Fenstad is a mathematical logician and occasional linguist. The book is filled with his own preferences and experience, but it is very forward thinking, perhaps venturing into the crackpot, but that's OK with me. It is inspiring.

Fenstad states "you can never be more than the mathematics you know," something I heartily agree with. Like him I will continue to seek out new mathematics to apply in linguistics. I'm tired of formal language theory and logic, these tools cannot be everything for linguistic theory. Fenstad's suggestion is that Geometry can provide a direct link between linguistic theory and cognitive modeling. He points to the Conceptual Spaces framework of Gärdenfors (MIT Press 2000) for a geometrical theory of semantics which he suggests could be connected to syntax using attribute-value grammar. He also discusses several works in cognitive neuroscience, suggesting how geometry could play a role in a bridging model connecting neuronal assemblies to conceptual spaces. It's an interesting book that pointed me to a lot of other readings.

What makes languages learnable?

A recent paper by Ed Stabler (in Language Universals, Christiansen et al. eds. 2009) puts the focus on an important question that is rarely formulated in the literature. What are the structural properties of natural languages which guarantee learnability? We know from a variety of negative results going back to the famous Gold theorem that such properties have to go far beyond the defining principles of the Chomsky hierarchy, since none of the traditional language classes except the finite languages are strictly learnable, in the sense of identifiability in the limit. Because we presume to model natural languages as some sort of infinite languages, something else must be going on. There must be some restrictions on the possible forms of natural language that permit learnability in some sense. Stabler says that to address this question, we need a proposal about how human learners generalize from finite data. There is as yet no complete answer to this problem, and indeed very little research seems to be currently motivated by such questions.

While I do not generally use this forum to highlight my own published work, in this case I believe that my 2010 paper in the Journal of Logic, Language and Information (together with an erratum published this year) does address Stabler's question directly. The title of my paper is Grammar Induction by Unification of Type-Logical Lexicons, and therein the basic proposal is given. Human learners are proposed to generalize from finite data by unification of the sets of syntactic categories that are discovered by an initial semantic bootstrapping procedure. The bootstrapping procedure extracts basic category information from semantically annotated sentence structures (this is highly enriched data, to be sure, but I argue for the plausibility of that in general terms). The basic system of categories is then unified by a two-step process that takes into account the distribution of the words (usage patterns, expressed structurally), and is able to generalize to "recursively redundant extensions" of the learning data. This is then a specific proposal of the sort invited by Stabler. The resulting learnable class of languages is highly restricted, including only those that are in some sense closed under recursively redundant extensions. This is in accord with general thinking about human languages, in that it is normally the case that a recognized recursive procedure (such as the appending of prepositional phrases to expand noun phrases, in rough terms) can always be applied indefinitely to yield grammatical sentences of increasing length.

I hope in the future to highlight my findings in a more general journal like Cognitive Science. It is helpful that Stabler conveniently provided the leading question to my purported answer.

Bare Grammar

I was recently reminded (thanks to the comment from Emilio on my previous post) of some great work that was done by my former professors Ed Keenan and Ed Stabler, that I first learned about when I was a PhD student in the 1990s. The key publication for reference is their book Bare Grammar, published by CSLI in 2003. It is actually rather embarrassing that I had largely forgotten about this work while preoccupied with my own efforts in learning theory the past several years, since I should have used some of their ideas to further my own.

Bare Grammar is a largely atheoretical framework for describing syntactic structure. I won't present any details here while trying to highlight the major points. Keenan and Stabler begin by pointing out that human languages always have lexicons in which most elements are grouped into classes (like parts of speech) whose members are in some sense intersubstitutable in the "same" positions within the "same" structures. A simple example is afforded by the English sentences:

Trevor laughed.

Nigel cried.

It is observed that Trevor can substitute for Nigel (and vice versa) without "changing structure," yielding equally grammatical sentences. The same can be said of the verbs laughed and cried. This means that the two sentences "have the same structure," in that they are each obtainable from the other by a sequence of "structure-preserving transformations." Hmm, this is starting to remind me of my previous post where I ruminated about gauge theory, as was suggested by the comment there.

