Regulated Finite Index Language Families Collapse - Semantic Scholar

Regulated Finite Index Language Families Collapse Henning Fernau and Markus Holzer

WSI-96-16

Henning Fernau und Markus Holzer Wilhelm-Schickard-Institut fur Informatik Universitat Tubingen Sand 13 D-72076 Tubingen Germany E-Mail: ffernau,[email protected] Telefon: (07071) 29-7569 (07071) 29-7568 Telefax: (07071) 68142

c Wilhelm-Schickard-Institut fur Informatik, 1996 ISSN 0946-3852

Regulated Finite Index Language Families Collapse Henning Fernau? and Markus Holzer Wilhelm-Schickard-Institut fur Informatik, Universitat Tubingen Sand 13, D-72076 Tubingen, Germany email: ffernau,[email protected]

Abstract. We prove normal form theorems for programmed, ordered, and random context, context-free grammars as well as context-free grammars with regular context conditions of nite index. Based on these normal forms, we reprove that the family of programmed, ordered, permitting, and forbidding random context context-free languages of nite index coincide, regardless whether erasing productions, i.e., rules of the form A ! , where  denotes the empty word, are allowed or not. These constructions lead to new results on cooperating distributed (CD) grammar systems, contained in another paper [6]. Furthermore, we consider conditional context-free grammars that generate languages of nite index. Thereby, we solve an open problem stated in Dassow and Paun's monograph on regulated rewriting. Moreover, we show that conditional context-free languages with context-free context conditions of nite index are more powerful than conditional context-free languages with regular context conditions of nite index.

1 Introduction Regulated rewriting is one of the main and classic topics of formal language theory [3, 15], since there, basically context-free rewriting mechanisms are enriched by dierent kinds of regulations, hence generally enhancing the generative power of such devices compared to the context-free languages. In this way, it is possible to describe more natural phenomena using contextindependent derivation rules, see [3]. ?

Supported by Deutsche Forschungsgemeinschaft grant DFG La 618/3-1.

In this paper, we are interested in the relation between formal languages which are build up by rewriting mechanisms|we restrict ourselves to contextfree core rules|that generate languages of nite index. Loosely speaking, the index of a grammar is the maximal number of nonterminals simultaneously appearing in a sentential form during a terminal derivation (considering the most economical derivation for each string). Originally the nite index restriction was investigated by Brainerd [1]. He introduced this notion in order to generalize the statement: \If L is an in nite language generated by a context-free grammar, then L contains a sequence, fwng, of strings such that the sequence of lengths fjwnjg is a (nontrivial) arithmetic progression," which is, e.g., a corollary from the existence of pumping lemmata for contextfree languages in combination with Van der Waerden's Theorem, to a class of languages that meets, but perhaps does not contain, the context-free languages. Brainerd showed that a similar statement as above is valid in case of matrix grammar of nite index. Later on, several authors have investigated nite index restrictions also to other rewriting mechanisms, as to, e.g., programmed, ordered, random context, conditional grammars with context-free core rules. For a summary of known results for these types of grammars (in general and in the nite index case) we refer to the monograph of Dassow and Paun [3]. Interestingly, we came across this topic via a very dierent route: we considered internal and external hybridization in cooperating distributed (CD) grammar systems, a very modern part of formal language theory aiming at a grammatical model of multi-agent systems introduced as a concept in Arti cial Intelligence, see [2]. We noticed in our work [6] that several special cases of such CD grammar systems characterize the family of languages generated by context-free matrix grammars of nite index. Other special cases of such CD grammar systems are equally powerful as special forms of ordered grammars with the nite index restriction. Unfortunately, it was obviously unknown whether these special forms are really normal forms for ordered grammars with the nite index restriction or not. Even worse, in [3, page 161, Open problems] it was stated as an open problem whether the family of languages generated by ordered contextfree grammars of nite index is a proper subset of the family of languages 2

generated by context-free matrix grammars of nite index or not. So, we got a two-fold motivation for studying nite index grammars in more detail. We were able to prove that the family of programmed, ordered, random context, and conditional context-free languages of nite index coincide, regardless whether erasing productions, i.e., rules of the form A ! , where  denotes the empty word, are allowed or not. The proofs are mainly based on normal form theorems for each of these grammars. In this way, we solved three problems marked as open in [3, page 161, Open problems]. When browsing the corresponding literature in parallel to our own research, we came across a sequence of papers of Rozenberg and Vermeir [11, 12, 13, 14], where they showed that the family of programmed, ordered, and random context languages of nite index coincide. Unfortunately, they took the deviation via ET0L systems with the nite index restriction. Moreover, it has not become clear to us whether their constructions easily lead to the normal form required for our works on CD grammar systems or not. For these two reasons, we kept the corresponding theorems together with our proofs in this work. Moreover, giving direct constructions within the framework of regulated rewriting, we pursue the studies begun by Stotskii in [17], (also refer to [16]) and furthermore we supplement the monograph [3] where regulated rewriting is seen as the central topic. The paper is organized as follows. In the next section, we introduce the necessary notations. Then, we deal with programmed, ordered, (forbidding) random context, and conditional context-free grammars of nite index. As indicated in the headline of this paper, we show that all the corresponding language classes (as discussed in the monograph of Dassow and Paun) of nite index are indeed the same class. Therefore, it is natural to ask whether there are regulation mechanisms of grammars with context-free core rules such that the corresponding language family of nite index languages breaks this seemingly impenetrable barrier of \regulated nite index languages." Indeed, in the penultimate section, we prove the existence of such a class, and in the last section, we discuss our results.

