Graphs Encoded by Regular Expressions

Transcription

1 Grah Encoded by Regular Exreion Stefan Gulan Univerität Trier, FB IV Informatik Abtract In the converion of finite automata to regular exreion, an exonential blowu in ize can generally not be avoided. Thi i due to grah-tructural roertie of automata which cannot be directly encoded by regular exreion and caue the blowu combinatorially. In order to identify thee tructure, we generalize the cla of arc-erie-arallel digrah to the acyclic cae. The reulting digrah are hown to be reveribly encoded by linear-ized regular exreion. We further derive a characterization of our new cla by a finite et of forbidden minor and argue that thee minor contitute the rimitive cauing the blowu in the converion from automata to exreion ACM Subject Claification F.4.3: Formal Language, G.2.2: Grah Theory Keyword and hrae Digrah, Regular Exreion, Finite Automata, Forbidden Minor Digital Object Identifier /LIPIc.STACS Motivation A fundamental reult in the theory of regular language i the equivalent decritive ower of regular exreion and finite automata, a originally hown by Kleene [9]. While regular exreion come natural to human a a mean of denoting uch language, automata are the object of choice on the machine level. Conequently, converting between thee two rereentation i of great ractical imortance. There are everal linear-time algorithm to tranform regular exreion into automata with ize linear in that of the inut, a detailed overview i given by Waton [16]. We hall focu on the convere contruction which i coniderably more troubling. In articular, Ehrenfeucht & Zeiger [3] give a cla of automata for which the ize of any equivalent exreion i exonential in that of a given automaton. Thee automata are defined over an alhabet which grow with automaton ize, which led Ellul et al. to ak whether a imilar blowu in exreion ize can be hown for automata over a fixed alhabet [4]. An affirmative anwer wa given by Gruber & Holzer [5] for binary alhabet already. Thi motly rule out alhabet ize a a factor contributing to the exonential blowu, the modifier motly giving credit to the fact that automata over unary alhabet can be converted to exreion of quadratic ize via Chrobak normal form [1, 4]. Oberve that a finite automaton i merely a digrah with edge label, acceting the language which conit of all equence of label met on a directed walk from an initial to a final tate. Informally, the increae of exreion- over automaton-ize reult from automata being combinatorial object, wherea exreion are term, i.e., linear entitie, that mut reort to reeated ubterm in order to convey information encoded in the grah-tructure of an automaton. Thi wa oberved quite early by McNaughton [11], who remark that although every regular exreion can be tranformed into a grah that ha the ame tructure, the convere i not true. The reent work aim to identify the grah that cannot be tranformed into exreion that have the ame tructure. Stefan Gulan; licened under Creative Common Licene NC-ND 28th Symoium on Theoretical Aect of Comuter Science (STACS 11). Editor: Thoma Schwentick, Chritoh Dürr; Leibniz International Proceeding in Informatic Schlo Dagtuhl Leibniz-Zentrum für Informatik, Dagtuhl Publihing, Germany