Being mathematical linguists, Keenan and Stabler formalize all this. A structure-preserving map is defined as an automorphism of the grammar. An automorphism h can have fixed points x, such that h(x) = x. The syntactic invariants of the grammar are fixed points of every automorphism. In the lexicon, these are identified with the function words---words that cannot be substituted without changing structure. By using some fancier mathematical operations (power lifting and product lifting), it is shown how certain properties of higher order can also be fixed points, which correspond to such things like invariant properties of expressions, and invariant relations and functions. Much of the book is engaged in evaluating potential invariants of these kinds.

A major point is that natural languages all have such invariants---an interesting fact, to be sure, which is not at all necessary from the basic considerations of formal language theory. It would be easy to devise a formal language that didn't have anything like a function word, yet natural languages all have such things.

I hope to spend some more quality time with this book in the near future, and may have some more posts about it. In the meantime, I recommend it to anyone interested in a novel theoretical approach to a range of interesting linguistic facts that are not seriously dealt with elsewhere.

A gauge theory of linguistics?

Here is a bit of idle speculation. In my reading on theoretical physics, I have learned something about gauge theories. A concise description is found, naturally, on the Wikipedia page, where it is explained that the root problem addressed by gauge theory is the excess degrees of freedom normally present in specific mathematical models of physical situations. For instance, in Newtonian dynamics, if two configurations are related by a Galilean transformation (change of reference frame), they represent the same physical situation. The transformations form a symmetry group, so then a physical situation is represented by a class of mathematical configurations which are all related by the symmetry group. Usually the symmetry group is some kind of Lie group, but it need not be a commutative (abelian) group. A gauge theory is then a mathematical model that has symmetries (there may be more than one) of this kind. Examples include the Standard Model of elementary particles.

So far so great, but what does this have to do with linguistics? Well, it seems to me that mathematical models of language are often encumbered by irrelevant detail or an overly rigid dependence on conditions that are in reality never fixed. A simple example would be that a typical generative grammar of a language (in any theory) depends critically on the vocabulary and the categories assigned to it. In reality, different speakers have different vocabularies, and even different usages assigned to vocabulary items, although they may all feel they are speaking a common language. There is a sense in which we could really use a mathematical model of a language that is flexible enough to allow for some "insignificant" differences in the specific configuration assigned to the language. There may even be lurking a useful notion of "Galilean transformation" of a language.

This idea is stated loosely by Edward Sapir in his classic Language. He applies it to phonology, where he explains his conviction that two dialects may be related by a "vowel shift" in which the specific uses or identities of vowels are changed, but (in a sense that is left vague) the "phonological system" of the vowels in the two dialects is not fundamentally different. This idea may help to explain how American English speakers from different parts of the country can understand one another with relative ease even though they may use different sets of specific vowel sounds.

This is all a very general idea, of course. Gauge theory as applied in physics is really quite intricate, and I do not know yet if the specifics of the formalism can be "ported" to the particular problems of linguistic variation in describing a "common system" for a language. But what better place than a blog to write down some half-baked ideas?

Connectionism and emergence

I have not read too much mathematical linguistics lately, but I have been reading a lot of cognitive science and neuroscience, as well as connectionist research. Let me start off with connectionism. This is the approach involving artificial neural networks to employ "distributed processing" for computational purposes. I think that, in principle, such an approach to modeling language as a cognitive phenomenon will ultimately be the right approach. But there is a very large problem with current neural net modeling, chiefly that the neurons are too simple and the networks too small.

Neuroscience studies real neurons and their networks, although at present there are huge gaps in our understanding. While we are able to record signals from single neurons or very small groups, and we can also do "brain imaging" to track activity in huge (order of 10^9) numbers of neurons, we have no way to study activity in a few thousand neurons. It is precisely this "mesoscopic" regime where the phenomena of thought, memory, and knowledge are likely to be emergent from the nonlinear dynamical system known as the brain.

