Contextual graph grammars characterising Rational Graphs

Transcription

1 Contextul grph grmmrs chrcterising Rtionl Grphs Christophe Morvn To cite this version: Christophe Morvn. Contextul grph grmmrs chrcterising Rtionl Grphs. Henning Bordihn nd Rudolf Freund nd Mrkus Holzer nd Mrtin Kutri nd Friedrich Otto. Workshop on Non-Clssicl Models for Automt nd Applictions 21, 21, Jen, Germny. Österreichischen Computer Gesellschft, pp , 21, <inri-52549> HAL Id: inri Sumitted on 11 Oct 21 HAL is multi-disciplinry open ccess rchive for the deposit nd dissemintion of scientific reserch documents, whether they re pulished or not. The documents my come from teching nd reserch institutions in Frnce or rod, or from pulic or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de documents scientifiques de niveu recherche, puliés ou non, émnnt des étlissements d enseignement et de recherche frnçis ou étrngers, des lortoires pulics ou privés.

2 CONTEXTUAL GRAPH GRAMMARS CHARACTERISING RATIONAL GRAPHS Christophe Morvn Universit Pris-Est, INRIA Rennes Bretgne Atlntique Cmpus de Beulieu, 3542 Rennes, Frnce Emil: Astrct Deterministic grph grmmrs generte fmily of infinite grphs which chrcterise contextfree (word) lnguges. The present pper introduces context-sensitive extension of these grmmrs. We prove tht this extension chrcterises rtionl grphs (whose trces re contextsensitive lnguges). We illustrte tht this extension is not strightforwrd: the most ovious context-sensitive grph rewriting systems generte non recursive infinite grphs. 1. Introduction In 1956, nd then in 1959 Nom Chomsky wrote two rticles which defined the Chomsky hierrchy. This hierrchy hs hd tremendous impct on the development of the theory of forml lnguges. Since the erly sixties, it hs served for the clssifiction, nd evlution of the expressive power of forml lnguges nd forml models. There is deep connection etween this hierrchy nd grphs. It is ovious for type 3 lnguges (regulr lnguges) which re chrcterised y finite utomt. But from the mid-eighties on, there hs een n incresing effort to chrcterise lnguge fmilies y grph fmilies. One of the first ttempts to estlish properties on structures chrcterising type 2 lnguges hs een done y Muller nd Schupp. They proved the decidility of the mondic second order theory of the grphs of pushdown utomt [MS85]. In their work, the vertices of the grphs re the configurtions of the pushdown utomton, nd the rcs re trnsitions etween configurtions. Such chrcteristion is clled internl, it thoroughly depends on the choice of the mchine, nd it gives n explicit nme to ech vertex. Courcelle, in [Cou9], extended the decidility of mondic second order theory to grphs generted y deterministic grph grmmrs. Indeed these grph grmmrs were clssicl grph rewriting systems used to chrcterise fmilies of grphs. In this pper they were used to define infinite grphs. Indeed these grphs re very close to grphs of pushdown utomt ([CK1]) ut it is n externl chrcteristion: ech grmmr genertes set of isomorphic grphs. The nme of the vertices re not explicitly stted long the genertion of the grph. This is relly importnt s it provides higher level of strction. Another slight extension of these two fmilies is formed y the