2 496 Grah Encoded by Regular Exreion For automata whoe underlying grah are arc-erie-arallel, Moreira & Rei [12] recently gave an efficient converion to exreion with ize linear in that of the inut. Thee grah are characterized by abence of a ingle minor-like ubtructure in acyclic grah, a wa hown by Valde et al. [15]. Korenblit & Levit [10] conjectured that thi ubtructure already caue a quadratic blowu in the ize of exreion contructed from acyclic automata. However, Moreira & Rei method i inherently confined to automata that accet finite language only. In order to accet an infinite language, an automaton need to contain cycle, which evade the cla of arc-erie-arallel digrah. While earately dealing with erie-arallel art of arbitrary automata ha been uggeted for converion-heuritic [6], no trict grah-theoretic analyi ha been conducted for the general cae a yet. Thi motivate our generalization of arc-erie-arallel digrah to the non-acyclic cae in Sec. 3, yielding a cla which i till efficiently recognizable. In Sec. 4 we how that uch grah can be reveribly encoded by regular exreion and that every regular exreion encode a grah of thi cla. Encoding and decoding i done in an automata-theoretic framework and can be immediately alied to the converion between automata and exreion. In Sec. 5 we derive a characterization of our new cla by a finite et of forbidden minor. Thi imlie that thee minor rereent the grah-tructural roertie of automata that cannot be encoded by regular exreion and thu caue the blowu oberved in the contruction of regular exreion from finite automata. 2 Preliminarie We conider finite directed grah with loo and multile arc. Thee are canonically known a directed eudograh but will be referred to a jut grah. Formally, a grah i a tule (V, A, t, h) with vertice V, arc A, tail-ma t : A V and head-ma h : A V. If G i not given exlicitly, let G = (V G, A G, t G, h G ). An xy-arc of G i any a A G with t G (a) = x and h G (a) = y; we write thi a a = xy A G. An xy-arc a leave x and enter y, and x and y are called the endoint of a. Ditinct xy-arc of a grah are arallel to each other. An xx-arc i an x-loo or jut loo, every other arc i a roer arc. The in-degree of x V G, denoted d G (x), i the number of arc entering x in G, the out-degree d+ G (x) i the number of arc leaving x. A contriction of G i any roer xy-arc where d + G (x) = 1 = d G (y). A vertex x V G i imle if d G (x) 1 and d+ G (x) 1. Subcrit are omitted if they are undertood. We write F G i F i a ubgrah of G. If F and G are ubgrah of H and a = xy A H with x V F and y V G, then a i called an (F, G)-arc, a well a an (x, G)- or an (F, y)-arc of H. A ath of length n, denoted P n i a grah on n + 1 vertice and n contriction. The ubdiviion of an arc a = xy i the relacement of a with an xy-ath of length 2. More generally, a ubdiviion of G, referred to a a DG, i any grah H.t. there are grah G 1,..., G n where G = G 1, G i+1 reult from ubdividing an arc of G i and G n = H. The lit of a vertex x i the relacement of x with two vertice x 1 and x 2 and an x 1 x 2 -arc and redirecting all arc that entered x to enter x 1, re. redirecting all arc that left x to leave x 2. Two vertice x, y V G are merged by being relaced with a new vertex z and redirecting all arc entering or leaving x or y to enter or leave z. A grah G i two-terminal if there are, t V G.t. every x V G lie on ome t-ath in G. The vertice and t are reectively called the ource and ink of G; we write G = (G,, t) to exre that G i two-terminal with ource and ink t. A two-terminal grah (G,, t) i a hammock if d G () = d+ G (t) = 0. Let x and y be vertice of (G,, t): x dominate y if x lie on every y-ath, and x co-dominate y if x lie on every yt-ath. Furthermore, x i a guard of y if x dominate and co-dominate y; alo, x i a guard of the arc a if x guard