2 De nitions We assume the reader to be familiar with the basic notions of formal language theory, as contained in Dassow and Paun [3] or Salomaa [15]. In addition, 3

we use  to denote inclusion, while  denotes strict inclusion. The set of positive integers is denoted by N, while N denotes the set of non-negative integers. The empty word is denoted by . We consider to languages L and L to be equal i L n fg = L n fg. The family of regular, context-free, context-sensitive, type-0 Chomsky languages, context-free matrix grammars, and context-free matrix grammars with appearance checking are denoted by L(REG), L(CF), L(CS), L(RE), L(M; CF), and L(M; CF; ac), respectively. For the convenience of the reader, we repeat some de nitions from the theory of regulated rewriting. A programmed grammar is a septuple G = (VN ; VT ; P; S; ; ; ), where VN , VT , and S 2 VN are the set of nonterminals, the set of terminals, and the start symbol, respectively. In the following we use VG to denote the set VN [ VT . P is the nite set of productions  ! , and  is a nite set of labels (for the productions in P ), such that  can be also interpreted as a function which outputs a production when being given a label;  and  are functions from  into the set of subsets of . For (x; r ), (y; r ) in VG   and (r ) = ( ! ), we write (x; r ) ) (y; r ) i either 1. x = x x , y = x x and r 2 (r ), or 2. x = y and rule  ! is not applicable to x, and r 2 (r ). In the latter case, the derivation step is done in appearance checking mode. The set (r ) is called success eld and the set (r ) failure eld of r . As  usual, the re exive transitive closure of ) is denoted by ). The language generated by G is de ned as 0

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L(G) = f w 2 VT j (S; r ) ) (w; r ) for some r ; r 2  g: 1

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The family of languages generated by programmed grammars containing only context-free core rules is denoted by L(P,CF; ac). When no appearance checking features are involved, i.e., (r) = ; for each label r 2 , we are led to the family L(P,CF). The special variant of a programmed grammar where 2

If there is no confusion, we sometimes write for a derivation of a programmed grammar D

instead of

D

:(

S; r1

: = S

)=(

w1

w 1 ; r1

)

) w2

==r2

== rn?1

) wn

=

(

)

(

)=(

==r1 )

w 2 ; r2

)




)


)

4

w n ; rn



w 2 VG

w; rn

)

2 V

 G



.

 =  is said to be a programmed grammar with unconditional transfer. Observe that due to our de nition of derivation, a production with empty goto- eld is never applicable. Hence, we assume without saying that grammars with unconditional transfer contain only productions with non-empty go-to elds. We denote the class of languages generated by programmed grammars with context-free productions and with unconditional transfer by L(P,CF,ut). An ordered grammar is a quintuple G = (VN ; VT ; P; S; ), where VN , VT , P , and S 2 VN are the nonterminal alphabet, terminal alphabet, set of productions, and axiom, respectively.  is a partial order on P . A production  ! is applicable to a string x 2 V  i x = x x for some x ; x 2 VG, and x contains no subword 0 such that 0 ! 0 2 P for some 0 and 0 ! 0   ! ; the application of  ! to x yields y = x x . As usual, the yield relation is denoted by ), and its re exive and transitive closure is denoted by ) . The family of languages generated by ordered grammars with context-free productions is denoted by L(O; CF). A random context grammar (RC grammar) is a system G = (VN ; VT ; P; S ) where VN ; VT ; S 2 VN are de ned as in a usual Chomsky grammar, and P is a nite set of random context rules, that is, triples of the form ( ! ; Q; R) where  ! is a rewriting rule over VG and Q and R are subsets of VN . For x; y 2 VG, we write x ) y i x = x x , y = x x for some x ; x 2 VG, ( ! ; Q; R) is a triple in P , all symbols of Q appear in x x , and no symbol of R appears in x x . Set Q is called the permitting context of  ! and R  is the forbidding context of this rule. As usual, ) is the re exive transitive closure of the yield relation ) and L(G) = f w 2 VT j S ) w g. The family of languages generated by RC grammars containing only context-free rules  ! is denoted by L(RC; CF; ac). Considering only random context rules with empty forbidding context, we are led to the language family L(RC; CF).In this case, no \appearance checking" is possible. The letters \ac" in L(RC; CF; ac) indicate possible appearance checks. Symmetrically, allowing only rules with empty permitting context, we come to the language family L(fRC; CF). A conditional grammar (K grammar) is a system G = (VN ; VT ; P; S ) where VN ; VT ; S are de ned as in a usual Chomsky grammar, and P is a nite set of rules of the form ( ! ; Q), where Q is a regular language over VG . The rule ( ! ; Q) is applicable to x = x x yielding y = x x i x 2 Q. Then, we write x ) y and de ne ) as the re exive transitive closure of ) and L(G) = f w 2 VT j S ) w g. 1

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The family of languages generated by conditional grammars containing only context-free rules  ! is denoted by L(K; CF). Up to now we have only de ned language families de ned via grammars with erasing rules. If we want to exclude erasing productions, we add ? in our notations, e.g., the family of languages generated by conditional grammars containing only context-free rules without erasing productions is denoted by L(K; CF ? ). We use bracket notations like L(M; CF[?])  L(M; CF[?]; ac) in order to say that the equation holds both in case of forbidding erasing productions and in the case of admitting erasing productions (neglecting the bracket contents). The length of a word w 2 VG, written as jwj is the number of letters in w. For a subset V of VG we denote the number of occurrences of letter of V in x 2 VG by jwjV . If V = fag, then we simply write jwja. Let G be an arbitrary grammar type (from those discussed above) and let VN , VT , and S 2 VN be its nonterminal alphabet, terminal alphabet, and axiom, respectively. For a derivation D : S = w ) w )


) wn = w 2 VT according to G we set ind (D; G) = maxf jwi jVN j 1  i  n g; and, for w 2 VT, we de ne ind (w; G) = minf ind (D; G) j D is a derivation for w in G g: The index of grammar G is de ned as ind (G) = supfind (w; G) j w 2 L(G) g: For a language L in the family L(X) of languages generated by grammars of type X we de ne ind (L) = inf f ind(G) j L(G) = L and G is of type X g: For a family L(X), we set Ln (X) = f L j L 2 L(X) and ind (L)  n g for n 2 N, and L n (X) = Ln (X): 1

2

X

X

[

n1

6

3 Programmed and Matrix Languages In this section we prove two normal forms for programmed context-free grammars (with appearance checking). Moreover, the normal forms directly imply that the family of programmed context-free languages (with appearance checking) coincides with the family of programmed context-free language with unconditional transfer. In the following we will refer to a grammar satisfying the two conditions below as bounded nonterminal form (BNF).

Lemma 3.1 For every (P; CF; ac) grammar G = (VN ; VT ; P; S; ; ; ) whose generated language is of index n 2 N, there exists an equivalent

(P; CF; ac) grammar G0 = (VN0 ; VT ; P 0; S 0; 0; 0; 0) whose generated language is also of index n satisfying the following two properties. 1. There exists a special start production with a unique label p , which is the only production where the start symbol S 0 appears. 0

 2. There exists a function f : 0 ! NV0 N such that, if S 0 ) v )p w is a 0 derivation in G , then (f (p))(A) = jvjA for every nonterminal A. 0

Moreover, we may assume that either G0 is a (P; CF) grammar, i.e., we have 0 = ;, or that G0 is a (P; CF; ut) grammar, i.e., we have 0 = 0.