3 2 Christophe Morvn prefix-recognizle grphs from [Cu96]. In this pper Cucl provides oth n externl nd internl chrcteristion of these grphs. He lso proves tht they hve decidle mondic second order theory, nd tht they chrcterise context-free lnguges. For type 1 lnguges (context-sensitive lnguges), there re lso severl grph chrcteristions: trnsition grphs of liner ounded Turing mchines [KP99], rtionl grphs [Mor], utomtic grphs [BG, Ris3] or liner ounded grphs [CM6]. All these works provide internl chrcteristions: the grphs re defined s sets of trnsitions etween vertices which re words on some lphet. To our knowledge there is no externl grph chrcteristion of context-sensitive lnguges. In prticulr there exists no context-sensitive extension of Courcelle s grph rewriting systems. [Sch97] is survey presenting severl contextul grph rewriting systems ut they re used exclusively to chrcterise fmilies of finite grphs, nd no connection is estlished with the Chomsky hierrchy. In this pper we propose n externl chrcteristion of rtionl grphs in terms of contextul rewriting system. It is orgnised in two min prts: the first reclls the definition of regulr nd rtionl grphs nd the second presents context-sensitive grph rewriting systems. We first exmine nturl, unrestricted, contextul extension of grph grmmrs. We show tht it is too generl, s it produces non-recursive grphs. Then, we propose restriction, which corresponds to rtionl grphs. Afterwrds we exmine possile relxtions of some constrints which yield fmilies of non-recursive grphs. We conclude this pper pplying our chrcteristion to reprove some fcts on context-sensitive lnguges. 2. Deterministic grph grmmrs nd rtionl grphs In this section we define clssicl nottions nd recll fundmentl definitions for deterministic grph grmmrs nd rtionl grphs Mthemticl preliminries Fundmentls For ny set E, its powerset is denoted y 2 E ; if it is finite, its size is denoted y E. Let the set of non-negtive integers e denoted y N, nd {1,2,3,...,n} e denoted y [n]. A monoid M is set equipped with n ssocitive opertion (denoted ) nd (unique) neutrl element (denoted ε). A monoid M is free if there exist suset A of M such tht M = A := n N An nd for ech u M there exists unique finite sequence of elements of A, (u(i)) i [n], such tht u = u(1)u(2) u(n); if A is finite, then is free, of finite rnk. Elements of free monoid of finite rnk re clled words. Let u e word in M, u denotes the length of u nd u(i) denotes its ith letter.

4 Contextul Grph Grmmrs Chrcterising Rtionl Grphs 3 Hypergrphs Let F e n lphet (finite set) rnked y mpping : F N, this mpping ssocites to ech element of F its rity. Given rnked lphet F, we denote y F n the set of symols of rity n. Now given V n ritrry set, hypergrph G is suset of n 1 F n V n. The vertex set of such hypergrph is the set V G = {v V FV vv G }, in our setting, this set is either finite or countle. A hyperrc of rity n is denoted y f v 1 v 2 v n. Hyper rcs of rity 2 re clled rcs. Given hypergrph G, n rc st in G is identified with the lelled trnsition s G t or simply s t if G is understood. Furthermore, is the lel, s the source nd t the trget. Elements of F 1 (lels of rity 1) re clled colours. Grphs A (simple oriented lelled) grph G over V, lelled y P is hypergrphs hving only hyperrcs of rity 1 or 2 (P = P 1 P 2 ). Given grph G, the set of sources nd trgets re respectively denoted y Dom(G) nd Im(G) (y convention, oth sets contin lso vertices elonging to rity 1 hyperrcs). AgrphGisdeterministic ifdistinctrcswiththesmesourcehvedistinctlel: r s r t s = t. Let G e grph, pth in etween vertices u Dom(G) nd v Im(G), lelled w, corresponds to the existence of vertices (u i ) i [n] such tht u = u 1, v = u n nd u i w(i) G u i+1 for i [n 1]. Such pth is denoted y w = G or simply w = if G is understood. A grph morphism g, etween grphs G nd G is mpping from Dom(G) Im(G) to Dom(G ) Im(G ) such tht if there is n rc u v, then there is n rc g(u) g(v). G G Such morphism is n isomorphism if g is ijection, nd its inverse is lso morphism Deterministic grph grmmrs A finite presenttion is n effective tool in order to mnipulte n infinite grph. Such presenttions llow to conceive lgorithms working on these grphs. Deterministic (hyperedge replcement) grph grmmrs re clssicl exmple of finite (externl) chrcteristions of infinite grphs. These grmmrs were initilly defined to e n extension to grphs of word grmmrs (see, e.g., [Roz97]). These grph grmmrs derived, from n xiom, n infinite fmily of finite grphs. Courcelle in [Cou9] considered the deterministic form of these grmmrs, ech grmmr, seen s set of equtions, dmits single (up to isomorphism) infinite grph s lest solution. In [Cu7], Cucl mde n extensive survey on deterministic grphs grmmrs. In prticulr, he gve uniform presenttion of severl results: he developed frmework of simple techniques, mking intensive use of colours nd colouring, to provide new proofs of these results.

