The First-Order Theory of Ordering Constraints over Feature Trees

Transcription

1 Discrete Mathematics and Theoretical Computer Science 4, 2001, The First-Order Theory o Ordering Constraints over Feature Trees Martin Müller 1 and Joachim Niehren 1 and Ral Treinen 2 1 Programming Systems Lab, Universität des Saarlandes, Saarbrücken, Germany Laboratoire de Recherche en Inormatique, Université Paris-Sud, Orsay, France. received April 19, 1999, revised February 2001, accepted Aug 15, The system FT o ordering constraints over eature trees has been introduced as an extension o the system FT o equality constraints over eature trees. We investigate the irst-order theory o FT and its ragments in detail, both over inite trees and over possibly ininite trees. We prove that the irst-order theory o FT is undecidable, in contrast to the irst-order theory o FT which is wellknown to be decidable. We show that the entailment problem o FT with existential quantiication is PSPACE-complete. So ar, this problem has been shown decidable, conp-hard in case o inite trees, PSPACE-hard in case o arbitrary trees, and cubic time when restricted to quantiier-ree entailment judgments. To show PSPACE-completeness, we show that the entailment problem o FT with existential quantiication is equivalent to the inclusion problem o non-deterministic inite automata. Keywords: eature constraints, logic o trees, automata 1 Introduction Ordering Constraints Expressiveness o the First-Order Theory Undecidability Results Entailment with Existential Quantiiers Correctness o the Entailment Test Completeness o the Entailment Test c 2001 Maison de l Inormatique et des Mathématiques Discrètes (MIMD), Paris, France

2 194 Martin Müller and Joachim Niehren and Ral Treinen 1 Introduction Feature constraints have been used or describing records in constraint programming [1, 30, 31, 36] and record-like structures in computational linguistics [14, 12, 23, 26]. Feature constraints also occur naturally in type inerence or programming languages with object types or record types [22, 5, 24]. Following [2, 4, 3], we consider eature constraints as predicate logic ormulas interpreted in the structure o eature trees. A eature tree is a tree with unordered edges labeled by eatures and with possibly labeled nodes. Features are unctional in that the eatures labeling the edges departing rom the same node must be pairwise dierent. The structure o eature trees gives rise to an ordering in a very natural way which is called weak subsumption ordering in [7]. Consider the ollowing example where an unlabeled node is indicated as : Here, the let tree τ 1 is said to weakly subsume the right tree τ 2 since τ 1 has ewer edges and node labels than τ 2. In other words, every positive assertion about the presence o labels or eatures that holds or τ 1 also holds or τ 2. In general, a tree τ 1 weakly subsumes a tree τ 2, written τ 1 τ 2, i every word o eatures in the tree domain o τ 1 belongs to the tree domain o τ 2 and the (partial) labeling unction o τ 1 is contained in the labeling unction o τ 2. We consider the system FT o ordering constraints over eature trees [18, 19, 17]. Its constraints ϕ are given by the ollowing abstract syntax ϕ :: x x!#" x$ % x!&" a' x()" ϕ * ϕ! where denotes a eature symbol and a a label symbol. The constraints o FT are interpreted in the structure o eature trees with the weak subsumption ordering. We distinguish two cases, the structure o inite eature trees and the structure o possibly ininite eature trees. A constraint x x! holds i the denotation o x weakly subsumes the denotation o x!, x$ % x! is valid i the denotation o x has an edge at the root that is labeled with the eature and leads to the denotation o x!, and a' x( means that the root o the denotation o x is labeled with a. The constraint system FT is an extension o the well-investigated constraint system FT [2, 4], which provides or equality constraints x y rather than more general ordering constraints x y. The system FT can be seen as a sub-system o FT since x y can be expressed as x y * y x thanks to anti-symmetry o the weak subsumption order. The ull irst-order theory o FT is decidable [4] and has non-elementary complexity [37]. The decidability question or the irst-order theory o FT has been raised in [17]. There, two indications in avour o decidability have been ormulated: its analogy to FT and its relationship to second-order monadic logic. However, we show in this paper that the the irst-order theory o FT is undecidable. Our result holds in the structure o possibly ininite eature trees and, more surprisingly, even in the structure o inite eature trees. Our proo is based on an encoding o the Post Correspondence Problem using a technique o [33]. Once the undecidability o the irst-order theory o FT is settled, it remains to distinguish decidable ragments and their complexity. It is well-known that the satisiability

3 The First-Order Theory o Ordering Constraints over Feature Trees 195 FT FT in Satisiability o n 3 [18] n 5 [7] positive constraints n 3 [18] Entailment w/o quantiiers n 3 [18] n 3 [18] Entailment with quantiiers Co-NP hard [17] PSPACE hard [17] PSPACE complete [here] PSPACE complete [here] Full theory undecidable [here] undecidable [here] Fig. 1: Fragments o the irst-order theories o FT and FT in problem o FT, its entailment problem ϕ " ϕ!, and its entailment problem with existential quantiiers ϕ " x 1 x n ϕ! can be solved in quasi-linear time [31]. The investigation o ordering constraints was initiated by Dörre [7] who gave an O' n 5 ( -algorithm or deciding satisiability o FT -constraints. This result was improved to O' n 3 ( in [18], where also the entailment problem o FT concerning quantiier-ree judgments ϕ " ϕ! was shown decidable in cubic time. The next step towards larger ragments o the theory o FT was to consider entailment judgments with existential quantiication ϕ " x 1 x n ϕ! which are equivalent to unsatisiability judgments ϕ * x 1 x n ϕ! with quantiication below negation. As shown in [17], this problem is decidable, conp-hard in case o inite trees, and PSPACE-hard in case o arbitrary trees. Decidability is proved by reduction to (weak) second order monadic logic (W)S2S. In a irst reduction step, it is shown how to substitute the structure o eature trees by the related structure o so-called suiciently labeled eature trees. We note that this step cannot be generalized to arbritrary irst-order ormulas beyond entailment with existential quantiiers. Since the ull irst-order theory o ordering constraints over suiciently labeled (inite) eature trees can easily be encoded in (weak) second order monadic logic, decidability o entailment o FT with existential quantiiers ollows rom the classical results on (W)S2S [32, 25]. This paper contributes the exact complexity o the entailment problem o FT with existential quantiication. We prove PSPACE-completeness, both in the structure o inite trees and in the structure o possibly ininite trees. This result is obtained by reducing the entailment problem o FT with existential quantiiers to the inclusion problem o non-deterministic inite automata (NFA), and vice versa. Our reduction o entailment is based on the ollowing idea: Given an existential ormula xϕ we construct an automaton that accepts all its consequences in orm o so called path constraints. An inverse reduction in the case o possibly ininite trees was irst presented in [17]. In this paper, we present another inverse reduction which also applies or inite trees. Applications and Related Work. The application domains o ordering constraints over eature trees are quite diverse. They have been used to describe so-called coordination phenomena in natural language [7] but also or the analysis o concurrent constraint programming languages [20]. The less general equality constraints over eature trees are central to constraint based grammars, and they provide record constraints or logic programming [31] or concurrent constraint programming [27, 15]. In concurrent constraint programming, entailment with existential quantiication is needed or deciding the satisaction o conditional guards. As mentioned above, our results are also relevant or constraint-based inerence o record types and object types. In this context, the entailment test has recently received some attention as a justiication or constraint simpliication and as a means to check type interaces [24, 5, 35, 16, 10, 11]. Originally, weak subsumption has been introduced as a weakening o subsumption. The subsumption ordering between eature structures [13, 28, 6] is omnipresent in linguistic theories like HPSG (head-driven phrase structure grammar) [23]. According to the more general view o [29, 7], the subsumption ordering and the weak subsumption ordering are deinable between elements o an arbitrary eature algebra (which include the structure o

