The number of occurrences of letters versus their distribution in some EOL languages

Abstract

A characterization theorem is given for a class of developmental languages. The theorem binds together the number of occurrences of letters in the words of the given language with the distribution of these letters.

Full text

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INFORMATION AND CONTROL 26, 256-271 (1974)
The Number of Occurrences of Letters Versus
Their Distribution in Some EOL Languages
A. E H R E N F E U C H T
Department of Computer Science, University of Colorado,
Boulder, Colorado 30302
AND
G. ROZENBERG
Department of Mathematics, Utrecht University, Utrecht-Uithof, The Netherlands
A characterization theorem is given for a class of developmental languages.
The theorem binds together the number of occurrences of letters in the words
of the given language with the distribution of these letters.
INTRODUCTION
This paper deals with a class of developmental languages. The theory
of developmental systems and languages originated in the works of
Lindenmayer (1968). This theory provided a useful theoretical framework
within which the nature of cellular behavior in development can be discussed,
computed and compared (see, e.g., Herman and Rozenberg, to appear,
Lindenmayer, 1968, and Lindenmayer and Rozenberg, 1972). It turned out
that developmental systems and languages are interesting and novel objects
from the formal language theory point of view. Especially in comparison
with Chomsky grammars and languages (see, e.g., Ginsburg, 1966) they
provided a lot of insight into the basic problems of formal language theory.
An important subclass of developmental systems are the so-called E0L
systems (see, e.g., Herman, 1974, or Herman, Lindenmayer, and Rozenberg),
which were devised to allow descriptions of development which take into
account the inaccuracy of our observations.
One of the basic open problems within the theory of E0L systems (and
in fact within the whole theory of developmental systems) is the charac-
256
Copyright © 1974 by Academic Press, Inc.
All rights of reproduction in any form reserved.

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E0L LANGUAGES 257
terization theorems that allow one, for example, to prove that some languages
are not E0L languages (i.e., languages generated by the E0L systems).
This paper provides such a characterization for a subclass of E0L
languages. The characterization theorem binds together the number of
occurrences (in the words of the given E0L language) of letters from a
given set of letters with the distribution of these letters.
The paper also discusses some applications of the main result.
1. PRELIMINARIES
We assume the reader to be familiar with the basics of formal language
theory (see, e.g., Ginsburg, 1966, whose notation and terminology we shall
mostly follow). In addition to this, we shall use the following notation:
(i) N denotes the set of nonnegative integers and N+ = N -- {0}.
If n is an integer, then abs(n) denotes its absolute value.
(ii) If x is a word over an alphabet Z, then I x I denotes the length
of x and Min(x) denotes the set of letters which occur in x. For a in Z,
#~(x) denotes the number of occurrences of the letter a in x and if B is a
subset of Z then #~(x) = ~2~B #~(x). If k is a positive integer, then x k
denotes x catenated h times with itself.
(iii) If A is a finite set, then #A denotes its cardinality. If B C A
and #B = 1 then B is called a singleton in A.
(iv) A coding is a letter to letter homomorphism. If h is a homo-
morphism from Z* into V* and L _C V* then h-l(L) = {x ~ Z*: h(x) = y
for some y in L}.
(v) If ~ = sl, s~, s a ,... is a sequence of objects and il, i2, ia .... are
such that i 1 < i~ < i a < -'-, then sil, si2, s<~ .... is called a subsequence of 7.
(vi) z denotes the empty set and A denotes the empty word.
(vii) If d is a (nondeterministic) finite automaton, then L(d) denotes
its language.
(viii) If A is an ultimately periodic sequence (set) of nonnegative
integers then thres(A) denotes the smallest integer j for which there exists
a positive integer q such that, for every i >/j, if i is in A then (i -t- q) is
in A. The smallest positive integer p such that, for every i/> thres(A),
whenever i is in A then also (i + p) is in A, is denoted by per(A).

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EHRENFEUCHTAND ROZENBERG
2. DEFINITIONS AND EXAMPLES OF E0L SYSTEMS
In this section we give basic definitions concerning developmental lan-
guages, which are relevant for this paper.
DEFINITION 1.
such that
An E0L system is a construct G = <VN, Vr, P, to)
V N is a finite alphabet (of nonterminal letters and symbols),
V T is a finite nonempty alphabet (of terminal letters or symbols), such
that VN n V T = 2J,
to is an element of (V N k_) VT) ÷ (called the axiom of G),
P is a finite nonempty set (called the set of productions of G) each
element of which is of the form a --+ ~, where the symbol "--+" is not in
V u td VT, a is in VN tJ VT, and ~ is in (V N k) Vr)*. Moreover, for every a
in VN U U r there exists a word a in (V u u fiT)* such that a ~ ~ is in P.
In the sequel we shall often write "a --+, £' rather than "a --+ c~ is in P."
Also a production of the form a ~ a is called a production for a in P.
DEFINITION 2.
0L system if, and only if, K~ = ~. (In this case we write G as < KT, P, co>).
0L systems are investigated, for example, in Rozenberg and Doucet (1971).
An E0L system G ~-<V w, VT,P, to) is called a
DEFINITION 3. Let G = < Vw, VT, P, to> be an E0L system.
(i) Let xa(Vlvt)VT)+,
V N td Vr, and let y c (V N tj Vr)*. We say that x directly derives y (in G),
denoted as x ~a Y, if there exists a sequence 7r 1 ,..., ~r t of productions from P,
such that, for everyiin{1 .... ,t},~r i =b i-+~iandy
say x =b l'-'b t for some b 1 .... ,b t in
~ c~ l'''c~ t.
(ii) As usual, ~+ denotes the transitive closure of the relation ~a
and ~a denotes the reflexive and transitive closure of the relation ~a.
If x *~a Y then we say that x derives y in G.
(iii) A finite sequence D =- (Xo, x 1 ,..., xr) of words from (Vw t.) VT)*
such that, r >~ 1 and, for each i in {1,., r}, xi-1 ~a xi, is called a derivation
(of xr from Xo) in G. If x o = to, then D is called a derivation of xr in G.
(iv) An infinite sequence D = (Xo, x I ,...) of words from (VN L; VT)+
such tha L for each i ~ 1, xi_ 1 ~c xl, is called an infinite derivation in G.
(v) If D = (Xo, xl ,..., xr) is a derivation in G, then its control sequence

