Convergence and Crash of a derivation (Chomsky)

Question

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.

Answer 1

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).

Answer 2

• 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.