Dassow, Jürgen

English

Dassow Jürgen

University of Szeged

On some extensions of indian parallel context free grammars B y J . DASSOW 1. Introduction In order to get better models for aspects of programming and natural languages some extensions of context free grammars are introduced, for instance matrix grammars, random context grammars, programmed grammars, time-variant grammars (see [1], [13], [7], [10], [11]), which are characterized by mechanisms regulating the use of the productions. The relations between the associated language families are studied by some authors (see [10], [4], [11]). In [12], R. SIROMONEY and K . KRITHIVASAN regard a parallel version of con-text free grammars. In this paper we introduce some of the above mentioned ex-tensions for these indian parallel context free grammars. An other generalization of the indian parallel grammars are the EDTOL systems (see [3]). We shall prove that all these language families coincide. The result has also another interesting aspect. The EDTOL systems work purely parallel, i.e. all occurrences of all letters are rewritten in a single derivation step; the other extensions of the indian parallel grammar have a sequential aspect because only all occurrences of one letter are rewritten in a single step. Therefore our result can be regarded as a sequential characterization of EDTOL languages. Thus, it is of interest in connection with the sequential characterizations of ETOL languages (see [14], [6], [9], [8], [2]). 2. Definitions and notations At first we recall the definition of the indian parallel context free grammar and its derivation process. Indian context free grammar. An indian context free grammar is a construct G=(VN, VT,P, S) where i) VN and VT are finite nonempty sets, VN fl K r = 0 , ii) P is a finite subset of VNX(VN{J VT)* (the elements of P are written as A£VN, W(I{VN\JVTY), iii) S£Vn. 1 Acta Cybcrnetica 304 J. Dassow Let V= VNUVT- Let V+, y£ V*. We say that x directly derives y iff i) x=x1Ax2Ax3...x„-1Ax„, AeVN, Xi€(V\{A})*, ii) y=x1wx2wx3...xn-1wxn, iii) A ->-w£P. * Then we write x=>y. Let => be the reflexive and transitive closure of =>. The language L{G) generated by G is defined as L(G) = {x: S^>x,x£Vl}. Now we define some extensions of this grammar by certain mechanisms regulating the derivation process. In all cases we use the alphabet VN of nonterminals, the alphabet VT of terminals, the axiom VN, productions A-»w, A£VN, w£V*, the application of a rule is in all cases defined as above, and if it is not stated other-wise then the associated language is defined in the way given above. We give the regulating mechanisms. Indian matrix grammar. G=(VN, VT, M, S) is an indian matrix grammar iff M is a finite set of finite sequences of productions, M = {m1,m2, ..., mr}, ™i = »V Al2 - wi2, ..., Ais - w,J for i = 1, 2, ..., r. The elements of M are called matrices. To apply such a matrix one has to apply the productions A ^ w ^ , . . . , A is-^w is in the given order. Only those words of Vj are in L(G) which are obtained by applications of the matrices. Indian periodically time-variant grammars. An indian periodically time-variant grammar is a construct G=(VN, VT,P, S, / ) , where / is a mapping N—ip(i>) such that f(i+j)=f(i) where m and j are fixed and i>m is arbitrary. The deriva-tion is regulated by the condition that the production used in the Ar-th step has to be in the set / (k). Indian random context grammar. The productions of an indian random con-text grammar G=(V1¥, VT, P, S) are of the form A+w,R,Q where R and Q are subsets of VN. Such a production is only applicable on a word x=x1Ax2Ax3...x„_1Ax„ if contains no letter of R and contains all letters of Q. Indian programmed context free grammar. The productions of an indian pro-grammed grammar G=(VN, VT,P, S) are of the form (/) A w, F, S where / is the label of the production, F and S are sets of labels. If A—w is ap-plicable to x, then the next production has to be a rule with a label contained in the success field S. If A~w is not applicable then the next production has to have a label contained in the failure field F. All these grammars work in a sequential-parallel way, i.e. only one letter is rewritten in a single'derivation step, but all occurrences of this letter are rewritten. Starting from biological motivations EDTOL languages are defined which are On some extensions of indian parallel context free grammars 305 also a generalization of indian parallel context free languages. The associated grammars work purely parallel as it is seen from the following definition. EDTOL system. An EDTOL system is a construct G—(V, VT, { P I , P2, . . . , P R } , S) where i) V is a finite set, VT is a nonempty subset of V, ii) S € F \ F r , iii) each Pt is a finite subset of FX V*, the projection of P, on the first coor-dinate is V, and if A—w1, A-+w2 are in P, then w1 = iv2. (Usually only S£ V + is required. It is easy to prove that K \ K r does not restrict the generative power.) Let V+ and yd V*. It is said that x directly derives y (also written x=>-.y) iff i) x = x1x2...x„, x£V, ii) y=yly2...y„, iii) there is a y£{l, 2, . . . , r} such that x ^ y ^ P j for / = 1 , 2 , . . . , « . The language L(G) is again defined as L(G) = {x: S^>x,x£Vf}. We use the following notations !F (IM) — family of indian matrix languages, !F (IPTV) — family of indian periodically time-variant languages, 3F (IRC) — family of indian random context languages, J5" (IPCF) — family of indian programmed context free languages, J5" (EDTOL) — family of EDTOL languages.. 3. Comparison of the language families In the following proofs we will often introduce new alphabets. We make the next convention: If UQV and V{, — {x': U) is a new alphabet then w' = = x'[x'2 ...x'^ for w—x1x2.:.xn where „ = (xl € U ~~ I*, U. Let min (vv) denote the set of letters occurring in w. Lemma 1 . 9 (EDTOL) g j ^ (IM) Proof. Let G=(V, VT, {P l5 P2 , . . . , Pr}, S) be an EDTOL system. Let wXtP be the right side of the production with the left side x in P^{P1,P2, . . . , P,}- Further WP put f(U,P)= U min (wx P) for UQ V. xiU For a subset UQ V we introduce a new alphabet Vu={xu: x£ U). Now we de^ne the following matrices for U={xh, xh, ..., x ; J Q V and P£ {Px, P2 , . . . , Pr}, Pu = [(*/i)i/ — (wxil,p)f(u,P)> (xi2)u "*" (wxj2,p)f(u,p)> •••» (x,k)u — (.wxik,p)nu,p)]> Qu — tC-^ii)C/ xhi (xiz)u xi2 > •••) (xik)u 1* 306 J. Dassow and consider the indian matrix grammar tf=i(F\Kr)U u Vv,VT, U U (PduV U- S a . s J . v ugv i=iugF i/sk ' The application of the matrix Pv models the application of P to words w with £/=min(»v), i.e. if w=>w' is in G then WV^>{W')V, is in H where £ /=min(w) , i / ' = m i n ( w ' ) . The application of the matrix Qv is a translation of wu in w. If wf Vy then we can apply only the matrices Pv and Qv. Now it is easy to see that L(G)=L(H). Therefore L{G)^{IM). Lemma 2. ^ ( I M ) ^ i ^ ( I R C ) . Proof. Let IM) and L=L{G) for the indian matrix grammar G = (VN, V T , M, S). L e t M={M1, M2, ...,MR), M—LA^W^, AIT-<*WIT, ..., . . . , AiM—w,J for i=\,2,...,r. We introduce new alphabets V',J = {xhJ: V}, l s a i ^ r , l ^ j m s - l . Let V'=\J(JV'-J\JVN. .