3 S. Gulan 497 x y x z y (a) erie exanion x y x y (b) arallel exanion x y l {x,y} (c) loo exanion Figure 1 Exanion of an xy-arc, re. the containing grah. l Figure 2 Contruction of an l-grah from P 1 by a equence of exanion. both t(a) and h(a). More generally, x guard a ubgrah F of (G,, t) if x guard every arc and vertex of F. 3 SPL - Grah Definition 1. The relation, and l are defined on grah a follow: Let G be a grah and a an xy-arc in G, then G H if H i obtained by ubdividing a in G G H if H i obtained from G by adding an arc which i arallel to a. G l H if a i a contriction and H i obtained by merging x and y in G. We ay that H i derived from G by mean of erie-, arallel- or loo-exanion if G H, G H or G l H, reectively. The local change in G uon exanion are ketched in Fig. 1. We write G H if the articular exanion i irrelevant, and G H if H i derived from G by a (oibly emty) finite equence of exanion. Definition 2. The cla of l-grah, denoted SPL, i generated by from P 1 a follow P 1 SPL Let G SPL, then H SPL if G H or G H, or if G l H where the l-exanded arc i not incident to the ource or the ink of G. We call P 1 the axiom of SPL. The retriction imoed on l-exanion enure that every l-grah i a hammock. An examle for the te-wie contruction of an l-grah i hown in Fig. 2. The acyclic l-grah coincide with the arc-erie-arallel grah invetigated by Valde et al. [15]; we reort to their reult whenever oible and elaborate only on roertie of SPL that arie from it non-acyclic member. To decide whether (G,, t) i an l-grah, we define a kind of dual to exanion. Some care mut be taken with the removal of loo, which i why the new oeration are retricted to hammock. Definition 3. The relation, G = (G,, t) be a hammock, then and l are defined on hammock a follow: Let i) G H if y i imle vertex of G, incident to a 1 = xy and a 2 = yz, and H i derived from G by removing y, a 1 and a 2 and adding an xz-arc. ii) G H if H i derived from G by removing one of two arallel arc. S TAC S 1 1

4 498 Grah Encoded by Regular Exreion iii) G l H if a i an x-loo in G.t. x doe not guard any arc beide a, and H i the lit of x in G \ {a}. If G c H for c {,, l} we call both H and the relacement oeration a c-reduction of G. A before we may imly write G H for any reduction, and G H if H can be derived from G by a equence of reduction. Exanion and reduction are not roer dual a the latter relation i retricted to hammock by definition. For hammock we find G c H iff H c G for c {, }; but while G l H imlie H l G, the convere i not true. The aymmetry i due to the fact that if l-exanion introduce an x-loo a, x might guard ome arc beide a, o the convere reduction i not enured. Thi, however, doe not haen within SPL. Prooition 4. Let C be a cycle of G SPL. Then exactly one vertex of C guard C. The intuition of Pro. 4 i that every cycle in an l-grah contain a vertex that erve a the unique entry and exit of thi cycle wrt. the ource and ink (ee the lat two te in Fig. 2). Alo note that a cycle might well be guarded by any number of vertice outide the cycle. Theorem 5. G SPL iff G P 1 Proof. Since G P 1 imlie P 1 G, reducibility i ufficient for memberhi. Neceity i hown by induction on the tructure of G. The claim hold for P 1, o uoe G SPL where G P 1 and let G H. We attend to l-exanion only, the other cae are trivial. Let a = uv be the relevant contriction of G and let l = xx be the loo introduced in H. If x guard ome ditinct arc a = yz in H, then G contain a cycle that defie Pro. 4, contrary to the aumtion G SPL. Therefore, H l G i a valid reduction; ince by aumtion G P 1, we find H P 1. While memberhi in SPL can be decided by reducing a hammock to the axiom of SPL, we do not know how to do o. Actually, there i no need for a trategy, ince the reduction-ytem exhibit unique normal-form. Uing a tandard argument from abtract rewriting (ee e.g. [14]), we firt how that