Some decision problems for parallel communicating grammar systems

Theoretical Elsevier Computer Science 365 134 (1994) 365-385 Some decision problems for parallel communicating grammar systems Ferucio Laurentiu Cecilia Magdalena Tiplea, Cristian Ene, Ionescu and Octavian Procopiuc zyxwvutsrqponmlkjihgfedcbaZ Department of Computer Science, “Al.I. Cuza” University , RO - 6600 Iasi, Romania Communicated by A. Salomaa Received March 1993 Revised October 1993 Abstract Tiplea, F.L., C. Ene, C.M. Ionescu and 0. Procopiuc, Some decision problems for parallel municating grammar systems, Theoretical Computer Science 134 (1994) 365-385. com- In this paper we investigate several decision problems for parallel communicating grammar systems: the enabling, circularity, centralizing, conflict-freeness, boundedness, membership, equivalence, inclusion, emptiness and finiteness problems. The first five problems are shown to be decidable for context-free PCS’s but undecidable for the context-sensitive case. The last five problems are only studied for the context-sensitive case and they are shown to be undecidable. 1. Introduction The modelling of parallel processing remains one of the major challenges of the Computer Science, both in theory and in practice. Many attempts have been made to find a suitable model and many of them are based on automata and formal language theory. For example, the Petri nets [19], the discrete event systems [lS], the cellular automata [7], the systolic trellis automata [2] are models based on automata theory, while the L-systems [20], the Russian and Indian parallel grammars [3], the cooperating grammar systems [lo], the cooperating/distributed grammar systems Cl], the parallel communicating grammar systems [ 151 are grammatical models. These models have been successfully used in a variety of domains, such as manufacturing, database systems, robotics, vehicular traffic, air traffic control, logistic (conveyance and storage of goods, organisation and delivery of services), computer and Correspondence to: F.L. Tiplea, Department of Computer Science, “AH. Cuza” University, Romania. 0304-3975/94/$07.00 0 1994-Elsevier SSDI 0304-3975(93)E0210-U Science B.V. All rights reserved RO-6600 Iasi, 366 F. Laurentiu et al. communication network, parallel architectures, developmental systems, neuronal systems, artificial intelligence, to name a few. The increasing complexity of man-made systems, made possible by the widespread application of computer technology, has taken such systems to a level of complexity where more detailed formal methods become necessary for their analysis and design. zyxwvutsrqponml Parallel communicating grammar systems (PCS, for short) are a new model of the parallel processing [IS] and they are aimed to combine the notions of parallelism and into a suitable model for theoretical investigations. These systems have evolved from the following considerations: communication l knowledge processing systems are characterized by an intimate cooperation between logic and functional programming, which requires an adequate communication discipline; l processing requirements for knowledge based problem solving are of a different nature, making heterogeneous parallel systems more appropriate; l although interprocess communication could be decided at process level, overall supervision is necessary for efficient task distribution, resource allocation and management. A PCS of degree n consists of n separated usual grammars working in parallel; each of them starts from its own axiom and, in well defined circumstances, communicates with each other. The moment of communication depends on the query symbols appearing in the current sentential forms generated by the grammars. Query symbols are special nonterminals, indexed from 1 to 12,to refer to the grammars. Such a symbol may belong to the nonterminal vocabulary of any grammar, except the one whose index it bears (a grammar cannot ask from itself). The appearance of a query symbol in any of the sentential forms imposes a communication, as the query symbols are nonterminals that cannot be rewritten. A communication consists of the replacing of all query symbols with the current strings of the grammars they refer to. However, a restriction is imposed: no replacing takes place for the sentential forms containing query symbols referring to strings containing further query symbols. Circular communications are not admitted. The theory of PCS’s is very young, therefore many questions are still unsettled in this area. Some results such as generative capacity, syntactic complexity, closure with respect to various operations, decision problems, have been studied in [6,12-l 7,231. The paper deals with decision problems concerning PCS’s For the context-free case, we introduce a finite coverability structure (Section 4) which permits us to prove the decidability of some decision problems for such PCS’s. For context-sensitive PCS’s we use a decision problem for monotonic grammars and the simulation of O-type grammars (Section 5) in order to establish undecidability results. 2. