Automata Theory and Infinite Transition Systems

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1 Automt Theory n Infinite Trnsition Systems Wolfgng Thoms Lehrstuhl Informtik 7, RWTH Ahen, Germny thoms@informtik.rwth-hen.e Astrt These leture notes report on the use of utomt theory in the stuy of infinite trnsition systems. This pplition of utomt is n importnt ingreient of the urrent evelopment of infinite-stte system verifition, n it provies n introution into the fiel of lgorithmi moel theory, stuy of infinite moels with n emphsis on omputility results. 1 Introution The nlysis of infinite trnsition systems is of entrl interest in infinite-stte system verifition n t the sme time one of the most promising pplition omins of utomt theory. These leture notes give n overview over some topis whih re urrently stuie in this fiel. The lssil set-up of lgorithmi verifition is efine y two items: trnsition system s moel of system (progrm, protool, ontrol unit), n speifition given y logil formul whih expresses some esire ehviour. The moel-heking prolem is the question Given trnsition grph G n formul ϕ, oes G stisfy ϕ?. Prominent logis in this ontext re the temporl logis LTL (liner-time temporl logi) n CTL (omputtion tree logi). An LTL formul expresses property of given infinite pth (exeution sequene) through the trnsition grph uner onsiertion, wheres CTL-formul expresses property of the infinite tree of exeution sequenes, strting from given initil stte. Common to oth logis is the view tht the non-terminting ehviour of the system (pture y its infinite pths) is to e nlyze. A oneptul sis for solving the moel-heking prolem is the frmework of utomt over infinite wors n infinite trees. For exmple, the eiility of the moel-heking prolem for LTL n simply e otine y trnsformtion of LTL-formuls into Bühi utomt. The moel-heking prolem is thus reue to prolem out the reltion of trnsition grph G n n Leture Notes for the EMS Summer Shool CANT (Comintoris, Automt n Numer Theory), My 2006, University of Liège. Reserh reporte in this pper ws prtilly supporte y the DAAD Proope Projet Finitely Presente Infinite Grphs in ollortion with the group of D. Cul, Rennes. 1

2 utomton A erive from the speifition ϕ. This reltion n e trete ompletely in the omin of utomt (trnsition systems). A wekness of this pproh is the ft tht the system grph uner onsiertion is ssume to e finite. Sine more thn ee, new tren in verifition is to onsier more moest mens of speifition ut to exten the omin of trnsition grphs to inlue ertin infinite strutures. Sine these strutures serve s inputs to the (verifition) lgorithms, finitry presenttion is neee. As it turne out, finite utomt re n essentil tool for this tsk, sine they offer very nturl wys to speify infinite strutures. The purpose of these letures is to report on some funmentl lsses of infinite moels (in our se: trnsition grphs) suh tht the moel-heking prolem is eile for interesting properties. So the emphsis is on infinite systems rther thn infinite ehviour. Our presenttion is fr from omplete; it is ise towrs stte-se moels. An lterntive n eqully funmentl pproh for introuing infinite moels, whih is not isusse here, is to exten finite trnsition grphs y infinite t strutures, for exmple over the nturl or rel numers (s in time systems) n to use the frmework of rithmeti (e.g., equlities or inequlities etween polynomils) for speifitions. 2 Tehnil Preliminries We onsier strutures in the formt of ege-lelle n vertex-lelle trnsition grphs: G = (V,(E ) Σ,(P ) Σ ) where V is the (t most ountle) set of verties (in pplitions: sttes ) where E V V (for symol from finite lphet Σ) is the set of - lelle eges, n where P V is the set of -lelle verties (in pplitions representing stte property) with Σ. We write E for the union of the E. More generlly, one n onsier reltionl strutures A = (A,R A 1,...,R A k ), where the R A i re reltions of possily ifferent rities over A, sy R A i of rity n i. In the sequel we sty with trnsition grphs for ese of nottion n for their signifine in verifition. Centrl exmples of trnsition grphs re Kripke strutures, whih re grphs of the form G = (V,E,(P )), where eh P ollets sttes whih stisfy the sme tomi propositions uner onsiertion, the orering (N,<) of the nturl numers, the inry tree T 2 = ({0,1},S 0,S 1 ) where S i = {(w,wi) w {0,1} } (nlogously, the n-ry tree T n = ({0,...,n 1},S n 0,...,S n n 1) is efine) Let us list the logil systems onsiere in these letures. First-orer logi FO over the signture with the symols E,P is uilt up from vriles x,y,... n tomi formuls x = y, E (x,y), P (x) where x,y re first-orer vriles, using the stnr propositionl onnetives,,,, n the quntifiers,. The rehility reltion over G is the reltion E efine y E (u,v) v 0...v k V : v 0 = u,(v i,v i+1 ) E (i < k),v k = v 2

3 It is well-known tht E is not FO-efinle. We ll FO(R) the logi otine from FO y joining symol for the rehility reltion E to the signture. Moni seon-orer logi MSO is otine y joining vriles X,Y,... for sets of elements (of the universe V uner onsiertion) n tomi formuls X(y), mening tht the element y is in the set X. We note tht MSO enompsses FO(R), sine we n express E (x,y) y the formul sying tht eh set whih ontins x n is lose uner E must ontin y. We use the stnr nottions; e.g. G = ϕ[v] inites tht G stisfies the formul ϕ(x) with the element v s interprettion of x. Given formul ϕ(x 1,...,x n ), the reltion efine y it in G is ϕ G = {(v 1,...,v n ) V n G = ϕ[v 1,...,v n ]} MSO enompsses most stnr temporl logis. We shll ress two methos for onstruting infinite trnsition grphs where moel-heking (with respet to MSO or FO(R)) is eile. First we explin regulr internl representtions of infinite trnsition grphs, using finite utomt over strings or trees for speifition. Seonly, we use funmentl moel onstrutions nmely, interprettion n unfoling to generte infinite moels. 3 Rtionl n Automti Grphs A onvenient wy to introue finitely presente infinite strutures is to use wors over some lphet s nmes of verties, regulr lnguges for stte properties, n utomton-efinle reltions over wors for the ege reltions. For the ltter, there is lrge reservoir of options, onsiering finite-stte trnuers or wor rewriting systems of ifferent kins. In this setion we onsier the two most prominent exmples: the rtionl reltions n the utomti (or: synhronize rtionl) reltions. 3.1 Rtionl Grphs A reltion R Γ Γ is rtionl if it n e efine y regulr expression strting from the tomi expressions O (enoting the empty reltion) n (u, v) for wors u, v (enoting the reltion {(u, v)}) y mens of the opertions union, ontention (pplie omponentwise), n itertion of ontention (Kleene str). An lterntive hrteriztion of these reltions involves moel of noneterministi utomton whih works one-wy from left to right, ut synhronously, on the two omponents of n input (w 1,w 2 ) Γ Γ. Exmple 1. Consier the suffix reltion {(w 1,w 2 ) w 1 is suffix of w 2 }. A orresponing utomton (noneterministi trnsuer) woul progress with its reing he on the seon omponent w 2 until it guesses tht the suffix w 1 strts; this, in turn, n e heke y moving the two reing hes on the two omponents simultneously, ompring w 1 letter y letter with the remining suffix of w 2. A rtionl trnsition grph hs the form G = (V,(E ) Σ,(P ) Σ ) where V n the sets P re regulr sets of wors over n uxiliry lphet Γ n where eh E Γ Γ is rtionl reltion. 3