reduction are locally confluent. Lemma 6. Let G be a hammock and uoe G H 1 and G H 2 hold. Then there i a hammock J.t. H 1 J and H 2 J hold. Each reduction decreae the number of arc or loo and none introduce loo, o every equence of reduction eventually terminate. Any grah derived from G by exhautive reduction i called normal-form of G and denoted R(G). A grah G that coincide with it normal-form, G = R(G), i called reduced. Alying Newman lemma [13, 14] yield Corollary 7. The normal-form R(G) of any hammock G i unique. Comuting the normal-form of a hammock can be realized by reeatedly running the reduction algorithm for arc-erie-arallel grah [15], interered with l-reduction, until no further reduction can be alied. To thi end ome bookmarking about the loo occurring in the intermediate grah i neceary. Teting whether x i a guard can be done in linear time by counting the comonent of G \ x. Overall, thi method comute R(G) in quadratic time. Theorem 8. Memberhi of G in SPL i effectively decidable due to G SPL iff G i a hammock and R(G) = P 1

5 S. Gulan Encoding by Regular Exreion Syntax and emantic of regular exreion (RE) follow Hocroft & Ullman textbook [8] excet that we do not allow for in RE. A for notation, L r denote the language decribed by the RE r and reg(σ) denote the cla of RE over Σ. An RE i imlified if it doe not contain ε a a factor. Any RE r can be converted to a imlified RE im(r), denoting the ame language, by relacing every ubexreion ε or ε with jut. An extended finite automaton (EFA) over Σ i a 5-tule E = (Q, Σ, δ, I, F ), whoe element denote the et of tate, the alhabet, the tranition relation, the initial and the final tate, reectively. Thee et are all finite and atify Q Σ =, δ Q reg(σ) Q, I Q, and F Q. The relation E i defined on Q Σ a (q, ww ) E (q, w ) if (q, r, q ) δ and w L r. The language acceted by E i L(E) := {w (q i, w) E (q f, ε) for q i I, q f F } Two EFA are equivalent if they accet the ame language. An EFA i normalized if I = F = 1 and the initial and final tate are ditinct; any EFA can normalized by adding a new initial (final) tate and ε-tranition from (to) thi new initial (final) tate to (from) the original one. The EFA E i trim if for every tate q of E there i a word w = w 1 w 2 L(E).t. (q i, w 1 ) E (q, ε) and (q, w 2) E (q f, ε) hold for ome q i I and q F F. Any EFA can be converted to a trim equivalent EFA by removing all tate that do not meet thi requirement and adjuting the tranition relation. A nondeterminitic finite automaton with ε-tranition (εnfa) i an EFA whoe tranition relation i retricted to δ Q (Σ {ε}) Q. The grah underlying E i G(E) := (Q, δ, t, h) where t : (, r, q) and h : (, r, q) q. It i eay to ee that E i trim and normalized iff G(E) i a hammock. An EFA dilay a comromie between the comlexity of it tranition-label and that of it underlying grah; RE and εnfa rereent the extreme in thi tradeoff: an RE can be conidered a an EFA whoe underlying grah i trivial, namely P 1, while an εnfa i an EFA with trivial label. Locally relaying information about a language between the grah-tructure of an EFA and it label lie at the heart of everal converion between RE and εnfa. 4.1 Exreion to Automata We conider a fragment of the relacement-ytem rooed by Gulan & Fernau [7]. Let E be an EFA with tranition τ = (, r, q) where r contain oerator, then τ can be relaced deending on the root of r, the out-degree of and the in-degree of q. The degree are only relevant if r i an iteration: in thi cae they determine whether and q hould be merged or a new tate hould be added (or neither) uon introduction of a loo. The rewriting rule, denoted, +, and 1 to 4 are