CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM

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1 Logicl Methods in Computer Science Vol. 2 (2:6) 2006, pp Sumitted Jn. 31, 2005 Pulished Jul. 19, 2006 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM ARNAUD CARAYOL AND ANTOINE MEYER Iris Cmpus de Beulieu Rennes Cedex Frnce e-mil ddress: Arnud.Cryol@iris.fr Lif Université de Pris 7 2 plce Jussieu, cse 7014, Pris Cedex 05 Frnce e-mil ddress: Antoine.Meyer@lif.jussieu.fr Astrct. We investigte fmilies of infinite utomt for context-sensitive lnguges. An infinite utomton is n infinite leled grph with two sets of initil nd finl vertices. Its lnguge is the set of ll words lelling pth from n initil vertex to finl vertex. In 2001, Morvn nd Stirling proved tht rtionl grphs ccept the context-sensitive lnguges etween rtionl sets of initil nd finl vertices. This result ws lter extended to su-fmilies of rtionl grphs defined y more restricted clsses of trnsducers. Our contriution is to provide syntcticl nd self-contined proofs of the ove results, when erlier constructions relied on non-trivil norml form of context-sensitive grmmrs defined y Penttonen in the 1970 s. These new proof techniques enle us to summrize nd refine these results y considering severl su-fmilies defined y restrictions on the type of trnsducers, the degree of the grph or the size of the set of initil vertices. 1. Introduction One of the cornerstones of forml lnguge theory is the well-known hierrchy introduced y Chomsky in [Cho59]. It consists of the regulr, context-free, context-sensitive nd recursively enumerle lnguges. This hierrchy ws originlly defined y imposing syntcticl restrictions on the rules of grmmrs generting the lnguges. These four fmilies of lnguges s well s some of their su-fmilies hve een extensively studied. In prticulr, they were given lterntive chrcteriztions in terms of finite cceptors. They re respectively ccepted y finite utomt, pushdown utomt, linerly ounded utomt nd Turing mchines. Recently, these fmilies of lnguges hve een chrcterised y fmilies of infinite utomt. An infinite utomton is lelled countle grph together with set of initil nd set of finl vertices. The lnguge it ccepts (or simply its lnguge) is the set of ll words lelling pth from n initil vertex to finl vertex. In [CK02], summry of four fmilies of grphs corresponding to the four fmilies in the Chomsky hierrchy ws given: they re respectively the finite grphs, prefix-recognisle 2000 ACM Suject Clssifiction: F.4.1. Key words nd phrses: lnguge theory, infinite grphs, utomt, determinism. LOGICAL METHODS Ð IN COMPUTER SCIENCE DOI: /LMCS-2 (2:6) 2006 c A. Cryol nd A. Meyer CC Cretive Commons

2 2 A. CARAYOL AND A. MEYER grphs [Cu96, Cu03], rtionl grphs [Mor00] nd trnsition grphs of Turing mchines [Cu03] (for survey, see for instnce [Tho01]). This work specificlly dels with fmily of infinite utomt for context-sensitive lnguges. The first result on this topic is due to Morvn nd Stirling [MS01], who showed tht the lnguges ccepted y rtionl grphs, whose vertices re words nd whose edges re defined y rtionl trnsducers, tken etween rtionl or finite sets of vertices, re precisely the context-sensitive lnguges. This result ws lter extended y Rispl [Ris02] to the more restricted fmilies of synchronized rtionl grphs, nd even to synchronous grphs. A summry cn e found in [MR04]. All proofs provided in these works use context-sensitive grmmrs in Penttonen norml form [Pen74] to chrcterize context-sensitive lnguges, which hs two min drwcks. First, this norml form is fr from eing ovious, nd the proofs nd constructions provided in [Pen74] re known to e difficult. Second, nd more importntly, there is no grmmr-sed chrcteriztion of deterministic context-sensitive lnguges, which forids one to dpt these results to the deterministic cse. Our min contriutions re new syntcticl proof of the theorem y Stirling nd Morvn sed on the thight correspondnce etween tiling systems nd synchronized grphs nd n in depth study of the trde off etween the structure of the rtionl grphs (numer of initil vertices nd out-degree), the trnsducers defining them, nd the fmily of lnguges they ccept, s summrized in Tle 1. Ech row of the tle concerns fmily or su-fmily of rtionl grphs, nd ech column corresponds to structurl restriction of tht fmily with respect to sets of initil vertices nd degree. The first cse is tht of rtionl (infinite) sets of initil vertices, while the second cse only considers the fixed rtionl initil set {} over single letter. The two remining cses concern grphs with unique initil vertex, with respectively ritrry nd finite out-degree. A cell contining n equlity symol indictes tht the lnguges ccepted y the considered fmily of grphs (row) from the considered set of initil vertices (column) re the context-sensitive lnguges. An inclusion symol indictes tht their lnguges re strictly included in context-sensitive lnguges. A question mrk denotes conjecture. When relevnt, we give reference to the proposition, theorem or remrk which sttes ech result. Fmily of grphs Set of initil vertices Rtionl set Set {} Unique vertex Unique vertex (finite degree) Rtionl [MS01] = = = = [4.6] Synchronized [Ris02] = = = [4.1] (?) [4.10] Synchronous [Ris02] = = [3.9] [4.2] Sequentil synchronous = [3.11] (?) [5.3] Tle 1: Fmilies of rtionl grphs nd their lnguges. Finlly, we investigte the cse of deterministic lnguges. A long-stnding open prolem in lnguge theory is the equivlence etween deterministic nd non-deterministic (or even unmiguous) context-sensitive lnguges [Kur64]. Thnks to our constructions, we