Proof. We construct a grammar G0 = (VN0 ; VT ; P 0; S 0; 0; 0; 0) of index n

which is equivalent to G = (VN ; VT ; P; S; ; ; ) and satis es the two requirements. Assume that VN = f A ; : : :; Am g, where S = A . Let VN0 = VN [ fS 0g, the set of labels 1

1

0  (  f (i ; : : : ; im) j 1

m

X

j =1

ij  n g) [ fp g; 0

and P 0 = P [ fS 0 ! S g. The start production S 0 ! S has a unique label p , with 0(p ) =   f(1; 0; : : : ; 0)g and 0(p ) = ;. For every label r 2  with (r) = Aj ! w, we de ne for the label (r; i ; : : :; im) 2 0: 1. 0((r; i ; : : : ; im)) = (r) = Aj ! w; 0

0

0

1

1

7

2. if ij = 0, then 0((r; i ; : : : ; im)) = ;; and 0((r; i ; : : : ; im)) = (r)  f(i ; : : :; im)g; 1

1

1

3. if ij > 0, then 0((r; i ; : : :; im)) = (r)  f(i ; : : : ; im) + (jwjA1 ? jAj jA1 ; : : : ; jwjAm ? jAj jAm )g ; if this label lies in 0 (otherwise, we set 0((r; i ; : : :; im)) = ;), and 0((r; i ; : : : ; im)) = ;. In such a way, the set of labels 0 is implicitly de ned as the set of labels obtained in the preceding procedure. Obviously, the idea of the construction is to count the number of nonterminals in the label. Therefore, the function f may be de ned as f ((r; i ; : : :; im))(Aj ) = ij ; and f ((r; i ; : : :; im))(S 0) = 0; f (p )(Aj ) = 0; and f (p )(S 0) = 1: Exploiting this counting mechanism, it is easy to modify the construction in order to obtain a grammar (1) with empty failure elds or (2) with unconditional transfer. In case (1), we alter the de nition of the label interpretation in case of a label (r; i ; : : :; im) 2 0 with (r) = Aj ! w and ij = 0. If (i ; : : :; im) is the all-zero vector, we have derived a terminal word. Therefore, we set 0((r; i ; : : : ; im)) = (r) and 0((r; i ; : : : ; im)) = fp g. Otherwise, there must be some k with ik > 0. We take 0((r; i ; : : : ; im)) = Ak ! Ak , and as the success eld, we take (r)  f(i ; : : : ; im)g. Observe that by this construction the production set P 0 given above is enhanced by unit productions of the form Ak ! Ak . In case (2), we alter the de nition of the label interpretation in case of a label (r; i ; : : :; im) 2 0 with (r) = Aj ! w in any case. 1. 0((r; i ; : : : ; im)) = (r); 2. if ij = 0, then 0((r; i ; : : : ; im)) = 0((r; i ; : : : ; im)) with 0((r; i ; : : :; im)) = (r)  f(i ; : : :; im)g; 1

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3. if ij > 0, then 0((r; i ; : : : ; im)) = 0((r; i ; : : : ; im)) with 1

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0((r; i ; : : :; im)) = (r)  f(i ; : : : ; im) + (jwjA1 ? jAj jA1 ; : : : ; jwjAm ? jAj jAm )g ; 1

1

if this label lies in 0 (otherwise, we set 0((r; i ; : : :; im)) = ;). 1

Moreover, the start production S 0 ! S has a unique label p with success and failure eld 0(p ) = 0(p ) =   f(1; 0; : : : ; 0)g. 2 Observe that the lemma above is of purely structural nature (only in the transformation into a programmed grammar without appearance checking there seems to be the need for unit productions). Therefore, it is true not only in the context of string rewriting, but for any rewriting system where \number of nonterminals in an intermediate (sentential) form of a derivation" can be meaningfully de ned. This observation is not true anymore in the following theorem, where we make extensive use of colouring techniques. 0

0

0

Theorem 3.2 For every (P; CF; ac) grammar G = (VN ; VT ; P; S; ; ; ) whose generated language is of index n 2 N, there exists an equivalent

(P; CF; ac) grammar G0 = (VN0 ; VT ; P 0; S 0; 0; 0; 0) whose generated language is also of index n satisfying the following three properties: 1. There exists a special start production with a unique label p , which is the only production where the start symbol S 0 appears. 0

 2. There exists a function f : 0 ! NV0 N such that, if S 0 ) v )p w is a 0 derivation in G , then (f (p))(A) = jvjA for every nonterminal A. 0

3. If D : S 0 = v0 =r) v =) v =)


=r) v = w is a derivation in G0, m m 1 1 r2 2 r3 then, for every vi, 0  i  m, and every nonterminal A, jvijA  1. In other words, every nonterminal occurs at most once in any derivable sentential form. Moreover, we may assume that either G0 is a (P; CF) grammar, i.e., we have 0 = ;, or that G0 is a (P; CF; ut) grammar, i.e., we have 0 = 0.

9

In the following, we will refer to a grammar satisfying the three conditions listed above as nonterminal separation form (NSF). Proof. Easy but tedious modi cations of the preceding lemma allows to colour and count each original nonterminal A as [A; i], where 1  i  n, such that all i-colours of all nonterminals occurring in a sentential form are dierent. Details are left to the reader. 2 Note that in the proofs above, we made heavy use of the possibility of one rule having multiple labels. This restriction is not necessary, since by a simple colouring technique of the nonterminal symbols, the uniqueness of a production label may be assured. It is possible to eliminate erasing productions from programmed grammars without changing the index of the language, as it is shown in the case of nite index matrix grammars in [3, Lemma 3.1.2]. Therefore, we may summarize for the case of programmed grammars: 3

Corollary 3.3 For every n 2 N, Ln(P; CF) = Ln (P; CF ? ) = Ln (P; CF; ut) = Ln (P; CF ? ; ut) = Ln (P; CF; ac) = Ln (P; CF ? ; ac):

2

It is easily seen that the constructions given in [3, Theorem 1.2.2] proving L(P; CF[?]) = L(M; CF[?]) preserve indexes (*). Therefore, we know by [3, Theorem 3.2.1] that each of the families Ln (P; CF) are full semi-AFL's. We will exploit this fact in the next section. Furthermore, the observation (*) allows us to improve [3, Lemma 3.1.3] somewhat, where the corresponding statement for non- xed nite-index matrix languages is stated. This improved version is also contained in [17, Teorema 1] referring to his construction [16, Teorema 5.7]. In addition, we refer to [13, Theorem 10] and [14, Theorem 9]. The last reference also claims the result for the case of unconditional transfer contained in our theorem above. Unfortunately, the quoted paper does not contain a proof of that fact.