4 ' 196 Martin Müller and Joachim Niehren and Ral Treinen eature trees and all eature structures). Following [8], ordering constraints interpreted with respect to the subsumption (resp. weak subsumption) ordering o arbitrary eature algebras are called subsumption (resp. weak subsumption) constraints. Syntactically, subsumption constraints, weak subsumption constraints, and FT constraints coincide but semantically they dier. As proved in [8], the satisiability problem o subsumption constraints is undecidable. The satisiability problem o weak subsumption constraints is equivalent to the satisiability problem o FT constraints [7, 18] and hence decidable in cubic time. Structure o the Paper. Section 2 reviews the deinitions o eature trees and weak subsumption constraints. We demonstrate the expressivity o the constraint language in Section 3 and introduce some ormulas used in later sections. Undecidability o the irst-order theory o weak subsumption constraints is shown in Section 4. Finally, we show the entailment problem o existentially quantiied constraints to be PSPACE-complete in Section 5. We prove the correctness o our algorithm in Section 6 and its completeness in Section 7. A short version o this paper has been published as [21]. 2 Ordering Constraints The constraint system FT is deined by a set o constraints, the structure o eature trees, and an interpretation o constraints over eature trees. We assume an ininite set V o variables ranged over by x y z, a set F o at least two eatures ranged over by g and a set L o labels ranged over by a b. 2.1 Feature Trees A path π is a word o eatures. The empty path is denoted by and the ree-monoid concatenation o paths π and π! as ππ!. We have π π π. A path π! is called a preix o π i π π! π!! or some path π!!. A tree domain is a non-empty preix closed set o paths. ' ' ' ' ' A eature tree τ is a pair D L( consisting o a tree domain D and a partial unction L : D L that we call labeling unction o τ. Given a eature tree τ, we write D τ or its tree domain and L τ or its labeling unction. For instance, τ 0 a( ( is a eature tree with domain D τ0 and L τ0 a(. A eature tree τ 0 a is inite i its tree domain is inite, and ininite otherwise. A node o τ is an element o D τ. A node π o τ is labeled with a i π a( L τ. A node o τ is unlabeled i it is not labeled with any a. The root o τ is the node. The root label o τ is L τ (, and F is a root eature o τ i D τ. A eature tree τ is ully labeled i L τ is a total unction with domain D τ. Given a tree τ with π D τ, we write as τ$ π% the subtree o τ at path π; ormally D τ π π! " ππ! D τ and L τ π ' π! a( " ' ππ! a( L τ. 2.2 Syntax and Semantics An FT constraint ϕ is deined by the abstract syntax ϕ :: x y " a' x()" x$ % y " ϕ 1 * ϕ 2 where a L and F. In other words, an FT constraint is a conjunction o basic constraints which are either ordering constraints x y, labeling constraints a' x(, or selection constraints x$ % y. We deine the structure FT over eature trees in which we interpret FT constraints. Its

5 The First-Order Theory o Ordering Constraints over Feature Trees 197 universe consists o the set o all eature trees. The constraints are interpreted as ollows: τ 1 τ 2 i D τ1 D τ2 and L τ1 L τ2 τ 1 $ % τ 2 i D τ2 π " π D τ1 and L τ2 a' τ( i ' a( L τ ' π a( " ' π a( L τ1 The substructure o FT whose universe contains only the inite trees is denoted by FT in. We will oten use the ollowing decomposition property without urther mention: Proposition 2.1 I τ 1 τ 2 and τ 1 $ % τ!1 and τ 2 $ % τ!2 then τ! 1 τ! First-Order Formulas I not speciied otherwise, a ormula is said to be valid (satisiable) i it is valid (satisiable) both in FT and FT in. Our intention here is to treat both cases simultaneously and to note a distinction when needed only. Let Φ and Φ! be irst-order ormulas built rom FT constraints with the usual irst-order connectives and quantiiers. We say that Φ entails Φ!, written Φ " Φ!, i Φ Φ! is valid, and that Φ is equivalent to Φ! i Φ Φ! is valid. We denote with V ' Φ( the set o variables occurring ree in Φ, and with F ' Φ( and L ' Φ( the set o eatures and labels occurring in Φ. 3 Expressiveness o the First-Order Theory In this section we introduce some abbreviations o ormulas needed in Section 4. We use the usual abbreviations or ordering constraints, or instance we write x y or x y, x y or x y * x y, x y or y x and x y z or x y * y z. 3.1 Minimal and Maximal Values We can construct, or any ormula ϕ, ormulas µx ϕ and νx ϕ expressing that x is minimal (maximal) with the property ϕ: µx ϕ : ϕ * y ' ϕ$ y x%* y x( νx ϕ : ϕ * y ' ϕ$ y x%* y x( Here, y is a resh variable not occurring in ϕ, and ϕ$ y x% denotes the ormula where every ree occurrence o x is replaced by y. Typically, x occurs ree in ϕ but this is not required. Note that, in contrast to x and x, µx and νx are no variable binders that restrict the scope o the variable x; hence x is ree in µx ϕ and in νx ϕ i it is ree in ϕ. The ormula µx ϕ expresses that x denotes a minimal tree satisying ϕ, which is not necassarily a smallest tree with this property. In analogy, νxϕ expresses that x denotes a maximal but not necessary greatest tree satisying ϕ. Example 1 The sentence x ' µx true( is valid in FT and in FT in (there even exists a smallest tree, namely ' (. The ormula νx true is not satisiable in FT in but is satisied in FT by any ully labeled tree with domain F. ' The dierence between smallest and minimal trees is important or the ormula atom' x( which expresses that x denotes an atom in the lattice-theoretic sense, i.e. that it is a tree strictly greater than the smallest tree ( but with minimal distance (one eature or one label more): one-dist' x y( : µy x y atom' y( : x '' µx true( * one-dist' x y((