4 Clerly, eh rtionl grph is reursive in the sense tht the ege reltions n the vertex properties re eile. However, very simple properties of rtionl grphs my e uneile. Theorem 2. For eh instne (u, v) of PCP (Post s Corresponene Prolem) one n onstrut rtionl grph G (u,v) suh tht (u,v) hs solution iff G (u,v) hs loop ege from vertex to itself. Proof. Given PCP-instne (u,v) = ((u 1,...,u m ),(v 1,...,v m )) over n lphet Γ, we speify rtionl grph G (u,v) = (V,E) s follows. The vertex set V is Γ. The ege set E onsists of the pirs of wors of the form (u i1...u ik,v i1...v ik ) where i 1,...,i k {1,...,m} n k 1. Clerly, n synhronously progressing noneterministi utomton n hek whether wor pir (w 1,w 2 ) elongs to E; silly the utomton hs to guess suessively the inies i 1,...,i k n t the sme time to hek whether w 1 strts with u i1 n w 2 strts with v i1, whether w 1 ontinues y u i2 n w 2 y v i2, et. So the grph G (u,v) is rtionl. Clerly, in this grph there is n ege from some vertex w k to the sme vertex w iff the PCP-instne (u,v) hs solution (nmely y the wor w). The existene of loop ege (w,w) is expressile y the first-orer formul x E(x,x). Hene we otin the following result. Corollry 3 (Morvn [15]). There is no lgorithm whih, given presenttion of rtionl grph G n first-orer sentene ϕ, eies whether G = ϕ. Let us onstrut single rtionl grph with n uneile first-orer theory. We give onstrution from [21]. Theorem 4. There is rtionl grph G with n uneile first-orer theory; in other wors, the prolem Given first-orer sentene ϕ, oes G stisfy ϕ? is uneile. Proof. We use universl Turing mhine M n the enoing of its uneile hlting prolem (for ifferent input wors x) into fmily of PCPinstnes. For simpliity of exposition, we refer here to the stnr onstrution of the uneiility of PCP s one fins it in textooks (see [13, Setion 8.5]): A Turing mhine M with input wor x is onverte into PCP-instne ((u 1,...,u m ),(v 1,...,v m )) over n lphet A whose letters re the sttes n tpe letters of M n symol # (for the seprtion etween M-onfigurtions in M-omputtions). If the input wor is x = 1... n, then u 1 is set to e the initil onfigurtion wor (x) := #q n of M; furthermore we lwys hve v 1 = #, n u 2,...,u m,v 2,...,v m only epen on M. Then the stnr onstrution (of [13]) ensures the following: M hlts on input x iff the PCP-instne (((x),u 2,...,u m ),(#,v 2,...,v m )) hs speil solution. Here speil solution is given y n inex sequene (i 2,...,i k ) suh tht (x)u i2...u ik = #v i2...v ik. 4

5 Let G e the grph s efine from these PCP-instnes s ove: The verties re the wors over A, n we hve single ege reltion E with (w 1,w 2 ) E iff there re inies i 2,...,i k n wor x suh tht w 1 = (x)u i2...u ik n w 2 = #v i2...v ik. Clerly G is rtionl, n we hve n ege from wor w k to itself if it is inue y speil solution of some PCP-instne (((x),u 2,...,u m ),(#,v 2,...,v m )). In orer to ress the input wors x expliitly in the grph, we further verties n ege reltions E for A. A (x)-lelle pth vi the new verties will le to vertex of G with prefix (x); if the ltter vertex hs n ege k to itself, then speil solution for the PCP-instne (((x),u 2,...,u m ), (#,v 2,...,v m )) n e inferre. The new verties re wors over opy A of the lphet A (onsisting of the unerline versions of the A-letters). For ny wor (x) we shll the verties whih rise from the unerline versions of the proper prefixes of (x), n we introue n E -ege from ny suh unerline wor w to w (inluing the se w = ǫ). There re lso eges to non-unerline wors: We hve n E -ege from the unerline version of w to ny non-unerline wor whih hs w s prefix. Cll the resulting grph G. It is esy to see tht G is rtionl. By onstrution of G, the PCP-instne (((x),u 2,...,u m ), (#,v 2,...,v m )) hs speil solution iff in G there is pth, lelle with the wor (x), from the vertex ǫ to vertex whih hs n ege k to itself. Note tht the vertex ǫ is efinle s the only one with outgoing E -eges ut without ny ingoing E -ege. Thus the ove onition is formlizle y first-orer sentene ϕ x, using vriles for the (x) + 1 verties of the esire pth. Altogether we otin tht the Turing mhine M hlts on input x iff G = ϕ x. These results show tht for pplitions in verifition, rtionl grphs in generl re too extensive to llow interesting lgorithmi solutions. 3.2 Automti Grphs In utomti (or synhronize rtionl) reltions more restrite proessing of n input (w 1,w 2 ) y n utomton is require thn in the synhronous moe s mentione for noneterministi trnsuers: We now require tht n utomton sns pir (w 1,w 2 ) of wors stritly in prllel letter y letter. Thus one n ssume tht the utomton res letters from Γ Γ if w 1,w 2 Γ. In orer to over the se tht w 1,w 2 re of ifferent length, one ssumes tht the shorter wor is prolonge y ummy symols $ to hieve equl length. Let [w 1,w 2 ] e the wor over the lphet (Γ Γ) ((Γ {$}) Γ) (Γ (Γ {$})) ssoite with (w 1,w 2 ). Thus, reltion R Γ Γ inues the lnguge L R = {[w 1,w 2 ] (w 1,w 2 ) R}. The reltion R is lle utomti if the ssoite lnguge L R is regulr. From this efinition it is immeitely ler tht the utomti reltions shre mny goo properties whih re fmilir from the theory of regulr wor lnguges. For exmple, one n reue noneterministi utomt whih reognize wor reltion synhronously to eterministi utomt, ft whih oes not hol for the trnsuers in the ontext of rtionl reltions. 5

