a b =) b a =) b b =) a b =) b b a b The same pobem aises hen Tuing machines ae simuated b gaph eiting. An eampe is a Tuing machine doing nothing ese t

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1 Fundamenta nfomaticae 33(2), 1998, 201{209 1 OS Pess Temination of gaph eiting is undecidabe Detef Pump Fachbeeich Mathematik und nfomatik Univesitat Bemen Bemen, Geman det@infomatik.uni-bemen.de Abstact. t is shon that it is undecidabe in genea hethe a gaph eiting sstem (in the \doube pushout appoach") is teminating. The poof is b a eduction of the Post Coespondence Pobem. t is aso agued that thee is no staightfoad eduction of the hating pobem fo Tuing machines o of the temination pobem fo sting eiting sstems to the pesent pobem. Keods: gaph eiting, temination, Post Coespondence Pobem 1. ntoduction n 1978, Huet and Lankfod shoed that it is undecidabe in genea hethe a tem eiting sstem is teminating, that is, hethe eve computation of a sstem eventua hats [4]. Moeove, thei poof impies that temination of sting eiting sstems is undecidabe. n the pesent pape, it is shon that temination of gaph eiting sstems in the so-caed doube pushout appoach [2, 1] is undecidabe, too. Huet and Lankfod simuated Tuing machines b tem eiting sstems such that a given machine hats on a inputs if and on if the coesponding tem eiting sstem teminates fo a tems. Thus, the obtained a eduction of the unifom hating pobem fo Tuing machines to the temination pobem fo tem eiting sstems. n the fameok of gaph eiting, hoeve, a staightfoad eduction of the hating pobem o of the temination pobem fo sting eiting is not possibe. n both cases, the pobem is caused b ccic gaphs. Conside, fo instance, the sting eite ue ab! ba hich ma be tansated into the fooing gaph eite ue fo edge-abeed gaphs: a b =) b a Cea, the sting eiting sstem fab! bag is teminating, that is, its ue can be appied to a sting on nite often. n contast, the gaph eiting sstem consisting of the above ue does not teminate as it admits the fooing innite eite sequence of ccic gaphs: Pat of this eseach as pefomed hie the autho as on eave at CW, Amstedam, suppoted b a gant of the HCM netok EXPRESS.

2 a b =) b a =) b b =) a b =) b b a b The same pobem aises hen Tuing machines ae simuated b gaph eiting. An eampe is a Tuing machine doing nothing ese than moving its head one ce to the ight afte eading the smbo a. This machine obvious teminates fo a input stings. But hen tansated into a gaph eiting sstem (anaogous to the tansation of sting eiting sstems sketched above), a ccic \tape" abeed ith a's i issue an innite eiting. The hating pobem fo Tuing machines easi can be educed to the temination pobem fo gaph eiting on e-fomed gaphs epesenting Tuing machine conguations. Likeise, thee is a staightfoad eduction of the temination pobem fo sting eiting to the temination pobem fo gaph eiting on \sting gaphs". Undecidabiit of temination fo these esticted gaph casses, hoeve, does not imp that temination is undecidabe fo genea gaph eiting, that is, fo gaph eiting on abita gaphs. Theefoe, in this pape, the Post Coespondence Pobem (PCP) is educed to the pobem of deciding hethe a given gaph eiting sstem is teminating. The basic idea of the encoding of the PCP is simia to the idea behind eductions of the PCP to the temination pobem fo tem eiting sstems, as given b Lescanne [5], Zantema [7] and Feeia [3]. But the encoding b gaph eiting is moe invoved as the non-inea ues used in those papes do not have countepats in gaph eiting sstems. The est of this pape is oganied as foos: Section 2 contains a bief evie of gaph eiting sstems, in Section 3 the main esut is poved, and Section 4 concudes b stating an undecidabiit esut fooing fom the poof of the main esut. 2. Gaph eiting Beo the doube pushout appoach to gaph eiting is bie evieed. Fo a compehensive suve, the eade ma consut [1] o [2]. A abe aphabet = h V ; E i is a pai of nite sets of vete abes and edge abes. A gaph ove is a sstem G = hv G ; E G ; s G ; t G ; G ; m G i consisting of to nite sets V G and E G of vetices (o nodes) and edges, to souce and taget functions s G ; t G : E G! V G, and to abeing functions G : V G! V and m G : E G! E. Given to gaphs G and H, G is a subgaph of H, denoted b G H, if V G and E G ae subsets of V H and E H, espective, and if s G, t G, G and m G ae estictions of the coesponding functions in H. A gaph mophism f: G! H beteen to gaphs G and H consists of to functions f V : V G! V H and f E : E G! E H peseving souces, tagets and abes, that is, s H f E = f V s G, t H f E = f V t G, H f V = G and m H f E = m G. The mophism f is an isomophism if f V and f E ae bijective. n this case G and H ae isomophic, hich is denoted b G = H. A ue = (L K R) consists of thee gaphs L, K and R such that K is a subgaph of both L and R. The gaphs L and R ae the eft- and ight-hand side of, and K is the inteface. Given a gaph G, a gaph mophism f: L! G satises the guing condition if the fooing hods: Danging condition. No edge in G? f(l) is incident to an node in f(l)? f(k). dentication condition. Fo a distinct items ; 2 L, f() = f() on if ; 2 K. 1 Given to gaphs G and H, and a set of ues R, thee is a diect deivation 2 fom G to H based on R, denoted b G ) R H, if thee is a ue = (L K R) in R and a gaph mophism f: L! G satisfing the guing condition, such that H is isomophic to the gaph M constucted as foos: 1 This condition is undestood to hod sepaate fo nodes and edges. 2 See [1, 2] fo an equivaent denition b a \doube pushout" of gaph mophisms.