6 ' 198 Martin Müller and Joachim Niehren and Ral Treinen Example 2 The ormula µx ' x$ 0% x * x$ 1% x( is satisied in FT by ' 0 1 (, that is the ull binary and everywhere unlabeled tree, and is not satisiable in FT in since FT in contains no ininite trees. 3.2 Label Restrictions The ormula x y reads x and y are consistent, that is whenever ' π a( L τ and ' π! a! ( L τ then a a! : x y : z ' x z * y z( ' * For any label a L we write x a to express that the root o x is either unlabeled or labeled with a: x a : y x y a' y(( The ollowing ormula expresses that the root o a tree is unlabeled: not-root-labeled' x( : x a * x b where a and b are two arbitrary dierent label symbols. We obtain a irst-class status o labels by encoding a label a as the eature tree ' ' a( (. label-atom' x( : atom' x( * not-root-labeled' x( We can now express that x and y either have the same root label or are both unlabeled at the root by: 3.3 Arity Restrictions same-root-label' x y( : z ' label-atom' z( ' x z y z(( We can simulate a irst-class status o eature symbols by encoding a eature by the tree /0(. eature-atom' x( : atom' x( * not-root-labeled' x( We can express that y has at least all the root eatures o x by z ' eature-atom' z( * z x z y( The ollowing ormula expresses that x has exactly the root eatures 1 n : x 1 n : x 1 x n ' x$ 1 % x 1 * x$ n % x n * y ' y$ 1 % x 1 * * y$ n % x n * same-root-label' x y( x y(( These so-called arity constraints have been introduced in [31]. A decidable eature logic where eature symbols have irst class status has been investigated in [34]. 3.4 Inductive Properties We start this section by a demonstration o the expressivity o FT and show that we can express in FT inductive properties o trees, that is properties that require an inductive construction (or instance an automaton) to deine. We conclude the section by the deinition o the predicate string-c' x( that we will need in the undecidabability proo o Section 4. In the case o ininite trees it is in act quite simple to express inductive properties o a tree. For instance, we can express that the domain o x contains the set 0 1 by y ' y$ 0% y * y$ 1% y * y x(

7 The First-Order Theory o Ordering Constraints over Feature Trees 199 The ollowing ormula says that the tree denoted by x has domain 0 1 and that exactely one o its nodes is labeled with a whereas all its remaining nodes are unlabeled: a-singleton' x( : y z ' µy ' y$ 0% y * y$ 1% y * not-root-labeled' y(( * µz ' z$ 0% z * z$ 1% z * a' z(( * y x z * one-dist' y x(( I a-singleton' x( is satisied then y denotes the complete binary, everywhere unlabeled tree (with domain 0 1 ), and z denotes the complete binary, everywhere a-labeled tree. The ormula b-singleton is deined analogously. We can now express that x denotes a tree with domain 0 1 and that all its nodes are labeled with either a or b by: µx y z ' a-singleton' y( * b-singleton' z( * y z ' y x z x(( The idea behind this ormula is the ollowing: an a-singleton and a b-singleton are inconsistent i they have their label at the same position. Hence, the ormula says that 0 1 D x and every node o x which is reachable via a 0 1 -path is either labeled with a or with b. The minimality o x yields D x 0 1. In case o inite trees we have to use another trick (which works also in case o ininite trees). The next ormula is crucial or our undecidability proo. A tree τ satisies this ormula i c c and all its nodes are unlabeled: D τ string-c' x( : x c * not-root-labeled' x( * y ' x$ c% y * y x( The correctness o this deinition o string-c' x( with respect to the above stated semantics ollows rom the ollowing lemma where we write c n or the word c c consisting o n letters c. ' * Lemma 3.1 The ormula y x$ c% y y x( is satisied by τ i c D τ and or all k m 0, D τ then whenever c m k τ$ c m k % τ$ c m % Proo. Let τ$ c% τ! and τ! τ. Obviously, c D τ. The inequality ollows by induction: For any m, i c m D τ then τ$ c m % τ$ c m %. Furthermore, or any k with c m k 1 D τ and τ$ c m k % τ$ c m % we have that τ$ c m k 1 % τ$ c% $ c m k % τ! $ c m k % τ$ c m k % τ$ c m % For the other direction, since c D τ there is a τ! such that τ$ c% τ!. From the above inequality we get by setting m 0 and k 1 that τ! τ$ c 1 % τ$ c 0 % τ$ % τ 4 Undecidability Results Theorem 4.1 The irst-order theories o FT in and o FT are undecidable. The result holds or arbitrary (even empty) L and or F o cardinality 2. For the sake o clarity we use in the proo distinct label symbols a b e and pairwise distinct eature symbols s c p l r. We prove Theorem 4.1 by reduction o the Post Correspondence Problem (PCP). The choice o PCP is motivated by the act that our proo works by simulation o an iterative construction, and that PCP uses a technically very simple iteration. This is

8 200 Martin Müller and Joachim Niehren and Ral Treinen dierent in nature to the technique in [8] or the proo o undecidability o the satisiability o strong subsumption constraints. There, Thue-systems could be used by exploiting a correspondence between word equations and the algebraic properties o eature structures. See [33] or a discussion o the proo technique employed in this chapter. An instance o PCP is a inite sequence P )'' p i q i (( i 1 m o pairs o words rom a b. Such an instance is solvable i there is a nonempty sequence ' i 1 i n (, 1 i j m, such that p i1 p in q i1 q in. According to a classical result due to Post, it is undecidable whether an instance o the PCP is solvable. In the ollowing, let P '' p i q i (( i 1 m be a ixed instance o PCP. We say that a pair ' v w( is P-constructed rom a pair o words ' v! w! ( i, or some j, v p j v! and w q j w!. We say that a set X o pairs o words is P-constructed i every pair in X is either ' ( or is P-constructed rom some other pair in X. To encode solvability o P into the theory o FT in, resp. FT, we employ the ollowing equivalent deinition o solvability: Proposition 4.2 P is solvable i there is a P-constructed set X o pairs o words containing a pair ' w w( with w. 4.1 Words and Trees Given a word w a b over labels a b L ixed above we denote its length by " w" and or a natural number 1 j " w" we write w j or the j th letter o w. There is an obvious one-to-one encoding unction γ rom words w a b to eature trees or which we use the eature symbol s and label e that we also ixed above: γ' w( ' D w L w ( where s s w, L w ' s j ( w j or 0 j " w" 1, and L w ' s w ( e (see Figure 2(a)). D w We deine a let-inverse unction γ, that is γ' γ' w(( w, rom eature trees to (possibly ininite) words in a b ω as ollows: I τ does not have root eature s, or i its root is unlabeled or has label dierent rom a and rom b then γ' τ(. Otherwise let τ! be such ( ( that τ$ s% τ!. We deine γ' τ( a γ' τ! i τ has root label a, and γ' τ( b γ' τ! i τ has root label b. To express that y denotes the ixed word π appended with the denotation o x, we deine or any π a b a ormula app π ' x y(, such that 1. i app π $ τ τ! % then π γ' τ( γ' τ! ( 2. app π $ γ' w( γ' πw( % is valid or all words w and eature trees τ τ!, by induction on π: app ' x y( : x y appaπ ' x y( : a' y( * z ' y$ s% z * app π ' x z(( app bπ ' x y( : b' y( * z ' y$ s% z * app π ' x z(( Furthermore, we deine eps' x(, expressing that x denotes a tree τ with γ' τ(, by eps' x( : y x$ s% y ' a' x( b' x(( Finally, the ollowing ormula expresses that x denotes a inite string: In case o FT in inite' x( : y ' y$ s% y * y x( this ormula is, o course, equivalent to true.