6 X 0 ε X Y XX XY Y Y XXX XXY XY Y Y Y Y Figure 1: An utomti grph A grph (V,(E ) Σ,(P ) Σ ) is lle utomti if V n eh P V re regulr lnguges over n lphet Γ n eh ege reltion E Γ Γ is utomti. Exmple 5. The infinite two-imensionl gri G 2 := (N N,E,E ) (with E - eges ((i,j),(i,j + 1)) n E -eges ((i,j),(i + 1,j))) is utomti: It n e otine using the wors in X Y s verties, whene the ege reltions eome E = {(X i Y j,x i Y j+1 ) i,j 0} n E = {(X i Y j,x i+1 Y j ) i,j 0}, whih oth re lerly utomti. Exmple 6. Consier the trnsition grph over Γ = {X 0,X,Y } where there is n -ege from X 0 to X n from X i to X i+1 (for i 1), -ege from X i Y j to X i 1 Y j+1 (for i 1,j 0), n -ege from Y i+1 to Y i (for i 0). We otin the utomti grph of Figure 1. If X 0 is tken s initil n ǫ s finl stte, the resulting infinite utomton reognizes the ontext-sensitive lnguge onsisting of the wors i i i for i 1. Exmple 7. Let T 2 = ({0,1},S 0,S 1,,λ) e the expnsion of the inry tree T 2 = ({0,1},S 0,S 1 ) y the prefix reltion = {(u,v) {0,1} u is prefix of v} n the equl length reltion λ = {(u,v {0,1} u = v}. Clerly T 2 is utomti. In the literture, the utomti reltions pper lso uner severl other nmes, mong them regulr, sequentil, synhronize rtionl. We give nother exmple whih illustrtes the power of utomti reltions. Exmple 8. Given Turing mhine M with stte set Q n tpe lphet Γ, we onsier the grph G M with vertex set V M = Γ QΓ, onsiere s the set of M-onfigurtions. By n pproprite tretment of the lnk symol, we n ssume tht the length ifferene etween two suessive M-onfigurtions is t most 1; thus it is esy to see tht the reltion E M of wor pirs whih onsist of suessive M-onfigurtions is utomti. So the onfigurtion grph G M = (V M,E M ) is utomti. Let us show tht first-orer properties of utomti grphs re eile: Theorem 9. The FO-theory of n utomti grph is eile. Proof. Let G = (V,(E ) Σ,(P ) Σ ) e grph with n utomti presenttion over Γ. We verify inutively over FO-formuls ϕ(x 1,...,x n ) tht the 6

7 following reltion is utomti: R ϕ = {(w 1,...,w n ) (G,w 1,...,w n ) = ϕ(x 1,...,x n )} For the tomi formuls, this is ler y the utomti presenttion of G. In the inution step, the Boolen onnetives re esy ue to the losure of regulr sets uner Boolen opertions. (Note tht the omplement is pplie with respet to the set of wors [w 1,w 2 ], i.e. the wors where the letter $ my our only in one omponent, n only t the en.) For the step of existentil quntifition, ssume s typil se tht the inry reltion R is reognize y the finite utomton A, sy with finl stte set F. We hve to verify tht lso S = {w 1 Γ w 2 : (w 1,w 2 ) R} is utomti (i.e. in this unry se: regulr lnguge). The utomton heking S is otine from A y projetion of the input letters to the first omponents n y n extension of F to set F. A stte is inlue in F if some (possily empty) sequene of letters ($,) les to F. (This overs the se tht the omponent w 2 is longer thn w 1.) The truth of n FO-sentene ϕ mounts to nonemptiness of n utomton whih results from this inutive onstrution over the suformuls of ϕ. For ϕ itself n input-free utomton (with unlelle trnsitions) is otine; the truth of ϕ is signlle y the existene of suessful run of this input-free utomton. A opy of this rgument shows tht Presurger rithmeti, the FO-theory of the struture (N, +), is eile ([3]). For this purpose, one oes n n-tuple of nturl numers y the n-tuple of the reverse inry representtions. The tomi formul x 1 + x 2 = x 3 efines ternry reltion over {0,1} whih is utomti, sine the usul hek tht n ition of inry numers is orret n e one y finite utomton. For the logil onnetives one proees s in the proof ove. If we exten the logi FO slightly y inluing the rehility reltion E, then this eiility result fils. Theorem 10. There is n utomti grph G suh tht over G the reltion E is uneile. Proof. Consier the onfigurtion grph G U of universl Turing mhine U, where the verties re onfigurtion wors in Γ QΓ (Γ is the tpe lphet of U, n Q is its set of sttes). Assume tht the mhine U hlts preisely in onfigurtions xq s y with stop stte q s. The one-step reltion of pirs (xqy,x q y ) of onfigurtion wors is utomti. Then n ritrry Turing mhine M epts the input wor w iff over G U, from the onfigurtion q 0 oe(m)w hlting onfigurtion n e rehe. This result is one of the min ostles in eveloping frmework for moelheking over infinite systems: The utomti grphs re very nturl frmework for moelling interesting infinite systems, ut most pplitions of moelheking involve some kin of rehility nlysis. Current reserh tries to fin goo restritions of the lss of utomti grphs where the rehility 7

8 prolem is still solvle. An importnt lss of suh grphs is presente in the next setion. Let us lso look t more mitious prolem thn rehility: eiility of the moni seon-orer theory of given grph (or equivlently: the moel-heking prolem with respet to MSO-efinle properties). Here we get uneiility lrey for utomti grphs with muh simpler trnsition struture thn tht of the grph G U of the previous theorem. The most prominent exmple is the infinite two-imensionl gri (introue s n utomti grph in Exmple 5). Note tht the rehility prolem over the gri (sy from given vertex to nother given vertex) is eile. Theorem 11. The moni seon-orer theory of the infinite two-imensionl gri G 2 is uneile. Proof. The ie is to oe the omputtions of Turing mhines in more uniform wy thn in the previous result. Inste of oing Turing mhine onfigurtion y single vertex n pturing the Turing mhine steps iretly y the ege reltion, we now use whole row of the gri for oing onfigurtion (y n pproprite oloring of its verties with tpe symols n Turing mhine stte). A omputtion of Turing mhine, sy with m sttes n n tpe symols, is thus represente y sequene of olore rows (using m + n olors), i.e., y oloring of the gri. (We n ssume tht even hlting omputtion genertes oloring of the whole gri, y repeting the finl onfigurtion infinitum.) In this view, the horizontl ege reltion is use to progress in spe, while the vertil one llows to progress in time. A given Turing mhine M hlts on the empty tpe iff there is oloring of the gri with m + n olors whih represents the initil onfigurtion (on the empty tpe) in the first row, respets the trnsition tle of M etween ny two suessive rows, ontins vertex whih is olore y hlting stte. Suh oloring orrespons to prtition of the vertex set N N of the gri into m + n sets. One n express the existene of the oloring y sying there exist sets X 1,...,X m+n whih efine prtition n stisfy the requirements of the three items ove. In this wy one otins effetively n MSO-sentene ϕ M suh tht M hlts on the empty tpe iff G 2 = ϕ M. 4 Prefix Rewriting n Pushown Systems The uneiility of the rehility prolem over utomti grphs is no surprise to nyone who hs her out Semi-Thue systems, i.e. rewriting systems tht llow to reple n infix w in wor y nother infix w. The one-step reltion over Turing mhine onfigurtion wors is efine y n infix rewriting system. It is well-kown tht the erivility reltion of infix rewriting systems is in generl uneile. As oserve lrey y Bühi in 1964, the sitution hnges when we use prefix rewriting inste. Bühi showe tht the wors whih re generte from fixe wor y finite prefix rewriting system form regulr lnguge. As n 8