5 4 Christophe Morvn Definition 2.1 (Hypergrph grmmr). A hypergrph grmmr (hr-grmmr for short) G, is 4-tuple (N,T,R,H ), where N nd T re two rnked lphets of respectively non-terminls nd terminls symols; H is the xiom, finite grph formed y hyperrcs lelled y N T, nd R is set of rules of the form f x 1 x (f) H where f x 1 x (f) is n hyperrc joining disjoint vertices nd H is finite hypergrph. Oserve tht usully, the vertices ppering in the left-hnd side of rule pper in the righthnd side (though the definition does not commnd it). Remrk 2.2. In this pper, we consider grphs, therefore, the terminl symols will hve either rnk one, or two. Such grph is seen s simple suset of T 2 VV T 1 V. A grmmr is deterministic if there is single rewriting rule per non-terminl: (X 1,H 1 ),(X 2,H 2 ) R X 1 (1) = X 2 (1) (X 1,H 1 ) = (X 2,H 2 ) Now, given set of rules R, the rewriting R is the inry reltion etween hypergrphs defined s follows: M rewrites into N, written M R N if there is non-terminl hyperrc X = Av 1 v 2...v p in M nd rule Ax 1 x 2...x p H in R such tht N is otined y replcing X y H in M: N = (M X) h(h) for some injection h, mpping v i to x i for ech i, nd every other vertices of H to vertices outside of M. This rewriting is denoted y M N. R,X Now, this rewriting oviously extends to sets of non-terminl, for E such set, this rewriting is denoted: M N. The complete prllel rewriting = is the rewriting reltive to the set of R,E R ll non-terminl hyperrcs of R. Now given deterministic grph grmmr G = (N,T,R,H ), nd hypergrph H, we denote y [H] := H (T V H V H T V H ) the set of terminl rcs, nd colours of H. We define G ω, the limit set of G, s the set of ll countle union of [H n ], derived from H (there re infinitely mny H n for ech n): } G ω = { n [H n ] n,h n = H n+1 R Formlly, G ω is set of grphs, nd we sy tht grph H is generted y G if it elongs to G ω. But ll these grphs re isomorphic, thus, we will often express tht G ω is the grph generted y G. Exmple 2.3. This is simple exmple of deterministic grph grmmr nd representtion of the resulting grph.

6 Contextul Grph Grmmrs Chrcterising Rtionl Grphs 5 (2) A rule An xiom A grph A A A c A A (1) (1) (2) c c Grph grmmrs chrcterise regulr grphs. This chrcteristion is very efficient to extend techniques working for finite grphs to these infinite grphs (for exmple computing the connected components of regulr grph is simply doing it inductively on the right-hnd sides of the rules). Furthermore these grphs (restricted to finite degree) correspond to trnsition grphs of pushdown utomt [CK1]. Yet, lgorithms defined on pushdown utomt often mke technicl ssumptions on the form of the utomton: for exmple tht ech stte reflects tht the configurtion elongs to certin regulr set. In most cses these ssumptions only ffect the internls of the utomton, it does not ffect the structure of its configurtion grph. More precisely, these trnsformtions do not ffect glolly the fmily of grphs ut some individul grph my e trnsformed switching representtion. This is not the cse for grph grmmrs: norml forms preserve the generted grph. Thus grph grmmrs re more suited when focussing on structurl properties. In the next susection we will define rtionl grphs with n internl chrcteristion, lter on, in Section 3 we will extend grph grmmrs in order to chrcterise rtionl grphs. The wy we enrich these grmmrs is similr to wht is done for word grmmrs: contextul rewriting is necessry. However, in order to void eing too generl, we will hve to seprte context rcs from non-terminl ones, nd the xiom will e regulr grph (defined y hr-grmmr) Rtionl grphs In this susection we present key elements on rtionl grphs. More detils cn e found in [Mor, MS1]. The fmily of rtionl susets of monoid (M, ) is the lest fmily contining the finite susets of M nd closed under union, conctention nd itertion. A trnsducer is finite utomton lelled y pirs of words over finite lphet X, see for exmple [Ber79]. A trnsducer ccepts reltion in X X ; these reltions re clled rtionl reltions s they re rtionl susets of the product monoid (X X, ). Now, let us consider the grphs of Σ X X (here, in order to seprte explicitly first nd second vertices, we use Crtesin product insted of conctention). Rtionl grphs re extensions of rtionl reltions, they re defined y lelled trnsducers. Definition 2.4. A lelled trnsducer T = (Q,I,F,E,L) over X nd Σ, is composed of finite