9 The First-Order Theory o Ordering Constraints over Feature Trees 201 a s p c b s a s a s e l r τ l 1 τ r 1 l p r τ l 2 τ r 2 c c l p r τ l n τ r n (a) The string abaa. (b) The solution sequence v i w i i. 4.2 P-Constructions Fig. 2: Representation o strings and o solution sequences. Provided an appropriate encoding o sets o pairs o words and a predicate in' x l x r s(, expressing that the pair ' x l x r ( is member o the set s, we can express that s is a P-constructed set o pairs o words and that P is solvable: construction P ' s( : y y!' in' y y! s( '' eps' y( * eps' y! (( z z! ' in' z z! s(* ' app p j ' z y( * app q j ' z! y! ((((( j 1 m solvable P : s ' construction P ' s( * x ' in' x x s(* eps' x( * inite' x((( Lemma 4.3 For any predicate in' x y z(, i solvable P is valid then the instance P o the Post Correspondence Problem is solvable. Proo. Let σ be a ixed value or s such that solvable P holds. In particular, construction P ' σ( holds. We can show or all inite trees τ τ! satisying in' τ τ! σ( that there exists a set containing ' γ' τ( γ' τ! (( which is P-constructed rom ' (. The proo is by induction on " γ' τ(" " γ' τ! (". Lemma 4.4 There is a predicate in' x y z( such that i the instance P o the Post Correspondence Problem is solvable then solvable P is valid. Proo. The crux o the proo is to deine 1. or any sequence o pairs o words σ '' v i w i (( i 1 n a eature tree ρ' σ( 2. a predicate in' y l y r x( such that in' τ l τ r ρ' σ(( holds i τ l γ' v i ( and τ r γ' w i ( or some 1 i n. There is, however, no need to deine a ormula expressing that a eature tree is the encoding o a sequence o words.

10 202 Martin Müller and Joachim Niehren and Ral Treinen c c l p r c c τ l i τ r i Fig. 3: A possible value or x such that one-branch x x, where x is as in Figure 2(b). Since we know already how to encode words as trees, we now have to deine an appropriate encoding o an arbitrary set o pairs o trees as a eature tree, together with a corresponding ormula in. The representation o a sequence '' τ l i τ r i(( i 1 n is given in Figure 2(b). We deine, or any ormula ϕ, a ormula µ!xϕ expressing that x denotes the smallest element satisying ϕ. This ormula is stronger than µxϕ in that it requires the existence o a smallest tree satisying ϕ in addition: µ!x ϕ : ϕ * y ' ϕ$ y x% x y( I x denotes a tree as given in Figure 2(b), then the ormula one-branch' x x! ( given below expresses that x! denotes a tree as given in Figure 3. one-branch' x x! ( : x c ' νx c ' string-c' x c ( * x c x( * x c x! x * νx!' z ' µ!z ' x c z x!(((( In this ormula, x! is smaller than x but is strictly greater than the c-spine x c o x. The tree x! can have only one o the p-edges o x since the set o trees between x c and x! must have a smallest element. By the maximality o x!, the tree x! contains x c plus exactly one o the subtrees o x starting with a p-edge (see Figure 3). The ollowing ormula select' τ l τ r σ(, where σ is as in Figure 3, expresses that τ l is the tree τ l i and τ r is the tree τ r i: select' y l y r x!( : x!!' µx!! ' x! x!! * x!!!' x!! $ c% x!!! * x!!! x!! (( * z ' x!! $ p% z * z$ l% y l * z$ r% y r (( From a tree σ! as given in Figure 3, we get the tree σ!! (denoted by x!! ) containing at all nodes c j with j i a pair ' τ l!j τ r!j( such that τ l i τ l!j and τ r i τ r!j (by Lemma 3.1). By the minimality o σ!! we get that τ l i τ l!j and τ r i τ r!j or all j i, hence in particular or j 0 (see Figure 4). Combination o the two ormulas yields in' y l y r x( : x! ' one-branch' x x!( * select' y l y r x! (( Now, it is easy to veriy the conditions announced at the beginning o the proo.

11 The First-Order Theory o Ordering Constraints over Feature Trees 203 l τ l i p r τ r i c l τ l i p r τ r i l τ l i p r τ r i c l p r c c τ l i τ r i Fig. 4: The value o x in the ormula select y l y r x where x is as in Figure 3. Note that this proo did not make use o the act that the eature trees considered here are partial. The proo o Theorem 4.1 transers immediately to the structures o completely labeled trees (both in the case o inite and o arbitrary trees), where a tree ' D L( is called completely labeled i L is a total unction with domain D. In this case, the trees depicted in Figures 2(b), 3 and 4 have to be completed by giving some label to the nodes. 5 Entailment with Existential Quantiiers In [17] it is shown that the entailment problem o FT with existential quantiiers ϕ " xϕ! is decidable, PSPACE-hard in the case o ininite trees and conp-hard in the case o inite trees. We settle the precise complexity o this entailment problem in both cases. " Theorem 5.1 Entailment o FT with existential quantiication ϕ xϕ! is PSPACEcomplete or both structures FT and FT in. In Section 5.3 we modiy the PSPACE-hardness proo given in [17] or the case o ininite trees such that it proves PSPACE-hardness or both cases (Theorem 5.2). In particular, we show that we can encode the Kleene-star operator without need or ininite trees. Containment in PSPACE is shown (Theorem 5.9) by reducing in polynomial time the entailment problem to an inclusion problem between the languages accepted by nondeterministic inite state automata (NFA). Language equivalence or NFA (and hence inclusion, since A B B A B) is known to be PSPACE-complete i the alphabet contains at least two distinct symbols [9]. 5.1 Path Constraints We characterize existential FT ormulas xϕ by equivalent sets o path constraints (where sets are interpreted as conjunctions). Feature path constraints or FT have been introduced in [29] and have been used in [4] or a quantiier elimination procedure or FT. The abstract syntax o path constraints ψ is deined as ollows, where π π! F and a L: ψ :: x$ π% " a' x$ π% ( " x?$ π% a " x?$ π% y?$ π! % " x?$ π% y?$ π! %