9 pplition one otins the well-known ft tht the rehle glol sttes of pushown utomton onstitute regulr set. In the theory of progrm verifition, the prolem usully rises in onverse (ut essentilly equivlent) form: Here one strts with (usully regulr) set T of verties s trget set, n the prolem is to fin those wors from whih wor in T is rehle y finite numer of prefix rewriting steps. We introue two types of grphs se on the ie of prefix rewriting. The first (n more restrite) version is the notion of pushown grph, with eges orresponing to moves of pushown utomton. The seon llows to pture infinitely mny instnes of prefix rewriting in single rule. A grph G = (V,(E ) Σ ) is lle pushown grph (over the lel lphet Σ) if it is the trnsition grph of the rehle glol sttes of n ǫ-free pushown utomton. Here pushown utomton is of the form P = (P,Σ,Γ,p 0,Z 0, ), where P is the finite set of ontrol sttes, Σ the input lphet, Γ the stk lphet, p 0 the initil ontrol stte, Z 0 Γ the initil stk symol, n P Σ Γ Γ P the trnsition reltion. A glol stte (onfigurtion) of the utomton is given y ontrol stte n stk ontent, i.e., y wor from PΓ. The grph G = (V,(E ) Σ ) is now speifie s follows: V is the set of onfigurtions from PΓ whih re rehle (vi finitely mny pplitions of trnsitions of ) from the initil glol stte p 0 Z 0. E is the set of ll pirs (pγw,qvw) from V 2 for whih there is trnsition (p,,γ,v,q) in. Then the ege reltion E oinies with the one-step erivtion reltion p 1 w 1 p 2 w 2 over P, n the trnsitive losure E with the erivility reltion. A more generl lss of grphs, whih inlues the se of verties of infinite egree, onsists of the prefix-reognizle grphs (propose y Cul in [5]). These grphs re introue in terms of prefix-rewriting systems in whih ontrol sttes (s they our in pushown utomt) re no longer use n where wor on the top of the stk (rther thn single letter) my e rewritten. Thus, rewriting step n e speifie y triple (u 1,,u 2 ), esriing trnsition from wor u 1 w vi letter to the wor u 2 w. The feture of infinite egree is introue y llowing generlize rewriting rules of the form U 1 U2 with regulr sets U 1,U 2 of wors. Suh rule les to the (in generl infinite) set of rewrite triples (u 1,,u 2 ) with u 1 U 1 n u 2 U 2. A grph G = (V,(E ) Σ ) is lle prefix-reognizle if for some finite system S of suh generlize prefix rewriting rules U 1 U2 over n lphet Γ, we hve V Γ is regulr set, E onsists of the pirs (u 1 w,u 2 w) where u 1 U 1, u 2 U 2 for some rule U 1 U2 from S, n w Γ. Exmple 12. The struture (N, Su, <) is prefix reognizle. We write the struture s (N,E,E ) n represent numers y sequenes over the one-letter lphet with the symol only. So V =, n the two reltions E,E re efine y the prefix rewriting rules ε n ε +. The prefix-reognizle grphs oinie with the pushown grphs when ε- ε rules re e to pushown utomt n eges re efine in terms of ε. 9

10 Let us ompre the four lsses of grphs introue so fr. Theorem 13. The pushown grphs, prefix-reognizle grphs, utomti grphs, n rtionl grphs onstitute, in this orer, stritly inresing inlusion hin of grph lsses. Proof. For the proof, we first note tht the prefix-reognizle grphs re lerly generliztion of the pushown grphs n tht the rtionl grphs generlize the utomti ones. To verify tht prefix-reognizle grph is utomti, we first proee to n isomorphi grph whih results from reversing the wors uner onsiertion, t the sme time using suffix rewriting rules inste of prefix rewriting ones. Given this formt of the ege reltions, we n verify tht it is utomti: Consier wor pir (u 1 w,u 2 w) whih results from the pplition of suffix rewriting rule U 1 U2, with regulr U 1,U 2 n u 1 U 1, u 2 U 2. A noneterministi utomton n esily hek this property of the wor pir y snning the two omponents simultneously letter y letter, guessing when the ommon prefix w of the two omponents is psse, n then verifying (gin proeeing letter y letter) tht the reminer u 1 of the first omponent is in U 1 n the reminer u 2 of the seon omponent is in U 2. The stritness of the inlusions my e seen s follows. The property of hving oune egree seprtes the pushown grphs from the prefix-reognizle ones (see Exmple 12). To istinguish the other grph lsses, one my use logil eiility results. It will e shown in Setion 5.1 tht the moni seon-orer theory of prefix-reognizle grph is eile, whih fils for the utomti grphs. Furthermore, the first-orer theory of n utomti grph is eile, whih fils in generl for the rtionl grphs. Let us solve the rehility prolem over pushown grphs (we leve the se of prefix-reognizle grphs s n exerise). For pushown utomton P = (P,Σ,Γ,p 0,Z 0, ) n T PΓ let pre (T) = {pv PΓ qw T : pv qw} We prove the following funmentl result whih goes k to Bühi [4]: Theorem 14. Given pushown utomton P = (P,Σ,Γ,p 0,Z 0, ) n finite utomton reognizing set T PΓ, one n ompute finite utomton reognizing pre (T). We n eie the rehility of onfigurtion p 2 w 2 from p 1 w 1 y setting T = {p 2 w 2 } n heking whether the utomton reognizing pre (T) epts p 1 w 1. The trnsformtion of given utomton A whih reognizes T into the esire utomton A reognizing pre (T) works y simple proess of sturtion, whih involves ing more n more trnsitions ut leves the set of sttes unmoifie. This onstrution, whih improves the originl one y Bühi regring effiieny, ppers in severl ppers, mong them [8] n [12]; we follow the ltter. It is onvenient to work with P s the set of initil sttes of A; so stk ontent pw of the pushown utomton is snne y A strting from stte p n then proessing the letters of w. This use of P s the set of initil sttes of A motivtes the term P-utomton in the literture. The P-utomt we use for speifying T o not hve trnsitions into P; we ll them normlize. 10

