From an e-mail from Paul Smolensky, March 28, 2015. Even though he wasn’t doing phonology in the mid-1980’s when he coined the term “Harmony Theory”, Paul had apparently taken a course on phonology with Jorge Hankamer and found vowel harmony fascinating.

 “Harmony” in “Harmony Theory” arises from the fact that the Harmony function is a measure of *compatibility*; the particular word was inspired by vowel harmony, and by the letter ‘H’ which is used in physics for the Hamiltonian or energy function, which plays in statistical mechanics the same mathematical role that the Harmony function plays in Harmony theory: i.e., the function F such that prob(x) = k*exp(F(x)/T).

(Although I took the liberty of changing the sign of the function; in physics, it’s p(x) = k*exp(–H(x)/T), in Harmony Theory, it’s p(x) = k*exp(H(x)/T). That’s because it drove me crazy working in statistical mechanics that that minus sign kept coming and going and coming and going from equation to equation, leading to countless errors; I just dispensed with it at the outset and cut all that nonsense off at the pass.)

I always thought the reason why the physicists had a minus sign in the exponential was that otherwise temperatures would have to be negative.  But I guess you can push the negation into the Hamiltonian, which is perhaps what Paul did.

Yes, that’s just what I did. Instead of minimizing badness I switched to maximizing goodness. I’m just that kind of guy.

Probabilities are never greater than one, so log probabilities are always less than or equal to zero.  So a negative log likelihood is always a positive quantity, and smaller negative log likelihood values are associated with more likely outcomes.  So one way to understand the minus sign in the Gibbs-Boltzmann distribution is that it makes H(x) correspond to a negative log likelihood.

But I think one can give a more detailed explanation.

In a Gibbs-Boltzmann distribution p(x) = k*exp(–H(x)/T), H(x) is the energy of a configuration x.

Because energies H(x) are non-negative (which follows from the definition of energy?), and given a couple of other assumptions (e.g., that there are an infinite number of configurations and energies are unbounded — maybe other assumptions will do?), it follows that probability must decrease with energy, otherwise the inverse partition function k would not exist (i.e., the probability distribution p(x) would not sum to 1).

So if the minus sign were not there, the temperature T (which relates energy and probability) would need to be negative.  There’s no mathematical reason why we couldn’t allow negative temperatures, but the minus sign makes the factor T in the formula correspond much closer with our conventional understanding of temperature.

In fact, I think it is amazing that the constant T in the Gibbs-Boltzmann formula denotes exactly the pre-statistical mechanics concept of temperature (well, absolute temperature in Kelvin).  In many other domains there’s a complex relationship between a physical quantity and our perception of it; what is the chance of a simple linear relationship like this for temperature?

But perhaps it’s not a huge coincidence.  Often our perceptual quantities are logarithmically related to physical quantities, so perhaps its no accident that T is inside the exp() rather than outside (where it would show up as an “exponential temperature” term).  And the concept of temperature we had before Gibbs and Boltzmann wasn’t just a naive perception of warmth; there had been several centuries of careful empirical work on properties of gases, heat engines, etc., which presumably lead scientists to the right notion of temperature well before the Gibbs-Boltzmann relationship was discovered.

Very nice!  I was thinking about kinetic energy, but yes, potential energy (such as gravitational energy) is typically conceived as negative (I remember my high school physics class, where we thought of gravitational fields as “wells”).  I never thought about how this is forced once kinetic energy is positive.

Continuing in this vein, there are a couple of other obvious questions once one thinks about the relationship between Harmony theory and exponential models in physics.

For example, does the temperature T have any cognitive interpretation?  That is, is there some macroscopic property of a cognitive system that T represents?

More generally, in statistical mechanics the number (or more precisely, the density) of possible states or configurations varies as a function of their energy, and there are so many more higher energy states than lower energy ones that the typical or expected value of a physical quantity like pressure is not that of the more probable low energy states, but instead determined by the more numerous, less probable higher energy states.

I’d be extremely interested to hear if Paul knows of any cases where this or something like it occurs in cognitive science.  I’ve been looking for convincing cases ever since I got interested in Bayesian learning!  The one case I know of has to do with “sparse Dirichlet priors”, and it’s not exactly overwhelming.

