Derived objects and derivation trees

I am pretty slow to understand things that are new to me. So I have a hard time rethinking things in new ways, being very comfortable with what I know, which is all a longwinded way of saying that when I started reading Stabler’s take on Minimalism (and Greg Kobele’s and Thomas Graff’s and Tim Hunter’s) that emphasized the importance of derivation trees (as opposed to derived objects) as the key syntactic “objects” I confess that I did not see what was/is at stake. I am still not sure that I do. However, thinking about this stuff led me to some thoughts on the matter and I thought that I would put them up here so that those better versed on these issues (and also more mentally flexible) could help set me (and maybe a few others) straight.

The way I understand things, derivation trees are ways of representing the derivational history of a sound/meaning pairing. A derivation applies some rules in some sequence and interpreting each step allows for a phonological and a semantic “yield,” (a phonetic form and a meaning correlate of these sequence of steps).  Some derivations are licit, some not. This divides the class of derivation trees into those groups that are G-ok and those that are not. However, and this seems to be an important point, all of this is doable without mentioning derived objects (i.e. classical phrase markers (PM)). Why? Because PMs are redundant. Derivation trees implicitly code all the information that a phrase marker does as the latter are just the products that applying rules in a particular sequence yield. Or, for every derivation tree it is possible to derive a corresponding PM.

Putting this another way: the mapping to sound and meaning (which every syntactic story must provide) is not a mapping from phrase markers to interpreted phrase markers but from sequences of rules to sound and meaning pairs (i.e. <s,m>). There is no need to get to the <s,m>s by first going through phrase markers. To achieve a pairing of articulation and semantic interpretations we need not transit through abstract syntactic phrase markers. We can go there directly. Somewhat poetically (think Plato’s cave), we can think of PMs as shadows cast by the real syntactic objects, the derivational sequence (represented as derivation trees), and though pretty to look at shadows (i.e. PMs) are, well, shadowy and not significant. At the very least, they are redundant and so, given Occamite sympathies for the cleanly shaved, best avoided.

This line of argument strikes me as pretty persuasive. Were it the case that derived objects/PMs didn’t do anything, then though they might be visually useful they are not fundamental Gish objects (sort of like linguistic versions of Feynman diagrams). But, I think that the emphasis on derivation trees misses one important function of PMs within syntax and I am not sure that this role is easily or naturally accommodated by an emphasis on derivation trees alone. Let me explain.

A classical view of Gs is that they sequentially map PMs into PMs. Thus, G rules apply to PMs and deliver back PMs. The derivations are generally taken to be Markovian in that the only PM that a G must inspect to proceed to the next rule application is the last one generated. So for the G to licitly get from PM_n to PM_n+1 it need only inspect the properties of PM_n. On this conception, a PM brings information forward, in fact all (and in the best case, no more) of the information you need to know in order to take the next derivational step. On this view, then, derived objects serve to characterize the class of licit derivation trees by making clear what kinds of derivation trees are unkosher. An illustration might help.

So, think of island effects. PMs that have a certain shape prohibit expressions within them from moving to positions outside the island. So an expression E within an island is thereby frozen. How do we code islandhood? Well, some PMs are/contain islands and some do not. If E is within an island at stage PM_n then E cannot move out of that island at stage PM_n+1. Thus, the derivation tree that represents such movement is illict. From this perspective, we can think of derived objects as bringing forward information in derivational time, information that restricts the possible licit continuations of the derivation tree. Indeed, PMs do this in such a way that all the information relevant to continuing is contained in the last PM derived (i.e. it supports completely markovian derivations). This is one of the things (IMO, the most important thing) that PMs (aka derived objects) brought to the table theoretically.

So the relevant question concerning the syntactic “reality” of PMs/derived objects is whether we can recapture this role of PMs without adverting to them. And the answer should be “yes.” Why? Well derived objects just are summations of previous derivational steps. They just code prior history. But if this is what they do, then derived objects are, as described above, redundant, and so, in principle eliminable. In other words, we can derive all the right <s,m>s by considering the class of licit derivation trees and we can identify these without peaking at the derived objects that correspond to them.[1]

This line of argument, however, is reminiscent of another one that Stabler (here) critically discusses. He notes that certain kinds of context free grammars (MCFGs) can mimic movement with the effect that all the same <s,m>s that a movement based theory derives such a CFG can also derive and in effectively the same way. However, he notes that these CFGs are far less compact than the analogous transformational grammars and that this can make an important difference cognitively. Here’s the abstract:

Minimalist grammars (MGs) and multiple context free grammars (MCFGs)

are weakly equivalent in the sense that they define the same languages, a

large mildly context sensitive class that properly includes context free languages.

But in addition, for each MG, there is an MCFG which is strongly

equivalent in the sense that it defines the same language with isomorphic

derivations. However, the structure building rules of MGs but not MCFGs

are defined in a way that generalizes across categories. Consequently,MGs

can be exponentially more succinct than their MCFG equivalents, and this

difference shows in parsing models too. An incremental, top-down beam

parser forMGs is defined here, sound and complete for allMGs, and hence

also capable of parsing all MCFG languages. But since the parser represents

its grammar transparently, the relative succinctness of MGs is again

evident. And although the determinants of MG structure are narrowly and

discretely defined, probabilistic influences from a much broader domain

can influence even the earliest analytic steps, allowing frequency and context

effects to come early and from almost anywhere, as expected in incremental

models.