11 The sturtion proeure is se on the following ie: Suppose pushown trnsition llows to rewrite the stk ontent pγw into qvw, n tht the ltter one is epte y A. Then the stk ontent pγw shoul lso e epte. If A epts qvw y run strting in stte q n rehing, sy, stte r fter proessing v, we enle the eptne of pγw y ing iret trnsition from p vi γ to r. The sturtion lgorithm performs suh insertions of trnsitions s long s possile. Sturtion Algorithm: Input: P-utomton A, pushown system P = (P, Γ, ) A 0 := A, i := 0 REPEAT: IF p p v n A i : p v q THEN (p,,q) to A i n otin A i+1 i := i + 1 UNTIL no trnsition n e e A := A i Output: A As n exmple onsier P = (P,Γ, ) with P = {p 0,p 1,p 2 }, Γ = {,,}, = {(p 0 p 1 ),(p 1 p 2 ),(p 2 p 0 ),(p 0 p 0 )} n T = {p 0 }. The P-utomton for T is the following: A: p 0 s 1 s 2 p 1 p 2 Exeution of the sturtion lgorithm introues eges s inite in the following figure. Insertion of p 0 p0 is se on the rule p 0 p 0 n ε A : p 0 p0, insertion of p 2 p0 on the rule p 2 p 0 n A : p 0 p0 (we enote y A here lwys the urrent utomton), insertion of p 1 s1 on the rule p 1 p 2 n A : p 2 s 1, insertion of p 0 s2 on the rule p 0 p 1 n A : p 1 s 2, n insertion of p 1 s2 on the rule p 1 p 2 n A : p 2 s 2. A : p 0 s 1 p 1 p 2 s 2 So for T = {p 0 } we extrt the following result. pre (T) = p 0 ( + ) + p 1 + p 1 + p 2 ( + ) 11

12 Proposition 15. The Sturtion Algorithm termintes n gives, for n input utomton A reognizing T, s output n utomton A reognizing pre (T). Proof. Termintion of the lgorithm is ler sine new trnsitions (p,, q) n e e only finitely often to the given utomton. Next we hve to show: pw pre (T) A : p w F For the iretion from left to right we use inution over the numer n 0 of steps to get to T: pw n ru T A : p w F The se n = 0 is ovious. In the inution step ssume pw n+1 ru n ru T. We hve to show tht A epts pw. Consier the eomposition of the step sequene to ru T: pw p vw n ru with w = w n pushown trnsition p p v. The inution ssumption gives A : p vw F. So, there exists n A -stte q with A : p v w q F. Consequently, the sturtion lgorithm proues the trnsition (p,,q) A, n pw is epte y A. For the iretion from right to left we show A : p w q = p w PΓ : A suh tht p w q pw p w For q F (the finl stte-set of A) we otin the lim; note tht A : p w q sys tht p w T. We enote y A i the P-utomton whih origintes from A fter i insertions of new trnsitions y the sturtion lgorithm. We show inutively over i: If A i : p w q, then p w PΓ suh tht A : p w q pw p w The se i = 0 ovious. For the inution lim ssume tht A i+1 : p w q. Consier n epting run A i+1 : p w q. Let j e the numer of pplitions of the (i + 1)-st trnsition. We prove the lim inutively over j. The se j = 0 is ovious (no use of the (i + 1)-th trnsition). For j + 1, onsier the eomposition of w in w = uu with A i : p u p 1, A i+1 : p 1 q1 }{{} (i+1)-st trnsition n A i+1 : q 1 u q By inution (on i) we hve pu p 1u 1 with A : p u 1 1 p1. Sine A is normlize, its initil stte p 1 hs no ingoing trnsitions, hene u 1 = ε n p 1 = p 1; thus pu p 1. The sturtion lgorithm s (p 1,,q 1 ) to A i. So, there re p 2 n v pushown rule p 1 p 2 v with A i : p 2 q1. Finlly, in the run on u, the (i + 1)-st trnsition is use j times, so y v u inution ssumption on j, we know for the run A i+1 : p 2 q1 q tht there is p w with A : p w q n p 2 vu p w. Altogether we hve pw = puu p 1 u p 2 vu p w ( T). 12

13 The ie of the sturtion lgorithm hs een trnsferre to mny relte prolems, for exmple for heking reurrent rehility over pushown grphs, for two-plyer rehility gmes plye on pushown grphs, n for rehility over trnsition grphs ssoite with tree rewriting systems (see elow). In the next setion we shll introue nother tehnique for showing eiility results. This tehnique overs more thn rehility properties, in ft ll properties whih re expressile in MSO-logi. The ie is to trnsfer the eiility of the MSO-theory from given stnr struture to other strutures. We shll two onsier types of moel trnsformtion: the MSOinterprettions n the proess of unfoling. 5 Opertions preserving eiility 5.1 Interprettions The strting point of this setion is eep n iffiult eiility result, lle Rin s Tree Theorem : Theorem 16. (Rin [18]) The MSO-theory of the infinite inry tree T 2 is eile. We o not enter the proof; self-ontine exposition is in [20]. Mny other theories were shown eile (lrey in Rin s lnmrk pper [18]) using interprettions in the tree T 2. To show tht the moel-heking prolem for struture S with respet to formuls of logi L is eile, one proees s follows: One gives n MSO-esription of S within the inry tree T 2, n using this, one provies trnsltion of L-formuls ϕ into MSOformuls ϕ suh tht S = ϕ iff T 2 = ϕ. Tking L = MSO, we see tht n MSO-interprettion (i.e., moel esription using MSO-formuls) preserves eiility of moel-heking with respet to MSO-formuls. Let us illustrte the ie of MSO-interprettion y showing tht the result lso hols for the strutures T n for n > 2. As typil exmple onsier T 3 = ({0,1,2},S 3 0,S 3 1,S 3 2). We otin opy of T 3 in T 2 y onsiering only the T 2 -verties in the set T = ( ). A wor in this set hs the form 1 i im 0 with i 1,...,i m {1,2,3}; n we tke it s representtion of the element (i 1 1)...(i m 1) of T 3. The following MSO-formul ϕ(x) (written in revite suggestive form) efines the set T in T 2 : Y [Y (x) y((y (y10) Y (y110) Y (y1110)) Y (y)) Y (ǫ)] It sys tht x is in the losure of ǫ uner 10-, 110-, n 1110-suessors. The reltion {(w,w10)w {0,1} } is efine y the following formul: ψ 0 (x,y) := z(s 1 (x,z) S 0 (z,y)) With the nlogous formuls ψ 1, ψ 2 for the other suessor reltions, we see tht the struture with universe ϕ T2 n the reltions ψ T2 i restrite to ϕ T2 is isomorphi to T 3. 13