12 204 Martin Müller and Joachim Niehren and Ral Treinen y y x y y z z z g x? g a y y z z x y z y g a z a z Fig. 5: Graphical Presentation o Example 4 The semantics o path constraints is given by extension o the structure FT through the ollowing predicates, which are deined on basis o the subtree selection unction τ$ π% introduced above. τ$ π% i π D τ a' τ$ π% ( i ' π a( L τ τ?$ π% a i π D τ implies τ$ π% a τ?$ π% τ!?$ π! % i π D τ and π! D τ imply τ$ π% τ! $ π! % τ?$ π% τ!?$ π! % i π D τ and π! D τ imply τ$ π% τ! $ π! % In the Section 5.2, we use path constraints or presenting typical examples o entailment judgements. Path constraints are also helpul or proving PSPACE-hardness in Section 5.3. In Section 5.5 we will construct a inite automaton that accepts all path constraints ψ entailed by xϕ and thereby reduce the entailment problem with existential quantiication to the inclusion problem o inite automata. 5.2 Examples A major diiculty in testing entailment with existential quantiiers is that there exist many equivalent FT constraints o quite distinct syntactic shape. This makes it very diicult (i not impossible) to apply a standard technique or deciding entailment, which perorms a comparison o constraints in some syntactic normal orm [2, 31, 18]. In this section we present some examples showing the diiculties o deciding entailment statement. We will come back to some o these examples in Section 5.5 to illustrate our solution. We start with a rather simple case: Example 3 The ormula y ' x y * a' y(( is equivalent to x?$ % a which is equivalent to y z ' x y * z y * a' z((. The next example o equivalent constraints with distinct syntactic shape is more complex. Example 4 (see Figure 5) Both o the ollowing ormulas are equivalent to x?$ g% hence equivalent to each other: a and ' * % * * * " " ' * % * * * y y! z z! x y y$ y! y! z z$ g% z! a' z! (( y y! z z! x y y$ y! z y! z$ g% z! a' z! (( In the next example, a constraint is given that entails x?$ g% a or all a. Note that this constraint thus also entails the constraints given in the previous example. Example 5 (see Figure 6) I b c then or all a the judgement x$ % x! * x! x!! * x!! $ g% x!!! * b' x!!! (* x y * y$ % y! * y! u * u z! * z! $ g% z!! * c' z!! ( " x?$ g% a

13 The First-Order Theory o Ordering Constraints over Feature Trees 205 x y x x y z g g b x c z x? g a Fig. 6: Graphical Presentation o Example 5 u x v x y u v y y Fig. 7: Graphical Representation o Example 6 holds. In other words, i α is a solution o the constraint displayed on the let hand side and i g D α x then the subtree o α' x( at g is compatible with any label a, and hence is unlabeled. Example 6 (see Figure 7) The ollowing situation illustrates a non-trivial example or entailment o selection constraints without existential quantiiers. ' y u! * u$ % u! * u x( * ' x v * v$ % v! * v! y( " x$ % y The right-hand side x$ % y is equivalent to the conjunction ' y? $ % x? $ % * x$ % ( * ' x?$ % y?$ % ( o path constraints which are entailed by the irst and second part o the let-hand side, respectively. 5.3 Entailment is PSPACE-hard In this section we show how the PSPACE completeness proo o [17] can be modiied such that it applies to the structure o inite eature trees as well. The ormulas used in the earlier proo require the existence x$ π% o all paths π in some regular language R; every solution o the ormula or an ininite language R has to map x to an ininite tree. Compared to this earlier proo, the trick is here to use conditional path constraints which may constrain ininitely many paths without requiring their existence. Theorem 5.2 The entailment problem or existentially quantiied FT -constraints is PSPACE-hard in both the inite and the ininite tree case. This ollows rom Proposition 5.6 (see below), which claims a polynomial reduction o the inclusion problem between regular languages over the alphabet F to an entailment problem between two existential FT ormulas. Notice that we have assumed F to contain at least two eatures. Our PSPACE-hardness proo is based on the act that a satisiable ordering constraint ϕ may entail an ininite conjunction o path constraints, even in case o inite trees: Example 7 1. or all n : x$ % y * y x * a' x( " x?$ n % a.

14 206 Martin Müller and Joachim Niehren and Ral Treinen 2. or all n m : x$ % y * y x " x?$ m n % x?$ n %. 3. or all π g : x x! * x$ % x! * x$ g% x! " x?$ π% x?$ %. For this reason the entailment problem or FT in does not necessarily reduce to an inclusion problem between inite regular languages (which is decidable in conp [9]). We ix a inite subset F F o eatures and consider regular expressions o the ollowing orm: R :: " " R " R 1 R 2 " R 1 R 2 ' where F( For encoding a regular expression R the main idea is to deine an existential ormula Θ' x R y( or resh variables x y such that Θ' x R y( is equivalent to π L R x?$ π% y?$ %. Once this is done, it will ollow immediately that L ' R! ( L' R( i Θ' x R y( " Θ' x R! y(. It is not obvious, however, how to deine such a ormula. The reader might notice, that a naive deinition o Θ' x R y( yields some unintended compatibility relations to be entailed too. Hence, we have to reine our main idea. We deine the ormula com F ' x( expressing that all subtrees o x reachable via an F-path are compatible with each other, i.e. they have a common upper bound: com F ' x( : y ' x y * F y! ' y$ % y! * y! y(( Lemma 5.3 (Comon upper bound) com F ' x( " " y π F x?$ π% y?$ %. For encoding a regular expression R, a reined idea is to deine an existential ormula Θ' x R y( such that Θ' x R y( is equivalent to com F ' x( * π L R x?$ π% y?$ %. We deine or all regular expressions R over F and variables x and y, the existential ormulas Θ' x R y( and Θ! ' x R y( recursively as ollows. Θ' x R y( com F ' x( * Θ! ' x R y( Θ! ' x y( x y Θ! ' x y( z ' x z * z$ % y( Θ! ' x R 1 R 2 y( Θ! ' x R 1 y( * Θ! ' x R 2 y( Θ! ' x R 1 R 2 y( z ' Θ! ' x R 1 z( * Θ' z R 2 y(( Θ! ' x R y( z ' x z * Θ! ' z R z( * z y( Apparently, Θ' x R y( has size linear in the size o R. Lemma 5.4 For all regular expressions R Proo. We proceed by induction on R. com F ' x( " Θ! ' x R y( x?$ π% y?$ % π L R : Θ! ' x y( x y x?$ % y?$ % π L x?$ π% y?$ %. : Θ! ' x y( z ' x z * z$ % y( x?$ % y?$ % π L x?$ π% y?$ %. R 1 R 2 : By induction hypothesis com F ' x( entails the equivalences Θ! ' x R 1 y( π L R 1 x?$ π% y?$ % and Θ! ' x R 2 y( π L R 2 x?$ π% y?$ %. Hence, com F ' x( entails Θ! ' x R 1 R 2 y( π L R 1 R 2 x?$ π% y?$ % also. R 1 R 2 : By deinition Θ! ' x R 1 R 2 y( z ' Θ! ' x R 1 z( * com F ' z( * Θ! ' z R 2 y((. By induction hypothesis, com F ' x( entails Θ! ' x R 1 z( π 1 L R 1 x?$ π 1 % z?$ % and