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E0L LANGUAGES 259
is any sequence r = (T 1 ,..., Tr) of subsets of P, such that, for each i in
{1,..., r}, x~_ 1 ~a xi "using" all and only productions from T~.
(vi) For x in (VN t.3 VT)+,y in (VN tO VT)* and a positive integer r we
write x ~ y if there exists a derivation D = (x o = x, x 1 .... , x r = y) in G.
We also write x 9 ° x, for every x in (V N W ~/'T) +.
(vii) max(G) is defined as max{I a I: a ~
and o~ in (V iv L3 VT)*}.
a for some a in VN V) V r
DEFINITION 4.
language of G, denoted as L(G), is defined by L(G) = {x c VT*: o, *~G X}.
Let G = (VN, Vr, P, ~o) be an E0L system. The
DEFINITION 5.
and only if, there exists an E0L system (0L system) G such that L(G) ~- K,
A language K is called an E0L language (0L language) if,
Remark 1.
D = (x o ..... x~) and its control sequence r = (T 1 .... , T,.), the pair (D, r),
in general, does not tell us which productions are used to rewrite the particular
occurrences of letters in the words x 0 ,..., x~_ 1 . However (to avoid cum-
bersome notation and to keep the size of this paper decent), we shall often
assume that the pair (D, r) provides such information. This should not
lead to confusion.
Given an E0L system G = (VN, Vr, P, w), a derivation
Remark 2.
for finite languages, hence we consider only infinite languages. Thus in the
sequel if we write "a language" (or "an E0L language") we mean an infinite
one, unless explicitly stated otherwise. Also whenever we write "an E0L
system" we mean one generating an infinite language.
The properties we are interested in (in this paper) are trivial
Remark 3.
sometimes consider P to be the "set of names for productions" rather than
the set of productions itself. In this sense we can talk about the words over P,
etc., and this should not lead to confusion.
We end this section with two examples of EOL systems.
Given an EOL system G = (VN, VT, P, oJ) we shall
EXAMPLE 1.
{S ---> a, S --+ b, a --+ a 2, b --+ b a} and w = S, is an E0L system such that
L(G) = {aS": n /> O} U {ba': n >~ 0}.
G = (VN, VT, P, ~), where 1/- N = {S}, Vr = {a, b}, P =-
EXAMPLE 2.
and co = ab, is a 0L system such that L(G) = {(ab)~=: n >/0}.
G = (2J, P, w), where 27 = {a, b}, P = {a --* (ab) 2, b --+ A}

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EHRENFEUCHT AND ROZENBERG
3. BASIC NOTIONS AND THEIR PROPERTIES
In this section we introduce basic notions describing the structure of OL
languages we are interested in and we prove some properties of these notions.
DEFINITION 6.
nonempty subset of Z. Let IL. B = {n E N: there exists a word w in L such
that #~(w) = n}.
Let L be a language over an alphabet Z and let B be a
(i) B is numerically dispersed (in L) if, and only if, IL, B is infinite
and for every positive integer k there exists a positive integer n7¢ such that,
for every ul, u 2 in IL. B , if u 1 :# u s , u 1 > n k and u 2 > n k then abs(u 1 -- us) > k.
(ii) B is clustered (in L) if, and only if, IL, B is infinite and there exist
positive integers k 1 , k s such that k s > l, k s > 1 and, for every word w in L,
if #B(w) >/k 1 , then w contains at least two occurrences of symbols from B
which are distant less than kz.
DEFINITION 7.
The symbol a is said to be frequent (in L) if, and only if, for every positive
integer n there exists a word w in L such that #a(W) > n; otherwise a is
called nonfrequent (in L).
Let L be a language over an alphabet Z and let a be in 2:.
DEFINITION 8.
subset of 27 and let a be in Z.
Let G = (Z, P, oJ) be a 0L system, let B be a nonempty
(i) We define a B-characteristic sequence of a (in G), denoted as
Seq(G, B, a), as an infinite sequence Z 1 , Z~ ,... of finite subsets of N such
that, for each i ~> 1 and every nonnegative integer n, n is in Zi if, and only if,
i
a :~a w for some w in Z* such that #B(w) = n.
(ii) Seq(G, B, a)= Z 1 , Z 2 .... is called unique if, and only if, for
every i >/ 1, #Zi = 1.
(iii) Seq(G, B, a) -- Z1, Z 2 ,... is called bounded if, and only if, there
exists a constant C such that, for every i/> 1, n < C for every n in Z~.
In this case we also say that a is B-bounded (in G) and that C bounds
Seq(G, B, a). We say that a is B-unbounded (in G) otherwise.
(iv) Seq(G, B, a)= Z1, Z 2 .... is called constant if, and only if, for
each i, j >~ 1, Zt = Zj. In this case we also say that a is B-constant (in G).
LEMMA 1. Let G = <Z, P, oJ) be a 0L system, let B be a nonempty
subset of Z and let a be a symbol in Z. Let Seq(G, B, a) = Z 1 , Z2 ,... and let