14 In generl, n MSO-interprettion of struture A in struture B is given y omin formul ϕ(x) n, for eh reltion R A of A, sy of rity m, n MSO-formul ψ(x 1,...,x m ), suh tht A with the reltions R A is isomorphi to the struture with universe ϕ B n the reltions ψ B restrite to ϕ B. Then for n MSO-sentene χ (in the signture of A) one n onstrut sentene χ (in the signture of B) suh tht A = χ iff B = χ. In orer to otin χ from χ, one reples every tomi formul R(x 1,...,x m ) y the orresponing formul ψ(x 1,...,x m ) n one reltivizes ll quntifitions to ϕ(x). As onsequene, we note the following: Proposition 17. If A is MSO-interpretle in B n the MSO-theory of B is eile, then so is the MSO-theory of A. As seon exmple of MSO-interprettion, onsier pushown utomton A with stk lphet {1,...,k} n sttes q 1,...,q m. Let G A = (V A,E A ) e its onfigurtion grph. Choosing n = mx{k, m}, we n exhiit n MSOinterprettion of G A in T n : Just represent onfigurtion (q j,i 1...i r ) y the vertex i r...i 1 j of T n. Note tht then the A-steps le to lol moves in T n, from one T n -vertex to nother, e.g. in push step from vertex i r...i 1 j to vertex i r...i 1 i 0 j. These moves re esily efinle in MSO, n so is rehility (from the initil vertex 11). Hene we otin the following result: Theorem 18. (Muller, Shupp [17]) The MSO-theory of pushown grph is eile. A strightforwr generliztion of the proof yiels orresponing sttements for the prefix-reognizle grphs. The ifferene to the proof ove is just refinement of the formul expressing the one-step erivtion reltion etween onfigurtions. Inste of esriing single move from one wor u 0 w to nother, sy v 0 w, we hve to esrie ll missile moves from wor uw to vw where rule U V exists with u U,v V. This n e one y esriing suessful runs of the orresponing utomt A U, A V on the pth segments from uw to w n from vw to w, respetively. The esription of run of n utomton sy with k sttes q 1,...,q k n e one in MSO logi y postulting k susets X 1,...,X k of the onsiere pth segments, the set X i ontining those verties where stte q i is ssume. Theorem 19. A prefix-reognizle grph is MSO-interpretle in T 2 ; thus, the moni seon-orer theory of prefix-reognizle grph is eile. For ompleteness we note n nlogous result for utomti grphs n firstorer logi: Theorem 20. An utomti grph is FO-interpretle in T 2. Proof. In these notes we only give the si ie of the proof. We onentrte on the se in whih the lphet Γ for the utomti presenttion of the given grph G is just {0,1}. We show how to give n FO-esription of regulr set V {0,1}, reognize, sy, y the utomton A, in the struture T 2. For this we shll evelop formul ϕ(x) whih expresses whether the vertex x {0,1} is epte y A. The letters of x n e reovere from the ourse of the pth leing to x; for exmple, the i-th letter of x is 0 (resp. 1) iff on level i 1 14

15 the pth towrs x rnhes left (resp. right). The verties on this pth re efinle with the prefix reltion of T 2. For simpliity, suppose tht A hs four sttes, so tht run on x n e ientifie with sequene of length x+1 of pirs of its. Seprting the two omponents we my oe the sequene y two it-wors z 1,z 2 of length x + 1. The existene of suh wors z 1,z 2 is iretly expressile in first-orer logi. Using the equl-length preite of the signture, it is now esy to express the neessry onitions out z 1,z 2, nmely tht the oe sequene strts with the initil stte of A, tht the step from level to the next respets the trnsition reltion of A (n the orresponing letter of x), n tht the lst oe stte is epting. It is remrkle tht the lst two theorems lso hol in the onverse iretion: Any grph tht is MSO-interpretle in T 2 is prefix-reognizle, n ny grph tht is FO-interpretle in T 2 is utomti [2]. In the ltter se, this yiels new wy to estlish the eiility of the FO-theory of n utomti grph: One just hs to omine eiility proof for the FO-theory of T 2 with n FO-interprettion of the onsiere grph G in T Unfolings In the previous setion we expline how to generte moel within given one, vi efining formuls. A more expnsive wy of moel onstrution is the unfoling of grph (V,(E ) Σ,(P ) Σ ) from given vertex v 0, yieling tree T G (v 0 ) = (V,(E ) Σ,(P ) Σ ): V onsists of the verties v 0 1 v 1... r v r with (v i 1,v i ) E i, E ontins the pirs (v 0 1 v 1... r v r,v 0 1 v 1... r v r v) with (v r,v) E, n P the verties v 0 1 v 1... r v r with v r P. The unfoling opertion hs no effet in isimultion invrint logis, ut is highly nontrivil for MSO-logi. Consier, for exmple, the singleton grph G 0 over {v 0 } with 1-lelle n 2-lelle ege from v 0 to v 0. Its unfoling is the infinite inry tree. While heking MSO-formuls over G 0 is trivil, this is eep result for T 2. A powerful result going k to Muhnik 1985 sys tht unrvelling preserves eiility of the MSO-theory. Theorem 21. (Muhnik 1985, Courelle n Wlukiewiz [9]) If the MSO-theory of G is eile n v 0 is n MSO-efinle vertex of G, then the MSO-theory of T G (v 0 ) is eile. The result hols lso for slightly more generl onstrution ( tree itertion ) whih n lso e pplie to reltionl strutures other thn grphs. We nnot go into etils here; goo presenttion is given in [1]. 6 Cul s Hierrhy MSO-interprettions n unfolings re two opertions whih preserve eiility of MSO moel-heking. Cul [6] stuie the strutures generte y pplying oth opertions, lternting etween unfolings n interprettions. He introue the following hierrhy (G n ) of grphs, together with hierrhy (T n ) of trees: T 0 = the lss of finite trees 15

16 e e e e e Figure 2: A grph, its unfoling, n pushown grph G n = the lss of grphs whih re MSO-interpretle in tree of T n T n+1 = the lss of unfolings of grphs in G n By the results of the preeing setions (n the ft tht finite struture hs eile MSO-theory), eh struture in the Cul hierrhy hs eile MSO-theory. By hierrhy result of Dmm on higher-orer reursion shemes, the hierrhy is stritly inresing. In Cul s orginl pper [6], ifferent formlism of interprettion (vi inverse rtionl sustitutions ) is use inste of MSO-interprettions. We work with the ltter to keep the presenttion more uniform; the equivlene etween the two pprohes hs een estlishe y Cryol n Wöhrle [10]. Let us tke look t some strutures whih our in this hierrhy (following [22]). It is ler tht G 0 is the lss of finite grphs, while T 1 ontins the solle regulr trees (lterntively efine s the infinite trees whih hve only finitely mny non-isomorphi sutrees). Figure 2 (upper hlf) shows finite grph n its unfoling s regulr tree. By n MSO-interprettion we n otin the pushown grph of Figure 2 in the lss G 1 ; the omin formul n the formuls efining E,E,E re trivil, while ψ (x,y) = ψ e (x,y) = z z (E (z,z ) E (z,y) E (z,x)) Let us pply the unfoling opertion gin, from the only vertex without inoming eges. We otin the lgeri tree of Figure 3, elonging to T 2 (for the moment one shoul ignore the she line). As next step, let us pply n MSO-interprettion to this tree whih will proue grph (V,E,P) in the lss G 2 (where E is the ege reltion n P unry preite). Referring to Figure 3, V is the set of verties whih re lote long the she line, E ontins the pirs whih re suessive verties long the she line, n P ontins the speil verties rwn s non-fille irles. This struture is isomorphi to the struture (N,Su,P 2 ) with the suessor reltion Su n preite P 2 ontining the powers of 2. To prepre orresponing MSO-interprettion, we use formuls suh s E (x,y) whih expresses 16

