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Basic Assumptions about a Sign's Life Cycle
for Mathematical Modelling of Language
System Evolution
A.A.Polikarpov D.V.Khmelev
24 August 2000
KEYWORDS: language evolution, mathematical modelling, branching pro-
cesses
AFFILIATION: Lomonosov Moscow State University, Russia; Heriot-Watt
University, Edinburgh, UK and Isaac Newton Institute for Mathematical
Sciences, Cambridge, UK.
POSTAL ADDRESS: 20 Clarkson Road, Cambridge, CB3 0EH, U.K.
FAX NUMBER: (01223) 330508
E-MAIL ADDRESS: polikarp@philol.msu.ru, D.Khmelev@newton.cam.ac.uk
Contents
1 Aim of the paper. 1
2 Basic assumptions. 2
3 3. Dying out, giving birth, and preserving of sign's mean-
ings. 2
4 Drawing a curve for a sign's polysemy dynamics. 3
5 Regularities for the dynamics of a sign's sense volume, fre-
quency of use, length, etc. 4
1

6 Arriving at the conclusion on general shape of momentary
word polysemy, frequency of use, length, etc. distributions
in language. 4
1 Aim of the paper.
Modelling of language evolution should be based on some assumptions con-
cerning its micro-level, i.e. level of its micro-units' development. A sign
(morphemic, lexemic and phraseologic) is an elementary (micro-) unit on
some certain level of language organisation. A sign's polysemy evolution in
time is the most fundamental ontological fact. That is why it has become the
starting point in the building of the mathematical model for the development
in time and synchronic correlation of the whole system of a sign's features.
Correspondingly, it can lead to building a theory of the organisation and
historical development of language systems as a whole.
2 Basic assumptions.
(1) A sign's polysemy development is a branching process of generating new
meanings from previously acquired (and, correspondingly, losing some previ-
ously generated) ones.
(2) According to the increase of the ordinal number i of meaning's gen-
eration within a sign there should proportionally grow the average degree of
meaning's abstractness A i (or, in other words, decrease the average degree of
meaning's lling by some number of semantic components B i ). This means
that A i = 1=B i .
(3) The more abstract, on the average, the meanings of some generation
of a sign are, the greater stability L i (length of life) specic to each of them.
(4) The more abstract, on the average, each meaning of some generation
of a sign, the lower generating activity G i (number of meanings of the next
generation produced from a meaning in its life) specic to each of them is.
(5) The more abstract meanings of some generation, the greater sense
volume V i (number of senses covered by each of them) that is specic, on the
average, to each of them.
(6) The greater sense volume of meanings of some generation, the higher,
on the average, the frequency of use U i for each of them is.
2

These assumptions provide us with the ability to draw some useful con-
clusions for modelling of some other functional dependences for any language
sign, as well as for ensembles of them, i.e. for a language system as a whole.
3 3. Dying out, giving birth, and preserving
of sign's meanings.
Consequence 1. From the fact of a nite number of features in any sign's
meaning it follows that maximal possible number of generations of meanings
in a sign can not exceed some n.
Consequence 2. From assumptions (1)-(4) it follows that L 1 L 2
L n and G 1 G 2 : : : G n 1 .
We shall consider evolution of a sign in continuous time. Let i = 1=L i
and i = G i =L i . Clearly, i is a decay rate of meanings of generation i in
a sign and i is a rate for generating new meanings (meanings of the next
generation i+1) by each meaning of a generation i. Let us assume that during
small intervals of time t every meaning of a generation i independently of
all other sign meanings does the following:
1) dies with probability i t + o(t),
2) if 1 i n 1 then it generates a meaning of the next generation
i + 1 with probability i t + o(t).
Otherwise a meaning just preserves itself, continues its existence (with
probability 1 ( i + i )t + o(t)).
It is easy to prove that within the model activity of a meaning belonging
to a generation i, i.e. average number of meanings of a generation i + 1
produced by a meaning of a generation i, equals G i .
4 Drawing a curve for a sign's polysemy dy-
namics.
It is much more diфcult to check a conclusion that a polysemy curve of a
typical sign should have only one global maximum. Assumption (7) reduces
the model to the branching process with the denite number of types of
particles (see [1]). Denote [1] by P i

the probability of the fact that a meaning
of a generation i will generate for the time t meanings determined by the
3

vector = ( 1 ; : : : ; n ): 1 meanings of a generation 1, . . . , n meanings of
a generation n. Dene generating functions
F i (s) =
X
0
P i

(t)s ;
where s = s 1
1 s n
n
. Also dene a vector generating function F (t; s) =
(F 1 (t; s); ; F n (t; s)) T . It follows from [1, p.119, theorem 3] and from as-
sumption (7) that F (t; s) satises the system of equations
@F (t; s)=@t = f(F (t; s)) (1)
with the initial conditions F (0; s) = s. Here f(s) = (f 1 (s); : : : ; f n (s))
where f i (s) = i ( i + i )s i + i s i s i+1 (we put n = 0). It is impossi-
ble to nd an explicit solution of the system (1) for all initial conditions
and all values of parameters. Nevertheless, behaviour of the average number
of sign meanings M(t) at the moment t is described by the system of lin-
ear dierential equations. It is possible to obtain the following formula for
M(t) = L 1 p 1 (t) +G 1 L 2 p 2 (t) + : : : +G 1 : : : G n 1 L n p n (t) where p i (t) for t 0
is a density for the sum of i exponentially distributed independent random
variables of means L 1 , . . . , L i .
Theorem 1. Under assumptions (1){(7) we have only two qualitatively
dierent kinds of behaviour for M(t) when t 0:
1. If G 1 > 1 then there exists a unique maximum at t > 0: M(t ) > M(t)
for all t 2 [0; 1] n ft g. Also M 0 (t) 0 for all t 2 [0; t ] and M 0 (t) 0 for
all t 2 [t ; 1].
2. If G 1 < 1 then M 0 (t) 0 for all t 0 and M(t) reaches its global
maximum at t = 0: M(0) = 1.
5 Regularities for the dynamics of a sign's
sense volume, frequency of use, length, etc.
These regularities are deduced on the basis of conclusions made earlier on the
polysemy quantitative dynamics and some other \qualitative" assumptions
(see "Basic assumptions").
4

6 Arriving at the conclusion on general shape
of momentary word polysemy, frequency of
use, length, etc. distributions in language.
For deducing the general shape of momentary distribution in language for
these features we assumed that some independent source generates signs
for a language with some constant rate 0 . It is possible to show that in
time this system arrives at some stationary state and to nd its numerical
characteristics.
Further details on this point, analytical deriving of other dependences,
as well as presenting of some empirical data for testing the deduced form of
polysemy distributions of lexemic signs (words) in various types of dictionar-
ies of languages of various types | Russian, English, Chinese, Vietnamese,
Mongolian, Hungarian, Estonian, Turkmen, Turkic, Tartar, Azerbaijan, etc.
| will be made in the extended version of this paper.
References
Sevast'janov B.A. (1976) Vetvyashchiesya protsessy. (Russian) [Branching
processes] Izdat. \Nauka", Moscow.
5