LINGUIST List 2.678

Thu 17 Oct 1991

Disc: Infinite Languages

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  1. , Infinite Languages
  2. Jacob Hoeksema, Re: 2.663 Is Language Finite? Polite Forms

Message 1: Infinite Languages

Date: Tue, 15 Oct 91 11:49:30 EDT
From: <Alexis_Manaster_Ramermts.cc.wayne.edu>
Subject: Infinite Languages
LINGUIST has once again made it possible for a significant issue
to be dissected in a way which does not seem possible in any
other forum I can think of! Avery Andrews and I seem
to disagree on the status of the claim that NLs are sets of
sentences without an upper bound on their lengths (but always
of finite length). Yet, as he points out in his latest, the
real issue he is concerned with is not cardinality but rather
the goals of linguistic theory as originally defined by Chomsky.
I, on the other hand, was concerned specifically with the question
of cardinality (and more generally with the issue of how
appropriate it is to treat mathematical idealizations as literal
claims about the real world).
Having said this, I would like to try and convince Avery and
everyone else that we must make some distinctions that are not
usually made in this area:
(1) To say that the limits on actual sentence length are undefined
is not the same thing as saying that there aren't any. Let me
use a simple analogy to illustrate this point. The set of citizens
of the United States is finite and reasonably well-defined. The
set of black citizens of the U.S. is not well-defined at all, yet
it is clearly a subset of the former, and hence also finite.
(2) The fact that sentences as they become longer and more complex
also become less acceptable can be captured in a number of ways,
one of which is to divide up your theory into two components,
one called competence, the other performance, and let the second
only worry about this fact. However, this is NOT the only
reasonable alternative. Another is to assume that the device (or
gadgetry, as Avery calls it) that we have is one that does things
in real time, and that competence is an idealization of it. Thus,
the theory of competence is not about a specific mental organ
separate from performance. It is about the one mental organ that
there is (but it is idealized in certain important ways, which is
perfectly reasonable, by the way. I have no quibble with
idealizations.)
(3) We are used to devices (automata) really which do not get
tired, i.e., which can do the same thing over and over without any
diminution of, let us say, acceptability (or grammaticality). This
is something that is commonly assumed in mathematical theories of
automata, but then ALL physical properties of automata are ignored
there. It would be reasonable to say that we can develop a theory
of automata that do get tired, which might offer a closer analogy
to human behavior (i.e., be less idealized in this respect).
(4) It is by no means obvious, as a linguistic fact, that people
reject long or involved utterances merely because they are not
comprehensible or whatever. I have often enough had the experience
of an informant who would understand perfectly what was intended
and yet opine that it was "too complicated" and offer a paraphrase.
The famous Dutch (and German) V-Raising constructions are a perfect
example. I have had Dutch informants who would understand
an utterance like:
 dat Jan Marie Hans heeft helpen laten zwemmen
but insist on a paraphrase. In general, it seems to me that there
are plenty of arguments that the division we tend to make between
competence facts and performance facts is arbitrary and that it
forces us to miss generalizations.
(5) One particular kind of problem that the usual idealization
causes is that it forces to assume that the set of morphemes
of a language is a finite list, whereas the set of sentences
is infinite. Yet in many respects the set of morphemes is also
what used to be called OPEN (as opposed to CLOSED).
 In a paper about to apper, Daniel Radzinski and I argue that
linguistics really needs the notions of open (alias productive) vs.
closed (alias unproductive) instead of infinite vs. finite, and we
offer a precise mathematical model of what we mean (on this view,
then, a language which reduplicates verb stems for some grammatical
function can be shown to be non-context-free, for example, whereas
on the traditional view, the set of stems being finite, the
language would be finite, hence regular, hence CF).
 As I alluded to at one point, some people in CS have explored
models in which the "language" is strictly speaking not a single
set of string but a set of sets of strings (each of the latter sets
being finite). It is difficult to distinguish this view from the
traditional one on factual grounds, though there are some cases
where there is a difference. This is essentially the approach
Radzinski and I employ: the set of morphemes is finite at any given
point in time, but it varies from one moment to the next (and there
infinitely many possible such lexicons).
 Again, my point is that there are, mathematically, many more
possibilities than linguists have tended to assume, and that some
of them may be better than what we have assumed (and in other cases
it does not matter much or at all what we assume).
?
?(7) Returning now to the question of infinite strings (which is I
think an example of where it does not matter much what we assume),
mathematicians and computer scientists have explored automata on
infinite strings (in fact, a talk was given on this at the last MOL
conference by a student of mine), which means that there is no
basis for the statement that "Infinite sentence lengths are on the
other hand not attainable
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Message 2: Re: 2.663 Is Language Finite? Polite Forms

Date: Wed, 16 Oct 91 16:30:39 MET
From: Jacob Hoeksema <hoeksemalet.rug.nl>
Subject: Re: 2.663 Is Language Finite? Polite Forms
Avery Andrews claims (following here Chomsky) that linguistics is concerned
with the structure of an actual device and that actual devices do not produce
infinite output. Therefore, sentences of infinite length can be ignored
and the idealization of choice for formal linguistics should be that
sentences are finite sequences with no upper-bound imposed on their length
(contra Langendoen, Postal).
However, following the footsteps of Chomsky a little further, one may ask
if this is not to confuse competence and performance. After all, while
I may be able to decide that
 n
I think (that I think)
is grammatical for any n, including perhaps omega, I certainly won't
be able to utter more than a small finite number of "that I think"'s.
Indeed, in various contexts it even may make sense to talk about
actually uttering infinite sequences. (For example, if time allows
loops long enough to say "that I think", or if the speed of talking
can be increased at ever shorter periods -- and why not, if we are
allowed to ignore speech rates in our linguistic theorizing -- or,
thirdly, if we allow one sentence to be uttered by more than one
speaker -- e.g. by consecutive generations (assuming that physics
does not dictate an end to our universe ;-)). I hope that these
objections suffice to show that unless Andrews is willing to forego
the compentence/performance distinction altogether, he cannot get
away with the claim that the very nature of the game of linguistics
prohibits the consideration of infinite sequences. The question
is rather, whether it is worthwhile to ponder about infinity in
linguistics. I for one consider it to be primarily of recreational
value.
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