17 e e e e e e e e Figure 3: Unfoling of the pushown grph of Figure 2 All sets whih ontin x n re lose uner E -suessors ontin y, n y hs no E -suessor As omin formul we use ϕ(x) = z(e (z,x) y(e (z,y) E (y,x))). The require ege reltion E is efine y ψ(x,y) = z z (ψ 1 (x,y) ψ 2 (x,y) ψ 3 (x,y)) where ψ 1 (x,y) = E (z,z ) E (z,x) E (z,y) ψ 2 (x,y) = E (z,z ) E e (z,x) E (z,y) ψ 3 (x,y) = E e (z,x) E e (z,y) Finlly we efine P y the formul χ(x) = z z (E (z,z ) E (z,x)). We infer tht the MSO-theory of (N,Su,P 2 ) is eile, result first prove y Elgot n Rin in 1966 with ifferent pproh. Let us isuss nother interesting struture of this kin, nmely the struture (N,Su,F) where F is the set of ftoril numers. We strt from simpler pushown grph thn the one use ove n onsier its unfoling, whih is the om struture inite y the thik rrows of the lower prt of Figure 4. We numer the verties of the horizontl line y 0,1,2... n ll the verties elow them to e of level 0, level 1, level 2 et. Now we use the simple MSO-interprettion whih tkes ll tree noes s omin n introues for n 0 new ege from ny vertex of level n + 1 to the first vertex of level n. This introues the thin lines in Figure 4 s new eges (ssume to point kwrs). It is esy to write own efining MSO-formul. Note tht the top vertex of eh level plys speil role sine it is the trget of n ege lelle, while the remining ones re trgets of eges lelle. Consier the tree otine from this grph y unfoling. It hs sutrees onsisting of single rnh off level 0, 2 rnhes off level 1, 2 3 rnhes off level 2, n generlly (n + 1)! rnhes off level n. Referring to the - lelle eges these rnhes re rrnge in nturl (n MSO-efinle) 17

18 Figure 4: Prepring for the ftoril preite orer. To pture the struture (N, Su, F), we pply n interprettion whih (for n 1) nels the rnhes strting t the -ege trget of level n (n leves only the rnhes off the trgets of -eges). As result, (n + 1)! n! rnhes off level n remin for n 1, while there is one rnh off level 0. Numering these remining rnhes, the n!-th rnh ppers s first rnh off level n. Note tht we trverse this first rnh off given level y isllowing -eges fter the first -ege. So tree shpe similr to Figure 3 emerges, now for the ftoril preite. Summing up, we hve generte the struture (N,Su,F) s grph in G 3. So fr we hve onsiere expnsions of the suessor struture of the nturl numers y unry preites. We now tret the expnsion y n interesting unry funtion (here ientifie with its grph, inry reltion). It is the flip funtion, whih ssoites 0 to 0 n for eh nonzero n tht numer whih rises from the inry expnsion of n y moifying the lest signifint 1-it to 0. We give n illustrtion of the grph Flip of this funtion It is esy to see tht the struture (N,Su,Flip) n e otine from the lgeri tree of Figure 3 y n MSO-interprettion. A Flip-ege will onnet vertex u to the lst lef vertex v whih is rehle y -pth from n nestor of u; if suh pth oes not exist, n ege to the trget of the -ege (representing numer 0) is tken. The grphs in the Cul hierrhy supply vst universe of strutures whih hs not een unerstoo very well on the higher levels (sy from level 3 onwrs). Mny interesting questions rise, for exmple the prolem whether 18

19 one n ompute the lowest level on whih given struture ours. It is lrey known tht the Cul hierrhy oes not fully exhust the omin of grphs with eile MSO-theory. 7 Gri Strutures n Tree Rewriting Grphs The trnsition grphs of the Cul hierrhy re still tightly onnete with infinite trees in ft, they n e generte for given level k from single tree struture vi MSO-interprettions. So these grphs re too restrite for mny purposes of verifition (exepting pplitions on the implementtion of reursion). A more flexile kin of moel is generte when the ie of prefix-rewriting is generlize in ifferent iretion, proeeing from wor rewriting to tree rewriting (whih we ientify here with term rewriting). Inste of moifying the prefix of wor y pplying prefix-rewriting rule, we my rewrite sutree of given tree, preisely s it is one in groun term rewriting. So rule t t llows to reple one ourrene of sutree t y t. To fix stte properties, we refer to the well-known onept of regulr sets of trees, efine y finite tree utomt. A groun term rewriting grph (GTRG) G = (V,(E ),(P )) hs vertex set V n susets P V whih re given y regulr tree lnguges, n eh ege reltion E is efine y finite groun term rewriting system. Usully one restrits V to ontin only trees whih re rehle from some regulr set of initil trees vi the ege reltions E. The onept is est introue y n exmple. Consier the grph generte from the tree f(,) y pplying the rules g() n g() whih proue the trees f(g i (),g j ()) in one-to-one orresponene with the elements (i,j) of N N (see Figure 5). We thus see tht the infinite N N-gri is GTRG. Hene the MSO-theory of GTRG n e uneile. However, for interesting properties the moel-heking prolem is still eile. It is possile to omine the tehniques of Setion 2 (on utomti grphs) n of Setion 3 (sturtion lgorithm), now pplie over the omin of finite trees rther thn wors. Sine the methoology oes not hnge, we only stte the result. (It shoul e note, however, tht in prt () of the theorem the opertor EFG ( there is pth with infinitely mny verties of ertin property ) uses lot of tehnil work whih is fr from strightforwr.) Theorem 22. (Duhet, Tison [11], Löing [14]) Over groun tree rewriting grph, the moel-heking prolem is eile for the logi FO(R) n for the rnhing-time logi with the following syntx: T (regulr) ϕ ϕ 1 ϕ 2 EX ϕ EFϕ EFGϕ. It is possile s for pushown grphs to generlize the rewriting rules without ffeting the eiility results: Inste of llowing replement of single sutree y nother one, one my use rules of the form T T for regulr tree lnguges T,T, mening tht n ourrene of sutree t T n e reple y ny t T. 19

20 f f g f g f g g f g g f g g g f g g f g g g f g g g g Figure 5: The gri s groun tree rwriting grph We now shll note tht slight extension of the logi of prt () ove les to uneiility. This extension n est e expline in terms of rnhing time temporl opertors in CTL-like nottion: While the opertors EF n EGF preserve eiility, this fils for the opertor AF ( on eh pth there is vertex with ertin property ). Theorem 23. (Löing [14]) The following prolem is uneile: Given groun tree rewriting grph G, vertex v n regulr set T of verties of G, oes every pth from v through G reh T? Proof. We give the min ie, whih is typil for uneiility proofs where the essentil logil opertor to e exploite is universl (rther thn existentil, s neee in iret oing of the hlting prolem). For reution of the hlting prolem for Turing mhines we represent Turing mhine onfigurtion 1... k q l... 1 s tree X 1. k X 1. l q 20