hown in Fig. 3. In order to convert an RE into an εnfa, we identify r reg(σ) with the trivial EFA A 0 r := ({q i, q f }, Σ, {(q i, r, q f )}, {q i }, {q f }), which obviouly atifie L(A 0 r) = L r. The language acceted by an EFA i invariant under each rewriting, hence exhautive alication of, + and i yield a equence A 0 r, A 1 r,... of equivalent EFA terminating in an εnfa which we denote A r. Lemma 9. Every A i r atifie G(A i r) SPL. Thu the grah underlying A r i an l-grah, too. It i hown in [7] that A r i unique. We thu define a ma α from reg(σ) to the cla of exreion-labeled l-grah (aka trim normalized EFA) by etting α(r) := A r. S TAC S 1 1

6 500 Grah Encoded by Regular Exreion t q t q + t q + q t q 1 {,q} (a) roduct (b) um (c) tar, d + ()=d (q)=1 q 2 ε q q 3 ε q q 4 ε ε q (d) tar, d + ()>1, d (q)=1 (e) tar, d + ()=1, d (q)>1 (f) tar, d + ()>1, d (q)>1 Figure 3 Relacing a tranition (, r, q) baed on it label and, in cae r =, the out-degree of and the in-degree of q in G(E). Either rule, 4 introduce a new tate between and q. 4.2 Automata to Exreion The l-reduction are augmented to handle exreion-labeled arc, which yield a econd rewriting-ytem on EFA. In order to meet the requirement for loo-reduction, we conider normalized EFA only. The labeled reduction, denoted, +, and, are hown in Fig. 4. Again, the acceted language i invariant under thee tranformation. t (a) labeled -reduction t + t + t (b) labeled -reduction (c) labeled l-reduction Figure 4 Labeled l-reduction Exhautive reduction of a normalized EFA E terminate in an equivalent EFA which we denote R l (E). The grah underlying R l (E) i the normal-form of the grah underlying E, G(R l (E)) = R(G(E)), and we further find Prooition 10. The label of R l (E) are unique u to aociativity and commutativity. In articular if G(E) SPL, we find G(R l (E)) = P 1, o the only label of R l (E) i an RE r with L r = L(E). By Pro. 10 thi RE i unique u to trivialitie, o we define a ma β from EFA with l-tructure to RE by etting β(e) := r, where r i the label of R l (E). 4.3 Duality of the Converion The converion between RE and εnfa with l-tructure are almot dual, ome extra effort arie with the treatment of ε-factor re. certain ε-labeled tranition. Thi i due to the fact that tar-exanion might introduce ε-tranition that have no correonding ubterm in the We write r = r if the exreion r and r are identical u to aociativity and commutativity of the regular oerator. Theorem im(r) = β(α(im(r))) for any RE r 2. A = α(im(β(a))) for any εnfa A with G(A) SPL Thu the encoding of labeled l-grah by imlified exreion i unique and reverible, and every imlified exreion encode a labeled l-grah. Hence every RE over a nonemty alhabet encode an l-grah and every l-grah can be encoded. Informally, we tate

7 S. Gulan 501 Corollary 12. G SPL iff G can be encoded by an RE 5 Forbidden Minor Characterization We adat the notion of toological minor, which i well-known for undirected grah (ee e.g. [2]), to our need. Definition 13. An embedding of F in G i an injection e : V F V G atifying that if a = xy A F, then G contain an e(x)e(y)-ath P a, and that P a and P a are internally dijoint for ditinct a, a A F. If an embedding of F in G exit, we call F a minor of G realized by the embedding. We write F G if F i a minor of G. If F G doe not hold then G i F -free; if M i a et of grah and G i F -free for every F M, then G i M-free. It i eaily een that ubdiviion allow for an equivalent characterization of minor: Prooition 14. F G iff G contain a DF Let F G be realized by e and x V F, we call e(x) a eg of F in G wrt. e; if G and e are known, we omit mentioning them. Oberve that the