3 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM 3 chrcterize two syntcticl su-fmilies of rtionl grphs respectively ccepting the unmiguous nd deterministic context-sensitive lnguges. Outline. Our presenttion is structured long the following lines. The definitions of rtionl grphs nd context-sensitive lnguges re given in Section 2. The results concerning lnguges ccepted y rtionl nd synchronous rtionl grphs re given in Section 3. In Section 4, we investigte rtionl grphs under structurl constrints, nd finlly Section 5 is devoted to deterministic context-sensitive lnguges Nottions 2. Definitions Before ll, we fix nottions for words, lnguges nd utomt, s well s directed grphs nd the lnguges they ccept. For more thorough introduction to monoids nd rtionlity, the interested reder is referred to [Ber79, Sk03] Lnguges nd Automt We consider finite sets of symols, or letters, clled lphets. In the following, Σ nd Γ lwys denote finite lphets. Tuples of letters re clled words, nd sets of words lnguges. The word u corresponding to the tuple (u 1,...,u n ) is written u 1... u n. Its i-th letter is denoted y u(i) = u i. The set of ll words over Σ is written Σ. The numer of letter occurrences of u is its length, written u = n. The unique word of length 0 is written ε. The conctention of two words u = u 1...u n nd v = v 1...v m is the word uv = u 1...u n v 1...v m. This opertion extends to sets of words: for ll A,B Σ, AB stnds for the set {uv u A nd v B}. By slight use of nottion, we will usully denote y u oth the word u nd the singleton {u}. A monoid is composed of set M together with n ssocitive internl inry lw on M clled product, with neutrl element in M. The product of two elements x nd y of M is written x y. An utomton over M is tuple A = (L,Q,q 0,F,δ) where L M is finite set of lels, Q finite set of control sttes, q 0 Q is the initil stte, F Q is the set of finl sttes nd δ Q L Q is the trnsition reltion of A. A run of A is sequence of trnsitions (q 0,l 1,q 1 )... (q n 1,l n,q n ). It is ssocited to the element m = l 1... l n M. If q n elongs to F, the run is ccepting (or successful), nd m is ccepted, or recognized, y A. The set of elements ccepted y A is written L(A). A is unmiguous if there is only one ccepting run for ech element in L(A). The str of set X M is defined s X := k 0 Xk with X 0 = {ε} nd X k+1 = X X k. Similrly, we write X + := k 1 Xk. The set of rtionl susets of monoid is the smllest set contining ll finite susets nd closed under union, product nd str. The set of ll words over Σ together with the conctention opertion forms the so-clled free monoid whose neutrl element is the empty word ε. Finite utomt over the free monoid Σ re known to ccept the rtionl susets of Σ, lso clled rtionl lnguges.

4 4 A. CARAYOL AND A. MEYER Grphs A leled, directed nd simple grph is set G V Γ V where Γ is finite set of lels nd V countle set of vertices. An element (s,,t) of G is n edge of source s, trget t nd lel, nd is written s t or simply s t if G is understood. G The set of ll sources nd trgets of grph form its support V G. A sequence of edges s 1 1 t1,...,s k u k tk with i [2,k], s i = t i 1 is clled pth. It is written s 1 t k, where u = 1... k is the corresponding pth lel. A grph is deterministic if it contins no pir of edges hving the sme source nd lel. The pth lnguge of grph G etween two sets of vertices I nd F is the set L(G,I,F) := { w s w t, s I, t F }. G If two infinite utomt recognize the sme lnguge, we sy they re trce-equivlent. In this pper, we consider infinite utomt: infinite grphs together with sets of initil nd finl vertices. We will no longer distinguish the notion of grph with initil nd finl vertices from the notion of utomton. However, s we will see in Section 3, with no restriction on the set of initil vertices nd on the structure of the grph this might not provide resonle extension of finite utomt Word trnsducers Automt cn e used to ccept more thn lnguges. In prticulr, when the edges of n utomton re lelled with pirs of letters (with n pproprite product opertion), its lnguge is set of pirs of words, which cn e seen s inry reltion on words. Such utomt re clled finite utomt with output, or trnsducers, nd they recognize rtionl reltions. We will now recll their definition s well s some of their importnt properties. For detiled presenttion of trnsducers, see for instnce [Ber79, Pri00, Sk03]. Consider the monoid whose elements re the pirs of words (u,v) in Σ, nd whose composition lw is defined y (u 1,v 1 ) (u 2,v 2 ) = (u 1 u 2,v 1 v 2 ), generlly clled the product monoid nd written Σ Σ. A trnsducer T over finite lphet Σ is finite utomton over Σ Σ with lels in (Σ {ε}) (Σ {ε}). Finite trnsducers ccept the rtionl susets of Σ Σ. We do not distinguish trnsducer from the reltion it ccepts nd write (w,w ) T if (w,w ) is ccepted y T. The domin Dom(T) (resp. rnge Rn(T)) of trnsducer T is the set {w (w,w ) T } (resp. {w (w,w ) T }). We lso write T(L) the set of ll vertices v such tht (u,v) T for some u L. A trnsducer ccepting function is clled functionl. In generl, there is no ound on the size difference etween input nd output in trnsducer. Interesting suclsses re otined y enforcing some form of synchroniztion. For instnce, length-preserving rtionl reltions re recognized y trnsducers with lels in Σ Σ, clled synchronous trnsducers. Such reltions only pir words of the sme size. A more relxed form of synchroniztion ws introduced y Elgot nd Mezei [EM65]: trnsducer over Σ with initil stte q 0 is left-synchronized if for every pth x 0 /y 0 x n/y n q 0 q 1...q n 1 q n, there exists k [0,n] such tht for ll i [0,k], x i nd y i elong to Σ nd either x j = ε for ll j > k or y j = ε for ll j > k. In other terms, left-synchronized reltion is finite union of reltions of the form S F where S is synchronous reltion nd F is either equl

5 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM 5 ε A A A/A B/B B AB AB B AB AB T = T = q 0 A/A q 0 ε/a ε/b q 1 B/B q 1 Figure 1: The grid nd its ssocited trnsducers. to {ε} R or R {ε} where R is rtionl lnguge. Right-synchronized trnsducers re defined similrly. In the following, unless otherwise stted, we will refer to left-synchronized trnsducers simply s synchronized trnsducers. The stndrd notion of determinism for utomt does not hve much mening in the cse of trnsducers ecuse it does not rely only on the input ut on oth the input nd the output. A more refined notion is tht of sequentility: trnsducer T with sttes Q is sequentil if for ll q,q nd q in Q, if q x/y q nd q x /y q then either x = x, y = y nd q = q, or x ε, x ε nd x x. Remrk 2.1. The stndrd determiniztion procedure pplied to synchronous trnsducer yields n equivlent unmiguous synchronous trnsducer (i.e for every pir of words (u, v) ccepted y the trnsducer there is exctly one ccepting run of the trnsducer lelled y u/v). This remins true for synchronized trnsducers. It is well-known tht there is close reltionship etween rtionl lnguges nd rtionl trnsductions. In prticulr, rtionl reltions hve rtionl domins nd rnges, nd re closed under restriction to rtionl domin or rnge. Moreover, the restriction of sequentil (resp. synchronous) trnsducer to rtionl domin is still sequentil (resp. synchronous) (see for instnce [Ber79]) Rtionl grphs The Chomsky-like hierrchy of grphs presented in [CK02] uses words to represent vertices. Ech of these grphs is thus finite union of inry reltions on words, ech reltion corresponding to given edge lel. In prticulr, the fmily of rtionl grphs owes its nme to the fct tht their sets of edges re given y rtionl reltions on words, i.e. reltions recognized y word trnsducers. Definition 2.2 ([Mor00]). A rtionl grph lelled y Σ with vertices in Γ is given y tuple of trnsducers (T ) Σ over Γ. For ll Σ, G hs n edge lelled y etween vertices u nd v Γ if nd only if (u,v) T. For w Σ + nd Σ, we write T w = T w T, nd u w v if nd only if (u,v) T w. Note tht T w T stnds for the set of ll pirs (u,v) such tht (u,x) T w nd (x,v) T for some x. Figure 1 shows n exmple of rtionl grph, the infinite grid, with the rtionl trnsducers which define its edges. By the properties of rtionl reltions, the support of rtionl grph is rtionl suset of Γ. The rtionl grphs with synchronized trnsducers