7 6 Christophe Morvn set of sttes Q, set of initil sttes I Q, set of finl sttes F Q, finite set of trnsitions (or edges) E Q X X Q nd mpping L from F into 2 Σ. An rc u v is ccepted y lelled trnsducer T if there is pth from stte in I to stte f in F lelled y (u,v) nd such tht L(f). Definition 2.5. The set of rtionl grphs in Σ X X, denoted y Rt(Σ X X ), is formed y the grphs ccepted y lelled trnsducers. Exmple 2.6. The grph depicted elow, on the right-hnd side, is generted y the lelled trnsducer on the left-hnd side. / 1/1 q 1 / p 1/ε q 2 /1 /1 ε/ 1/ 1/1 r 2 1/1 r 1 / r 3 c ε c 1 c 1 11 c The pth p / q 1 /1 r 2 1/1 r 2 ccepts the pir (1,11), the finl stte r 2 is lelled y thus there is n rc 1 11 in the grph. Rtionl grphs hve een introduced in order to define generl fmily of infinite grphs generlising previous fmilies. They hve few decidle properties, ut they chrcterise contextsensitive lnguges (csl) [MS1]. The rtionl trees (rooted connected cyclic rtionl grphs such tht ech vertex hs t most one predecessor) hve decidle first order theory [CM6]. Employing trnsducers to define fmily of grphs is n internl chrcteristion. A side-effect of such definition is tht the set of vertices of rtionl grph is regulr set of words. This implies tht some grphs re not rtionl simply ecuse of their vertex set, ut it is not relted to their structure. In contrst, for finite grphs, every lgorithm, every chrcteristion, is given up to renming of the vertices. Such externl chrcteristion is difficult to otin for infinite grphs. Still, chrcteristions like hr-grmmrs overcome this prolem. The next section defines contextul grph grmmrs chrcterising rtionl grphs. 3. Contextul grph grmmrs There re severl contextul grph rewriting systems (see Section 4 in [Sch97]). But like for hr-grmmrs, these systems hve not een used to chrcterise fmilies of infinite grphs. Especilly in the sense of deterministic grph grmmrs. We will see tht it is difficult to define such systems generting exclusively recursive sets of rcs.

8 Contextul Grph Grmmrs Chrcterising Rtionl Grphs Contextul grph rewriting systems Recursivity is understood differently for set of finite grphs, nd single infinite grph: in the first cse, it corresponds to checking tht given grph elongs to the fmily, wheres for systems tht produce single grph, recursivity corresponds to deciding, from the grmmr, the existence of n rc etween two given vertices. In the cse of sets of grphs, s soon s the grmmr does not remove terminl rcs, it is recursive. In the cse of single grph, it is not sufficient: the rules my use non-terminl rcs to perform complex computtions, which my eventully produce n rc depending on the result of this computtion, in such sitution, the grph itself my e non-recursive. In this first susection we present nive contextul rewriting system which genertes nonrecursive grphs. Let N R e finite rnked set of non-terminls, nd T R finite rnked set of terminls. We propose here nturl definition of contextul grph rewriting system. Definition 3.1 (Contextul grph rewriting system). A contextul grph rewriting system S, is set of rules of the form H c f x 1 x (f) H c H where f x 1 x (f) is non-terminl hyperrc, H c is finite context grph, nd H is finite hypergrph, tht cn shre some vertices with H c nd { x 1,, x (f) }. Furthermore, Hc is composed only of terminl hyperrcs, nd H c f x 1 x (f) forms connected hypergrph. The sketch of proof for Proposition 3.2 provides n exmple of contextul rewriting rules (rules r,r 1,r re explicitely depicted). Now, given rewriting rule H c f x 1 x (f) H c H, rewriting step in grph G consists in finding the non-terminl f in G, such tht there is morphism h of H c f x 1 x (f) into G then removing f from G, nd dding H to G ccording to the rule, nd to the morphism h. Given contextul grph rewriting system S, nd finite grph H, we define S ω (H) in the sme wy s G ω for hr-grmmr G. Recll tht this grph is restriction to terminl symols, in prticulr in our setting it only hs hyperrcs of rity 1 or 2. We sy tht such contextul rewriting system is deterministic, if there is, t most, one rule for ech non-terminl. This restriction only limits non-determinism, s there might e some situtions where the context of non-terminl cn e found more thn once. In such sitution, it mens tht the rewriting system my generte t lest two non-isomorphic grphs. Unfortuntely this nturl generliztion of hr-grmmrs is much too generl (even restricted to systems where there is lwys single morphism to mtch the context): the grph rewriting systems generte non recursive grphs. The first oservtion towrd this result is the following: the trnsition grph of Turing Mchine is recursive: given two configurtions of the mchine it is trivil to check tht they