in-/out-degree of a vertex in F doe not exceed the in-/out-degree of it correonding eg in G: Prooition 15. If e realize F G, then d F (x) d G (e(x)) and d+ F (x) d+ G (e(x)). Let e realize F G, a bya of F in G wrt. e i an e(x)e(y)-ath in G, where xy i not an arc of F. An embedding of F in G i bare if G contain no bya of F wrt. to the embedding; we then write M G. Oberve that F G might well be realized by variou in articular bare and non-bare embedding. Baed on Pro. 14, we alo call a DF in G bare if G contain no bya wrt. to the embedding realizing thi DF. The exitence of an xy-ath i invariant under l-exanion and -reduction if x and y are not ubject to the oeration. Prooition 16. Let G H or G H and {x, y} V G V H, then G contain an xy-ath iff H doe. Half of the ought characterization i given by the et of grah F = {C, C R, N, Q}, hown in Fig. 5. Note that Valde et al. roved that an acyclic hammock i arc-erie-arallel iff it i N-free [15]. Lemma 17. Every G SPL i F-free. Proof. Clearly, P 1 i F-free. Aume G SPL i F-free and let G H. Conider any F F: ince F i free of arallel arc, and the exitence of ath among vertice in V G V H i invariant under exanion (Pro. 16), F H imlie that a eg of F in H wa introduced uon exanion. Hence in cae G H, F i not a minor of H, i.e., H i F-free. The ame goe for G H: a the new vertex in H i imle, but no vertex of F i, Pro. 15 imlie that H i F -free and therefore F-free. If G l H, let a = xy be the relevant contriction of G and l = zz the loo of H introduced by exanion. If F H i realized by e, then z = e(q) for ome q V F, a wa dicued above. Let H = H \ l: ince F i free of loo, F H hold, too, and ince q i not imle in F, z i not imle in H. We actually find d H (z) 2 and d+ H (z) 2: if d H (z) = 0, then F G i realized by e, which i defined a e excet that e (q) = y contradicting the aumtion that G i F-free. If d H (z) = 1, there i exactly one arc entering z in H. Let S TAC S 1 1

8 502 Grah Encoded by Regular Exreion (a) C (b) C R (c) N (d) Q Figure 5 Grah contituting F. x t l x 2 x 1 t Figure 6 The grah N emerge a a ubgrah due to l-reduction of a hammock. Still, C i a minor of either ide. thi be a = z z, then F G i realized by e which i a e excet that, again, e (q) = y, contradicting our aumtion. A ymmetric argument how d + H (z) 2. In fact, we have alo hown that q, of which z i the eg, ha in- and out-degree at leat two. But ince d G (x) = d H (z) and d+ G (y) = d+ H (z), ome F F, contructed by litting q in F atifie F G contradicting the aumtion that G i F-free. Likewie, it can be hown in general that if H G and H i not F-free, then neither i G. However, there i a catch: the F-minor of G and H need not coincide. Thi i hinted at by the following lemma, and an exlicit examle i hown in Fig. 6. Lemma 18. If H G for hammock H and G, then H i F-free iff G i. More ecifically: i) F H iff F G for F {C, C R, Q} ii) N H only if N G, wherea iii) N G only if (N G or C G or C R G) Still, F-freene of a hammock i not ufficient for memberhi in SPL: for examle, the hammock Φ, hown in Fig. 7a, i F-free, but not included in SPL. The additional grah neceary for a characterization by forbidden minor are Φ, Ψ, and Ψ R, hown in Fig. 7. Lemma 19. Every G SPL i free of bare Φ-, Ψ-, and Ψ R -minor On the other hand, each of {Φ, Ψ, Ψ R } may well be a minor of certain l-grah. Examle for l-grah with Φ and Ψ a minor are given in Fig. 8, the reader might want to check that thee can be reduced to P 1. In the abence of F-minor an invariance-reult akin to Lem. 18 hold for bare ubdiviion of thee three grah. (a) Φ (b) Ψ (c) Ψ R Figure 7 Grah that do not allow for a bare embedding in any G SPL.