6 6 A. CARAYOL AND A. MEYER were lredy defined y Blumensth nd Grdel in [BG00] under the nme utomtic grphs nd y Rispl in [Ris02] under the nme synchronized rtionl grphs. It follows from the definitions (see Section 2.2) tht sequentil synchronous, synchronous, synchronized nd rtionl grphs form n incresing hierrchy. This hierrchy is strict (up to isomorphism): first, sequentil synchronous grphs re deterministic grphs wheres synchronous grphs cn e non-deterministic. Second, synchronous grphs hve finite degree wheres synchronized grphs cn hve n infinite degree. Finlly, to seprte synchronized grphs from rtionl grphs, we cn use the following properties on the growth rte of the out-degree in the cse of grphs of finite out-degree. Proposition 2.3. [Mor01] For ny rtionl grph G of finite out-degree nd ny vertex x, there exists c N, such tht the out-degree of vertices t distnce n of x is t most c cn. This upper ound cn e reched: consider the unleled rtionl grph G 0 = {T } where T is the trnsducer over Γ = {A,B} with one stte q 0 which is oth initil nd finl X/Y Z nd trnsition q 0 q 0 for ll X,Y nd Z Γ. It hs n out-degree of 2 2n+1 t distnce n of A. In the cse of synchronized grphs of finite out-degree, the ound on the out-degree is simply exponentil. Proposition 2.4. [Ris02] For ny synchronized grph G of finite out-degree nd vertex x, there exists c N such tht the out-degree of vertices t distnce n > 0 of x is t most c n. It follows from the ove proposition tht G 0 is rtionl ut not synchronized. Hence, the synchronized grphs form strict su-fmily of rtionl grphs Context sensitive lnguges In this work, we re concerned with the fmily of context-sensitive lnguges 1. Severl finite formlisms re known to ccept this fmily of lnguges, the most common eing linerly ounded mchines (LBM), which re Turing mchines working in liner spce. Less well-known cceptors for these lnguges re ounded tiling systems, which re not trditionlly studied s lnguge recognizers. However, one cn show tht these formlisms re equivlent, nd tht syntcticl trnsltions exist etween them. Since they re t the hert of our proof techniques, we now give detiled definition of tiling systems. For more informtion out linerly ounded mchines the reder is referred to [HU79]. Tiling systems were originlly defined to recognize or specify picture lnguges, i.e. twodimensionl words on finite lphets [GR96]. They cn e seen s normlized form of dominos systems [LS97]. Such sets of pictures re clled locl picture lnguges. However, y only looking t the words contined in the first row of ech picture of locl picture lnguge, one otins context-sensitive lnguge [LS97]. A (n,m)-picture p over n lphet Γ is two dimensionl rry of letters in Γ with n rows nd m columns. We denote y p(i,j) the letter occurring in the ith row nd jth column strting from the top-left corner, y Γ n,m the set of (n,m)-pictures nd y Γ the set of ll pictures 2. Given (n,m)-picture p over Γ nd letter # Γ, we denote y p # the (n + 2,m + 2)-picture over Γ {#} defined y: p # (i,1) = p # (i,m + 2) = # for i [1,n + 2], 1 In order to simplify our presenttion, we only consider context-sensitive lnguges tht do not contin the empty word ε (this is stndrd restriction). 2 We do not consider the empty picture.

7 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM 7 # # # # # # # # # # # # Figure 2: A tiling system ccepting n n (Cf. Exmple 2.7). p # (1,j) = p # (n + 2,j) = # for j [1,m + 2], p # (i + 1,j + 1) = p(i,j) for i [1,n] nd j [1,m]. For ny n,m 2 nd ny (n,m)-picture p, T(p) is the set of (2,2)-pictures ppering in p. A (2,2)-picture is lso clled tile. A picture lnguge K Γ is locl if there exists symol # Γ nd finite set of tiles such tht K = {p Γ T(p # ) }. To ny set of pictures over Γ, we cn ssocite lnguge of words y looking t the frontiers of the pictures. The frontier of (n,m)-picture p is the word fr(p) = p(1,1)... p(1,m) corresponding to the first row of the picture. Definition 2.5. A tiling system S is tuple (Γ,Σ, #, ) where Γ is finite lphet, Σ Γ is the input lphet, # Γ is frme symol nd is finite set of tiles over Γ {#}. It recognizes the locl picture lnguge P(S) = {p Γ T(p # ) } nd the word lnguge L(S) = fr(p(s)) Σ. A tiling system S recognizes lnguge L Σ + in height f(n) for some mpping f : N N if for ll w L(S) there exists (n,m)-picture p in P(S) such tht w = fr (p) nd n f(m). We cn now precisely stte the following well-known equivlence result. Theorem 2.6. The following simultions link linerly ounded mchines nd tiling systems: (1) A linerly ounded Turing mchine T working in f(n) reversls cn e simulted y tiling system of height f(n) + 2. (2) A tiling system of height f(n) cn e simulted y linerly ounded Turing mchine working in f(n) reversls. Exmple 2.7. Figure 2 shows the set of tiles of tiling system S over Γ = {,, }, Σ = {,} nd the order symol #. The lnguge L(S) is exctly the set { n n n 1}. A context-sensitive lnguge is clled deterministic if it cn e ccepted y deterministic LBM or tiling system, where tiling system is deterministic if one cn infer from ech row in picture single possile next row.