21 Now we set up rewriting rules whih simulte Turing mhine omputtions. The min prolem for orret upte of trees is the ft the left n right rnh involve inepenent rewritings for single Turing mhine step. We enfore orret mth y the requirement tht we implement sequene of tree rewriting steps n require tht ll intermeite trees elong to ertin regulr tree lnguge T. As n exmple onsier the Turing mhine instrution q L p. In the figure elow sequene of rewriting steps is presente whih trnsform tree oe of the onfigurtion q into tree oing the next onfigurtion p. In the first two steps the new entry to the right-hn rnh is guesse; the ft tht ws e is signlle y the mrker 1. In the left-hn rnh the symol re 1 inites tht the symol there is to e remove. The next steps introue n inrese of the - n re-inies (for tehnil resons not expline here). A wrong guess (i.e., mismth etween the symol to e remove n the one to e e) n e esrie in terms of regulr tree lnguge Err whih ontins the trees showing suh mismth. The tree lnguge Hlt ontining the oes of hlting onfigurtions is lso regulr. X X q X X p 1 X X re 1 p 1 X X re 1 p 2 X X re 2 p 2 X X re 2 p 3 X X p 3 X X p The rewriting steps s inite ensure tht Turing mhine hlts iff in the groun tree rewriting grph uilt up with the inite rules, eh pth strting from the tree for the initil onfigurtion meets either Err or Hlt. 8 Completing the Piture 8.1 Struturl hrteriztions We hve isusse four si types of infinite trnsition grphs: the rtionl, utomti, prefix-reognizle n the groun tree rewriting grphs. As speiliztion of the prefix-reognizle grphs we onsiere the pushown grphs, n s generliztion the grphs of the Cul hierrhy. For the efinition of these strutures, two pprohes were pursue: the internl presenttion in terms of utomton efinle sets n reltions of wors, respetively trees, the externl presenttion y mens of moel trnsformtions, strting from ertin funmentl strutures (in our se, the strutures T 2,T 2). 21

22 Some results t the en of Setion 3 (on prefix-reognizle n utomti grphs) inite tht the two pprohes n e merge. A mth of internl n externl presenttions hs lso een hieve for the other types of grphs presente here: the rtionl grphs [15], the grphs of the Cul hierrhy (see e.g. [10]), n the groun tree rewriting grphs [7]. The omintion of oth views (internl n externl) is neessry for eveloping the lgorithmi theory of infinite strutures. Usully, the internl esription is helpful in evising effiient lgorithmi solutions, n the externl presenttion gives onvenient wy of generting moels without entering too muh into etils of implementtion. In lssil mthemtis, these two views re stnr n omplement eh other. For exmple, if we speify vetor spe y sis (n the rule tht liner omintions over the sis generte the elements of the spe), we give n internl representtion. If we tke ll liner mps over some vetor spe to onstrut new vetor spe, we re uiling n externl presenttion. The seprtion of these lsses of grphs is not esy in generl. We o not yet hve mny mngele struturl hrteriztions whih n e pplie to grphs without referene to their presenttions. A mster exmple of suh hrteriztion is result of Muller n Shupp, onerning pushown grphs. Let G = (V,(E ) Σ ) e grph of oune egree n with esignte origin vertex v 0. Let V n e the set of verties whose istne to v 0 is t most n (vi pths forme from eges s well s reverse eges). Define G n to e the sugrph of G inue y the vertex set V \ V n, lling its verties in V n+1 \ V n the ounry verties. The ens of G re the onnete omponents (using eges in oth iretions) of the grphs G n with n 0. In [17], Muller n Shupp estlishe eutiful hrteriztion of pushown grphs in terms of the isomorphism types of their ens (where n en isomorphism is ssume to respet the vertex property of eing ounry vertex): Theorem 24. (Muller, Shupp [17]) A trnsition grph G of oune egree is pushown grph iff the numer of istint isomorphism types of its ens is finite. As n pplition, we see tht the infinite N N-gri is not pushown grph. The ens G n exlue ll verties from the origin up to istne n. The verties of istne preisely n form ounter-igonl from vertex (0, n) to vertex (n, 0). This ounter-igonl shows in prtiulr tht no two grphs G m,g n for m n re isomorphi. For the other grphs isusse in these letures, suh nie struturl hrteriztions re still missing. 8.2 Reognize lnguges Any infinite trnsition grph G = (V,(E ) Σ,I,F) with unry preites I,F V (of initil n finl verties) my e use s n eptor of wors in the ovious wy: A wor is epte if it ours s lelling of pth from vertex in I to vertex in F. If V is finite, we otin the usul moel of noneterministi finite utomton (here with severl initil sttes), whih yiels the regulr lnguges s orresponing lss of lnguges. It is not surprising tht the pushown grphs (n: s it is esily verifie, lso the prefix-reognizle grphs) yiel preisely the ontext-free lnguges: 22

23 Theorem 25. (Muller-Shupp [17], Cul [5]) A lnguge L is ontext-free iff L is reognize y pushown grph (with regulr sets of initil n finl sttes) iff L is reognize y prefix-reognizle grph (with regulr sets of initil n finl sttes). This trk of reserh ws ontinue y surprising results regring the rtionl n utomti grphs: Theorem 26. (Morvn-Stirling [16], Rispl [19]) A lnguge L is ontext-sensitive iff L is reognize y n utomti grph (with regulr sets of initil n finl sttes) iff L is reognize y rtionl grph (with regulr sets of initil n finl sttes). The grphs of the Cul hierrhy lso orrespon to known lnguge lsses whih hve een introue in terms of higher-orer pushown utomt. For instne, the lnguges reognize y Cul grphs of level 2 oinie with the inexe lnguges introue in the 1960 s y Aho. It is n open prolem to provie orresponing esription for the lnguges reognize y groun tree rewriting grphs. 9 Retrospetive n Outlook These letures gve n introution to funmentl lsses of infinite trnsition grphs, with some emphsis on the question whih types of moel-heking questions n e solve lgorithmilly. Let us summrize the entrl ies n proofs: the reution of the Post Corresponene Prolem n of the Hlting Prolem for Turing mhines to simple questions out rtionl n utomti grphs, the eiility of the FO-theory of n utomti grph using n inutive onstrution of utomt for efinle reltions, the rehility nlysis for pushown systems y the sturtion lgorithm, the metho of interprettions, use to show tht the MSO-theory of pushown grph is eile, n the omintion of interprettions n unfolings for uiling up the Cul hierrhy, the role of the infinite gri, s struture with n uneile MSO-theory ut s groun tree rewriting grph shring still some eiility properties, the uneiility of properties over groun tree rewriting grphs tht involve universl pth quntifition. The sujet of finitely presente infinite strutures using utomt theoreti ies is fstly eveloping. Mny trks of reserh re open. We mention just few. 23