3 1. Remove a nodes and edges in f(l)? f(k) to obtain a subgaph D of G. 2. Add disjoint to D a nodes and edges in R? K to obtain M, hee a added items keep thee abes and hee the souce of an edge e in R? K is dened b The taget t M (e) is dened anaogous. fv (s s M (e) = R (e)) if s R (e) 2 V K, s R (e) otheise. Given some n 0, a deivation of ength n fom G to H based on R is a sequence of the fom G = G 0 ) R G 1 ) R : : : ) R G n = H. The eation ) n R is dened as foos: G )n R H if thee eists a deivation of ength n fom G to H based on R. The eation ) R ()+ R ) is dened b: G ) R H (G )+ H) if thee is some R n 0 (n > 0) such that G )n H. R A gaph eiting sstem G = (; R) consists of a abe aphabet and a nite set R of ues ith gaphs ove. The sstem G is teminating if thee does not eist an innite eite sequence of the fom G 0 ) R G 1 ) R : : : 3. Undecidabiit of temination This section is devoted to the poof of the main esut, hich is stated net. Theoem 3.1. t is undecidabe in genea hethe a gaph eiting sstem is teminating. n hat foos, Theoem 3.1 is poved b educing the Post Coespondence Pobem (PCP) to the pobem of deciding hethe a given gaph eiting sstem is teminating. Eve instance of the PCP i be encoded as a gaph eiting sstem that is non-teminating if and on if the given instance has a soution. Reca that the PCP is the pobem to decide, given a nonempt ist L = h( 1 ; 1 ); : : : ; ( n ; n )i of pais of ods ove some nite aphabet?, hethe thee eists a sequence ; : : : ; i k of indices such that i1 : : : i k = : : : i k. The ist L is an instance of the PCP, and a sequence ; : : : ; i k ith the above popet is a soution of this instance. t is e-knon that it is undecidabe in genea hethe an instance of the PCP has a soution [6]. n the fooing encoding of the PCP, a sting a 1 : : : a m (ith m 0) i be encoded as a gaph consisting of m consecutive edges abeed b a 1 ; : : : ; a m : a 1 a 2 a m Such a gaph i be depicted aso as foos: a 1 :::am } { Let no L = h( 1 ; 1 ); : : : ; ( n ; n )i be an abita instance of the PCP. t is assumed that fo each pai ( i ; i ), not both i and i ae empt ods. (The PCP ith this estiction cea emains undecidabe.) Constuct the gaph eiting sstem G(L) = (; R) as foos: V = f; ; g, E =? + f1; : : : ; ng + f; ; g 3 and R = R 1 [ R 2 [ R 3 [ R 4, hee R 1 ; : : : ; R 4 ae dened beo. 3 Hee + denotes the disjoint union of sets.