9 8 Christophe Morvn re connected in the trnsition grph. The second oservtion is tht the ccessile restriction (restriction to rechle configurtions) of this grph is non-recursive; if it were recursive it would imply the decidility of ccessiility for Turing mchines. Proposition 3.2. Let M e Turing mchine; u, nd u 1 two configurtions of M, there exists effectively contextul grph rewriting system S which genertes, from finite xiom A, set of grphs isomorphic to the ccessile component of the trnsition grph of M, from its initil configurtion. Furthermore ech vertex in ijection with u nd u 1 is mrked y colour c nd c 1 respectively (ech grph in S ω contins single pir of such vertices). Proof (sketch). Let M e Turing mchine, with tpe lphet Σ = {,1},nd sttes in the finite set Q. For simplifiction we ssume the tpe is filled initilly y zeros, nd tht M opertes on such tpe, we lso ssume tht the tpe is only infinite on the right, finlly configurtion ends with 1 or stte symol. A trnsition of M is tuple (p,a,q,b,δ) where p,q Q re respectively the current nd next stte of M, A Σ is the tpe letter immeditely on the right of the reding hed, δ { 1,+1} represent the movement of the hed to the left or right, nd B is the letter replcing A fter the move. Let = (p,q,1,, 1) e trnsition of M. 1 X 1 X 1 r r 1 1 p 1 1 p n X 1 X 1 X X 1 1 # root end p 1 q r, root p 1 end nxt q X 1 X 1 Rules r nd r 1 generte complete tree corresponding to configurtions of M, ech pth leding from the root to n rc lelled # corresponds to configurtion of the mchine. For ech trnsition of M, there is one or two rules 2 similr to r, simulting the rewriting step from configurtion to the next one. The context of such rule is formed y the rcs,p,1 nd q,,1 the rewriting consist simply in removing the non-terminl nd producing nxt. Before pplying these rules the trnsition is simply moving long single pth (reclling the root nd the end). The rule corresponding to nxt performs the sme pth long the second nd forth vertices. The rule tht ensures termintion nd thus the restriction to single connected component is the following: root chk # end τ root # r chk # end # new The left hnd-side of this rule witnesses tht the computtion hs reched the end of the originl configurtion, the non-terminl new records the position of the root nd the position of the extremity of the new configurtion, the right hnd-side from new strts over new 2 There re two rules when δ = 1 depending on the symol preceding the reding hed.