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Preliminaries We assume the reader is familiar with basic concepts example, from [21]); we only specify some notations in formal language and the definitions theory (for concerning PC%. Some decision problems for parallel communicating 367 grammar systems For an alphabet V, we denote by V* the free monoid generated by operation of concatenation and the null element 2. Let weV* and U s denotes the length of w and #(U, w) denotes the length of the string erasing from w all symbols not in U. When U = {a} we simply write #(a, #((4 4. We denote by CS, CSn, RE the classes of context-sensitive grammars I’ under the V. Then 1WJ obtained by w) instead of (i.e., whose rules are monotonic), arbitrary context-sensitive grammars (i.e., the grammars contain the rule S--+2, S being the axiom), and O-type grammars, respectively. may A PCS of degree n, n2 1, is a system Y=(G r,...,G,) where Gi=( VN,i, l’r,i, Si, Pi), 1 <id n, are Chomsky grammars such that Vr,i s I’r, 1 for all 2 < i < n, and there is a set K E {Qi, . , Q,,} of special symbols (query symbols), KGUl=,VN,i. zyxwvutsrqponmlkjihgfedcbaZYXWVUT Xi,yi~ Vz*,ifor all The n-tuple (x1, ...,x,) directly derives in the n-tuple (yi, . . . , y,), zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF if one of the next two cases holds: denoted (xi ,..., x,)*(yl ,... , y,), (i) #(K, xi) = 0 for all 1~ i< n, and xijyi in the grammer Gi, or xiE I’,*,i, Xi = yi, l<i<n, ldidn; (ii) if #(K, xi) > 0 for some i, 1~ i < n, then for each such i we write with zjEI’z,i and # (K,zj)=O for all 1 <j<t+ 1. If #(K, xi,)=O, 1 <j<t, then yi=z1xi,zzxi, ...z,xi,z,+1 and yi,=Sij for all 1 <j<t; when for some j, 1 <j < t, # (K, Xi,) > 0 then yi = Xi. For all i, 1 < i < n for which yi was not defined above, we put yi = Xi. In words, a derivation step consists either of a rewriting step (i) or of a communication step (ii). The communication has priority over rewriting. The above definitions refer to returning synchronized PCS’s (returning, because after communicating each component whose string has been sent to another component returns to axiom, and synchronized, because each grammar uses exactly one rule in a rewriting step, the only components which may “wait” being the terminal ones). When in point (ii) of the previous definition we erase the words “and yi,=Si, for all 1 <j < t” we obtain a nonreturning PCS (after communicating, the grammar Gi, does not return to Si,, but continues to process the current string), and when in point (i) of the previous definition we erase the words “XiE VT*,;‘, then we obtain an unsynchronized PCS (the grammar may wait without restrictions). Moreover, if Kn VN, i = 0, 2 d i < n, then the system is called centralized (only Gr, the master grammar, is allowed to introduce query symbols); a system without this restriction is called noncentralized. 3. Some useful notations and conventions on PCS’s To express the properties of the PCS’s in an adequately notations and conventions that follow. formal way, we use the 368 F. Laurentiu et al. Let y=(G1, . . . , G,) be a PCS of degree II. A conjiguration of y is any n-tuple w=(wl, . ..) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA w ,) where WiE I’,*,i for all zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI 1 < i < TV( VG,i = VN,iu VT,i). The configuration of y. We say that the configuration v is (S 1, ... 3S,) is called the initial con$guration reachablefrom the configuration u (in y) if u & U,where % is the reflexive and transitive closure of the relation *. If u is the initial configuration of y then we simple say that u is reachable (in y). The configuration w =(wl, . . . , w,) is called circular the pairwise distinct integers il, iZ, . . . , ikE { 1,2, . . . , n} such that: Let us remark that a circular configuration string on the first component (the definition if there exist may occur after generating a terminal of the relation 3 permits such a case). The productions in the grammar Gi (the ith component of y) are considered to be labelled (it is not necessary to have distinct labels for distinct components of y). The set of all labels of the ith component of y will be denoted by Lab+, or Labi when y is understood from the context. In a rewriting step in y there are cases when no production is applied on some components. For example, one case is that of a terminal string and an other case is that of an unsynchronized PCS. For technical reasons we want to apply exactly one production in a rewriting step, from each grammar. Thus, we consider a phantom production which can be applied to any string, in the synchronized case, and only to terminal strings, in the unsynchronized By extension we put: #(x,LHS(O))= case. In any case, 0 does not change the string. #(x,RHS(O))=O for any symbol x, where, for a rule r, LHS(r) and RHS(r) denote the left hand side and the right hand side, respectively, of the rule r. Now we can say that in any effective rewriting step of y it is applied an n-tuple of productions, n-tuple which will be called a transition of y. For uniformity we assume that the communication steps are performed by a special transition denoted ,4. This transition will cumulate all consecutive communication steps between two rewriting steps. Thus, a transition of y is either il or any n-tuple