4 =) =) R 1 contains the ues 4 a 1 a p b 1 b q i fo i = 1; : : : ; n, hee i = a 1 : : : a p and i = b 1 : : : b q. R 2 contains the ues i i fo i = 1; : : : ; n. R 3 contains the ues i a 1 a p a 1 a p fo i = 1; : : : ; n, hee i = a 1 : : : a p. R 4 contains the fooing ue: No the task is to sho that the instance L has a soution if and on if G(L) is not teminating. The \on if"-diection, hich is the easie pat, is given b the net emma. Thee futhe emmas i be needed to sho the convese. Lemma 3.1. f L has a soution, then G(L) is not teminating. Poof: f ; : : : ; i k is a soution of L, then G(L) admits the fooing ccic deivation: :::ik } { { } :::ik k =)R 1 i k R 4 R 2 :::ik } { { } :::ik k (= R 3 i k 4 n the fooing pictues,, and ae node names hich ae used to depict subgaph incusions. ut

5 n the fooing, a node abeed ith o is caed a conto node. As each ue in R has a unique conto node in its eft- and ight-hand side, a diect deivation G ) R H i be consideed as an appication of a ue \to a conto node", and the images of the eft- and ight-hand conto node in G and H i be consideed as \the same" node (athough the abes ma be dieent). Lemma 3.2. Eve innite eite sequence ove G(L) contains a conto node to hich the ue in R 4 is appied innite often. Poof: t suces to sho that the sstem G = (; R? R 4 ) is teminating. Fo then eve innite eite sequence ove G(L) contains innite man appications of the ue in R 4. As no ue in R inceases the numbe of conto nodes, it foos that thee is a conto node to hich the ue in R 4 is appied innite often. To sho that G is teminating, suppose the conta. Ca an edge a?-edge if it is abeed ith a smbo fom?. Caim: n eve eite sequence ove G, no?-edge poduced b R 3 is eve emoved. Poof: Given a step G ) R3 H, thee is no diected path in H fom a node abeed ith to the souce node of an?-edge poduced b the step. Hence none of these edges can be emoved b apping a ue to H. Moeove, it is eas to see that a ues in R 1 [ R 2 [ R 3 peseve the absence of a diected path fom a node abeed ith to a ed?-edge. Thus none of the?-edges poduced b G ) R3 H can be emoved in a futue step. B the caim and the fact that the ues in R 1 [R 2 do not poduce?-edges, a?-edges emoved b an R 1 -step in a eite sequence ove G have been pesent aead in the stat gaph of the sequence. t foos that eve innite eite sequence ove G contains on a nite numbe of R 1 -steps: each of these steps emoves some?-edges hie neithe R 1 no R 2 poduces?-edges. Hence, if G is not teminating, the sstem (; R 2 [ R 3 ) must admit an innite eite sequence. But this is impossibe since a ues in R 2 [ R 3 decease the numbe of nodes and edges ith abe in fg [ f1; : : : ; ng. Thus G is teminating. This concudes the poof of Lemma 3.2. ut Lemma 3.3. f an innite eite sequence ove G(L) stats fom a connected gaph, then a gaphs in the sequence contain eact one conto node. Poof: Fo eve gaph G, ca a subgaph C an inde chain of ength k, k 0, if C has the fom i k hee ; : : : ; i k 2 f1; : : : ; ng and hee on the conto node ma be incident to edges not beonging to C. The fooing caim foos immediate fom the shape of the ues in R. Caim: Let G ) R H be a diect deivation and C be an inde chain of ength k in G. Then (1) C is (up to isomophism) aso an inde chain in H, o (2) C is tansfomed into an inde chain of ength k + 1, o (3) G ) R H is an appication of a ue fom R 2 to the conto node in C. Let no G 0 ) R G 1 ) R : : : be an innite deivation such that G 0 is connected. Since a ues in R peseve the numbe of conto nodes, it suces to sho that thee is some gaph in the eite sequence that contains eact one conto node. B Lemma 3.2, thee is a conto node c to hich the ue in R 4 is appied innite often. Hence thee ae 0 s < t such that G s ) R4 G s+1 ) + R G t ) R4 G t+1, hee G s ) R4 G s+1 and G t ) R4 G t+1 ae appications of R 4 to c and hee G s+1 ) + R G t does not contain an appication of R 4 to c. B the shape of the ue in R 4, c beongs to an inde chain of ength 0 in G s+1. Moeove, thee must be some s 0 ith s < s 0 < t such that G s 0 ) R2 G s 0 +1 is an appication of R 2 to c. Hence, b the above obsevation, c beongs to an inde chain in G s 0. B the shape of the ues in R 2, if c is connected ith anothe conto node d in G s 0, then a nodes of the inde chain containing c ae connected ith d via a