E-mail from Paul Smolensky, March 27, 2016

The absolute magnitude of T has no significance unless the absolute magnitude of H does, which I doubt. So I’d take Mark’s question about T to come down to something like: what’s the cognitive significance of T —> 0 or T —> infinity or T ~ O(1)?

And I’d look for answers in terms of the cognitive role of different types of inference. T —> 0 gives maximum-likelihood inference; T —> infinity gives uniform sampling; T ~ O(1) gives sampling from the distribution exp(H(x)).  Mark, you’re in a better position to interpret the cognitive significance of such inference patterns.

As for the question of density of states of different Harmony/energy, the (log) density of states is essentially the entropy, so any cognitive significance entropy may have — e.g., entropy reduction as predictor of incremental sentence processing difficulty à la Hale — qualifies as cognitive relevance of density of states. As for the average value of a quantity reflecting less-probable-but-more-numerous states more than more-probable states, I’m not sure what the cognitive significance of average values is in general.

I’ve recently had some useful discussion with people about the nature of representations in OT, and how they did or did not (or should or should not) change from a theory with inviolable constraints (= principles and parameters theory). I’d like to summarize my thoughts, and would very much welcome further discussion.

In our discussion following Tobias Scheer’s mfm fringe presentation, I brought up the point that when one switches to violable constraints, it’s not obvious that the representations should stay the same. Tobias asked for an example, and I didn’t have a good one right away, but then realized that a particularly clear and worked out one is in the discussion of extrametricality vs. nonfinality in Prince and Smolensky 1993/2004. Gaja Jarosz also reminded me of underspecification: since markedness is expressed in output constraints in OT, it’s not obvious that one also requires a theory of input underspecification for that purpose.

I think my own experience of working on my *NC project in the mid nineties illustrates some more aspects of what happened to representations as OT was being extended to segmental phonology, and brings up some further issues. When I started that project, I was looking for an explanation for the facts in terms of feature markedness and positional licensing. I wanted to get the directionality of postnasal voicing, which unlike most other local assimilation processes is L-to-R, instead R-to-L. I also wanted to get conspiracies amongst processes that resolve nasal-voiceless obstruent clusters. I tried hard and failed, and eventually “gave up” and used the formally stipulative but phonetically grounded *NC constraint. I later realized that this failure was part of a more general issue: it doesn’t seem to be the case that positional markedness can always be derived from the combination of general context-free markedness constraints and general positional licensing constraints. The best I could do in terms of those assumptions was to say that the coda nasal [voice] wanted to be licensed in onset position and hence spread, but that didn’t explain why it was just nasals that did this, nor did it deal sufficiently well with directionality. I have some more discussion of the general problems with deriving positional markedness from prosodic licensing, and further references, in a discussion of local conjunction on p. 10 of this paper (also in my review of the Harmonic Mind).

Taking the approach to segmental phonology in the *NC proposal, we can ask what that commits us to in terms of a theory of representations. It looks to me like what we would need is a feature set that is sufficiently expressive to formalize our constraints, but that’s it. Phonetic grounding is expressed as restrictions on the universal set of constraints (or as restrictions on possible rankings, in work like Steriade’s on perceptual grounding). And this set of features could be universal, with no language-specific choices (thanks to Pavel Iosad for a question on this) – contrast and its absence can be captured by ranking alone. Furthermore, I don’t think there is any sense in which this theory has the concept of a natural class (thanks to Kristine Yu for a question on that). So this perhaps at least partially explains why the nature of segmental representations has not been a big topic in at least some variants of OT.

Now I should say that I can see lots of reasons why you would want to say that features do differ from language to language, and why the particular feature set you choose could have consequences for predictions about learning and generalization. But in terms of the particular theory I’ve just described, I don’t see any arguments for language-specific differences in feature specification. I should probably also say that I see reasons why one might not want to model the role of phonetics in phonology as just stipulations about the constraint set with some armchair phonetic justification – there are clearly plenty of alternatives. But the proposal is a natural extension to general approach to encoding of substance in OT as stipulations about constraints: e.g. that there is Onset and NoCoda, but not NoOnset and Coda.

