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.

Niyogi's analysis of Principles and Parameters learning

In this post I will summarize Chapter 4 of Niyogi (1998) The Informational Complexity of Learning. In it, Niyogi emphasizes the importance of going beyond theoretical learnability when analyzing a grammatical paradigm. "One also needs to quantify the sample complexity of the learning problem, i.e., how many examples does the learning algorithm need to see in order to be able to identify the target grammar with high confidence."

He sets his sights upon the Triggering Learning Algorithm, put forth by Gibson and Wexler as a learning scheme for the grammatical "parameters" within the Chomskyan Principles and Parameters framework. For those unfamiliar with the background, this is a theory of language that posits a Universal Grammar underlying all natural languages (the principles), and then a finite set of variable parameters which account for the differences among languages. The "parameter setting" is really the sole learning task for the developing child on this account.

I think that in the beginning, this theory was put forth in an effort to address the supposed "poverty of the stimulus," with the hope that the resulting learning problem would be tractable, even easy. Niyogi, however, manages to demonstrate that Gibson and Wexler's assumption of the existence of "local triggers," i.e. a path through the parameter-setting space from the initial hypothesis to the target, is not even sufficient to guarantee learnability at all (though it was believed sufficient by Gibson and Wexler), much less tractability. He further demonstrated the surprising theorem that, for all its carefully thought out design, the Triggering Learning Algorithm is less optimal than a random walk on the parameter space!

At the time of Niyogi's writing, he judged that the Triggering Learning Algorithm was a preferred explanation of language learning in psycholinguistics. His results should really have killed it, but as far as I can see they have had no such effect. In fact, Google Scholar finds only 12 literature citations of his entire book, most of which are due to the author himself. This is hardly a flurry of activity; only one other author writing on problems of natural language learning appears to be among the citations.

Pregroup grammars' generative capacity

There has been a movement within mathematical linguistics toward Lambek's pregroup grammars, which were mentioned in one or two earlier posts. The book Computational Algebraic Approaches to Natural Language (Casadio & Lambek eds., 2008) collects a number of papers on this subject, and this is recommended for those who wish to catch up on the trend. The book is also available as a free download directly from the publisher Polimetrica. Myself, I am somewhat ambivalent about the framework, but there does seem to be some confusion in the literature about its generative capacity.

The first paper in the mentioned volume is "Pregroup grammars and context-free grammars" by Buszkowski and Moroz. This paper relies on a result by Buszkowski (published in the Logical Aspects of Computational Linguistics proceedings in 2001) showing the weak equivalence between pregroup grammars and context-free grammars. Yet Greg Kobele and Marcus Kracht did a paper (unpublished) showing that pregroup grammars generate the recursively enumerable languages. I asked Greg about this, and he explained (as does his paper with Kracht) that key elements of the Buszkowski result are excluding the empty string from the context-free languages, and using only free pregroups. Kobele and Kracht showed, on the other hand, that by allowing all pregroups and also allowing the empty string, one achieves Turing-equivalence, generating all r.e. languages.

This is all esoteric stuff which is nevertheless important to have nailed down when one is working with a grammar formalism. Another issue with pregroup grammars is that they deny the existence of syntactic constituents in the normal sense. But that discussion has to wait for another post.

Russell's "No Class" theory

A fine paper by Kevin Klement on Bertrand Russell's "No Class" theory is published in the Dec. 2010 issue of the Review of Symbolic Logic. Klement outlines a sympathetic reading of Russell's attempts to eliminate actual "classes" as objects in his logical theory, showing along the way how the many high-profile criticisms of Russell (which were mostly swallowed whole by the community) fail to dislodge the No Class theory on any philosophical grounds.

Without going into technical details, Russell tried to define "propositional functions" as being open sentences of logic, such as a predicate applied to a variable as in Mortal(x). While a complete logical sentence such as Mortal(Socrates) makes the statement that Socrates is mortal, the open sentence Mortal(x) has usually been construed as equivalent to a function which maps things (substituted for x) to the statement that they are mortal. Because such a function essentially classifies things according as the resulting statement is true or false, it ends up with classes. Russell wanted to see such an expression as not involving a function or class literally, but rather as just an open sentence at face value.

I wanted to point this out here because the dichotomy between realism and nominalism is central to Klement's discussion of the No Class theory. Klement argues that while a realist philosophy compels one to accept that an open sentence is something, a nominalist position permits the consistent disavowal of that assertion. The realist, being thus compelled, is then forced to identify an open sentence with the function classifying things that could instantiate the variable, since the two are extensionally equivalent. For the realist, then, the No Class theory of open sentences is a chimera because it collapses into the same thing as having classes in the first place. The nominalist, however, permits himself to have an expression like an open sentence that doesn't correspond to something which "exists", and then is not led to an identity between open sentences and functions as maps.

The moral, for linguistic theory, is that it can be very important foundationally whether one construes a theoretical construct as representing something which exists.