t =(I-~, . . . , r,), where riELabiu{O} for all 1 <id n. By TR(y) we denote the set of all transitions of y. Those transitions which produce circular configurations will be called circular transitions; they can be effectively determined by the inspection of the productions. Let w=(wl, . . . , w,) be a configuraiton of y and t =(I-~, . . . , r,) a transition. We say that t is enabled at w if w does not contain query symbols and the rule ri can be applied to the string Wi. If t is enabled at w then t may occur yielding a new configuration, say u, computed by applying each rule ri to the string Wi. We denote this by zyxwvutsrqponmlkjihgfed w 2 U. The transition /1 is only enabled at configurations containing query symbols but not circular ones. The occurrence of /1 yields a new configuration solving all communications imposed by the query symbols appearing configuration. computed by in the current Some decision problems 4. On context-free for parallel communicating 369 grammar systems PCS’s In this section we only deal with nonreturning synchronized context-free PCS’s and we present a method to associate a finite coverability structure to such PCS’s a careful analysis of it we shall derive some useful properties of PCS’s. By The structure will be a tree and it will be obtained by analysing the increment and decrement of the number of nonterminals in all derivations of the grammar system. To do this we associate to each configuration w of a PCS y a vector which retains the number of occurrences of nonterminals in w (the image of w by the Parikh mapping associated to I’,,,). To each derivation in y will correspond a computation with vectors of natural numbers. 4.1. The coverability We follow then the ideas from [8]. tree of a context-free We begin with some notations, Notation 4.1. The set N is extended operations conventions, and definitions. by a special symbol w to the set N,z= N u(o). “ + “, “ - “, “.“, and the relation o+o=w+n=n+o=o, PCS “<” over N are extended w-n=o, o.n=n.w=o, The to N, by: n<o, for all nEN. The operations and relations over N (NJ are extended to Nk (Ni) by applying them componentwise (for a set A and a natural number k> 1, Ak denotes the set of all k-dimensional vectors over A). 4.2. In this section the PCS’s are considered as follows: synchronized context-free PCS of degree n, n 3 1; y=(Gi, . . . , G,) is a nonreturning Gi = (VN, i, Vr, if Si, Pi) is the ith component of y, 1 < id n; K={Q1,... , Q,,} is the set of all query symbols of y; the set VN,?- K of all nonterminals of y, excepting the query symbols, is ordered, Notation l l l l A 1, ... such that A,=S,, Notation vector 3 A,+, (m30), . . . ,A,=&. 4.3. (1) Let w=(wi, . . . , w, ) be a configuration Mw=K#(X,,w,)> ... , #(xzn+m,wA of y. By M, we denote the . ,(#(X~,W,), . .. , #(x,,+,,w,))), where Xi=Aiforall 1 <i<n+m and Xn+m+j= Qj for all 1 <j d n. M,(i, j) denotes the element #(Xj, Wi) and M,(i, -) denotes the vector zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR (#(x19wi)2..., where l<iQn #(XZn+m,W i)), and l<j<2n+m. 370 F. Laurentiu et al. (2) Let t =(rI, . . . , r,) be a transition zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON of y (t # A). By At we denote the vector At=((a:,...,cr:,+,),...,(crl,...,cr2,+,)) where M:= #(Xj, The notations RHS(ri))At(i,j) #(Xj, LHS(ri)) and At(i, -) for all 1 <idn and 1 dj62n+m. are as those from (1). In Notation 4.3 we have introduced an “encoding” which associates a vector of natural numbers with each configuration of a PCS. We show now how we can associate a computation with such vectors to each derivation in a PCS. zyxwvutsrqponmlkjihgfedcbaZ Definition 4.4. Let y be a PCS (as in Notation 4.2), U~(kIc+~)n and t a transition of y. (1) The transition t is enabled at U (in y), denoted U[t),, if one of the next two cases holds: and n +m+ 1 <j<2n +m (U does not “contain” (a) U(i,j)=O for all 1 <i<n queries), and t = (r 1, . . . , r,) # A, and for each i, 1~ id n, we have: if ri = 0 then U(i,j) = 0 for all 1 <j d n + m (0 can be only applied LHS(ri)) for all 1 <j<n+m; U(i,j)# O for some l<idn and n+m+ 1 <j<2n+m, (b) not exist any sequence of natural numbers U(i,j)3 to terminal strings); else # (A,, i, ,..., i/E{1 ,..., and t is /1, and does n} such that: U(iI,n+m+i,)31, U(i2,n+m+i3)3 1, .. U(i,-1,n+m+i,)31, U(i(, n + m + iI) 3 1 (i.e., there is no circular query). (2) If t is enabled at U then t may occur yielding a new vector WE(~J$‘~~)“, denoted U[t),W , and given by: (4 if.U(i,j)=O then: for all 1 <i<n W (i, -)= and n+m+ U(i, -)+At(i, -), U(i, -), 1 <j<2n+m, and t=(rl, . . ..r.)# A, if ri# O, if ri= 0, for all 1 <i < n; (b) if U(i,j)# O for some i and j, l<i<n, computed by the following algorithm: Pl. wI=u; n+m+l<j<2n+m, then W is Some decision problems for parallel communicating grammar sy stems 371 P2. Compute the largest set A E {1, . . . , n} such that for any SEA there is je(1,2 ,..., n}-A,suchthat W(i,n+m+j)>Oandforallp~(l,2 ,..., rr> we have W (j,n+m+p)=O. Let B= {jl 1 <j<n, 3i~A such that W(i,n+m+j)>O and W (j,n+m+p)=O for any l<p<n}. If A #Q, then goto P3 else goto P4. P3. for (any SEA) do for (any jEB such that W(i, n + m + j) > 0) do begin for p:= 1 to n+m do W (i,p)~= W (i,p)+ W (i,n+m+j)W (j,p); W (i,n+m+j)+O; end; got0 Pl; P4. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA stop (W is computed). We remark that we have associated a computation step with vectors of natural numbers either to each rewriting step (Definition 4.4 (2(a))) or to all consecutive communication steps between two rewriting steps (Definition 4.4 (2(b))). Let us exemplify the above constructions. Example 4.5. Let y =(Gr, Gz, G3) be the nonreturning synchronized PCS given below (we only specify the rules of the grammars Gi; the nonterminals will be denoted by capital letters). Gr: rl : S1-+uQ2 G1: rl : S2+aQ3 G3: rl : S,- +aABbB r2 : A- +bQ3 r2 : S2+aAB r2 : B+AAbB r3 : A- - +a r3 : B+bbB r3 : A+aBQ,. r4: B-+b r5 : B- rAB The following are derivations in y: 372 F. Laurentiu et al. wheretl =(rlt rl, rl), t2 =(r2, r3, r2), t3 =(r3,r3,r3), t4 =(rl,r2, rd, and t5 =(r2, r-3, r3). If we denote by wo, wl, w2, w3, w4, w5, w6 (resp., wo, ul, u2, uj) the configurations from derivation Dl (resp., D2) then we have: and (considering the order Sl, S2,S3,A, B, Ql, Q2, Q3, the vectors M , and At, w being a configuration and t a transition, are constructed as in Notation 4.3; for example, MW6 =((0,0,0,2,4,0,0,0,), (O,O, 0,1,2,0,0,0),(0,0,0,~> 2,0,0,0)) and Atl=((-1,0,0,0,0,0,1,0,),(0, -l,O,O,O,O,O, l),(O,O, -1,1,2,0,0,0))) We remark that the derivation D2 is blocked by circularity. Let A and B be two arbitrary sets. If F( V, E,tl,t2) is an (A, B)-labelled tree (i.e. /, : V-tA is the node labelling function and /,: E- B is the edge labelling function), by d,-(u, v’) we denote the set of all nodes on the path from u to v’. We are now in a position to introduce the notion of coverability tree of a context-free PCS y. It is based on the tree F’(y) of all reachable configurations of y. The The branches root of F’(J) is labelled by M ,o, w. being the initial configuration. starting at a node o labelled by M are constructed as follows: if the occurrence of a transition t at M yields the vector M ’, the tree F’(y) has a t-labelled edge that start at u and ends at an M’-labelled node. This construction yields of course, in general, infinite trees F’(y). But we can derive from F’(y) a finite tree F(y) without loosing too much information about the reachable configurations. The idea is to skip “regularly structured” paths and to indicate this regularity in the inscription of leafs. Assume a path ~~-!&~&a~. of Y’(y) with two nodes v,, up such that tl< p and up does not contain “query symbols” and for all i and j, d,(v,) (i,j) <E,(v,) (i,j). F’(y) has in this case an infinite path which infinitely often repeats the transition sequence t a+1 ... tS. This infinite sequence is in F(‘J) replaced by a single leaf. Its label may contain the symbol w which says us that the value on some component unboundedly increases within the replaced path. 373 Some decision problems for parallel communicating grammar sy stems zyxwvutsrqponmlk Definition 4.6. Let y be a PCS (as in Notation 4.2). tree, F = (V, E, t’,, f,), is called a zyxwvutsrqponmlkjihgfedcbaZYXWVU coverability tree of y if A (W 2n+m)n, TR(y))-labelled the following (1) the hold: root, by vO, is labelled denoted by MWO where w0 =(S1, . . . , S,) (i.e., [1(vo)= M,J; (2) for any node UE V, !I there 0, if either IU+J= is not any transition the number of transitions enabled at e,(u), otherwise, t which is enabled (3) for any DE V with Iv+1 >O and any transition at e,(v), uf u’ and e,(u)=el(u’), is u’~d~(u~, u) such that or there enabled at L,(u) there is a node u’ such that: (a) (u, u’)E E, (b) lz(u, 0’) = t, (c) d,(u’) is given by: - let A4 be such that f,(u)[t),M; _ if A4 contains “queries” then /,(u’)= M else w, e,(u’)(U) if there dl(u” )dM = M(i,j), for all ldidn exists and u”~d~(u~,u) f,(u” )(i,j)< such that M(i,j), otherwise, and ldj<2n+m. If F1 and Y-2 are two coverability tress of a PCS y then F1 and F2 are isomorphic (i.e. there is an isomorphism of directed trees which preserves the labels of the nodes and edges). Hence, we can talk about the coverability tree of a PCS y which will be denoted by F(y). The coverability tree of a PCS is always finite. This fact can be shown following the same line as in [8] where the finiteness of the coverability tree of vector addition system was established. Theorem 4.7. For any nonreturning synchronized finite and can be efectiuely context-free PCS y, the tree T(y) is constructed. Proof. The next two results are used: (1) Let uO,ul, . . . , II,,, . be an infinite sequence of elements of NP,, p 2 1. Then there is an infinite subsequence Vi,,Ui,, . . . , Ui.3. . . such that Ui, < Ui, < . . . < ui. < . . . (for a proof see zyxwvutsrqponm C81); 314 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA F. Laurentiu et al. (2) (Ktinig Infinitary Lemma [9]) Let T be a rooted tree in which each node has only a finite number of successors and there is no infinite path directed away from the root. Then T is finite. The first remark is that each node in Y(y) has only a finite number of successors. Now if we assume that there is an infinite path v~,u~, . . . .v,, . . . in Y(y), then ~,(u~),~,(u,), . ,tl(un), . . . is an infinite sequence of elements of (N$““)“. For such vectors, (1) holds too. Hence, there is an infinite subsequence Ui,, Ui,, . . . ,Uin, . such that e,(vi,)~e,(Ui,)~ ... <d,(Oi.)