8 8 A. CARAYOL AND A. MEYER Rtionl grphs Prop. 3.2 Prop. 3.3 Synchronous grphs Tiling systems Prop. 3.7 Sequentil synchronous grphs Prop Figure 3: Ech edge represents n effective trnsformtion preserving lnguges. 3. The lnguges of rtionl grphs In this section, we consider the lnguges ccepted y rtionl grphs nd their sufmilies from nd to rtionl set of vertices. We give simplified presenttion of the result y Morvn nd Stirling [MS01] stting tht the fmily of rtionl grphs ccepts the context-sensitive lnguges. This is done in severl steps. First, Proposition 3.2 sttes tht the rtionl grphs re trce-equivlent to the synchronous rtionl grphs. Then, Proposition 3.3 nd Proposition 3.7 estlish very tight reltionship etween synchronous grphs nd tiling systems. It follows tht the lnguges of synchronous rtionl grphs re lso the context-sensitive lnguges (Theorem 3.9). The originl result is given s Corollry Finlly, Proposition 3.11 estlishes tht even the smllest su-fmily we consider, the fmily of sequentil synchronous rtionl grphs, ccepts ll context-sensitive lnguges. The vrious trnsformtions presented in this section re summrized in Figure From rtionl grphs to synchronous grphs We present n effective construction tht trnsforms rtionl grph G with two rtionl sets I nd F of initil nd finl vertices into synchronous grph G trce-equivlent etween two rtionl sets I nd F. The construction is sed on replcing the symol ε in the trnsitions of the trnsducers defining G y fresh symol #. Let (T ) Σ e the set of trnsducers over Γ chrcterizing G nd let # e symol not in Γ. For ll, we define ā to e equl to if Γ, nd to ε if = #. We extend this to projection from (Γ #) to Γ in the stndrd wy. We define G s the rtionl grph defined y the set of trnsducers (T ) Σ where T hs the sme set of control sttes Q s T nd set of trnsitions given y { / p q p ā/ } { #/# } q T p p p Q. By definition of ech T, G is synchronous rtionl grph. Let I nd F e the two rtionl sets such tht I = {u ū I} nd F = {v v F } (the utomton ccepting I (resp. F ) is otined from the utomton ccepting I (resp. F) y dding loop leled y # on ech control stte). We clim tht G ccepts etween I nd F the sme lnguge s G etween I nd F. For exmple, Figure 4 illustrtes the previous construction pplied to the grph of Figure 1. Only one connected component of the otined grph is shown. Before we prove the correctness of this construction, we need to estlish couple of technicl lemms. Let B e the set of ll mppings from N to N. To ny mpping δ B, we ssocite mpping from (Γ {#}) to (Γ {#}) defined s follows: for ll w = # i 0 1 # i 1... n # in with 1,..., n Γ, let δw = # i 0+δ(0) 1 # i 1+δ(1)... n # in+δ(n).

9 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM 9 ### ##A #A# A## #AA A#A AA# AAA ##B #B# B## #BB B#B BB# #AB A#B AB# ABB AAB T = A/A #/# q 0 A/A #/# #/A B/B #/# q 1 B/B #/# BBB T = q 0 #/B q 1 Figure 4: Synchronous grph trce-equivlent to the grid (1 connected component). Before proceeding, we stte two properties of these mppings with respect to the sets of trnsducers (T ) nd (T ). Lemm 3.1. We hve the following properties: u,v Γ, (u,v) T δ u,δ v B, (δ u u,δ v v) T, (3.1) (u,v) T, δ B, δ B, (δu,δ v) T, (3.2) nd dully (u,v) T, δ B, δ B, (δ u,δv) T. (3.3) We cn now prove the correctness of the construction: word w is ccepted y G etween I nd F if nd only if it is ccepted y G etween I nd F. Proposition 3.2. For every rtionl grph G nd rtionl sets of vertices I nd F, there is synchronous grph G nd two rtionl sets I nd F such tht L(G,I,F) = L(G,I,F ). Proof. We show y induction on n tht for ll u 0,...,u n Γ, if there is pth u 0 w(1) G u 1...u n 1 w(n) G u n, then there exist words u 0,...,u n (Γ {#}) such tht for ll i, ū i = u i, nd u 0 w(1) G u 1...u n 1 The cse where n = 0 is trivil. Suppose the property is true for ll pths of length t most n, nd consider pth u 0 w(1) G... w(n) G w(n) G u n. u n w(n+1) G u n+1.

10 10 A. CARAYOL AND A. MEYER T / /, /, /, / #/ #, # #/ #, #/ #, T / / #, #, / /,, / /, / Figure 5: Trnsducers of synchronous grph ccepting { n n n 1}. By induction hypothesis, one cn find mppings δ 0,...δ n such tht δ 0 u 0 w(1) G... w(n) G δ n u n. We now use the properties of mppings stted in Lemm 3.1. By (3.1), there exist δ n nd w(n+1) δ n+1 such tht δ n u n δ n+1 u n+1 G. Let γ n nd γ n e two elements of B such tht δ n γ n = δ n γ n. By Lemm (3.2) nd (3.3), we cn find mppings γ n+1 nd γ 0 to γ n 1 such tht: w(1) γ 0 δ 0 u 0... w(n) γ G G n δ n u n = γ nδ nu w(n+1) n γ G n+1δ n+1u n+1 which concludes the proof y induction. If we suppose tht u 0 I nd u n F, then necessrily u 0 I nd u n F. It follows tht for every pth in G etween I nd F, there is pth in G etween I nd F with the sme pth lel. Conversely, y (3.1), for ny such pth in G, ersing the occurrences of # from its vertices yields vlid pth in G etween I nd F. Hence L(G,I,F) = L(G,I,F ) Equivlence etween synchronized grphs nd tiling systems The following propositions estlish the tight reltionship etween tiling systems nd synchronous rtionl grphs. Proposition 3.3 presents n effective trnsformtion of tiling system into synchronous rtionl grph. Proposition 3.3. Given tiling system S = (Γ,Σ,#, ), there exists synchronous rtionl grph G nd two rtionl sets I nd F such tht L(G, I, F) = L(S). Proof. Consider the finite utomton A on Γ with set of sttes Q = Γ {#}, initil stte #, set of finl sttes F nd set of trnsitions δ given y: F : such tht # δ : # A, A for ll, # # (respectively). #

11 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM 11 I [ # # ] [ ] [ F ] # # # # # Figure 6: The synchronous rtionl grph ssocited to the tiling system of Figure 2 whose trnsducers re presented in Figure 5. Cll M the lnguge recognized y A, M represents the set of possile lst columns of pictures of P(S). Note tht this does not imply tht ech word of M ctully is the lst column of picture in P(S), only tht it is comptile with the right order tiles of. Let us uild synchronous rtionl grph G nd two rtionl sets I nd F such tht L(G,I,F) = L(S). The trnsitions of the set of trnsducers (T e ) e Σ of G re: (#, #) c/d Td (c,d) for ll c d, d # (,) c/d Te (c,d) for ll c d,,d #, e Σ where (#, #) is the unique initil stte of ech trnsducer nd the set of finl sttes F of ech trnsducer is given y: F : (,) (Γ {#}) Γ such tht. A pir of words (s,t) is ccepted y the trnsducer T e if nd only if e is the first letter of t, nd either s nd t re two djcent columns of picture in P(S) or s # nd t is the first column of picture in P(S). As consequence, L(S) = L(G, #,M). Exmple 3.4. Figure 5 shows the trnsducers otined using the previous construction on the tiling system of Figure 2. They define rtionl grph whose pth lnguge etween # nd is { n n n 1}. Figure 6 presents the corresponding synchronous grph whose vertices re the rtionl set of words # , the set of initil vertices is # 2 nd the set of finl vertices is +. Remrk tht in this exmple, the set of vertices ccessile from the initil vertices is rtionl: this is not true in the generl cse. Remrk 3.5. The correspondence etween tiling system S nd the synchronous grph G constructed from S in Proposition 3.3 is tight: ech picture p with frontier w cn e mpped to unique ccepting pth for w in G (nd conversely).