-=i j=i Now we can model the application of the matrix m l by the following sequence of productions of an indian random context grammar AH - K ) ' - 1 , 0 x - x i ' 1 , K ' \ ( ^ j v U K i - 1 ) ) m i n ( ( w i l ) ' ' - 1 ) fo r X£VN, Atf - (w^)'-2, 0 3th1 - x>'\ VXiV'^UV'-2), min((vv12)''-2) for x'-^V'-1 A'R/-1 - (w I s), V\VI,S~1, 0 ^ K M F ' ^ - ^ F j v ) , min ( w j for x ^ " 1 ^ 1 ' ' 5 " 1 . If we consider only such productions in our indian random context grammar then we can have also only such derivations which model the application of matrices. Therefore, we generate the same language. Thus L £ ^ ( I R C ) . The above construction works correctly only if we have no rule of form A^—X in the matrices (A denotes the empty word). If we have a matrix M such that m = [A1-~ w1; ..., Aj — X, ..., Ak — wfc] we use m' = [Ax - wu ..., Aj - Bj, ..., Ak - wk, Bj - X] instead of m, where B} is a new nonterminal. It is easy to see that this modification do not change the generative capacity and that our construction works also in the modified case. Lemma 3. ^ ( I R Q i J ^ C I P C F ) . Proof. Let G=(VN, V t , P. S), VN={ALT A2, ..., AN}. We give a possibility to model a production of G by rules of an indian programmed context free grammar. 307 J. Dassow: On some extensions of indian parallel context free grammars Let Aj-w, {Ah,Ah, ...,A,s}, {Ah, Ah, ..., AJt) be a rule of G. Consider a new alphabet V = \x': x€ VN} associated with the rule and the following diagram of IPCF-productions (an arc labelled by F connects a production with its failure field, and an arc labelled by S connects it with its success field). It is obvious that we can only simulate rules of G because we have associated the primed alphabets with the productions of G. This proves £(G)£J5"(IPCF). Lemma 4. J ^ ( I P C F ) g ^ ( E D T O L ) . Proof. Let Z,G^(IPCF), L = L(G) for an indian programmed context free grammar G=(VN, VT, P, S). Let R be the set of labels of the productions of P. With each subset I of R we associate new alphabets Vf = {xt: VN} and V[ = = x£ VN}. We define tables P I > M for any /6/, IQR, /€ {1, 2, 3} in the following way: If (/) A-*w, F, S is a rule and / € / then put Pi,i,i = {*/ - * ^ A}U{Aj / S - S}U{x - x: x£VT), Pi,i,2 = {*/ - * * A}{J{Aj - Ai, f - / , S - S}U{x - x : x£VT}, = {*/.-> *s = x^A}U{Ai-~ws,f~f, S - S } U { x - x : x£VT). Using productions from sets of these types we can only generate words which contain only letters of VT and Vr for a certain I with exception of one letter which can be in K/. If we have such a word we can apply only tables PIJJ. Further Pl<lt 1 models the case that A, does not occur and produces a word which consist of termi-nals and nonterminals of the alphabet associated with the failure field. The other two tables model the application of A—w, and we get a word with nonterminals of the alphabet associated with the success field. Let / 1 , / 2 , . . . , / r be the sets containing labels whose production has the left sidft S, and put a = {S-S/1,/-/}U{*-*: x€VTU U (VjUVi)}. IQR Then the EDTOL system H = ( { S , / } U r T U U (VjUVi), VT, {Phly. I ^ R , ¿€{1, 2, 3}, / € /}U IQR U{&: 1 ^ r ) , S) generates L. Thus LG^(EDTOL). Lemma 5. ^ ( I M ^ J ^ I P T V ) . Proof. The proof of [10], Theorem 11 works also in the indian parallel case. Lemma 6. ^ ( I P T V ) g J^(EDTOL). Proof. Let L—L(G) for the indian periodically time-variant grammar G=(Vn, V t , P, S , f ) where f{i+j)—f(i) for />w?. We introduce new alphabets K<» ={*<*>: xeF,v} for l g / S m + ; / - l . For A-*w=p£P, p£f(i), l S / < m + j - l we define the tables P. p = {x(0 - jcO+D; x ^ A}U{Aw - w ( i+1)}U {x - x : x€VT} 308 J. Dassow A2 — A'z W i s F{ | S All rules with left side Al and a primed version of Ax on the right side F Fi iS * < A'n-An S Fig.1 309 J. Dassow: On some extensions of indian parallel context free grammars and Pm+J.liP = {x(m+-'-1> - x(m): x 7i A}U{A(m+J~1) - w(m,}U{jc - x: x£VT}. -It is easy to see that the EDTOL system ( m+j-l \ H=\VTU ¿ J y{k\ VT, {Pi,P: 1 ^ ^ m+j-l, pfj(i)}, 5 (1 )J generates!,. We say that a grammar is A-free if it contains no production with the empty word X at the right side. The family of X-languages generated by 1-free A'-grammars is denoted by Fx{X). As usual we identify languages which differ only in the