10 Contextul Grph Grmmrs Chrcterising Rtionl Grphs 9 computtion. There is τ-lelled rc etween the two vertices corresponding to the respective configurtions. Ultimtely the grph is formed simply y τ lelled rcs. The xiom is this finite grph: X new root p # end Colours c nd c 1 re simply dded y single rule hving for context tree corresponding to oth configurtions. If it were decidle whether there is n τ-rc etween c nd c 1, it would enle to decide the existence of n rc in the (non-recursive) ccessile component rechle in the trnsition grph of M from the initil configurtion. Proposition 3.2 my e reformulted into following corollry. Corollry 3.3. Deterministic contextul grph rewriting systems generte non recursive grphs. Oviously from Turing mchine, M, nd two configurtions, the existence of n rc etween these configurtions in the ccessile component of the trnsition grph of M might e checked if it could e checked in the grph generted y the system derived from Proposition Contextul hyper-edge-replcement grph grmmrs We present, now, more restrictive contextul rewriting system which will e used to chrcterise rtionl grphs. Definition 3.4. A contextul hyper-edge-replcement hypergrph grmmr (chr-grmmr for short) is tuple (C,N,T,R c,h ), where C,N nd T re finite rnked lphets of respectively contextul, non-terminl nd terminl symols; R c is deterministic contextul grph rewriting system where, for ech rule H c fx 1...x (f) H c H, the grph H c is formed only y rcs lelled in C, nd H y rcs lelled in T N, nd H is the xiom: deterministic regulr grph formed y rcs with lels in C, nd single non-terminl hyperrc. Given chr-grmmr G = (C,N,T,R c,h ), the set G ω is defined similrly s for hr-grmmr s R ω c(h ). Oserve tht, for ech rule in R c, the context grph (H c ) is not modified: no rc is removed, nd no rc is dded, thus successive rewritings preserve the xiom H. Furthermore, this definition imposes tht the xiom is deterministic regulr grph. This ensures tht for ech rule r R c (with non-terminl A) nd ech occurrence of A in the grph, there is t most single morphism which mps the context of the left-hnd side of r to the neighourhood of A. This restriction my e checked, since verifying tht given grph grmmr genertes deterministic grph is decidle (direct consequence of Proposition 3.13 in [Cu7]). Lter in this pper we will discuss other structurl restrictions (for the xiom), nd see tht it is difficult to llow grphs tht re not trees. We will lso hve to impose stronger restrictions on the rules R c in order chieve chrcteristion.

11 1 Christophe Morvn We first show tht n X -ry tree s xiom is sufficient to otin ech grph of Rt(Σ X X ). Proposition 3.5. Ech rtionl grph is generted y chr-grmmr. Like for Proposition 3.2, the proof is in the full pper. We sketch this strightforwrd construction. Proof (sketch). Let G e rtionl grph in Σ X X (nd T trnsducer representing it), let H e the complete X -ry tree lelled on X (with non-terminl p on the root). For ech stte p of T, we hve the following rule r p. p L(p) u 1 u 2 u n v 1 v 2 v n r p u1 u 2 u n v 1 v 2 v n q 1 q 2 q m Here, we suppose tht there re trnsitions p u i/v i qi for some sttes (q i ) i [m], nd lso L(p) represent ll lels produced t stte p (if p is terminl stte). Now ech pir of pth in H correspond to pth of pirs in T. Thus oth grphs re the sme Tree-seprted contextul grmmrs generte rtionl grphs This susection identifies restrictions on chr-grmmr which enle converse inclusion for Proposition 3.5. A chr-grmmr (C,N,T,R c,h ) is clled tree-chr-grmmr if the xiom H is tree, nd the left-hnd side of ech rule of R c is formed y trees rooted in the vertices of the non-terminl (some vertices of this non-terminl my e non-root vertices of theses trees). Furthermore, if ech of these trees possesses single vertex elonging to the non-terminl (its root) this grmmr is clled tree-seprted-chr-grmmr. Grphs generted y the ltter re cptured y rtionl grphs: Proposition 3.6. Any grph generted from tree-seprted-chr-grmmr, is isomorphic to grph in Rt(Σ X X ). There re severl difficulties to estlish this result: the xiom is not complete tree, so we need to ensure tht n rc is never set on vertex tht does not exist. Hyperrcs of rity greter thn 2 need to e tken into ccount. Finlly, there my e twists: in the right-hnd side of rule, the pth from the gol of the originl non-terminl my led to the source of some other non-terminl.