< .... By definition of Y(y), each vector e,(u,+,) has more o-coordinates than e,(U,) (since none of those nodes is an end, we never have d,(Ui,)=&1(Ui,+,)); a contradiction with the finite number of coordinates. Thus, all paths in r(y) are finite and Y(y) is finite by Kiinig’s lemma. Since r-(y) is finite, its construction, using the (recursive) effective. definition, is clearly 0 Example 4.8. A fragment pictorially represented of the coverability tree of the PCS given in Example 4.5 is in Fig. 1. The letters in circles represent the nodes of the tree and, t6 and t, are the transitions: t6 =@3, y3T r2), c7 = (r4, r3, r2 1. The node vlo is a leaf node because its label is identical with a previous one (the node v8), and u7 is a leaf node because no transition is enabled at e,(v,) (in fact, el(u7) contains a circular query). The dot line in Fig. 1 specifies an “o-breakpoint”. Fig. 1 375 Some decision problems for parallel communicating grammar sy stems Definition 4.9. Let y be a PCS and ME(N~+~)“. (1) The vector A4 is called coverable in y if there is a derivation: such that M, > M (wO being the initial configuration of y). (2) The vector M is called coverable in F(y) if there is a node v or F(y) such that e,(v) > M. Some information about the set of all reachable configurations of a PCS y cannot gained from the coverability tree F(y). But, it will be shown in the following coverability tree F(y) covers all reachable configurations of the system y. be that the Theorem 4.10. Let y be a nonreturning sy nchronized context- free PCS and M E(N’“+~) such that M (i, n + m + j) = 0 for all 1~ i, j < n. Then, M is coverable in y iff M is coverable in F(y ). Proof. Suppose t1 first that M is coverable wg*w~=E-..~~wh t2 in y. There is a derivation in y th such that M ,I 2 M (3 specifies that the step of the derivation transition ti). We consider the sequence of vectors is performed by the Mwo,M,,, . . . , M,b and we shall prove that there is a sequence vo,ut,..., vh (not necessarily of nodes in F(y) pairwise distinct) such that e,(Ui) > M ,! for all 0 <i < h. Thus we shall obtain coverable in F(y). el(uh) 2 M and hence M is The node v. is the root of F(y), and we have ~,(v~)=M ,~. Suppose that we have determined the nodes u0, . . . , vi, i< h, such that e,(Uj)a M , for all 0 <j< i, and we want to determine the node Ui+l. We have two cases to consider: (1) vi is not a leaf node. Then ti+ 1 is enabled at vi and vi + 1 is uniquely is that node v with the property ez(Oi, v) = ti+ 1; determined: it (2) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Vi is a leaf node. Then it is easy to see that only possible case is that in which there is vEd,(,) (vo, vi), v # vi, such that zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF l,(ui)=/,(u). The transition ti+ 1 is enabled at e,(v) and hence there is a unique node v’ such that e2(v, v’)= ti+ 1. We choose then Vi+l=V’. In both cases it is easy to see that e,(vi+l)>M ,Z+I. 316 F. Laurentiu et al. Conversely, suppose that A4 is coverable in F(y). Let v be a node of r(y) such that e,(v) > zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA M and let the path from v. to v have the successive nodes vo, 01, ... ) Oh = v. Let ti, 1 did h, be the transition which labels the arc (Vi_ 1, vi). If e,(v) does not contain o-components then the derivation has the property M,6=81(v), and hence M is coverable in y. In the case that e,(v) contains o-components, the idea is that there are vectors M, (w being a reachable configuration and can be made arbitrarily large sequence of transitions which led tion involve some calculation and from [S]). 0 4.2. Decidable of y) which agree with e,(v) in its finite coordinates, in the coordinates equal to o by repetition of the to the occurrence of o. The details of the constructhey are omitted (they follow a line similar to that questions We apply now the results obtained decidability PCS’S. of some properties about in the previous section in order to establish the nonreturning synchronized the context-free The first problem is: can we decide whether a transition of a nonreturning context-free PCS y is enabled in y? The answer to this question will be obtained from the following theorem. Theorem 4.11. Let y be a nonreturning context-free enabled in y ifs there is at least an arc in F(y) Proof. Suppose first that t is enabled PCS and tETR(y) - {A}. Then t is labelled in y. Hence S %Y w &-? w’ (the last step in the above derivation by t. there exist w and w’ such that is performed by t). Let ME(N 2n+m)nbe the smallest vector such that t is enabled at it. Clearly M, 2 M, and from this fact it follows that M is coverable in y (M does not contain query symbols! ). From Theorem 4.10 it follows that M is coverable in r(y), i.e. there is a node in Y(y) such that M Gus. Without loss of generality we may assume that ll(v) does not contain queries. Now, if w is not a leaf node then t is enabled at k,(o) and hence v will have a successor o’ such that /,(v, v’) = t. In the case that v is a leaf node, there is v’ sd ~~y~(vo, v), v # v’ , and k’,(u) = e,(v’). We obtain that t is