LINGUIST List 3.730

Mon 28 Sep 1992

Disc: Parameter Setting

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1. , Parameter Setting and the Garden of Eden. II

 Parameter Setting and the Garden of Eden. II

 I recently published a statement and discussion of a result in the
mathematical theory of languages. In brief, I show that under very general
conditions, the group of human languages (=parameter settings in a
parameter-setting model of human language) will have defined on it a very
specific type of metric structure called an ultrametric topology. Intuitively,
using a natural measure of separation between languages, the existing languages
should form a tree structure.

 My posting of this result on LINGUIST has led to many useful and interesting
discussions. Some forced me to modify the statement of the result. In
particular, my result concerns the variation of linguistic typology, not
lexical relatedness. I was misled in my understanding of this last point
by the involvement of Joe Greenberg in such projects. My ignorance on this
point is now being cured through references kindly suppled by readers.
(More than one respondent regarded it as a virtue of
my result that it was in fact UN-related to the work on Proto-Nostratic.)
Other respondents encouraged
me to modify the presentation of the result. At any rate, I enclose here a
re-statement of my original argument. It is, as before, in the form of a logical
implication: granting certain premises, certain highly nontrivial, testable
results follow. As with any such result, one is free to reject the validity for
the study of language of the hypotheses I employ.

 If hypotheses of a general type are accepted, I show that ultrametric
structures may be a natural tool in the study of language. Since these are
quite new even to physicists (who delight in sifting through mathematics for
items of use) I note here a pedagogical review article that explains
ultrametric structures to those who have but a rudimentary background in basic
algebra. The article is:
 "Ultrametricity for Physicists", by R. Rammal, G. Toulouse, and M.A.
 Virasoro, Reviews of Modern Physics 58, 765 (1986)

 Again, thanks to all who helped me refine the result stated below.

 J.A. Given
 SUNY Stony Brook

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I ASSUME the following two statements are true:

1. A "principles and parameters" approach to grammar is possible, with the
possible human languages corresponding to points in a parameter space.

2. Some quantity which I here term "communicative efficiency" is optimized or
near-optimized by each of the languages, equivalently by each of the sets of
parameters, that humans do in fact actually employ. I call this quantity
"communicative efficiency" just to give it a name. All I require is that SOME
QUANTITY be maximized (or nearly maximized), and that that quantity have a
specific quality which I here term FRUSTRATION. The term FRUSTRATION is here
being used in a specific technical manner:
 I say that a function defined on a parameter space is FRUSTRATED or
 has FRUSTRATION if it can be written as the sum of many independent
 terms or contributions, such that no set of parameters gives an
 extremal value to a large fraction of the separate terms. The
 picture implied here is that of a quantity which involves a
 weighting on the performance of a large number of competing subgoals
 such that the sets of parameters which maximize one particular term
 will seldom maximize other terms. Under these conditions, the sets
 of parameters which maximize the entire "communicative efficiency"
 function will do so by assigning moderate values to many different
 terms.

The quantity just characterized as FRUSTRATION is found empirically to be a
characteristic of many utility functions, cost functions, and free energy
functions appropriate to complex systems in engineering, economics, and
physics.

 "Communicative efficiency" functions displaying the trait called above
"frustration" have MANY SETS of parameter values for which they are maximized
or nearly maximized. This collection of efficient sets of parameter values
(to be identified with human languages according to postulate 2 above)
forms a HIERARCHICAL TREE-LIKE STRUCTURE under an overlap metric which is a
natural way to compare two settings for the parameters. Because there are many
near maxima in this collection, the last statement is highly nontrivial. Its
chance of being true by accident is vanishingly small. The existence of such a
tree-like structure is verified numerically in detail in many cases studied
by computer. The tree-like structure can be rigorously constructed
mathematically in certain paradigm models called spin glasses which have been
studied in detaill by theoretical physics. A good introduction to these topics
that doesnot require advanced mathematical background is:

 "Ultrametricity for Physicists", by R. Rammal, G. Toulouse, and M.A.
 Virasoro, Reviews of Modern Physics 58, 765 (1986)

 The tree-like structure just described is called by mathematricians an
"ultrametric topology". How much "frustration" must be present in a
"communicative efficiency" function in order for one to expect this structure?
One has no general theory about this, but only a body of empirical experience
that leads one to expect it under the conditions described. Mathematicians
studying neural networks use the spin glass systems mentioned above as a
paradigm. They have proven theorems corresponding to the statements made above
for certain classes of model systems. However, the above characterization is, I
think, best regarded as "empirical mathematics", i.e., as a group of
statements which numerical and computational experience show to be true in many
cases that have been studied.

 In other words, if an explicit parametrization were available for human
grammar, one could explicitly test the implications I discuss here. The
result implied is quite nontrivial and rather general. I emphasize that the
parameters involved may take discrete values; no postulates of "smoothness",
"continuity", etc. need be made.

 How to verify the extent to which such statements are true? We need to see a
large family of languages mapped out in parameter space. I emphasize that the
tree-like structures predicted here occur in the variation of human languages
as a function of their TYPOLOGY, not as a function of any lexical similarity.
So the recent well-publicized studies of language super-families are not
relevant to the results obtained here.

 The above is an argument that, assuming conditions 1 and 2 the set of
linguistic typologies actually realized will form a "family tree structure".
These results make no assumptions about any contact between languages,
influence of one language upon another, etc.