12 Contextul Grph Grmmrs Chrcterising Rtionl Grphs 11 In order to prove Proposition 3.6 we first stte two technicl lemms (whose proofs re postponed to the ppendix). Lemm 3.7. The set of pth lels from the root of regulr tree leding to some colour (rc of rity 1) is regulr set of words. Lemm 3.8. Any tree-seprted-chr-grmmr G cn e effectively trnsformed into treeseprted-chr-grmmr G generting the sme set of grphs nd such tht, for ech rule r of G, ech non-terminl ppering in the right-hnd side of r hs t most one vertex in ech tree of the context grph. We re now le to sketch the proof of Proposition 3.6. Proof (sketch). We first suppose the xiom is some complete n-ry tree. Lemm 3.8 enles tht ech non-terminl hs t most single vertex in ech sutree of the right-hnd side of ech rule. Then the sttes of the trnsducer re defined y ordered pirs of vertices of the non-terminls, nd the trnsitions from the pths in the right-hnd sides. Terminl sttes re defined y pirs of vertices of terminl rcs (source, trget). Second, if the regulr tree is not complete tree, we use Lemm 3.7 to compute regulr set of pths reching leves: P. Then, it is not sufficient to mke n intersection with the set P P, since some rewriting my not occur from the sence of some vertex long the rewritings. Thus, we synchronize the trnsducer with the set P: ech stte A (u1,u 2 ) checks tht the complete context for the rule of A is ville. Comining Proposition 3.6 nd 3.5 we stte the min result of this pper (oserve tht the grmmr which proves Proposition 3.5 is tree-seprted-chr-grmmr). Theorem 3.9. The fmily of rtionl grphs, nd the grphs generted y tree-seprted-chrgrmmrs coincide. Now, slightly modifying the rules presented in the proof Proposition 3.2 enles to extend it. The rules r nd r 1 generte the xiom, nd the only rule tht does not stisfy tree-seprtedchr-grmmr restrictions is r chk. From this oservtion we otin the following result. Proposition 3.1. Tree-chr-grmmrs generte non-recursive grphs. This result proves tht in some sense tree-seprted-chr-grmmrs re the most generl context-sensitive grph rewriting system cpturing rtionl grphs nd nothing more.

13 12 Christophe Morvn 4. Applictions nd conclusion Applictions We first reformulte Theorem 4.12 of [MS1] in the context of tree-seprted-chr-grphs. Theorem 4.1. [MS1] The set of pths etween two colours in tree-seprted-chr-grphs coincide with context-sensitive lnguges. A mjor interest of hving grph chrcteristion of lnguges is tht it enles simple proofs for lnguge properties. In this susection we provide direct pplictions of tree-seprtedchr-grmmrs to reprove simple results for csl. Proposition 4.2. Let G 1 nd G 2 e two tree-seprted-chr-grphs: the itertion of G 1, the conctention nd synchronistion product of G 1 nd G 2 re tree-seprted-chr-grphs. These results re proved constructing compound xiom nd compound rules. Using Theorem 4.1, Proposition 4.2 proves the following results: Corollry 4.3. The csl re closed under conctention nd Kleene str; the intersection of two csl is csl. One of the most stunning result for csl is closure under complementtion [Sze88, Imm88]. It would e interesting to reprove this result using grph pproch. Unfortuntely, proving closure under complementtion with grphs is relted to closure under deterministion. But here, lnguges recognised y deterministic tree-seprted-chr-grphs re deterministic csl. If they contined ll csl it would prove Kurdod s conjecture [Kur64] on the equivlence etween deterministic nd non-deterministic csl. Our conjecture is tht deterministic rtionl grphs do not recognise ll csl. Thus, using this grph chrcteristion to prove closure under complement would e similr to proving the closure under complement of Bchi utomt: it would not use the grph structure. Conclusion In this pper we hve presented fmily of infinite grphs generted y contextul grph grmmrs. This fmily chrcterises rtionl grphs. We lso hve exmined severl vrints which generte non-recursive grphs. Elorting on techniques from hr-grmmr we hope to provide toolox for mnipulting these grphs nd then eing le to improve the knowledge of csl. An interesting question is the chrcteristion of the grphs of higher order pushdown utomt in terms of grph rewriting systems. Such chrcteristion could enle deeper understnding of the structure of these ojects.

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