3 A corresponding normal form for matrix grammars of nite index is proved in [3, Lemma 3.1.4].

10

Corollary 3.4 For every n 2 N, Ln (M; CF) = Ln (M; CF ? ) = Ln (M; CF; ac) = Ln (M; CF ? ; ac) = Ln(P; CF) = Ln (P; CF ? ):

2

It is also possible to formulate our last corollary involving matrix grammars with unconditional transfer as de ned in [5], but we do not want to go into these details here.

4 Ordered Languages

Lemma 4.1 For every n 2 N, Ln (O; CF[?])  Ln (P; CF[?]; ut).

Observe that the following proof is of purely structural nature. Proof. Let G = (VN ; VT ; P; S; ) be an ordered context-free grammar, and let  be the set of unique labels of the rules in P . Then we can construct an equivalent programmed context-free grammar with unconditional transfer G0 = (VN [ fF g ; VT ; P 0; S; 0; ; ) in the following way: for each labelled rule r : A ! w from P with F (r) = f (r; i) j 1  i  k (r) g denoting the set of labels of the rules pr;i greater than r with respect to the order relation , in G0 we set 0 = f (r; i) j r 2  ^ 1  i  k (r) + 1 g. Assume that pr;i 2 F (r), 1  i  k (r) is the label of the rule Ai ! wi. Then, let 0((r; i)) = Ai ! F , and ((r; i)) = ((r; i)) = f(r; i + 1)g. Moreover, we set 0((r; k(r) + 1)) = A ! w, and ((r; k(r) + 1)) = ((r; k(r) + 1)) = f (r; 1) j r 2  g. If there exists a successful derivation v =w S = v =r) v =r) v


=r) m m 1 2 0

1

2

of a word w 2 VT via grammar G using subsequently the productions r ; : : :rm, such that every intermediate sentential form vi, 1  i  m, does not contain more than n nonterminal symbols, then S = v == ) v


======= ) v ======= )v r1 ; r1 ;k r1 r1 ;k r1 == ) v


======= ) v ======= ) v


======= ) vm = w r; r ;k r r ;k r r ;k r 1

0

(

1)

(

2 1)

0

1

(

(

2

(

))

(

2 ))

0

1

(

(

11

2

(

)+1)

(

2 )+1)

1

2

(

m

(

m )+1)

is a successful derivation of w 2 VT via grammar G0, such that every intermediate sentential form vi, 1  i  m, does not contain more than n nonterminal symbols. On the other hand, every label string describing a successful derivation in G0 is of the form D0 = (p ; 1) : : : (p ; k(p ) + 1)(p ; 1) : : : (p ; k(p ) + 1) : : : : Therefore, we have a corresponding sequence D = p p : : : of labels in the original grammar G, such that a successful derivation of G is described, which furthermore does contain the same sentential forms as the ones occurring in G0 in the derivation D0 . Especially, no more than n nonterminals occur in each of the sentential forms generated in D0. 2 1

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Lemma 4.2 For every n 2 N, Ln (P; CF[?])  Ln (O; CF[?]). Proof. Due to our theorems above, it remains to be shown how a (P; CF[?])language L  VT of index n can be generated by an ordered grammar whose

generated language is of index n. Let the (P; CF[?]) grammar G = (VN ; VT ; P; S; ; ; ) generating L obey NSF. Especially, p is the unique label of a production with S as left-hand side. We construct an ordered grammar G0 = (VN0 ; VT ; P 0; S 0; ) generating L such that L is seen to be of index n. Let VN0 = VN [ (VN    f0; 1; 2g) [ fF g. The idea of the simulation is to keep track of the label of the production of the programmed grammar which we are going to simulate hidden inside one marked nonterminal symbol. Therefore, we start with S 0 = (S; p ; 1). The order relation (implicitly de ning the set of productions P 0) is given in the following. For each p 2 , (p) = A ! w, we can assume without loss of generality (f (p))(A) = 1 (i.e., actually there exists one occurrence of the nonterminal A to be rewritten with the harmless exception of an already obtained string of terminal symbols). 1. If we have marked the symbol A with label p, then we consider two cases: (a) if jwjVN = 0, then we add (A; p; 1) ! w  D ! F for D 2 VN [ (VN  f0g); 0

0

12

(b) if jwjVN > 0, then let w = w B w B : : : wmBmwm , such that w : : : wm 2 VT and B ; : : : ; Bm 2 VN . For every label q 2 (p) and every 1  i  m, we take each production of the form (A; p; 1) ! w B : : :wi? Bi? wi(Bi; q; 1)wi Bi : : : wmBm wm into the production set P 0 (without any greater rules). 2. Otherwise, we have marked an arbitrary nonterminal B 6= A (this inequality is due to the NSF) with the label p. This case is more involved. We add a number of new productions: (a) A ! (A; p; 0)  D ! F for D 2 f (C; q; 1) 2 VN   f1g j q 6= p _ A = C g[ VN   f0; 2g: 1

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+1

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+1

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+1

+1

+1

(b) (B; p; 1) ! (B; p; 2)  D ! F for D 2 fAg [ VN    f2g g; Observe that, due to the NSF property of G, by induction the presence of a nonterminal (B; p; 1) in a sentential form derived in G0 testi es the presence of one occurrence of an A or (A; p; 0), respectively, in this sentential form. (c) (A; p; 0) ! w  D ! F for D 2 f C 2 VN j jwjC > 0 g [ VN    f0; 1; 2g n f(A; p; 0)g [ VN  fpg  f2g : 

!