12 12 A. CARAYOL AND A. MEYER Conversely, Proposition 3.7 sttes tht the lnguges ccepted y synchronous rtionl grphs etween rtionl sets of vertices cn e ccepted y tiling system. To mke the construction simpler, we first prove tht the sets of initil nd finl vertices cn e chosen over one-letter lphet without loss of generlity. Lemm 3.6. For every synchronous rtionl grph G with vertices in Γ nd rtionl sets I nd F, one cn find synchronous rtionl grph H nd two symols i nd f / Γ such tht L(G,I,F) = L(H,i,f ). Proof. Let G = (K ) Σ e synchronous rtionl grph with vertices in Γ. For i,f two new distinct symols, we define new synchronous rtionl grph H chrcterized y the set of trnsductions ( T = (T I K ) K (K T F ) ) Σ where T I = {(i n,u) n 0, u I, u = n} nd T F = {(v,f n ) n 0, v F, v = n}. For ll vertices u I, v F we hve u w v if nd only if i u w H f u, i.e. L(G,I,F) = L(H,i,f ). G We re now le to estlish the converse of Proposition 3.3, which sttes tht ll the lnguges ccepted y synchronous rtionl grphs etween rtionl sets of vertices cn e ccepted y tiling system. Proposition 3.7. Given synchronous rtionl grph G nd two rtionl sets I nd F, there exists tiling system S such tht L(S) = L(G, I, F). Proof. Let G = (T ) Σ e synchronous rtionl grph with vertices in Γ (with Σ Γ). By Lemm 3.6, we cn consider without loss of generlity tht I = i nd F = f for some distinct letters i nd f, nd tht neither i nor f occurs in ny vertex which is not in I or F. Furthermore y Remrk 2.1, we cn ssume tht T is non-miguous for ll Σ. We write Q the set of control sttes of T. We suppose tht ll control stte sets re disjoint, nd designte y q0 Q the unique initil stte of ech trnsducer T, nd y Q F the set of finl sttes of ll T. Let,,c,d Σ, x,x,y,y,z,z Γ, nd p,p,q,q,r,r,s,s Σ Q. We define tiling system S = (Γ,Σ, #, ), where is the set of tiles from Figure 7. By construction, P(S) is in exct ijection with the set of ccepting pths in G with respect to I nd F. Let φ e the function ssociting to picture p P(S) with columns 1 w 1,..., n w n, the pth i w 1 1 w1... n wn where w is otined y removing the control sttes from w. By construction of S, the function φ is well defined. It is esy to check tht φ is n onto function. As the trnsducers defining G re non-miguous, two distinct pictures hve distinct imges y φ nd therefore φ is n injection. Hence, the tiling system (Γ, Σ, #, ) exctly recognizes L(G, I, F). Remrk 3.8. As in Remrk 3.5, the set of pths in G from I to F nd the set of pictures P(S) ccepted y S re in ijection, nd the length of the vertices long the pth is equl to the height of the corresponding picture. Putting together Propositions 3.3 nd 3.7 nd Theorem 2.6, we otin the following result concerning the pth lnguges of synchronous rtionl grphs. Theorem 3.9 ([Ris02]). The lnguges ccepted y synchronous rtionl grphs etween rtionl sets of initil nd finl vertices re the context-sensitive lnguges. Note tht this formultion of the theorem could e mde it more precise y reclling tht initil nd finl sets of vertices only of the form x, where x is letter, re sufficient

13 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM 13 # c d # # # xp c y q z r d # f s # with q 0 i/x T p, q c 0 y/z Tc r, s Q d # xp y q z r f s # # x p y q z r fs # i/x with,,c, p p y /z, r r, s,s Q c T T # xp y q z r f s # with p,q,r,s Q F Figure 7: Tiling system ccepting the lnguge of synchronous grph. to ccept ll context-sensitive lnguges, s stted in Lemm 3.6. By Proposition 3.2, this implies s corollry the originl result y Morvn nd Stirling [MS01]. Corollry The lnguges ccepted y rtionl grphs etween rtionl sets of initil nd finl vertices re the context-sensitive lnguges. If we trnsform rtionl grph into Turing mchine y pplying successively the construction of Proposition 3.2, Proposition 3.7 nd Theorem 2.6, we otin the sme Turing mchine s in [MS01] Sequentil synchronous grphs re enough Theorem 3.9 shows tht when considering rtionl sets of initil nd finl vertices, synchronous grphs re enough to ccept ll context-sensitive lnguges. Interestingly, when considering rtionl sets of initil nd finl vertices, the even more restricted clss of sequentil synchronous trnsducers re sufficient. Proposition The lnguges ccepted y sequentil synchronous rtionl grphs etween rtionl sets of initil nd finl vertices re the context-sensitive lnguges. Proof. Thnks to Proposition 3.7, it suffices to prove tht ny context sensitive lnguge L Σ is ccepted y synchronous sequentil rtionl grph. By Theorem 2.6, we know tht there exists tiling system S = (Γ,Σ, #, ) such tht L(S) = L. Let Λ = Γ {#} nd [ nd ] e two symols tht do not elong to Λ. We ssocite to ech picture p Λ with rows l 1,...,l n the word [l 1 ]...[l n ]. We re going to define set of sequentil synchronous trnsducers tht, when iterted, recognize the words corresponding to pictures in P(S). First, for ny finite set of tiles, we construct trnsducer T which checks tht word in ([Λ 3 ]) 2 represents picture with tiles in. The checking is done column y column, nd we introduce mrked letters to keep trck of the column eing checked. Let Λ e finite lphet in ijection with ut disjoint from Λ. For ll x Λ we write x Λ the mrked version of x. For every word w = u xv Λ ΛΛ, we write π(w) the word uxv Λ nd ρ(w) = u + 1 designtes the position of the mrked letter in the word.