9 S. Gulan 503 t (a) (b) t Figure 8 Examle for l-grah with Φ and Ψ a (non-bare) minor Lemma 20. Let H be an F-free hammock and aume H G, then F H iff F G for F {Φ, Ψ, Ψ R }. Proof. Each of Φ, Ψ, and Ψ R i free of arallel arc, o Pro. 16 yield the claim if all eg occur in V G V H ; in articular, nothing need to be done for H G. We rove the claim for Φ, the rocedure i the ame for Ψ and Ψ R. In the following, let H be F-free. Let Φ H be realized by e. If G H remove a eg x = e(q), q i one of the two imle vertice of Φ; here, let q be the unique vertex with d Φ (q) = 0 (the other cae i ymmetric). Since -reduction i alicable due to x, an arc a = yx exit in H, with y alo occurring in G. Let e be an embedding of Φ in G,.t. e (q) = y and e a e for the other vertice. If e i bare, the claim follow for Φ and -reduction, o aume it i not. Then G contain a bya of Φ wrt. e, which i necearily a ath leaving y, otherwie H would contain a bya of Φ wrt. e, contradicting the aumtion that e i bare. We find C G, if the other endoint of the bya i the eg of the vertex in Φ cycle that i not adjacent to q. If the bya i from e (q) to the eg of the vertex with out-degree 0 in Φ, we get Q G. In both cae Lem. 18 imlie that H i not F-free, contradicting our aumtion. Proving that -reduction doe not introduce new bare DΦ i trivial. Again let Φ H be realized by e with eg x V H. Conidering H l G, let a = xx be the loo that allow for reduction, and let x 1 x 2 denote that contriction ariing from it. A in the roof of Lem. 18 our argument i baed on the fact that a i irrelevant for the DΦ in H and that d G (x 1) = d H\a (x) and d+ G (x 2) = d H\a (x) hold. Since every of Φ ha either in- or out-degree 1, we can contruct an embedding e of Φ in G by aigning the role of x to either x 1 or x 2. Definition 21. A kebab i a connected grah coniting of three arc-dijoint ubgrah: a trong comonent B, called the body, and two nonemty vertex-dijoint ath S 1 and S 2, called the ike of the kebab. We name ome unique vertice in a kebab: the endoint of a ike connecting that ike to the body i the uncture of thi ike, the other endoint i it ti. A ike which enter the body of a kebab i an in-ike, one that leave the body i called an out-ike. If both ike of a kebab K enter (leave) the body, K i alo called an in-kebab (out-kebab) if one enter and the other leave the body, K i called an inout-kebab. In order to rove two lemma concerning kebab, ome auxilliary rooition are neceary. Prooition 22. Let G be a reduced hammock with ditinct arc a 1, a 2.t. h(a 1 ) = v = t(a 2 ). Then v i incident to a third roer arc. Prooition 23. Let x and y be ditinct vertice of a hammock G. Then exactly one of the following i true: 1) x dominate y, 2) y dominate x, or 3) for ome z V G \ {x, y}, G contain internally dijoint zx- and zy-ath. S TAC S 1 1

10 504 Grah Encoded by Regular Exreion Prooition 24. Let x and y be ditinct vertice of a trong grah G, then there i a cycle C G and ditinct z x, z y V C,.t. G contain an xz x - and a yz y -ath that are dijoint. Lemma 25. Let (G,, t) be an l-reduced hammock and uoe G contain a kebab. Then F G for ome F F or Φ G. Proof. We chooe a bigget kebab K G with the following roertie 1. the body of K i arc-maximal, i.e., no kebab of G ha a body with more arc 2. the ike of K are incluion-maximal in G, i.e., they are not ub-ike of a bigger kebab with the ame body a K but longer ike than K. We need to ditinguih whether K i in an in-, an out- or an inout-kebab. Due to ace retriction we only treat the firt cae, however note that the firt and econd cae are ymmetric. Let K G be an in-kebab and let B denote the body of K, S 1 and S 2 the ike, with ti t 1 and t 2, and uncture 1 and 2, reectively (Fig. 9a). A (G,, t) i a hammock, according to Pro. 23 either one of t 1 and t 2 dominate the other, or G contain a vertex x and internally dijoint xt 1 - and xt 2 -ath. 1. If t 2 dominate t 1 (the convere cae i ymmetric), let