Finally, let me emphasize that what I’ve said about a lack of interest in the nature of segmental representations based on my *NC work should not be taken as representative of a lack of interest in segmental features in OT as a whole (or even in my mind!). For example, there are extremely interesting questions about the nature of the representations needed for `spreading’: Bakovic (2000: diss.) takes the position that there is no spreading, while others have defended autosegmental representations or adopted gestural representations, and yet others (e.g. Cole and Kisseberth) have proposed domain-based representations.

I’d like to raise as a discussion topic the question of what the data are that we are trying to explain in generative phonology. In my view, the lack of clarity about this issue is a bigger foundational issue in our field than the lack of clarity about the goals we are pursuing, one of the discussion points I raised in my mfm fringe workshop presentation. It’s also a foundational issue not only for what I called Classical Universal Phonology in that presentation, but for just about any approach to phonology one can imagine. I should be clear that I don’t think that there needs to be a uniform set of data or goals. Rather, I think we’d be making quicker progress towards our broader shared goals of understanding the formal structure of phonologies, and explaining learning and typology, if we made our commitments in these respects more explicit in our work.

To get the discussion going, let me repeat the worry I expressed in the mfm fringe discussion, and mention some other data-related points that came up. When Marc van Oostendorp pressed me on my assertion that data issues were foundational issues, I brought up the lack of a definition of productivity as an example. It’s unfortunately too common that when an analysis or theory fails to capture some data pattern, the claim is made that the pattern is unproductive (e.g. that there are exceptions, that there are no alternations or that they are limited in some way, etc.), without applying the same scrutiny and criteria to the data that the theory is capturing. Probably even more common is that exceptions or variation are abstracted from, again without any clear criteria on when that can be done. My own belief is that productivity is gradient (see Hayes’ textbook ch. 9), and that we need theories that capture that gradience. But whether we are working with theories that are categorical or gradient in this respect, we need to define productivity if we are going to use it as a criterion for what data we need to explain.

In his question period at the fringe, Michael Becker pressed his interlocutors to provide evidence that the generalizations they saw in existing alternations were in fact encoded as generalizations in speakers’ minds. Becker’s approach, like that of a lot of other current work, is to test productivity experimentally. I’m on board with that program, but I’m also on board with good old analysis of corpus data (where ‘corpus’ includes the data from grammars and dictionaries that phonologists typically study), and I’m starting to get worried about what to do when the two sets of data point in very different directions. For example, the ‘stress heavy if penult’ part of the Latin stress rule is a nearly exception-free pattern in unsuffixed nouns in English. But as Claire Moore-Cantwell (p.c.) reports, it seems that it’s not particularly productive in nonce word productions/judgments. Claire has some good ideas about how to relate the corpus data to the judgments via learning, but it’s clear that the grammar is going to look very different from those posited for English from Chomsky and Halle (1968) onwards.

Wendell Kimper mentioned in his talk the issue that the set of attested human languages appears to be a small sample from the space of possible human languages. There are various kinds of statistical measures and data controls that we can use to determine how robust the typological generalizations are that we observe. But Kimper also reports that vowel harmony looked at that way may provide relatively little information, since many of the patterns of each type of harmony come from the same language families. My gut feeling, like Wendell’s I think, is that in those circumstances we should still keep going with the usual practice of just making an attested/unattested cut, and hoping that we are modeling signal rather than noise. But it is a worry, and probably one of the reasons that it’s good that we’re not putting all of our eggs in the typology modeling basket. A possible strategy is to focus on typological claims with a relatively large scope, for example, the size of stress windows, or the absence of sour grapes-style harmony (and the presence of spreading up to a blocker).

I’d very much welcome comments or questions on my handout from the Manchester fringe session, held May 27th, 2015  [also: prepared introduction not on handout]. I’m not planning on turning it into a paper, though further discussion might change my mind! I also hope that this page can serve as a place for discussion of other issues raised at the workshop – links to handouts and other relevant materials would be very much appreciated.