14 14 A. CARAYOL AND A. MEYER # # # ã # # ã # # # # # # # # ã # # # # # # # # # # # # # # # # # # # # # # # # # Figure 8: Connected component of sequentil synchronous grph ccepting { n n n 1}. We consider words in [Λ ΛΛ ] 2. Let Shift e the reltion tht shifts ll mrks in word one letter to the right. More precisely, Shift stisfies Dom(Shift) = ([Λ ΛΛ + ]) 2, nd Shift([w 1 ]...[w n ]) = [w 1 ]... [w n ] with π(w i ) = π(w i) nd ρ(w i ) = ρ(w i)+1 for ll i [1,n]. The rtionl reltion Shift cn e relized y synchronous sequentil trnsducer T Sh. Consider the following rtionl lnguge: x i 1y i 1 R = [w 1x 1 ỹ 1 w 1 ]... [w nx n ỹ n w n ] n 2 nd i [2,n], x i y i. The trnsducer T otined y restricting T Sh to the domin R is oth synchronous nd sequentil. For ll w = [w 1 ]...[w n ] ([Λ ΛΛ ]) 2, if w = T N(w) then w = [w 1 ]...[w n] with π(w i ) = π(w i ) nd ρ(w i ) = N + 2 for ll i [1,n]. Let r i e the word contining the N + 1 first letters of w i, strightforwrd induction on N proves tht the picture p formed of the rows r 1,...,r n only hs tiles in. In prticulr, T N(w) elongs to ([Λ Λ]) R if nd only if π(w) represent picture p of width N + 2 such tht T(p). We now define more precisely the sequentil rtionl grph G = (T ) Σ ccepting L. For ll Σ, the trnsducer T is otined y restricting the domin of T to the set of words representing pictures whose mrked symol on the second row is, i.e. to the set [(Λ Λ) ][(Λ ãλ ][(Λ Λ) ]. T cn e chosen synchronous nd sequentil. The set of initil vertices I is [# ## ]([# ΓΓ #]) [# ## ] nd the set of finl vertices F is [# #]([#Γ #]) [# #]. Exmple Figure 8 shows prt of the result of the previous construction when pplied to the lnguge { n n n 1} s recognized y the tiling system of Figure 2. Ech vertex is represented y the corresponding picture, insted of the word coding for it. Also, only one connected component of the grph is shown. The other connected components ll hve the sme liner structure: the degree of the grph is ounded y 1. The leftmost vertex elongs to the set I, nd the rightmost to the set F, hence the word 2 2 is ccepted. Remrk In the cse of synchronized trnsducers, it hs een shown in Lemm 3.6 tht I could e tken over one letter lphet without loss of generlity. This does not seems to hold for sequentil trnsducers s the proof we present relies on the expressiveness of the initil set of vertices. In fct, s shown in Proposition 5.3, the lnguges recognized y sequentil synchronous grph from i re deterministic context-sensitive lnguges.

15 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM Rtionl grphs seen s utomt The structure of the grphs otined in the previous section (propositions 3.3 nd 3.11) is very poor. Synchronous grphs re y definition composed of possily infinite set of finite connected components. In the cse of Proposition 3.11, we otin n even more restricted fmily of grphs since oth their in-degree nd out-degree is ounded y 1. However, when considering ccepted lnguges from possily infinite rtionl set of vertices, even this extremely restricted fmily ccepts the sme lnguges s the most generl rtionl grphs, nmely ll context-sensitive lnguges. This is why, in order to compre the expressiveness of the different su-fmilies of rtionl grphs nd to otin grphs with richer structures, we need to impose structurl restrictions. We first consider grphs with single initil vertex, ut this restriction lone is not enough. In fct, oth synchronized nd rtionl grphs with rtionl set of initil vertices ccept the sme lnguges s their counterprts with single initil vertex. Lemm 4.1. For every rtionl grph (resp. synchronized grph) G nd for every pir of rtionl sets I nd F, there exists rtionl grph (resp. synchronized grph) G, vertex i nd rtionl set F such tht L(G,I,F) = L(G, {i},f ). Proof. Let G = (T ) Σ e rtionl grph with vertices in Γ nd let i e symol which does not elong to Γ nd Γ = Γ {i}. For ll Σ, let T e trnsducer recognizing the rtionl reltion T {(i,w) w T (I)}. Remrk tht if T is synchronized then T cn lso e chosen synchronized. If ε L(G,I,F) then F = F else F = F {i}. It is strightforwrd to show tht L(G,I,F) = L(G, {i},f ). It follows from Proposition 3.3 nd Lemm 4.1 tht the synchronized rtionl grphs with one initil vertex ccept the context-sensitive lnguges [Ris02]. Remrk 4.2. It is firly ovious tht this result does not hold for synchronous grphs: indeed, the restriction of synchronous rtionl grph to the vertices rechle from single vertex is finite. Hence, the lnguges of synchronous grphs from single vertex re rtionl. Similrly, s ny rtionl lnguge is ccepted y deterministic finite grph, it cn lso e ccepted y sequentil synchronous grph with single initil vertex. Note tht the construction of Lemm 4.1 relies on infinite out-degree to trnsform synchronous grph with rtionl set of initil vertices into rtionl one with single initil vertex. In order to otin more stisfctory notions of infinite utomt, we now restrict our ttention to grphs of finite out-degree with single initil vertex Rtionl grphs of finite out-degree with one initil vertex. We present syntcticl trnsformtion of synchronous rtionl grph with rtionl set of initil vertices into rtionl grph of finite out-degree with unique initil vertex ccepting the sme lnguge. The construction relies on the fct tht for synchronous grphs to recognize word of length n > 0, it is only necessry to consider vertices whose length is smller thn c n (where c is constnt depending only on the grph). We first estlish similr result for tiling systems nd conclude using the close correspondence etween synchronous grphs nd tiling systems estlished in Proposition 3.7.