15 The First-Order Theory o Ordering Constraints over Feature Trees 207 com F ' z( entails Θ! ' z R 2 y( π 2 L R 2 z?$ π 2 % y?$ %. Hence, com F ' x( entails that Θ! ' x R 1 R 2 y( is equivalent to (1): z % * ' * % x?$ π 1 z?$ % com F z( z?$ π 2 y?$ % (1) π 1 L R 1 π 2 L R 2 It remains to show that com F ' x( entails the equivalence between (1) and (2): x?$ π% y?$ % (2) π L R 1 R 2 Since (1) obviously entails (2), it is suicient to prove the validity o com F ' x( " ' 2( ' 1(. Let α be an FT -valuation which satisies both com F ' x( and (2). We deine a tree τ such that α z τ satisies the matrix o (1). For this deinition we use a least upper bound operator on eature trees denoted by : τ α' x($ π 1 % π 1 L R 1 D α x Since α solves com F ' x( there exists an upper bound o α' x($ π% " π F as stated by Lemma 5.3 and thus the least upper bound τ exists. We next demonstrate that α z τ satisies the matrix o (1). The deinition o τ yields α' x($ π 1 % τ or all π 1 L ' R 1 ( D α x, i.e. the variable assignment α z τ satisies the irst conjunction in (1). From com F ' α' x(( it ollows that com F ' τ( holds, i.e. α z τ satisies com F ' z(. Furthermore, all π 2 α' x($ π 1 π 2 %. % ' ( % ' ( % D τ satisy: τ$ π 2 % π 1 L R 1 D α x Since α is a solution o (2), α' x($ π 1 π 2 α' y( is satisied by all π 2 L R 2. Thus τ$ π 2 α' y( is valid or all π 2 L R 2, i.e. α z τ satisies π 2 L R 2 z?$ π 2 y?$ %, the remaining conjunct in (1). R : By deinition Θ! ' x R y( z ' com F ' z(* x z * Θ! ' z R z( * z y(. The induction assumption yields that com F ' z( entails Θ! ' z R z( π L R z?$ π% z?$ %. Hence, com F ' x( entails that Θ! ' x R y( is equivalent to (3): z com F ' z( * x z * z?$ π% z?$ % * z y (3) π L R It remains to show that com F ' z( entails the equivalence between (3) and (4): x?$ π% y?$ % (4) π L R In order to show the non-trivial implication, we assume an FT -valuation α which satisies both com F ' x( and (4). We deine a tree τ such that α z τ satisies the matrix o (3) as ollows: τ α' x($ π% π L R D α x Note that τ is well-deined or the same reason as in the preceeding case. Our assumptions on the choice o α yields: com F ' τ(, α' x( τ (since L ' R ( ) and τ α' y(. In order to show that α z τ is a solution o (3) it remains to prove τ$ π! % τ or all π! L ' R ( D τ : τ$ π! % π L R D α x α' x($ π% $ π! % π L R D α x α' x($ π!! % τ

16 208 Martin Müller and Joachim Niehren and Ral Treinen Lemma 5.5 For all regular expressions R 1 and R 2 L ' R 1 ( L ' R 2 ( i com F ' x( " x?$ π% y?$ % π L R 2 x?$ π% y?$ % π L R 1 Proo. The implication rom the let to the right is trivial since ' ( is a sub-conjunction o ' ( i L ' R 1 ( L ' R 2 (. For the other direction, we assume L ' R 1 ( L ' R 2 ( and show how to contradict the entailment judgment to the right. We ix a word π F rom L ' R 1 ( L ' R 2 ( and a new eature h F F (which exists since F is inite whereas F is not). We construct values τ or x and τ! or y such that ' ( is satisied but ' ( is not. Both trees are completely unlabeled; hence com F ' τ( holds. We deine the domain D τ to be the preix closure o the word πh and the domain D τ to be the suix closure o D τ with the exception o the word h. For illustration, we display the trees τ and τ! or the word π g below: τ τ! It is easy to check that $ x τ y τ! % satisies ' and h D τ π but h D τ. h g h g ( but not ' h g ( since π L ' R 1 ( L ' R 2 ( Proposition 5.6 For all variables x y and or every pair o regular expressions R 1 and R 2 : Θ' x R 1 y( " Θ' x R 2 y( is equivalent to L ' R 2 ( L ' R 1 (. Proo. This ollows rom Lemmas 5.4 and Satisiability Test In this section we recall the satisiability test or FT introduced in [18], which we will also need as a preprocessing step in our entailment test in Section 5.5. Clearly, satisiability (and hence entailment) depends on the choice o inite or ininite trees. For instance, x$ % x is unsatisiable in FT in but satisiable in FT. Let an extended constraint be a conjunction o constraints ϕ and (atomic) compatibility constraints x y. From now on, we will only deal with extended constraints and reely call them constraints or simplicity. In the case o ininite trees, we say that an (extended) constraint ϕ is -closed i it satisies the ollowing properties or all x y z x! y! a b. ' % % % % 1 x x ϕ i x V ϕ( 2 x z ϕ i x y ϕ and y z ϕ x! y! ϕ i x$ x! ϕ x y ϕ y$ y! ϕ 1 x y ϕ i x y ϕ 2 x z ϕ i x y ϕ and y z ϕ 3 x y ϕ i y x ϕ x! y! ϕ i x$ x! a b i a' x( ϕ x y ϕ b' y( ϕ