 



(d) (B; p; 2) ! (B; q; 1)  D ! F for D 2 VN    f0; 1g for every q 2 (p). It can be seen that the productions A ! (A; p; 0), (B; p; 1) ! (B; p; 2), (A; p; 0) ! w and (B; p; 2) ! (B; q; 1) can only be applied in this order. In our simulation, many derivations are blocked since the \wrong" nonterminals have been coloured. This is no problem, because there must be one \last" production A ! w which is applied in a derivation of G, and this rule is the only one to which the rst subcase of case (i) applies. This also determines which nonterminals carry the label information, if we proceed the derivation trees upwards to the root S , or (S; p ; 1), respectively. 2 0

13

Theorem 4.3 For every n 2 N, Ln (O; CF) = Ln (O; CF ? ) = Ln (P; CF) = Ln (P; CF ? ):

2

The theorem above was already proven in [13, Theorem 3]. Our proof does not leave the \world" of regulated rewriting deviating via ET0L systems. Moreover, our constructions deliver the following normal form results for ordered languages of nite index which is used in [6] to prove some results on cooperating distributed context-free grammar systems. This corollary does not follow directly from the construction leading to [13, Theorem 3]. Corollary 4.4 Every (O; CF) language is generated by an (O; CF) grammar G = (VN ; VT ; P; S; ) such that: 1. VN contains a special failure symbol F not appearing on any left-hand side of a production in P . 2. P contains failure productions B ! F for every B 2 VN , B 6= F . 3. For each production A ! w 2 P , either w 2 VT [fF g or there exists a nonterminal B 2 VN n fF g such that w = w Bw and A ! w  B ! F. 4. All nonterminals occurring in some sentential form v (not containing the failure symbol), which are derivable in a certain number of steps from the start symbol S , are pairwise distinct. Therefore, no production with non-failure symbols on its right-hand side can be applied twice in immediate sequence. 2 1

2

5 Random Context Languages In the following, we turn our attention towards (forbidding) random context grammars. The results of this section were rst proved in [11] (again using the path over ET0L systems of nite index), answering questions raised in [17, Section 4]. In case of forbidding random context grammars Dassow and Paun [3, page 128, Theorem 2.3.4] have shown that L(fRC; CF[?]) and L(O; CF[?]) coincide. Since their construction preserves the nite index property, we already have the following result. 14

Theorem 5.1 For every n 2 N, Ln (fRC; CF[?]) = Ln (O; CF[?]).

2

Example 5.2 We show that the language Ln = f b(aib) n j i 2 N g 2 Ln (O; CF) = Ln (fRC; CF) for n 2 N: +1

2( +1)

+1

+1

This language has been used in the monograph of Dassow and Paun [3, page 160, Theorem 3.1.7] as separator of the index-n-hierarchy of matrix language families. Let G = (VN ; VT ; P; S ) be a forbidding random context grammar, where VN = fS g [ f Ai; A0i; A00i j 1  i  n + 1g, VT = fa; bg, and P contains the following productions: The start production is S ! bA1bA2 : : : bAn+1 : ;; ;. For every 1  i  n + 1, we have rules of the form 8 > < f Aj j 1  j < i g [ Ai ! aA0ia : ;; > f A0j j i + 1  j  n + 1 g [ : f A00j j 1  j  n + 1 g and 8 0 > < f Aj j 1  j < i g [ 0 Ai ! Ai : ;; > f Aj j i + 1  j  n + 1 g [ : f A00j j 1  j  n + 1 g: The terminating productions are 8 > < f Aj j 1  j < i g [ 00 Ai ! Ai : ;; > f A00j j i + 1  j  n + 1 g [ : f A0j j 1  j  n + 1 g and 8 00 > < f Aj j 1  j < i g [ A00i ! aba : ;; > f Aj j i + 1  j  n + 1 g [ : f A0j j 1  j  n + 1 g: For the other special case, the random context grammars with only permitting context we have to do a little bit more.

Theorem 5.3 For every n 2 N, Ln(RC; CF) = Ln(RC; CF ? ) = Ln (P; CF) = Ln(P; CF ? ): 15

Proof. The inclusion Ln (RC; CF ? )  Ln (RC; CF) is trivial, and Ln (RC; CF[?])  Ln(P; CF[?]) is quite an easy exercise. It is known that Ln (P; CF ? ) = Ln (P; CF). It remains to be shown that Ln (P; CF ? ) is contained in Ln (RC; CF ? ) to nish our proof. Let G = (VN ; VT ; P; S; ; ; ;) be a (P; CF ? ) grammar satisfying NSF. Especially, there is a function f :  ! f0; 1gVN delivering the number of

occurrences of the nonterminals in the sentential form in question when a production labelled p is applied. Without loss of generality, we may assume that some order relation < is de ned on VN . We give a simulating (RC; CF ? ) grammar G0 = (VN0 ; VT ; P 0; S 0) with VN0 = VN    f0; 1g, S 0 = (S; p ; 0), where p is the unique label of the production with S as left-hand side in G, and P 0 contains the following productions. For every label p 2  with (p) = A ! w, we take into P 0: 1. (A; p; 0) ! (A; p; 1); f (B; p; 0) j (f (p))(B ) = 1 g; ; ; 0

0







2. (B; p; 0) ! (B; q; 0); f(A; p; 1)g [ f (C; q; 0) j C < B ^ C 6= A ^ (f (p))(C ) = 1 g; ; for every q 2 (p); 





3. (A; p; 1) ! hq (w); f (C; q; 0) j C 6= A ^ (f (p))(C ) = 1 g; ; for every q 2 (p). Here, the morphism hq : (VT [ VN ) ! (VT [ VN    f0g) is de ned by hq (a) = a for a 2 VT , hq (X ) = (X; q; 0) for X 2 VN . Observe that this construction heavily exploits one subtlety in the de nition of nite index, namely the fact that there may be successful derivations in G0 which do introduce more than n nonterminals (in case a production p : A ! w with w 2 VT is simulated, occurrences of symbols of type (A; p; 1) may remain unchanged without disturbing the \counting protocol" of the function f ), but it is at least possible to take a derivation obeying the indexn-restriction. 2

Corollary 5.4 All the following language families are equal to the family L n (P; CF) = L n (P; CF ? ): 1. L n (RC; CF; ac), L n (RC; CF ? ; ac), 2. L n (RC; CF), L n (RC; CF ? ), and 16

3. L n (fRC; CF), L n (fRC; CF ? ).

2

6 Conditional Languages Finally, we consider nite index conditional context-free languages. Similarly to nite index programmed and ordered context-free languages we rst show a normal form result for conditional grammars that generate nite index languages. Theorem 6.1 For every (K; CF) grammar G = (VN ; VT ; P; S ) whose generated language is of index n 2 N, there exists an equivalent (K; CF) grammar G0 = (VN0 ; VT ; P 0; S 0) whose generated language is also of index n satisfying the following two properties: 1. There exists a special start production, which is the only production where the start symbol S 0 appears. 2. If D : S 0 = v ) v ) v )