enabled at Ll(v’) and v’ is not a leaf node. Hence, there is a node v” such that e2(v’, v”) = t. 371 Some decision problems for parallel communicating grammar sy stems Conversely, let (u, u’) be an arc in Y(y) labelled by t. We have e,(o)[t) and considering the vector M as above we obtain e,(u) 3 M , i.e. M is coverable in Y(y). From Theorem 4.10 it follows that M is coverable in y, i.e. there is a reachable w such that M , B M . Thus, t is enabled configuration enabled in y. From Theorem Corollary. Corollary 4.10 and 4.12. It is decidable sy nchronized at w and hence the transition t is 0 context- free Theorem whether 4.11 one a transition PCS y is enabled easily t# A obtain the following of a given nonreturning in y. A PCS y is called circular if it has some circular the circular configurations are yielded obtain the following Corollaries. can reachable by applying circular Corollary 4.13. It is decidable whether a given nonreturning configurations. However, transitions and so, we sy nchronized context- free PCS y is circular. Corollary 4.14. It is decidable whether a given nonreturning sy nchronized context- free PCS y works like a centralized one. Proof. A PCS y works like a centralized y has the property: #(K,RHS(t(i)))=O The corollary 4.12. 0 one iff any transition t # ,4 which is enabled in for any 2<i<n. follows now from the finiteness of the tree Y(y) and from Corollary We say that there is conflict in a PCS y if there is a configuration y, and there exist i, j, k such that i # j # k # i and # (Qk, wi) > 1 and If in y there is not any conflict we say that y is conflict-free. Corollary 4.15. It is decidable whether a given nonreturning w = (wi, . . . , w,) in # (Qk, wj) 2 1. sy nchronized context- free PCS y is conflict- free. Proof. In a PCS y there is conflict iff there is a transition t #A which is enabled in y, and there is j and k such that j# k# i# j and #(Qk, RHS(t(i)))> 1 and #(Qk, RHS(t(j)))> 1. The corollary and from Corollary 4.12. 0 follows now from the finiteness of the tree 5(y) 378 F. Laurentiu et al. There have been studied in [6] the PCS’s with a bounded tions. But, is it decidable whether a given PCS has a bounded number number of communicaof communica- tions? Corollary 4.16 It zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA is decidable whether a given nonreturning sy nchronized context- free PCS has a bounded number of communications. Proof. A nonreturning synchronized PCS y is not a bounded PCS iff there is a path in F(Y) Vl,V2, ... ,u,, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (k32) such that: (ii) there is 1 did k 5. On context-sensitive PCS’s This section is dedicated to context-sensitive are undecidable for such PCS’s 5.1. Some undecidable The enabling, 0 such that ez(vi, vi+ 1) = n. PCS’s, As we shall see many problems questions circularity, centralizing, conflict-freeness and boundedness problems defined for context-free PCS’s are similarly defined for context sensitive PCS’s. For example, the enabling problem is the problem to decide whether a transition t # A of a given context-sensitive PCS y is enabled in y. Using an undecidability result for context-sensitive grammars we shall the above problems are undecidable for context-sensitive PCS’s. Let c be an arbitrary but fixed symbol. The c- reachability problem for Chomsky grammars 3 consist of giving an algorithm that will tell whether symbol c is reachable in a given grammar GE% (i.e., if there is a word c derivable from the axiom of G). We shall prove that this problem is undecidable using the Post correspondence problem ([21]). machines, can be found in [4]. Theorem 5.1. The c- reachability Proof. Assuming the existence Post correspondence problem. show that a family of or not the containing for context-sensitive grammars by Another proof, based on counter problem is undecidable for context- sensitive of such an algorithm, grammars. we show the decidability of the Some decision problems for parallel communicating 379 grammar systems Consider an arbitrary instance PCP of the Post correspondence problem, given by an alphabet C not containing c and two arbitrary lists of words over Z, c1r,. . . , cc,and P 1, . . . , fl,,. Without loss of generality we may assume that there is no 1 <i < n such that both C(iand pi are the null string. Consider the grammar G = (I’,, VT,S, P) given by: 0 VN={S,A,X}; n}, where @ is a new symbol and C is disjoint with 0 V,=cu{c,@}u{l,..., (1, . . ..n}. l P contains the next groups of rules; (1) S-AX; n, if we suppose (2) for each 1~ i < zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA tLi = Ui, /Ji=bir ai, . . . aik,, biz . . . him,, and then ki<mi, If ki>,mi, then @ occur on the second component in a similar way (the nonterminal A tries for a solution of PCP); zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM (3) [:IM+M [:I~ for any x,y~C and z~Cu(@] (the words cli and pi are compacted to the right); (4) for any x~Cu{@} [:]+:]~ (the nonterminal X verifies the correspondence); iX- +ic, for any 1 d i < n (when a solution of PCP have been found, the symbol c is generated). Clearly, G is context-sensitive and PCP possesses a solution iff the symbol c is reachable in G. 0 (5) Corollary 5.2. sy nchronized The enabling problem or unsynchronized) is undecidable context- sensitive for PCs’s (returning of degree or nonreturning, n, n > 2. F. Laurentiu et al. 380 Proof. Let c be an arbitrary symobi and G be a context-sensitive grammar such that there exists at least one rule containing c in its right hand side (we assume c is not the axiom of G). Let zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA rl, . . . , rk be all rules such that #(c, RHS(ri)) > 0, 1~ i < k. Consider y=(G1, . . . , G,), n 22, where G, = G and Gj, 2 <j,< n, only contains the rule rl: Clearly, c is reachable in G iff at least one of the transitions 1 did k, is enabled in y. The corollary follows now from Theorem 5.1. 