16 16 A. CARAYOL AND A. MEYER Lemm 4.3. For ny tiling system S = (Γ,Σ,#, ), if p P(S) then there exists (n,m)-picture p such tht fr (p) = fr(p ) nd n Γ m. Proof. Let p e (n,m)-picture with n > Γ m, nd suppose tht p is the smllest picture in P(S) with frontier fr (p). Let l 1,...,l n e the rows of p. As n > Γ m then there exists j > i 1 such tht l i = l j. Let p e the picture with rows l 1,...,l i,l j+1,...,l n. It is esy to check tht T(p #) T(p #), we hve tht p P(S) nd s p hs smller height thn p ut the sme frontier, we otin contrdiction. We know from Remrk 3.8 tht for every synchronous rtionl grph G = (T ) Σ nd w two rtionl sets I nd F, there exists tiling system S such tht i f with i I nd G f F if nd only if there exists p K such tht fr(p) = w nd p hs height i = f. Hence, s direct consequence of Lemm 4.3, one gets: Lemm 4.4. For every synchronous rtionl grph G nd rtionl sets I nd F, there exists k 1 such tht: w w L(G,I,F), i I,f F such tht i f nd i = f k w. G We cn now construct of rtionl grph of finite out-degree ccepting from single vertex the sme lnguge s synchronous grph with rtionl set of initil vertices. Proposition 4.5. For every synchronous rtionl grph G nd rtionl sets I nd F such tht I F =, there is rtionl grph H of finite out-degree nd vertex i such tht L(G,I,F) = L(H, {i},f). Proof. According to Lemm 3.6, there exists synchronous rtionl grph R descried y set of trnsducers (T ) Σ over Γ such tht L(G,I,F) = L(R, #,F). Note tht for ll w # nd w Γ, if w w then w does not contin #. We define grph H such R tht L(G,I,F) = L(H, {i},f) for some vertex i of H. Let k e the constnt involved in Lemm 4.4, T nd T two trnsducers relizing the rtionl reltions { (# n, # kn ) n N } nd {# n, # m m [1,n]} respectively. For ll,,c Σ nd u Σ, H hs edges: w (Γ \ {#}), w (Γ \ {#}), n N, u # n u T T(# n ) (Type 1) n N, u # n u T T T (# n ) (Type 2) n N, cu # n u T T T c (# n ) (Type 3) cu w w # u T T c (w) (Type 4) T T (w) (Type 5) T T T (#) (Type 6) The grph H is clerly rtionl nd of finite out-degree. We tke i = # s initil vertex. Remrk tht in H n edge of type 2 or 3 cnnot e followed y edges of type 1, 2 or 3, nd t most one edge of type 2 or 3 nd of type 5 or 6 cn e pplied. Moreover, n edge of type 1 increses the length of the left prt of the word y one, nd n edge of type 4 decreses it y one. Also, in ny ccepting pth, the lst edge is of type 5 or 6. Figure 9 illustrtes the structure of the otined grph. It is technicl ut strightforwrd to show correspondence etween ccepting pths in H nd R, nd to conclude tht L(R, #,F) = L(H, {i},f).

17 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM 17 (1) cu # n (4) cu T (# m 4n ) (2) (4) w (3) (4) (5) w f # (1) T 2 (#) (1) (1) u T T c(# m n ) (4) (4) (4) cu w u T T c(w) (Type 1) (guessing the size) Type 2 or 3 (strting computtion) (Type 4) (ctching up) Type 5 (ccepting) Figure 9: Schem of the construction in Proposition 4.5. From Proposition 3.2 nd Proposition 4.5, we deduce tht the rtionl grphs of finite out-degree with one initil vertex ccept ll context-sensitive lnguges. This result ws proved in [MS01] using the Penttonen norml form of context-sensitive grmmrs [Pen74]. Theorem 4.6. The pth lnguges of rtionl grphs of finite out-degree from unique initil vertex to rtionl set of finl vertices re the context-sensitive lnguges Synchronized grphs of finite out-degree with one initil vertex We now consider the lnguges of synchronized grphs of finite out-degree with one initil vertex. First, we chrcterize them s the lnguges recognized y tiling systems with squre pictures (i.e. for which there exists c N such tht for every word w L(S), there exists (n,m)-picture in P(S) with n cm nd with frontier w). A slight dpttion of the construction of Proposition 4.5 gives the first inclusion. Proposition 4.7. Let S = (Γ,Σ,#, ) e tiling system with squre pictures. There exists synchronized rtionl grph of finite degree ccepting L(S) from one initil vertex. Proof. Let G = (T ) Σ e the synchronized grph otined from S in Proposition 3.3. In the construction from the proof of Proposition 4.5, if we replce the trnsducer T y trnsducer S relizing the synchronized reltion {(# n,# n+c ) n N}, we otin synchronized grph H, vertex i nd set F such tht L(H,i,F) = L(S). Before proceeding with the converse, we stte result similr to Lemm 4.4 for synchronized grphs of finite out-degree tht sttes tht when recognizing word w from unique initil vertex i, the vertices involved hve length t most liner in the size of w. Lemm 4.8. For ny synchronized rtionl grph G of finite out-degree with vertices in Γ nd for every vertex i, there exists constnt k such tht for ll w in L(G, {i},f), there exists pth from i to some f F, leled y w, nd with vertices of size t most k w.

18 18 A. CARAYOL AND A. MEYER Proof. It follows from the definition of synchronized trnsducers tht for every synchronized trnsducer of finite out-degree there exists c N such tht (x,y) T implies tht x y +c (see [Sk03] for proof of this result). We tke k to e the mximum over the set of trnsducers defining G of these constnts. The result follows y strightforwrd induction on the size of w. The converse inclusion is otined y remrking tht the composition of the construction of Proposition 3.2 nd Proposition 3.7 gives tiling system with squre pictures when pplied to synchronized grph of finite out-degree. Proposition 4.9. Let G = (T ) Σ e synchronized grph of finite out-degree. For every initil vertex i nd set of finl vertices F, there exists tiling system S with squre pictures such tht L(S) = L(G, {i},f). Proof. Let G, I nd F e the synchronous grph nd the rtionl set of initil nd finl vertices otined y pplying the constructions of Proposition 3.2 to G, {i} nd F. It is esy to show tht for every word w L(G,I,F ), there exists i I nd f F such tht i w = f with i = f k w where k is the constnt of Lemm 4.8 for G. We conclude y Proposition 3.7, tht sttes the existence of tiling system S such tht L(S) = L(G,I,F ). By Remrk 3.5, S is tiling system with squre pictures. Putting together Proposition 4.9 nd Proposition 4.7 nd with the use of the simultion result from Theorem 2.6, we otin the following theorem. Theorem The lnguges ccepted y synchronized grphs of finite out-degree from unique vertex to rtionl set of vertices re the context-sensitive lnguges recognized y non-deterministic linerly ounded mchines with liner numer of hed reversls. We conjecture tht this clss is strictly contined in the context-sensitive lnguges. However, few seprtion results exist for complexity clsses defined y time nd spce restrictions (see for exmple [vm04]). In prticulr, the digonliztion techniques (see [For00]) used to prove tht the polynomil time hierrchy (with no spce restriction) is strict do not pply for lck of suitle notion of universl LBM Bounding the out-degree It is nturl to wonder if the rtionl grphs still ccept the context-sensitive lnguges when considering ounded out-degree. This is difficult question, to which we only provide here prtil nswer concerning the synchronized grphs of ounded out-degree. It follows from Lemm 4.8 tht the vertices used to ccept word w in synchronized rtionl grph hve length t most liner in the length of w nd therefore, cn e stored on the tpe of LBM. Moreover if the grph is deterministic, we cn construct deterministic LBM ccepting its lnguge. Proposition The lnguge ccepted y deterministic synchronized grph from unique initil vertex is deterministic context-sensitive. Proof. Let G = (T ) Σ e deterministic synchronized grph over Γ, i vertex nd F rtionl set of vertices. We define deterministic LBM M ccepting L(G, {i},f). When ccepting w = 1... w, M strts y writing i on its tpe. It successively pplies T 1,..., T n 1 nd T n to i. If the imge of the current tpe content y one of these trnsducers