) vm = w is a derivation in G0 , then, for every vi, 0  i  m, and every nonterminal A, jvijA  1. In other words, every nonterminal occurs at most once in any derivable sentential form. Additionally, one can require that for every vi , 0  i  m, jvijVN  n, i.e., there are at most n nonterminals in any derivable sentential form. Proof. We construct a grammar G0 = (VN0 ; VT ; P 0; S 0) of index n which is equivalent to G = (VN ; VT ; P; S ) and satis es the two requirements. Let VN0 = VN f1; : : : ; ng[fS 0g and de ne the morphism h from VG nfS 0g to VG as follows: h(a) = a if a 2 VT ; h((A; i)) = A if (A; i) 2 VN  f1; : : : ; ng. The start production is S 0 ! (S; 1); fS 0g. Thus, the required property one is ensured. Then for every production (A ! w; Q) of the original grammar G we construct for every i, 1  i  n, and for every v 2 h? (w) a production of the form: (A; i) ! v; h? (Q) \ R \ T; 0

1

2

0

0

1

1

17

where R and T are the regular sets

R =











VG n f(A; i)g
fg [ f(A; i)g
VG n f(A; i)g ; 0

0

A;i)2VN f1;:::;ng i [  VTVN0
VT: 0in

(

T =







\

Note that the second property required is controlled by the regular sets R, since R is the set of words such that every nonterminal occurs at most once except S 0. Language T ensures the additional property that only words that contain at most n nonterminals are derivable. 2 Now we are ready to prove that Ln (P; CF[?]) and Ln(K; CF[?]) coincide.

Theorem 6.2 For every n 2 N, Ln (P; CF[?]) = Ln (K; CF[?]). Proof. By Theorem 5.3 it suces to prove the inclusion Ln (K; CF[?])  Ln (P; CF[?]; ac), since Ln (P; CF[?]) = Ln (RC; CF[?]) and the inclusion Ln (RC; CF[?])  Ln (K; CF[?]) is an easy exercise, see [3, page 121].

Let G = (VN ; VT ; P; S ) be a (K,CF)-grammar of the normal form described above that generates a language of nite index n. Assume that every production of P has a unique label r, 1  r  m = jP j, and that the start production is labeled by 1. In this way, we can refer to the regular language Qi of the ith production. Furthermore, we assume that Qi is represented by some deterministic nite automaton Mi = (Ki; VG ; i; q ; Fi), where Ki is the nite set of states, VG the input alphabet, i : Ki  VG ! Ki the transition function, q 2 Ki the start state, and Fi  Ki the set of nal states. For each Mi , 1  i  m, we consider state mappings
i : Ki ! Ki . Note that the number of distinct mappings
i is the nite number jKi jjKij. Let Di be the set of all mappings
i, and we use D to denote D  D 


 Dm . Of course, every d 2 D can be interpreted as a function mapping K 


 Km into itself. For d ; d 2 D with di = (
i; ; : : :;
i;m), 1  i  2, let d  d = (
; 
; ;
; 
; ; : : : ;
;m 
;m); hence  denotes the usual composition of functions, i.e., for two functions f; g let f  g (x) mean g(f (x)). 0

0

1

2

1

1

1

2

2

1

11

21

12

18

22

1

2

To each v 2 VG one associates mappings
v;i, 1  i  m, with the property that for every 1  i  m and for every p 2 Ki,
v;i(p) = i(p; v), where i is the extension of i to domain Ki  VG. Further, for v 2 VG let dv denote (
v; ; : : :;
v;m) 2 D. Now we are ready to brie y describe the construction of an equivalent programmed grammar G = (VN0 ; VT ; P 0; S 0; ; ; ) of same nite index n. Set VN0 = VN and let 1

  fp g [ f1; : : : ; mg  (

[

0

kn

1

(D  VN )k )  D:

A label (r; d ; A ; d ; A ; : : :; dk ; Ak; dk ), 1  r  m, 1  k  n, with di = (
i; ;
i; ; : : : ;
i;m), 1  i  k + 1, that satis es the properties 1. r(
;r (q ); A ) = p , 2. r(
i ;r (pi ); Ai ) = pi , 1  i < k, and 3.
k ;r (pk ) 2 Fr , is said to be valid with respect to automaton Mr . The start production is S 0 ! S has the unique label p , 1

1

1

2

2

+1

2

1

0

1

1

+1

+1

+1

+1

0

(p ) = f(1; id ; S; id )g and (p ) = ;; 0

0

where id = (id ; : : :; id m) and id i, 1  i  m, denotes the identity mapping on state set Ki . For each rule r of the form (A ! w; Q), we construct a bunch of labels with the corresponding  and  elds. Let (r; d ; A ; d ; A ; : : :; dk ; Ak; dk ) be a label that (1) contains exactly one Ai, 1  i  k with Ai = A (this is ensured by the required normal form of the underlying (K; CF) grammar), and (2) that is valid with respect to automaton Mr . For the rule r we have to distinguish two cases: 1. If w = w B w : : : wsBsws , with w : : :ws 2 VT and B ; : : : ; Bs 2 VN , we de ne for that speci c label: (a) ((r; d ; A ; d ; : : :; dk ; Ak; dk )) = A ! w; 1

1

1

1

1

2

+1

1

1

2

+1

19

1

+1

2

2

+1

1

(b) ((r; d ; A ; d ; : : : ; dk ; Ak ; dk )) consists of exactly those labels 1

1

2

+1

(r0; d ; A ; : : : ; di? ; Ai? ; di  dw1 ; B ; dw2 ; : : : : : : ; dws ; Bs; dws+1  di ; Ai ; : : : ; Ak; dk ) 1

1

1

1

+1

1

+1

+1

that are valid with respect to automaton Mr , and that satisfy the inequality k + s ? 1  n; (c) ((r; d ; A ; d ; : : :; dk ; Ak; dk )) = ;. 2. The construction for a terminating rule, i.e., w 2 VT is as follows: (a) ((r; d ; A ; d ; : : :; dk ; Ak; dk )) = A ! w; (b) ((r; d ; A ; d ; : : : ; dk ; Ak ; dk )) consists of exactly those labels 0