0 Sj + Sj. Corollary 5.3. The circularity synchronized problem or unsynchronized) is undecidable context-sensitive Proof. Consider y = (G,, . . , G,), n > 2, where l G, is a context-sensitive grammar containing l all rules of G2 are SZ-+S2, S2+Q1; l Gi, 3 <i<n, only contains the rule Si-‘Si. Corollary 5.4. The centralizing synchronized problem or unsynchronized) a unique n, n 2 2. query symbol a circular is undecidable context-sensitive for (returning or nonreturning, PCS’ s of degree The symbol Q2 is reachable in G1 iff y reaches conclusion follows from Theorem 5.1. 0 ti =(ri, rl, . . . , rl), Q2; configuration, and the for (returning or nonreturning, PCs’ s of degree n, n>2. Proof. Consider y = (G,, . . . , G,), n > 2, where l Gi zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (1 < i<n1) has the unique production Si+Si; l G, is a context-sensitive grammar which contains the query symbol Q1. 0 The symbol Q1 is reachable in G, iffy is noncentralized. Corollary 5.5. The conflict-freeness ing, synchronized problem or unsynchronized) is undecidable for (returning or nonreturn- context-sensitive PCS’ s of degree n, n > 3. Proof. Consider y = (G,, . . . , G,), n 3 3, where l G1 is a context-sensitive grammar which contains the unique query symbol l the rules of G2 are SZ+S2, S2-+Q3; l Gi, 3 <i<n, has the unique rule Si-+Si. 0 The symbol Q3 is reachable in GI iffy is not conflict-free. Corollary 5.6. The boundedness synchronized or unsynchronized) problem Q3; is undecidable for (returning or nonreturning, context-sensitive PCS’ s of degree Proof. Let Q2 be an arbitrary symbol and G be a context-sensitive contains Q2. Consider y = (G,, . . . , G,), n 2 2, where n, n>,2. grammar which Some decision problems for parallel communicating grammar l l G1 contains in G); Gi, 2<i<n, all rules from G and in addition The symbol Q2 is reachable tions. only contains sy stems 381 the rule S2-+Q2 (S, does not appear the rule Si-‘Si. in G iffy has an unbounded number of communica- 0 5.2. The simulation of O- ty pe grammars and other undecidable questions The O-type grammars can be simulated by very “simple” context-sensitive PCs’s. Theorem 5.7. For any O- ty pe grammar G which cannot generate the null string 1 we can e&tively construct a returning sy nchronized PCS y o of degree 3 such that: (1) yc has two regular components (without using A- rules) and one context sensitive component (the master grammar is regular); (2) L(G)= W C). Proof. Let G =( V,, V,, modify any rule r: a-b (a) if (CIId IpI, then we r is aAb+aaB then we S, P) be a O-type grammar such that I$L(G). The idea is to as follows: concatenate to p the query symbol Q1 (for example, if the rule construct the new rule aAb+aaBQ,); (b) if ICI\> IpI, then we concatenate to fi the string Q’;, where m= I@I-IflI (for example, if the rule r is aAb+a then we construct the new rule aAb- +aQ,Q,). These new rules will be in the second component of our system. We assume V,=(aI,..., a,} and y =(G1, Gz, G3) where: G,: S1+S;, S’+Q3, A- A’, A’+Q3, Bk- +ak for any l<k<p; Gz: &+CXS#QI, u+vQ1 for any U+VEP u*vQ{ for any U+UEP such that j = JuJ- Iv]> 0, S;a+aQ1 such that JuI<(v(, for any zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB a El/,uV ,u(# }, CXak+CX’ak for any 1 <k < p, a,X+a,QJX, akAXakai+akAakXai, BkXak # - SBkakX# , for any 1 <k<p; G,: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA s3 +s3 , S3+akA, S3+BL for any l<k<p 382 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA F. Laurentiu et al. PI, S;, SZ, S3, A, A’, C, # , X, &, 1 <k < p, are new symbols; they are not members of the set V,u V,). The system y simulates the derivations I Sl in G as follows: Sl S2 S3 ‘$XS#S> 3 S3 : (the markers c and #, the nonterminal second component); X, and the axiom of G, are introduced s; Sl ... j 1 =a.. ~Xucm # (S;)’ S3 1 a CXuPQ( u # (s;) i s3 on the Ii = Sl CXu/qS;)ju#(S;)’ s3 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI CXug~+(sl)i+j S3 (it is simulated j = a rewriting step in G using the rule a-j?; if IM]= I/J1 then j= 1 else I4 - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA IDI); Sl ... * i CXUl . . . 4t#(S;)’ I s3 I 1 I * : CQ 3XU, s3 I UlA zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML Q3 = s; CX’U, . ..u.#(sJ ~u,AXulu2 . . . u,# (S;)’ UlA . ..un#(S.)’ j ... s3 (the system y guesses that a terminal string has been generated on the second component. Then y generates this string on the third component in a linear manner and verifies it step by step. The master grammar asks for this string); zyxwvutsrqponmlkjihgfedcb alQ3 Cu,Au,Q3Xu, aA ...u.