This brings me to the subject of "emergent phenomena," which refers to things that happen in a nonlinear dynamical system as a result of huge numbers of interactions among nonlinear dependencies. An emergent phenomenon on the ocean is a "rogue wave." An emergent phenomenon cannot be directly simulated through deterministic calculation, because it happens at a scale where there is not enough computing power in the world to run the simulation, there are too many interdependent variables.

Meanwhile, connectionism involves running simulations of neural networks that can be deterministically calculated. There are no emergent phenomena (so far as I know) in standard connectionist networks. So, this means they are not even able to manifest the most important thing happening in the brain in principle. So there is not any question that artificial neural networks do not model anything about the brain in the slightest sense.

Meanwhile in linguistics, a 'hot' idea is that classical linguistic categories like phonemes and parts of speech are "emergent" in a similar sense to an emergent phenomenon. The "emergentist" view of language holds that a phoneme emerges as an element of knowledge only after broad experience with "exemplars" in real speech. I am not exactly clear on the sense in which emergentist linguists think that such categories are emergent; do they mean statistically somehow, or do they mean "emergent" in the nonlinear chaos theory sense?

Conventional mathematical linguistics is looking quite far behind these newer developments and directions, but there is no question that better mathematical analysis would really help everyone to understand the new ideas like emergent linguistics.

Formal cognitive reasoning

As a book selector for my university library, I find out about numerous books that would remain unknown to me otherwise. One interesting book which we recently bought is Cognitive Reasoning: A Formal Approach by Anshakov and Gergely. This book presents a massive effort to systematize a formal logical framework which is seriously intended to model cognitive reasoning. It is, in one sense, an amplification of the typical efforts to devise a "logic of knowledge and belief" within AI, but this description does not really do any justice to the efforts documented.

The book relies on "a long prehistory" of work carried out in Russia and Hungary over many years, founded on the plausible reasoning methods of so-called JSM systems (named after John Stuart Mill because of the close affinity with his logical proposals). The formal logics underlying the technique are versions of "pure J-logics." These are multi-valued logics with two sorts of truth values called "internal" and "external." The external truth values are just the usual two, but the internal truth values can be a very rich (countably infinite) set. The logics contain J-operators, which are characteristic functions of subsets of the internal truth values. All this machinery is used to devise a sophisticated logic of knowledge updating which "permits one to describe the history of cognitive processes." The objective is to model the reasoning process which moves a cognitive agent "from ignorance to knowledge," and so the system is a dynamic logic as well. The formalism also involves a new kind of syntactic entity in the logic, a "modification inference" which can add new formulae by nondeductive rules and thereby modify constituents which are already established in the inference.

I have not finished reading this book, but I believe that it is very important for anyone interested in logic-based AI or cognitive science. It provides a rich and technically interesting set of formal methods for seriously trying to model cognitive reasoning.

Robot language acquisition

I discovered an interesting research program going on currently in the Adaptive Systems Group at the University of Hertfordshire, UK. A representative paper is "An integrated three-stage model towards grammar acquisition" by Yo Sato and colleagues, that appeared in the 2010 IEEE International Conference on Development and Learning. The paper documents an experiment in "cognitive robotics" where a robot is situated in a realistic language-learning environment.

According to the abstract, "the first, phonological stage consists in learning sound patterns that are likely to correspond to words. The second stage concerns word-denotation association. . . The data thus gathered allows us to invoke semantic bootstrapping in the third, grammar induction stage, where sets of words are mapped with simple logical types." This work is especially interesting to me because the grammar induction uses a semantic bootstrapping algorithm related to one which I developed, and published in 2005 (Journal of Logic, Language and Information).

In a discussion following my previous post, I offered the opinion that as computing power increases, we will (I hope) see more efforts to implement theoretically inspired learning algorithms that are quite intractable. This robotics paper represents one such effort, which I am very pleased to see. Yo Sato tells me that they are now looking at incorporating the improvements I have recently made to the original semantic bootstrapping algorithms. It's always gratifying to see an application inspired by my theoretical developments, since this is really why I pursue the work, but I am not sufficiently capable or interested to carry out the applied work that is then called for.