P be a hortet t 2 t 1 -ath in G. If P and B are dijoint, then P contain a egment P from S 2 to S 1. Uing Pro. 24 we now find C G (Fig. 9b). So let P go through B, then the lat egment of P i a (B, t 2 )-ath outide B. By the choice of K and P thi egment conit of a ingle arc a = bt 1 for b V B (Fig. 9c). According to Pro. 22, t 1 i incident to a further arc a, a G i reduced. Our choice of K require that the other endoint z of a lie in K, ince B, S 1 and a form a trong comonent bigger than B. It i now eay to ee (from Fig. 9c), that z V S2 yield C G (regardle of a orientation), and that z V B yield C G or C R G (deending on a orientation), o let z V S1 \ { 1 }. Thi leave two oibilitie: If a = zt 1 we find Q G, with eg t 1, 1,b and z (Fig. 9d). On the other hand, a = t 1 z lead to a contradiction: Since G i -reduced, there i at leat one vertex z between t 1 and z on S 1 ; omitting the t 1 z -egment of S 1 let u identify an in-kebab with ti z and t 2 and a body roerly containing B (Fig. 9e), contradicting maximality of B. 2. Let G contain a zt 1 -ath P 1 and a zt 2 -ath P 2 which are internally dijoint. If both P i are dijoint with B, we find C G with hel of Pro. 24 (Fig. 9f, where x i denote the firt vertex on P i that i alo in S i ). If wlog. P 1 interect B, let b denote the lat vertex on P 1 that i in B and x the firt vertex on P 1 that i in V Si \ { i }. If x t 1, we find a kebab in G with a body containing B, contradicting our choice of K. A the claim wa already roven for x = t 1 (ee Fig. 9c), the tatement follow for in-kebab. Lemma 26. Let G P 1 be a reduced hammock with cycle. Then F G for ome F F or F G for ome F {Φ, Ψ, Ψ R }. We have thu found a characterization of SPL by forbidden ubgrah. Theorem 27. Let G be a hammock, then G SPL iff G i F-free and no F {Φ, Ψ, Ψ R } i a bare minor of G.

11 S. Gulan 505 t 1 S 1 1 t 2 S 2 2 B P b (a) in-kebab (b) C G (c) bt 1-arc a z z z x 1 b b z x 2 (d) Q G (e) a = t 1z (f) C G Figure 9 Cae occurring in the roof of Lem. 25 for K being an in-kebab. Solid arrow rereent arc, dahed arrow rereent ath. Proof. Let G SPL, then Lem. 17 tate that G i F-free, while Lem. 19 tate that none of {Φ, Ψ, Ψ R } i a bare minor of G. Converely if G / SPL then Cor. 7 yield R(G) P 1. By Valde reult and Lem. 26, we know F R(G) for ome F F and/or F R(G) for ome F {Φ, Ψ, Ψ R }. If G = R(G), i.e., G i already reduced, the claim follow immediately; otherwie, induction on the length of the reduction uing Lem. 18 and 20 rovide the tatement. 6 Concluion We generalized the cla of arc-erie-arallel grah by augmenting the tandard contruction with a rule that allow for loo. Member of the new cla can be reveribly encoded by regular exreion which rereent the recurive tructure of a grah; naturally, the ize of uch an encoding i linear in that of the inut. Moreover, any regular exreion rereent an l-grah under thi encoding. Modulo iomorhim of grah, re. modulo aociativity and commutativity of oerator in exreion, the encoding and decoding are unique; thu they rovide u to trivialitie a bijection between l-grah and regular exreion. The encoding i done by contructing a erie of arc-labeled l-grah. A an automaton can be interreted a an arc-labeled grah, thi can be immediately alied to the converion of finite automata with l-tructure to equivalent regular exreion whoe ize i linear wrt. to the automaton. Thi generalize a recent reult for acyclic automata. We further characterized our new cla by mean of 7 forbidden minor. Therefore the exonential increae of exreion ize over automaton ize, which cannot be avoided in the general cae, i due to grah-tructural roertie of automata that are not reent in l-grah. The forbidden minor can be conidered a being the rimitive of thee non-exreible roertie, they hould be further invetigated in order to imrove on current converion from automata to exreion. S TAC S 1 1

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