#(S;)’ Some decision problems for parallel communicating 383 zyxwvutsrqponmlk grammar systems a, . . . a,- ,A alaJzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA = CalAalazAXa,a3 i * . ..a.# (S;)’ $ s3 1 CalAal ...a,_lAXa,_la,# (S~)’ i s3 a, . . . an_1A’ a, . . . an-IQ3 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO ca,Aal CatAal . ..a.- ,Aa,- ,Q3Xa,# (S;)’ . . . a,_,Aa,_,Xa,# (S;) s3 B” J a, . . . a,- 1B, + Ca,Aal . . . a,, - 1Aa, - 1B,Xa, # (S’J s3 a, . . . am- la, = Cal Aa, . . . a,_,Aa,- 1B,a,X# (S;)’ s3 It is easy to see that the derivation is blocked in other cases. Moreover, we have: Sg,;l a, . . . a, iff 3 20 such that Sl s2 i 1 . ..a._,Aa,_,B,a,X# (S;)’ s3 The theorem is completely proved. 0 Remark 5.8. The condition “A$L(G)” permits us not to use A-rules in the regular components of YG.Otherwise, with the same construction, point (2) of Theorem 5.7 becomes (2’)L(G)- {A>= UYG). If we use I-rules, the O-type grammars can be simulated by PCS’s of degree 2. Theorem 5.9. Any O- ty pe grammar can be simulated by a (returning or nonreturning) sy nchronized PCS of degree 2 with one context sensitive component and one regular component. 384 F. Laurentiu et al. Proof. If G =( VN,VT,S, P) is a O-type grammar, let yG =(Gi, G2) where: G,: u-+v for any U-+VEP such that Iuj 6 (VI, u+vQT for any U+VEP such that zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ m=lul-Ivl>O; G2: S,-+1. It is easy to see that G1 works like G but its rules are monotonic. Corollary 5.10. The membership, lems are all undecidable degree at least text-sensitive 3, and for PCs’ s equivalence, both for returning (returning of degree inclusion, emptiness synchronzied or nonreturning) 0 and finiteness context-sensitive synchronized prob- PCs’ s arbitrary of con- at least 2. Proof. These problems all are undecidable for the family of O-type grammars theorem follows from this fact and from the Theorems 5.7 and 5.9. 0 and the 6. Final remarks The results established in Section 4.2 hold also true for the unsynchronized What we have to do is to consider that the phantom rule 0 can be applied string (not only to terminal strings). The circularity problem remains open for returning context-free case. to any PCS’s. Acknowledgements The authors wish to thank Dr. Gheorghe and for suggesting the proof of Theorem Paun both for his encouraging 5.1 based on the Post comments correspondence problem. References [l] E. Csuhaj-Varju and J. Dassow, On cooperating/distributed grammar systems, J. Inform. Process. Cy bern. EIK 26 (l/2) (1990) 49- 63. [Z] K. Culik, J. Gruska [3] J. Dassow and A. Salomaa: Systolic trellis automata, Internat. .I. Comput. M ath. 15 (1984) 16. and Gh. Paun, Regulated Rewriting in Formal Language Theory, (Akademie-Verlag. Berlin, 1989). [4] C. Ene, A decision problem for monotonic grammars, submitted. [S] M. Hack, Petri Net Languages: CSG Memo 124, Project MAC (MIT Press, Cambridge, MA, 1975). [6] C.M. Ionescu and 0. Procopiuc, Bounded communication in parallel communicating grammar systems, submitted. [7] J. Kari, Games Played on the Plane: Solitaire and Cellular Automata, EATCS Bull. 40 (1990). Some decision problems for parallel communicating grammar systems 385 [S] R. Karp and R. Miller, Parallel program schemata, J. Comput. System Sci. 3 (1969) 147-195. [9] D. Kiinig, Theorie der Endlichen und Unendlichen Graphen (Akademische Verlagsgesellschaft, Leipzig 1936). [lo] R. Meersman and G. Rozenberg, Cooperating grammar systems, in: Lecture Notes in Computer Science 64 (Springer, Berlin, 1978) 364-373. [11] M.L. Minsky, Recursive unsolvability of post’s problem of “Tag” and other topics in the theory of Turing machines, Ann. of Math. 74 (3) (1961) 437-455. [12] Gh. Paun, Parallel Communicating grammar systems: the context-free case, Found. Control Engrg. 14 (1) (1989) 39-50. [13] Gh. Paun: Non-centralized parallel communicating grammar systems, EATCS Bull. 40 (1990) 257-264. 1141 Gh. Paun and L. Santean, Further remarks on parallel communicating grammar systems, Internat. .I. Comput. Math. 34 (1990) 187-203. [lS] Gh. Paun and L. Santean, Parallel communicating grammar systems: the regular case, Ann. Univ. Bucharest, Math.-lnformatics Ser. 37 (2) (1989) 55-63. [16] Gh. Paun, S. Vicolov and A. Salomaa, On the generative capacity of parallel communicating grammar systems, Internat. J. Comput. Math. 45 (1992) 49-59. [17] 0. Procopiuc, CM. Ionescu and F.L. Tiplea, Parallel communicating grammar systems: the context-sensitive case, Internat. J. Comput. Math. 51 (1) and (2), to appear. [18] P.J.G. Ramadge and W.M. Wonham, The control of discrete event systems, in: Proc. of the IEEE, Vol. 77 (1) (1989). [19] W. Reisig, Petri Nets - An Introduction (Springer, Berlin, 1985) [20] G. Rozenberg and A. Salomaa, The M athematical Theory ofL System (Academic Press, New York, 1980). [21] A. Salomaa, Formal Languages (Academic Press, New York, 1973). [22] L. Santean, Parallel communicating systems, Bull. EATCS 42 (1990) 160-171. [23] F.L. Tiplea and C. Ene, A converability structure for parallel communicating grammar systems, J. Inform. Process. Cybern. EIK 29 (5) (1993) 303-315.