empty word. Theorem 1. (EDTOL) = # ' ( I M ) = J i r ( [ R C ) = J r ( I P C F ) = ^ ( I P T V ) = — J5";. (EDTOL) = SF-t, (IM) = J ^ (IRC) = (IPCF) = (IPTV). Proof. The first row follows directly by Lemma 1—6. Further 2F (EDTOL) = =3Fk (EDTOL) is known and all our constructions in Lemma 1—4 and 6 preserve A-freeness. If the matrix grammar in the proof of [10], Theorem 11 is 1-free then we modify the proof in the following way: The new symbols Yj are not catenated with Pj, the last letter of Pj has to be in a new "primed" alphabet and the last rules have to change the letter into a "not primed" letter. By Theorem 1, we get some information on properties of the extensions of indian parallel context free grammars, because we have knowledge on 2F (EDTOL). — In [3], closure properties under AFL-operations are given. — There are context free languages which are not ' in (EDTOL). — It is known that the families of matrix languages, programmed context free languages, random context languages and periodically time-variant languages prop-erly contain (EDTOL). Thus the indian parallel restriction reduces the generative capacity of the considered extensions. — The proof of v. SOLMS [14] works also in the indian parallel and deterministic case. This proves that indian random context grammars of special type generate already all indian random context languages. A further language family which is equal to the above families is given in [2], Theorem 2. Finally we want to mention without proof that all our language families also coincide with the family of indian unordered scattered context languages, which are the indian parallel version of the unordered scattered context grammars of [5]. TECHNOLOGICAL UNIVERSITY OTTO V O N GUE RI CKE D E P A R T M E N T OF MATHEMATICS A N D PHYSICS M A G D E B U R G , G R D References [1] ABRAHAM, S . , Some questions of phrase structure grammars, Comput. Linguistics, v. 4 , 1 9 6 5 , p p . 6 1 — 7 0 . [2] DASSOW, J . , E T O L systems and compound grammars, Rostock, Math. Colloq., v. 1 1 , 1 9 7 9 , p p . 4 1 — 4 6 . 310 J. Dassow: On some extensions of indian parallel context free grammars [3] EHRENFEUCHT, A. & G. ROZENBERG, On images of inverse homomorphisms, Automata, Lan-guages, Development, Ed. A. Lindenmayer, G. Rozenberg, North-Holland, 1976. [4 ] MAYER, O . , Some restrictive devices for context-free grammars, Inform, and Control, v. 20, 1972, pp. 69—92. [5] MILGRAM, D . & A . ROZENFELD, A note on scattered context grammars, Inform. Process. Lett., v. 1, 1971, pp. 47—50. [6] PENTTONEN, M., ETOL-grammars and N-grammars, Inform. Process. Lett., v. 4, 1975, pp. 11—13. [7] ROSENKRANTZ, D. J., Programmed grammars and classes of formal languages,/. Assoc. Comput. Mach., v. 10, 1969, pp. 107—131. [8] ROZENBERG, G., More on ETOL systems versus random context grammars, Inform. Process. Utt., v. 5, 1976, pp. 102—106. '91 ROZENBERG, G. & D. VERMEIR, Context-free programmed grammars and ETOL systems, Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, v. 45, 1976, pp. 482—487. [ 1 0 ] SALOMAA, A . , On grammars with restricted use of productions, Ann. Acad. Sci. Fenn. Ser. A I Math., No. 454, 1969. [11] SALOMAA, A., Formal languages, Academic Press, 1973. [12] SIROMONEY, R. & K. KRITHIVASAN, Parallel context-free languages, Inform, and Control, v. 24, 1974, pp. 155—162. [13] VAN DER WALT, A . P . J . , Random context languages, Information Processing 1971, North-Holland, 1972, pp. 68—66. [14] VON SOLMS, S . H., On TOL languages over terminals, Inform. Process. Lett., v. 3, 1975, pp. 69—70. (Received June 6,1979) 

On some extensions of indian parallel context free grammars

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On some extensions of indian parallel context free grammars

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