Convergence and Crash of a derivation (Chomsky)

I would like to pose you a terminological question, regarding the following quotes from Noam Chomsky's work:

"A strong feature must be eliminated (almost) immediately upon its introduction into the phrase marker; otherwise, the derivation cancels."

"A strong feature that is not checked (and eliminated) in overt syntax causes a derivation to crash at LF." [LF= Logical Form]

" ‘‘strong’’ features are visible at PF and ‘‘weak’’ features invisible at PF. These features are not legitimate objects at PF; they are not proper components of phonetic matrices. Therefore, if a strong feature remains after Spell-Out, the derivation crashes."

"Alternatively, weak features are deleted in the PF component so that PF rules can apply to the phonological matrix that remains; strong features are not deleted so that PF rules do not apply, causing the derivation to crash at PF." [PF = Phonetic Form]

"The language L determines a set of derivations (computations). A derivation converges at one of the interface levels [PF, LF] if it yields a representation satisfying FI [FI = Full Interpretation] at this level, and converges if it converges at both interface levels, PF and LF; otherwise, it crashes. "

Chomsky is taking some kind of computer-science terminology, but I am not sure whether he makes some systematic and standard use of the terms or whether he just loosely hovers around them. The idea of a computation crashing (or being blocked with no chance of going forward, or of the system collapsing, whatever) seems quite intuitive, but, does it refer to a technical use from his part? I am more interested, though, in the rationale behind "convergence". Does it mean the derivation converges to a numerical value $0$?

Thanks in advance for your replies.

• and which work of Chomsky are you citing? – miracle173 Dec 27 '14 at 16:15
• The Minimalist Program, MIT Press, 1995. – Javier Arias Dec 27 '14 at 16:16

2 Answers

From a theoretical computer science point of view, to "crash" is not standardly a technical term. (Of course computer scientists speak informally about programs crashing, but usually not as a technically well-defined thing).

"Covergence" in the sense of the quote sounds less informal, but is still not quite standard. What is standard, however, is to speak of a computation "diverging" if it goes on forever without reaching a result, so using "converge" about a computation that does yield a result is not too far off. (The more conventional word choice would be "terminate", however).

• Does this use of convergence have anything to do with convergence to zero in functional analysis and the like? – Javier Arias Dec 27 '14 at 16:08
• @JavierArias: Only indirectly, and not, I'd think, in any terribly enlightening way. – Henning Makholm Dec 27 '14 at 16:11
• @ Henning Makholm. Thanks a lot. Wasn't there in functional analysis some requirement for a kernel to be symmetric that i some cases its only value be zero or something like that? I am telling that because of the use linguists make of kernels, albeit in a different context. – Javier Arias Dec 27 '14 at 16:15
• @Javier: Doesn't ring any bell for me. (Also, I suspect functional analysis doesn't mean what you think it does; it would make more sense for you to be talking about ordinary real analysis here). – Henning Makholm Dec 27 '14 at 16:30
• @JavierArias: I don't completely understand what he's saying based on your quotes, but yes, it sounds like you're just guessing randomly. In particular, it sounds (both in this question and in your other ones) that you have a tendency to think that just because the same word is used in different areas of mathematics, there must necessarily be a deep analogy between their uses. But reality is not so nice and tidy -- people chose words for their technical concepts willy-nilly and without coordinating with anyone. – Henning Makholm Dec 29 '14 at 13:25
• What could be a "crash" in the domain of computation? One of the questions in computation theory is if the mathematical machine under consideration is able to terminate on its input or not, this means: Is it able to finish its computation in a finite number of steps?

For those inputs where a machine does not terminate, it will go through all it its states, without ever reaching a final state. For example it traverses an endless loop.

So that is more a "hang" than a "crash".

• What could Chomsky mean?

What I read so far looks similar to the derivation processes of formal languages (link). There, a similiar problem to the termination of a machine function shows up in the question of generating / deriving a sentence from some start symbol using the rules of the grammar in finite many steps.

That cited derivation of the minimalist program (link) seems about natural language, processed by a brain. Some part of the brain seems to produce sentences via similar derivations steps and a talk part (PF) and a think part (LF) of the brain watch this and if they accept the result, it is called convergence.

I personally never heard the term "convergence" used for a successful (= terminating) derivation in the context of formal languages.

The term convergence in the mathematical context is usually associated with an successful approximation process of some kind, where one gets closer to some object, starting with sequences of numbers usually.

The derivation of formal languages is all about deciding if a sentence belongs to language or not, if it is well formed. The "is element of" relation $\in$ just has the values true ($1$) or false ($0$).

So I wonder if that MP-derivation-convergence is about the convergence of some sequence $(\in_k)_{\in\mathbb{N}}$ on $\{ 0, 1 \}$ or $\{ 0, 1 \}^2$ (the set of pairs of numbers from $\{ 0, 1\}$), the $k$-th member related to the $k$-th derivation step.

• Well, but keep in mind that the analogy has its limits, since we are talking about the human mind / brain computing a sentence that has effectively heard or just milliseconds before uttering it....so the fact that computation is not deterred for eternity is a given....crash in this sense would mean collapse and then move on (to trying to produce another sentence, for instance), and not an aeternal loop. In other words, collapse would mean some crucial violation of well-formedness has been detected at some point of the derivation. – Javier Arias Dec 30 '14 at 19:31
• @JavierArias I gave my thoughts, but I lack the knowledge of that linguistic MP derivation to be of substantial help. – mvw Dec 30 '14 at 21:13
• @nwm That is fine. – Javier Arias Dec 30 '14 at 22:16