Can we apply the same logic to the discussion above? Well maybe. Even if the derivation trees contain all the same information that a theory with PMs does, does it make it available in the same way that PMs do? Or, if we all agree that certain information must be “carried forward” (Kobele’s elegant term) derivationally, might it make a cognitive difference how this information is carried forward; implicitly in a derivation tree or explicitly in a PM? Well, here is one place to look: One thing that PMs allow is for derivation to be markovian. Is this a cognitively important feature, analogous to being compact? I can imagine that it might be. I can imagine that Gs being markovian has nice cognitive properties. Of course, this might be false. I just don’t know. At any rate, I have no problem believing that how information is carried forward can make a big computational difference. Consider an analogy.

Think of adding a long column of multi-digit numbers. One useful operation is the “carry” procedure whereby only the last digit of a column is registered and the rest is carried forward to the next column. But is “carrying” really necessary? Is adding all but the last digit to the next column necessary? Surely not, for the numbers carried can be recovered at every column by simply adding everything up again from earlier ones. Nonetheless, I suspect that re-adding again and again has a computational cost that carrying does not. It just makes things easier. Ditto with PMs. Even if the information is recoverable in derivation trees, PMs make accessing this information easy.

Let me go further. MPs of the kind that Stabler and his students have developed don’t really have much to say about how the kinds of G restrictions on derivations are to be retrieved in derivation trees without explicit mention of the information coded in the structural properties PMs exhibit. The only real case I’ve seen discussed in depth is minimality, and Stabler-like MGs (minimalist grammars) deal with minimality effects by effectively supposing that they never arise (it is never the case in Stabler MGs that a licit derivation allows two objects to have the same accessible checkable features). This rendering of minimality works well enough in the usual cases so that Stabler MG formalizations are good proxies for minimalist theories “in the wild.”  However, not only does this formalization not conform to the intuition that most syntactitians have about what the minimality condition is, it is furthermore easy to imagine that this is not the right way to formalize minimality effects for there may well be many derivations where more than one expression carries the requisite features in an accessible way (in fact, I’ve heard formalists discussing just this point many times (think multiple interrogation or multiple foci or topics or even case assignment). This is all to say, that the one case where a reasonably general G-condition does get discussed in the MP literature leaves it unclear how MP should/does treat other conditions that do not seem susceptible to the same coding (?) trick. Or, minimality environments are just one of the conditions that PMs make salient. It would be nice to see how other restrictions that we explain by thinking of derivations as markovian mappings from PMs to PMs is handled with derived objects.[2] Take structure dependence or, Island/ECP effects or the A-over-A condition for example. We know what needs doing: we need to say that some kinds of G relations are illicit between some positions in a derivation tree so that some extensions of the derivation tree are G-illicit. Is there a nice compact description of what these conditions are that make no mention, however inadvertently, of PMs?

That’s it. I have been completely convinced that derivation trees are indispensible. I am convinced that derived objects (aka PMs) are completely recoverable from derivation trees. I am even convinced that one need not transit through PMs to get to the right <s,m> pairs (in fact, I think that thinking of the mapping via PMs that are “handed over” to the interpretive components is a bad way of thinking of what Gs do). But this does not yet imply, I don’t think, that PMs are not important G like objects. At the very least they describe the kinds of information that we need to use to specify the class of licit derivation trees. Thus, we need an account of how information is brought forward in derivational time in derivation trees and, more importantly, what is not. Derived objects seem very useful in coding the conditions on G-licit derivation tree continuations. And as these are the very heart of modern GG theory (I would say the pride of what GG has discovered) we want to know how these are coded with PMs.

Let me end with one more historical point. Syntactic Structures argues that transformations are necessary to capture evident generalizations in the data. The argument for affix hopping and Aux movement was not that a PSG couldn’t code the facts, but that it did so in such a completely ugly, ad hoc and uninformative way. This was the original example for the utility of compact representations. PMs proved useful in similar ways: inspecting their properties allowed for certain kinds of “nice looking” derivations. The structure of a given PM constrained what next derivational was possible. That’s what PMs did well (in addition to feeding <s,m>s). Say we agree that PMs are not required (or really add much) in understanding the mapping between sounds and meaning (i.e. in deriving <s,m> pairs) what of the more interesting use to which OMs were made (i.e. stating restrictions on derivations). Is this second function as easily, insightfully discharged without PMs? I’d love to know.

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[1] It is perhaps noteworthy that there is not a clear match between grammaticality and semantic interpretability. Thus, there are many unacceptable sentences that are easlity interpreted and, in fact, have only one possible interpretation (e.g. The child seems sleeping, or Which man did you say that left). This, IMO, is an important fact. We odn’t want our theories of linguistic meaning to go off the rails if a sentence is ungrammatical for that would (seem to) imply that it has no licit meaning. Now there are likely ways to get around this, but I find nothing wrong with the idea that an expression can be semantically well formed even if syntactically illicit and I would like a theory to allow this. Thus, we don’t want a theory to not yield a licit <s,m> just because there is no licit derivation. Of course, how to do this, is an open question.

[2] See here and here for another possible example of how derivation trees handle significant linguistic details that are easy to “see” if one adopts derived objects. The gist is that bounding “makes sense” in a theory where PMs are mapped into PMs in a computationally reasonable way. It is less clear why sticking traces into derived objects (if understood as yields (i.e. ss or ms in <s,m> pairs) makes any sense at all given their interpretive vacuity.