19 CONTEXT-SENSITIVE LANGUAGES, RATIONAL GRAPHS AND DETERMINISM 19 is not defined, the mchine rejects. Otherwise, it checks whether the lst tpe content represents vertex which elongs to F. We now detil how the mchine M cn pply one of the trnsducers T of G to word x in deterministic mnner. As T hs finite imge, we cn ssume without loss of generlity tht T = (Γ,Q,i,F,δ) is in rel-time norml form: δ Q Γ Γ Q (see for instnce [Ber79] for presenttion of this result). The mchine enumertes ll pths in T of length less thn c x in the lexicogrphic order where c is the constnt ssocited to G in Lemm 4.8. For ech such pth ρ, it checks if it is n ccepting pth for input x, nd in tht cse replces x y the output of ρ. The spce used y M when strting with word w is ounded y (2c+1) w. Moreover if M ccepts w, then there exists pth from i to vertex F in G leled y w. Conversely, if w elongs to L(G, {i},f) then y Lemm 4.8, there exists pth in G from i to F with vertices of length t most c w nd y construction M ccepts w. Hence, M is deterministic linerly ounded Turing mchine ccepting L(G, {i},f). Remrk The result of Proposition 4.11 extends to ny deterministic rtionl grph stisfying the property expressed y Lemm 4.8. The previous result cn e extended to synchronized grphs of ounded out-degree thnks to uniformiztion result y Weer. First oserve tht rtionl grph is of outdegree ounded y some constnt k if nd only if it is defined y trnsducers which ssocite t most k distinct imges to ny input word. The reltions relized y these trnsducers re clled k-vlued rtionl reltions. Proposition 4.13 ([We96]). For ny k-vlued rtionl reltion R, there exist k functionl rtionl reltions F 1,...,F k such tht R = i [1,k] F i. Note tht even if R is synchronized reltion, the F i s re not necessrily synchronized. However, they still stisfy the inequlity y x + c for ll (x,y) F i. To ny synchronized grph G with n out-degree ounded y k defined y set of trnsducers (T ) Σ, we ssocite the deterministic rtionl grph H defined y (F i ) Σ,i [1,k] where for ll Σ, (F i ) i [1,k] is the set of rtionl functions ssocited to T y Proposition According to Proposition 4.11 nd to Remrk 4.12, L(H, {i},f) is deterministic context-sensitive lnguge. Let π e the lpheticl projection defined y π( i ) = for ll Σ nd i [1,k], it is strightforwrd to estlish tht π (L(H, {i},f)) = L(G, {i}, F). As deterministic context-sensitive lnguges re closed under lpheticl projections, L(G, {i}, F) is deterministic context-sensitive lnguge. Theorem The lnguge ccepted y synchronized grph of ounded out-degree from unique initil vertex is deterministic context-sensitive. The converse result is not cler, for resons similr to those presented in the previous section for synchronized grphs of finite degree. A precise chrcteriztion of the fmily of lnguges ccepted y synchronized rtionl grphs of ounded degree would e interesting. 5. Notions of determinism In this lst prt of the section on rtionl grphs, we investigte fmilies of grphs which ccept the deterministic context-sensitive lnguges. First of ll, we exmine the fmily

20 20 A. CARAYOL AND A. MEYER yielded y the previous constructions when pplied to deterministic lnguges. Then, we propose glol property over sets of trnsducers chrcterizing su-fmily of rtionl grphs whose lnguges re precisely the deterministic context-sensitive lnguges Unmiguous context-sensitive lnguges When pplying the construction of Proposition 3.3 to deterministic tiling system S, one otins synchronous rtionl grph G (which is non-deterministic in generl) nd two rtionl sets of vertices I nd F such tht L(G,I,F) = L(S), with the prticulrity tht for every word w in L, there is exctly one pth leled y w leding from some vertex in I to vertex in F: G is unmiguous with respect to I nd F. However, the converse is not grnted: given grph G nd two rtionl sets I nd F such tht G is unmiguous with respect to I nd F, we cnnot ensure tht L(G,I,F) is deterministic context-sensitive lnguge. Rther, the otined lnguges cn e ccepted y unmiguous linerly ounded mchines. This clss of lnguges is clled USPACE(n), nd it is not known whether it coincides with either the context-sensitive or deterministic context-sensitive lnguges. Theorem 5.1. Let L e lnguge, the following properties re equivlent: (1) L is n unmiguous context-sensitive lnguge. (2) There exist rtionl grph G with unmiguous trnsducers nd two rtionl sets I nd F with respect to which G is unmiguous, such tht L = L(G,I,F). This result only holds if one considers unmiguous trnsducers, i.e. trnsducers in which there is t most one ccepting pth per pir of words. The reson is tht miguity in the trnsducers would induce miguity in the mchine. However, since synchronized trnsducers cn e mde unmiguous (Cf. Remrk 2.1), we cn drop this requirement in the cse of synchronized grphs. Note tht the unmiguity of rtionl or synchronized grphs with respect to rtionl sets of vertices is undecidle. However, since ny rtionl function cn e relized y n unmiguous trnsducer [Ko69, Sk03], the lnguge of ny deterministic rtionl grph is, y to Theorem 5.1, unmiguous. Corollry 5.2. The lnguges of deterministic rtionl grphs from n initil vertex i to rtionl set F of vertices re unmiguous context-sensitive lnguges Glolly deterministic sets of trnsducers We just sw n ttempt t chrcterizing nturl fmilies of grphs whose lnguges re the deterministic context-sensitive lnguges, which ws sed on restriction of previous constructions to the deterministic cse, ut filed to meet its ojective ecuse of slight nunce etween the notions of determinism nd unmiguity for tiling systems. First, we nturlly consider the clss of sequentil synchronous utomt with n initil set of the form {}, where is letter of the vertex lphet (in other words, given initil vertex does not code for ny informtion esides its length). It is esy to check tht when pplying the construction of Proposition 3.7 to one of these utomt, we otin deterministic tiling system. Proposition 5.3. The lnguges of sequentil synchronous grphs from {} re deterministic context-sensitive lnguges.