1

1

2

+1

1

1

2

+1

1

1

2

+1

(r0; d ; A ; : : : ; di? ; Ai? ; di  dw  di ; Ai ; : : :; Ak ; dk ) 1

1

1

1

+1

+1

+1

that are valid with respect to automaton Mr , and that satisfy the inequality 1  k ? 1. If k = 1, then the success eld contains label p , only; (c) ((r; d ; A ; d ; : : :; dk ; Ak; dk )) = ;. This completes our construction. It is seen that the index of the language is preserved, and that the constructed programmed grammar G0 is equivalent to the original given one. More precisely, by induction we have 0

0

1

1

2

+1

 S0 ) )v p0 S ) u ============== r;d ;A ;d ;:::;d ;A ;d (

1 1 2

k k k+1 )

in G0 if and only if word u equals u A u : : : uk Ak uk with u u : : :uk 2 VT, A ; : : : ; Ak 2 VN , and di = (
i; ;
i; ; : : : ;
i;m), 1  i  k + 1, that satis es for every 1  j  m: 1. j (
;j (q ); A ) = p ;j = j(q ; u A ), 2. j (
i ;j (pi;j ); Ai ) = pi ;j = j(pi;j ; ui Ai ), 1  i < k, and 3.
k ;j (pk;j ) = pk ;j = j(pk;j ; uk ). 1

1

1

1

1

+1

+1

0

1

1

+1

+1

2

+1

2

0

1

1

+1

+1

+1

20

+1

1

2

+1

Furthermore, pk ;r 2 Fr, i.e., u 2 L(Mr ) = Qr . Hence the desired label (r; d ; A ; d ; : : :; dk ; Ak ; dk ) is valid with respect to automaton Mr . In other words, di = dui , 1  i  k + 1, and for every 1  j  m, every p 2 Kj , and every Ai 2 VN : +1

1

1

2

+1





(dui  dAi ) Kj (p) =
i;j  (dAi Kj )(p) = j(p; uiAi) for 1  i  k and duk+1 Kj (p) =
k ;j (p) = j(p; uk ):

+1

+1

Therefore, for 1  j  m, j(q ; u A u A u : : : uk Ak uk ) = = j( j(q ; u A ) ; u A u : : : uk Ak uk ) 0

1

1

2

|

= = = = = =

2

0

3

{z

+1

1

1

2

}

2

3

+1

(du1 dA1 )jKj (q0 )=p1;j  j (j(j(q0; u1A1); u2A2); u3 : : : uk Ak uk+1) {z } | (du2 dA2 )jKj (p1;j )=p2;j

::: j(j(: : : (j(j(q ; u A ); u A ); : : :); uk Ak ); uk ) duk+1 Kj  j(: : : (j(j(q ; u A ); u A ); : : :); uk Ak ) ::: (du3 


 duk  dAk  duk+1 ) Kj  j(j(q ; u A ); u A ) (du2  dA2 


 duk  dAk  duk+1 ) Kj j(q ; u A ) (du1  dA1  du2  dA2 


 duk  dAk  duk+1 ) Kj (q ):

0

1

1

0

2

2

1

1

+1

2

2



0



0

1

1

1



2

2

1

0

In case j = r, (du1  dA1  du2  dA2 


 duk  dAk  duk+1 ) Kr (q ) 2 Fr :

0

2

Together with our previous theorems, this immediately yields:

Corollary 6.3 All the language families L n (P; CF), L n (P; CF ? ), and L n (K; CF), L n (K; CF ? ) coincide. 2 Observe that the last corollary answers an open question listed in [3]. 21

7 Beyond the Finite Index Barrier We have seen that dierent context-free rewriting mechanisms which generate languages of nite index coincide. Thus, the question arises whether one can think about \natural" rewriting mechanisms (based on context-free core productions), which generate languages of nite index which are not programmed context-free nite index languages. Since in the non- nite index case, conditional context-free languages are a superset of programmed, random context, and ordered context-free languages, one way to break the programmed context-free nite index barrier might be to use conditional grammars with enlarged condition sets. Conditional grammars with dierent Chomsky type-i languages for core rules and condition sets were already investigated in [9]. In all cases, characterizations of context-sensitive and recursively enumerable languages were obtained; the only exceptions are given below: L(REG) = L(K; REG)  L(CF)  L(K ; REG)  L(CS): Here K denotes a conditional grammar with condition sets from L(X). Thus, e.g., L(K ; REG) denotes the family of languages generated by conditional grammars with context-free condition sets and regular productions. Observe, that L(K ; REG) obviously contains only languages of nite index by de nition for any X. We show that in the nite index case we can go beyond L n (P; CF) if one uses, e.g., conditional grammars with context-free core rules and context-free condition sets. CF

X

CF

X

Theorem 7.1 L n (K; CF)  L n (K ; CF[?]). Proof. The inclusion is obvious; it remains to proof the strictness. Since every one-letter language in L n (K; CF) is regular by [3, page 159, Corollary], CF

it is sucient to prove that a conditional grammar with context-free core rules and context-free condition sets can generate a non-context-free oneletter language. The conditional grammar G = (fag; fS; A; B g; P; S ) with the productions 1: S ! AB; aS , 1a: S ! a; aS , 22

2: B ! aB; f anAamB j n; m  0 and n > m g, 3: A ! a; f anAamB j n; m  0 and n = m g, and 4: B ! S; aB generates the language f a n j n  0 g. This is seen as follows: Note that the productions 1, 3; 4; 1; 2; 3; 4; : : : ; 1; 2; 3; 4; 1a must be applied in this order and rule 2 can be repeated several times. Therefore, a successful derivation in G has the structure 2

S ) AB ) aB ) aS ) aAB ) aAaB ) aaaB ) aaaS ) aaaAB )


) aaaAaaaB ) aaaaaaaB ) aaaaaaaS ::: ) a n? AB )


) a n? Aa n? B ) a n a n? B ) a n a n? S )a a n a n = a n+1 : 1

3

1

4

2

1

3

2

2

2

2

2

2

3

1

1

2

4

1

2

2

4

1

2

3

2

1

2

2

1

4

2

1

By induction one proves that L(G) equals the desired language. Obviously, language L(G) has nite index 2. This proves our claim. 2 The previous proof shows a little bit more than only the separation of L n (K; CF) and L n (K ; CF[?]). Since the condition sets of G are even linear context-free languages we obtain on the one hand: CF

Corollary 7.2 If L(X) is a language family that contains the linear contextfree languages, then L n (K; CF)  L n (KX ; CF[?]). 2 On the other hand, the language of the previous proof even separates the classes L(K ; REG) and L n (K ; CF[?]), since every one-letter language in L(K ; REG) is regular by [9, page 184, Lemma 2]. CF

CF

CF

Corollary 7.3 L(CF)  L(K ; REG)  L n (K ; CF[?]). CF

CF

2

Finally, we nd it quite surprising that conditional grammars with context-free condition sets and context-free core rules (even admitting -rules) that generate languages of nite index already belong to L(CS). Here the 23

presence of -rules is not crucial, because by the nite index restriction in every sentential form there are at most a constant number of nonterminals that can derive the word . Therefore, by a direct simulation of a (K ; CF) grammar of nite index n using a Turing machine, only a linear number (in the length of the input word) of additional cells on the work-tape may be used. Thus, we have: CF

Theorem 7.4 L n (K ; CF[?])  L(CS).