17 The First-Order Theory o Ordering Constraints over Feature Trees 209 The rules o and require that ϕ is closed with respect to relexivity, transitivity, and decomposition o. The rules in and require that ϕ contains all compatibility constraints that it entails (this is proved in [18]), and requires ϕ to be clash-ree. In the case o inite trees, we say that a constraint ϕ is -closed i it satisies - and the additional occurs check property or all n 1, x 1 x n 1 y 1 y n 1 n : x 1 x n 1 ϕ i x i $ i % y i * x i 1 y i ϕ or all 1 i n The ollowing result is proved in [18] (Theorem 1 and Proposition 4). It holds in both cases, or inite trees and or possibly ininite trees, but with the respective notion o -closedness. Proposition 5.7 There exists a cubic time algorithm that, given a constraint ϕ, computes an -closed constraint containing ϕ or proves its unsatisiability. Every -closed constraint is satisiable. 5.5 An Automaton or Path Constraints In this section we show that or every -closed constraint ϕ there is a non-deterministic automaton A ϕ o size polynominal in the size o ϕ which accepts the set o all path constraints which are entailed by ϕ and which mentions only symbols rom a ixed set o variables, labels, and eatures. Note that -closedness is a necessary assumption or our automaton construction. Note also that the automaton does not dier in the case o inite and ininite trees, only the assumed version o -closedness diers. The algorithm o Dörre [7] can be seen in this perspective. There, the non-satisiability o a (in some sense normalised) weak subsumption constraint ϕ was equivalent to the act that two labeling path constraints a' x$ π% ( and b' x$ π% ( or dierent label symbols a and b are entailed by ϕ, which could be checked by inspection o the automaton that describes all the labeling path constraints entailed by ϕ Path Constraints as Words The automaton accepts words ψ associated with a path constraint ψ over some inite subalphabet o F L V?$ %' (. In irst approximation, let ψ be the concrete syntax o ψ. There is however a serious problem with recognizing the concrete syntax o entailed path constraints: Example 8 1. The set o words representing a path constraint entailed by x x is not regular (when restricted to the variables in x x): ψ " x x " ψ x?$ π% x?$ π% " π F x?$ π% x?$ π% " π F 2. The set o words representing a path ordering constraint entailed by y ' x$ % y * x y( is not regular: ψ " y ' x$ % y * x y( " ψ x?$ m % x?$ n % " 0 m n x$ n % " n 0 x?$ m % x?$ n % " m n 0 We thereore have to alter the deinition o ϕ slightly but undamentally. The trick is to actor out the maximal common suix o the two paths in a path constraint o the orm x?$ π 1 % y?$ π 2 %. More exactly, we add the symbol # to the alphabet and alter the deinition o ψ such that: x?$ π 1 % y?$ π 2 % x?$ π% y?$ π! % #π!! x?$ π 1 % y?$ π 2 % x?$ π% y?$ π! % #π!!

18 210 Martin Müller and Joachim Niehren and Ral Treinen where π!! is the longest common suix o π 1 and π 2 such that π 1 ππ!! and π 2 π! π!!. Hence, either one o π or π! is the empty path, or π and π! end with distinct eature symbols. This solves the regularity problem o Example 8, i.e., the ollowing sets are regular: ψ " x x " ψ ψ " y ' x$ % y * x y( " ψ " " % " % " % " % " x?$ % x?$ % #π π F x?$ % x?$ % #π π F x?$ % x?$ n # m n m 0 x$ n n 0 x?$ n x?$ % # m n m 0 x?$ % x?$ n # m n m 0 The deinition o ψ also adjusts some simple but tedious regularity problems raised by the validity o the ollowing entailment judgement: x?$ π% y?$ π! % " x?$ ππ!! % y?$ π! π!! % " % % " Example 9 The set ψ x?$ g y?$ ψ restricted to words with eatures g and variables x y is regular: % " " " x?$ g% y?$ # π π g z?$ % z?$ % #π z x y π z?$ % z?$ % #π z x y π The Alphabet o the Automaton g g For each constraint ϕ we will deine a non-deterministic inite automaton A ϕ whose alphabet is the set: F ' ϕ( L ' ϕ( V ' ϕ(? $ % ' ( # Given a sequence o variables x, we will also deine another automaton A x ϕ or the existential ormula xϕ, which is obtained rom A ϕ by removing the local variables in x rom the alphabet, i.e. by removing all transitions labeled with a symbol rom x. Note that the local variables in x matter or the deinition o the states (but not the alphabet) o A x ϕ i they occur in V ' ϕ(. To solve an entailment problem o the orm ϕ " xϕ! we construct the automata A ϕ and Aϕ x and test or language inclusion. In order to avoid that A ϕ x accepts tautological constraints not accepted by A ϕ we will require in Proposition 5.10 that F ' ϕ! ( F ' ϕ( and V ' xϕ! ( V ' ϕ(, which can be imposed w.l.o.g. Furthermore, we assume throughout the paper that bound variables are renamed apart, i.e. when considering an entailment problem ϕ " xϕ! we assume x V ' ϕ( /0. Every automaton A ϕ (and thereby A x ϕ) alls into ive parts (sharing only the initial state q s and the accepting state q ), corresponding to the ive kinds o path constraints. The construction o the automaton A ϕ is given in Figures 8, 9, 10 and 11. It is completely spelled out except or one additional symmetry (rule 6) which can be expressed through a dozen o urther transitions. In the rest o this section we explain this construction Constraints as Graphs Our construction o the automaton is motivated by considering constraints as graphs. For instance, the constraint x x! * x! $ % y * a' y( * z y * z$ g% y

19 The First-Order Theory o Ordering Constraints over Feature Trees 211 x 1 1 q s x 1 2 x y x y ϕ % % 1 3 x y x$ y ϕ 1 4 x q a x 2 1 q s x:a 2 2 x:a y:a x y ϕ 2 3 x:a y:a x$ y ϕ 2 4 x:a q a' x( ϕ Fig. 8: The sections o the automaton A ϕ or path constraints x π and a x π. can be depicted as the ollowing graph, where variables are represented as nodes. x x z a y g Intuitively, when the automaton A ϕ accepts a word ψ it traverses the constraint graph associated with ϕ where ψ is associated a certain traversal pattern. We will depict such traversal patterns graphically; or instance, the above constraint entails x? $ gggg% a and its associated graph allows or the ollowing traversal: x y a gggg In these pictures, the horizontal dimension corresponds to the ordering the vertical one corresponds to eature selection (top to bottom). (let to right) and Path Existence and Labeling Constraints (Fig 8). The subautomaton comprising rules recognizes all the path existence constraints x$ π% entailed by ϕ. Analogously, the rules serve to recognize the path labeling constraints a' x$ π% ( entailed by ϕ. The associated patterns look as ollows. y π x x π a' y( π x Rules dier rom rules in that its states o the orm y:a memorize the label a read at the beginning o some input word a' x$ π% ( (rule 2.1) in order to check it against a labeling constraint in ϕ (rule 2.4). a x π