6 path that does not contain c. But this contadicts the denition of an inde chain. That is, c cannot be connected ith anothe conto node. Since G 0 is connected and a ues in R peseve connectedness, it foos that c is the on conto node in G s 0. ut Lemma 3.4. f G(L) is not teminating, then thee is an innite eite sequence that stats ith a deivation of the fom } { { } =)R 4 } { { } m j m =)R 1 =)R j m 2 m =)R 3 } { { } hee m 1 and ; : : : ; j m 2 f1; : : : ; ng. Poof: Since the eft- and ight-hand sides of a ues in R ae connected, eve diect deivation G ) R H takes pace ithin some connected component C of G and tansfoms C into a connected component of H. Theefoe eve innite eite sequence ove G(L) contains a connected component hich is subject to innite man ue appications. B teing out a ue appications to this component and esticting a gaphs to this component, one obtains an innite eite sequence of connected gaphs. B Lemma 3.2, fom some point on this sequence has the fom G 1 ) R4 H 1 ) G R 2 ) R4 H 2 ) : : : ith R = R? R R 4. B Lemma 3.3, a gaphs in this sequence contain eact one conto node. Thus, taking into account the shape of the ues in R, the deivation G 1 ) R4 H 1 ) G R 2 must be of the fom G 1 ) R4 H 1 ) m R 1 P 1 ) R2 Q 1 ) m R 3 G 2, hee m 1 and hee G 1, H 1, P 1, Q 1 and G 2 ae as in the above pictue. (The conto node to hich R 4 is appied in G 1 can be subject to an R 4 -appication in G 2 on if H 1 ) G R 2 is as depicted. Note that, b the danging condition fo diect deivations, nodes being emoved ae incident on to edges having a peimage in the eft-hand side of the appied ue.) ut Using Lemma 3.1 and 3.4, it is staightfoad to pove the main esut.

7 Poof of Theoem 3.1: B the undecidabiit of the Post Coespondence Pobem, it suces to sho that the instance L has a soution if and on if the gaph eiting sstem G(L) is not teminating. Lemma 3.1 shos the \on if"-diection of this equivaence. Fo the convese, suppose that G(L) is not teminating. Then, b Lemma 3.4, thee is an innite eite sequence containing a gaph of the fom } { { } hee is a some od ove?. B the shape of the ues in R, the net steps in the sequence have the fom } { { } =) R 4 } { { } k =) R 1 i k hee k 1 and ; : : : ; i k 2 f1; : : : ; ng. B the shape of the ues in R 1, this impies i1 : : : i k = = i1 : : : i k. Thus, ; : : : ; i k is a soution of L Concusion B a eduction of the Post Coespondence Pobem, it has been shon that temination of gaph eiting is undecidabe in genea. Somehat supising, the possibe pesence of cces in gaphs pevents a staightfoad eduction of the hating pobem fo Tuing machines o of the temination pobem fo sting eiting sstems to the pesent pobem. t is oth noting that the given eduction of the PCP aso shos the undecidabiit of the fooing pobem. Ca a gaph eiting sstem ccic if it admits a deivation in hich some gaph occus tice. (So eve ccic sstem is non-teminating, but the convese does not hod in genea.) Fom the poof of Theoem 3.1 one obtains the fooing esut. Theoem 4.1. t is undecidabe in genea hethe a gaph eiting sstem is ccic. Fo, the poof of Lemma 3.1 shos that the sstem G(L) is ccic heneve an instance L of the PCP has a soution, and the pemise of Lemma 3.4 hich equies that G(L) is not teminating cea hods tue if G(L) is ccic. Acknoedgement. The autho ishes to thank Jugen Mue fo discussions about temination of gaph eiting, and Anneget Habe fo comments on a pevious vesion of this pape.

8 Refeences [1] Andea Coadini, Ugo Montanai, Fancesca Rossi, Hatmut Ehig, Reiko Hecke, and Michae Loe. Agebaic appoaches to gaph tansfomation Pat : Basic concepts and doube pushout appoach. n Gego Roenbeg, edito, Handbook of Gaph Gammas and Computing b Gaph Tansfomation, voume 1, chapte 3, pages 163{245. Wod Scientic, [2] Hatmut Ehig. ntoduction to the agebaic theo of gaph gammas. n Poc. Gaph- Gammas and Thei Appication to Compute Science and Bioog, voume 73 of Lectue Notes in Compute Science, pages 1{69. Spinge-Veag, [3] Maia C.F. Feeia. Temination of tem eiting. Dissetation, Univesiteit Utecht, Facuteit Wiskunde en nfomatica, [4] Gead Huet and Daas Lankfod. On the unifom hating pobem fo tem eiting sstems. Repot no. 283, NRA Rocquencout, [5] Piee Lescanne. On temination of one ue eite sstems. Theoetica Compute Science, 132:395{401, [6] Gego Roenbeg and Ato Saomaa. Conestones of Undecidabiit. Pentice Ha, [7] Hans Zantema. Tota temination of tem eiting is undecidabe. Jouna of Smboic Computation, 20:43{60, 1995.