2

CF

Unfortunately, we have to leave as open the question whether the inclusion of the last theorem is proper or not. Without the nite index restriction it is known from [9] that

L(CS) = L(K ; CF ? ) and L(RE) = L(K ; CF): CF

CF

Employing a non-constant number of nonterminals during the simulation of a Turing machine (with a linear space bound) seems to be inherent in both cases. A natural idea for separating L n (K ; CF[?])  L(CS) might be to look at closure or decidability properties. Let us nally remark that, since it is possible to encode a variant of the language of valid computations of a Turing machine within the (K ; CF[?])framework, L n (K ; CF[?]) is not closed under arbitrary homomorphisms, nor does it possess a decidable emptiness problem. This observation proves once more that L n (K ; CF[?]) breaks the nite index barrier given by the other mechanisms. CF

CF

CF

CF

8 Conclusions We investigated programmed, ordered, (forbidding and permitting) random context, and conditional context-free grammars that generate languages of nite index. We proved normal form theorems for these grammars, and showed that all above mentioned classes coincide, even when considering language families of nite index n, regardless whether erasing productions are allowed or not. This is quite surprising, since we nd many inequalities regarding the power of these regulation mechanisms without the nite index 24

restriction. We just mention (we refer to [3, 4]) L(CF[?])  L(O; CF[?]) = L(fRC; CF[?])  L(P; CF[?]; ut)  L(P; CF[?]; ac)  L(K; CF[?]); and the inclusions L(CF[?])  L(RC; CF[?])  L(P; CF[?])  L(RC; CF[?]; ac) = L(P; CF[?]; ac): Moreover, L(K; CF) (L(K; CF ? ), respectively) describes just the class of recursively enumerable (context-sensitive, respectively) languages. Finally, let us mention that \recursively enumerable languages" (de ned by arbitrary phrase-structure grammars) even of nite index 1 just characterize the recursively enumerable languages. A similar statement is also valid for the context-sensitive languages. This can be easily seen by a technique using g-grammars as elaborated in [7] in the case of bidirectional cooperating distributed grammar systems. We have seen that (many) nite index languages have nice properties from the formal language theoretical point of view, but nally, let's turn back to the original motivation of Brainerd [1] for introducing the notion of nite index. As already said in the introduction, he proved that the length set of each in nite context-free matrix language of nite index contains an in nite arithmetic progression. It is natural to ask whether this is a property special to languages of nite index in the regulated rewriting case with context-free core rules. There is a negative answer to this question in two senses: 1. The family L(M; CF) (without the nite index restriction) has the property that every in nite language in L(M; CF) contains an in nite arithmetic progression, because the class L(M; CF) is closed under homomorphisms, and every one-letter-language in L(M; CF) is semilinear, as proved in the famous paper of Hauschildt and Jantzen [8]. 4

5

This inclusion is strict if and only if erasing productions are forbidden. In these cases, one has to be a bit careful about the de nition of type-0- and type-1grammars. For a detailed discussion, we refer the reader to [10] and [14, Section 4]. 4 5

25

2. In the preceding section, we have found a language class of nite index, n namely L n (K ; CF[?]), which contains the language f a j n  0 g that does not have an in nite arithmetic progression. 2

CF

References [1] B. Brainerd. An analog of a theorem about context-free languages. Information and Control (now Information and Computation), 11:561{ 567, 1968. [2] E. Csuhaj-Varju et al. Grammar Systems: A Grammatical Approach to Distribution and Cooperation. London: Gordon and Breach, 1994. [3] J. Dassow and Gh. Paun. Regulated Rewriting in Formal Language Theory, volume 18 of EATCS Monographs in Theoretical Computer Science. Berlin: Springer, 1989. [4] H. Fernau. A predicate for separating language classes. EATCS Bulletin, 56:96{97, June 1995. [5] H. Fernau. On unconditional transfer. In MFCS'96, 1996. To appear. [6] H. Fernau, R. Freund, and M. Holzer. External versus internal hybridization for cooperating distributed grammar systems. Technical Report TR 185{2/FR{1/96, Technische Universitat Wien (Austria), 1996. [7] H. Fernau and M. Holzer. Bidirectional cooperating distributed grammar systems. Technical Report WSI{96{1, Universitat Tubingen (Germany), Wilhelm-Schickard-Institut fur Informatik, 1996. [8] D. Hauschildt and M. Jantzen. Petri net algorithms in the theory of matrix grammars. Acta Informatica, 31:719{728, 1994. [9] Gh. Paun. On the generative capacity of conditional grammars. Information and Control (now Information and Computation), 43:178{186, 1979. [10] Gh. Paun. Length-increasing grammars of nite index. Rev. Roumaine Math. Pures Appl., XXVIII(5):391{403, 1983. 26

[11] G. Rozenberg. More on ET0L systems versus random context grammars. Information Processing Letters, 5(4):102{106, 1976. [12] G. Rozenberg and D. Vermeir. On ET0L systems of nite index. Information and Control (now Information and Computation), 38:103{133, 1978. [13] G. Rozenberg and D. Vermeir. On the eect of the nite index restriction on several families of grammars. Information and Control (now Information and Computation), 39:284{302, 1978. [14] G. Rozenberg and D. Vermeir. On the eect of the nite index restriction on several families of grammars; Part 2: context dependent systems and grammars. Foundations of Control Engineering, 3(3):126{142, 1978. [15] A. K. Salomaa. Formal Languages. Academic Press, 1973. [16] E. D. Stotskii. Formal~nye grammatiki i ograniqeni na vyvod. Problemy peredaqi informacii, VII(1):87{101, 1971. [17] E. D. Stotskii. O nekotoryh strogih ierarhih zykov. Nauqnotehniqeska informaci, Seri 2, (4):40{45, 1972.

27