20 212 Martin Müller and Joachim Niehren and Ral Treinen x? 3 1 q s x: 3 2 x:h y:h x y ϕ 3 3 x:h y: x$ % y ϕ y? 3 4 x:h y: x h 3 5 x: y h g x! : y h g x x! ϕ 3 6 x: y h g x! : y h x$ % x! ϕ 3 7 x: y h g # y x h g h g 3 8 x y x! y! x x! y y! ϕ 3 9 x y 3 10 x x x! y! x$ % x! y$ % y! ϕ F ϕ q Fig. 9: The section o the automaton A ϕ or path constraints x? π 1 syntax or x? π 1 π 3 y? π 2 π 3. y? π 2 #π 3 which is concrete Example 10 The constraint x$ % y * y y! * y! $ g% z * a' z( entails a' x$ g% (. This constraint is accepted by the ollowing transitions: q s a x x:a y:a y! :a g z:a q Ordering Path Constraints (Fig. 9) The next group o rules serves to recognize constraints o the orm x?$ π% y?$ π! %. Note that ϕ " x?$ π% y?$ π! % i π π 1 π 3 π 4 and π! π 2 π 3 π 4 or some π 1, π 2, π 3, and π 4, and there exists x! y! z such that where we may assume that ϕ " x?$ π 1 % x!?$ % (5) ϕ " x!?$ π 3 % z?$ % (6) ϕ " z?$ % y!?$ π 3 % (7) ϕ " y!?$ % y?$ π 2 % (8) π 1 and π 2 have no common suix except (10) The associated graph pattern is as ollows, where a dashed line indicates a paths o the constraint graph and a dotted an arbitrary path. (9) x π 1 π 2 x! y! π 3 z π 3 π 4 y x? π 1 π 3 π 4 y? π 2 π 3 π 4

21 The First-Order Theory o Ordering Constraints over Feature Trees 213 x? 4 1 q s x x 4 2 x x! y y! x y x! y! ϕ 4 3 x x! y y! x$ % y x! $ % y! ϕ 4 4 x x! y x! x y ϕ 4 5 x x! y y! x y x! y! ϕ 4 6 x x! y y! x$ % y x! $ % y! ϕ 4 7 x x! x y! x! y! ϕ 4 8 x x! y y! x y x! y! ϕ 4 9 x x! y y! x$ % y x! $ % y! ϕ a 4 10 x x! q a' x( ϕ c 4 11 x x! q a' x( b' x! ( ϕ a b Fig. 10: The section o the automaton A ϕ or path constraints x? π a. Note that π 3 π 4 is the maximal common suix o π 1 π 3 π 4 and π 2 π 3 π 4. Consequently, the concrete syntax o the constraint x?$ π 1 π 3 π 4 % y?$ π 2 π 3 π 4 % as checked by the automaton is x?$ π 1 % y?$ π 2 % #π 3 π 4 Rule 3.1 starts reading x?$ π% y?$ π! % which is continued by rules 3.2 and 3.3 veriing condition (5). Rule 3.4 switches to the veriication o condition (8) by rules 3.5 and 3.6. Rule 3.7 switches to the veriication o conditions (6) and (7) which is done jointly by rules 3.8 and 3.9. The respective last symbols o π 1 and π 2 are memorized in the state (the symbols h and g in the state x: y h g), allowing rule 3.7 to veriy condition 10. In order to allow or π 1 and π 2 to be, the automaton also memorizes whether or not a eature symbol has been consumed (rules 3.1 and 3.4). Slightly abusing notation, we allow or h and g in these rules to denote either a eature symbol or. Example 11 The constraint rom Example 6 entails x$ % y. This selection constraint is equivalent to the conjunction o the three path constraints x$ %, x?$ % y?$ %, and y?$ % x?$ %. The words corresponding to these constraints are accepted by the ollowing transitions o the automaton (or the constraint in Example 6): x q s x u u! q x? q s x: v: v! : y? y: u! : x? x: u! u: u! q s y: y? y: y # y y q u! : u! # u! u! q Label Compatibility (Fig. 10). Rules check constraints o the kind x? $ π% a. Note that ϕ " x?$ π% a i there are y y! z z! v v! b c and π 1 π!1 π 2 π!2 with π π 1π 2 such that: π!1 π! 2 " % " " % " " * ( ' ϕ x?$ π 1 y?$ %* y z (11) ϕ x?$ π!1% y!?$ %* y! z! (12) ϕ v?$ % z?$ π 2 (13) ϕ v!?$ % z!?$ π!2% (14) ϕ b' v( c' v! and b c or a b c( (15)

22 214 Martin Müller and Joachim Niehren and Ral Treinen y? 5 1 x:h y: x h 5 2 x: z h g y: z h g x y ϕ 5 3 x: z h g y: z h x$ % y ϕ 5 4 x: z h g # z x:# h g h g 5 5 x: z h g y: z h g x y ϕ 5 6 x: z h g y: z h g x y ϕ 5 7 x: z h g y: z h x$ % y ϕ 5 8 x: z h g # z x h g h g 5 9 x x! :# y y! :# x y x! y! ϕ 5 10 x x! :# 5 11 x x! :# 5 12 x x! y y! :# x$ % y x! $ % y! ϕ x y! x! y! ϕ y y! x y x! y! ϕ % $ % %! ' ( %! ' ( 5 13 x x! y y! x$ y x! y! ϕ F ϕ 5 14 x x q 6 x?$ π% y?$ π! #π! L A ϕ y?$ π! x?$ π% #π! L A ϕ Fig. 11: The section o the automaton A ϕ or path constraints x? π 1 syntax or x 1? π 1 π 3 x 2? π 2 π 3. y? π 2 #π 3 which is concrete The associated pattern looks as ollows. x x z y π 1 π!1 z! π 2 π!2 y! b' v( c' v! ( " x?$ π 1 π 2 % a i ' b c or a b c( and π 1 π 2 π!1 π! 2 We check the conditions (11) and (12) as well as (13) and (14) in parallel where we assume, by symmetry, that π 1 is a preix o π!1. With the names used above, the automaton consumes π 1 by rules , switches y to z with rule 4.4, then consumes π 2 minus its suix π!2 (which is identical to π! 1 minus its preix π 1) by rules , switches rom y! to z! in rule 4.7 and consumes π!2 in rules Finally rules check the label constraints (15). Example 12 In the case o Example 5 we obtain q s x? x x x y x! y! x!! y! x!! z! g x!!! z!! a q Path Compatibility (Fig. 11). Rules , in conjunction with rules , check or constraints x 1?$ π 1 % x 2?$ π 2 %. One possible justiication or ϕ " x 1?$ π 1 % x 2?$ π 2 % is that