Ordering Constraints over Feature Trees

Transcription

1 Orderig Costraits over Feature Trees Marti Müller 1, Joachim Niehre 1 ad Adreas Podelski 2 1 Uiversität des Saarlades, 2 Max-Plack-Istitut für Iformatik, podelski@mpi-sb.mpg.de Saarbrücke, Germay Abstract. Feature trees have bee used to accommodate records i costrait programmig ad record like structures i computatioal liguistics. Feature trees model records, ad feature costraits yield extesible ad modular record descriptios. We itroduce the costrait system FT of orderig costraits iterpreted over feature trees. Uder the view that feature trees represet symbolic iformatio, the relatio correspods to the iformatio orderig ( carries less iformatio tha ). We preset a polyomial algorithm that decides the satisfiability of cojuctios of positive ad egative iformatio orderig costraits over feature trees. Our results iclude algorithms for the satisfiability problem ad the etailmet problem of FT i time O 3. We also show that FT has the idepedece property ad are thus able to hadle egative cojucts via etailmet. Furthermore, we reduce the satisfiability problem of Dörre s weaksubsumptio costraits to the satisfiability problem of FT. This improves the complexity boud for solvig weak subsumptio costraits from O 5 to O 3. Keywords: feature costraits, tree orderigs, weak subsumptio, satisfiability, etailmet, complexity. 1 Itroductio Feature costraits have bee used for describig records i costrait programmig [2, 24, 23] ad record like structures i computatioal liguistics [13, 12, 20, 18, 19]. Followig [3, 5, 4] we cosider feature costraits as predicate logic formulae that are iterpreted i the structure of feature trees. A feature tree is a possibly ifiite tree with uordered labeled edges ad with possibly labeled odes. Edge labels are fuctioal; i.e., the labels of the edges departig from the same ode must be pairwise differet. Uder the view that feature trees represet symbolic iformatio, the feature tree τ 1 represets less iformatio tha the feature tree τ 2 if τ 1 has fewer edges ad ode labels tha τ 2. The relatio that we defie correspods to the iformatio orderig i precisely this sese. Algebraically, τ 1 τ 2 if there is a homomorphic embeddig from τ 1 to τ 2 (i.e., a mappig from odes i τ 1 to odes i τ 2 uder which the ode labelig is ivariat). A example is give i the picture. We itroduce the costrait system FT of iformatio orderig costraits over feature trees. The system FT is obtaied by addig orderig costraits to the costrait

2 system FT [3]. The sytax of FT costraits ϕ is defied by ϕ :: x x x a x a x ϕ ϕ where x ad x are variables ad a is a label. The sematics of FT is give by the iterpretatio over feature trees where the symbol is iterpreted as iformatio orderig o feature trees. The sematics of x a y ad a x are defied as i FT. For istace, both trees depicted above are possible values for x i solutios of the costrait x x x x. It is clear that FT is more expressive tha FT sice the iformatio orderig is atisymmetric (i.e., x x x x x x is valid). As we show i the paper, FT is strictly more expressive tha FT. For istace, o costrait i FT ca be equivalet to x x. Also, we do ot kow of ay formula over FT (eve with existetial quatifiers) equivalet to x x 1 x x 2 x x x 1 x x 3 x ; this FT formula expresses that x 1 is uifiable with both x 2 ad x 3 (but does ot imply uifiability of x 2 ad x 3 ). We show that the satisfiability problem of cojuctios of positive ad egative FT costraits ϕ ϕ 1 ϕ is decidable i O 3. This result icludes a decisio procedure for the etailmet problem of the form ϕ ϕ sice a formula ϕ ϕ is valid if ad oly if the formula ϕ ϕ is usatisfiable. To establish our result, we prove that FT has the fudametal idepedece property (similar to its relatives RT [6], FT [3], ad CFT [24]). We reduce the satisfiability problem of Dörre s weak-subsumptio costraits [7] over feature algebras liearly to the oe i FT. Thereby, our algorithm improves o the best kow satisfiability test for weak subsumptio costraits which uses fiite automata techiques ad has a O 5 -complexity boud [7]. Pla of the Paper. Sectio 2 surveys related work. Sectio 3 defies FT. Sectio 4 presets the satisfiability test for FT costraits. Sectio 8 cotais the completeess proof. Sectio 5 presets the etailmet test for FT costraits, ad proves the idepedece property of FT. Sectio 6 defies weak subsumptio costraits ad reduces their satisfiability problem to the oe of FT costraits. Sectio 7 shows that FT is strictly more expressive tha FT. 2 Related Work Ies Costraits. I previous work [17], we have itroduced the costrait system INES of iclusio costraits over o-empty sets of trees ad a cubic satisfiability test. The satisfiability test for FT is ispired by ad subsumes the oe for INES. However, the etailmet problems for FT ad INES costraits are differet. The etailmet problem of INES costraits is conp-hard [16]. Ituitively, the etailmet problem of FT is less expressive tha the oe of INES because a FT costrait ϕ caot uiquely describe a sigle feature tree (i absece of arity costraits); i cotrast, INES costraits (which are iclusios betwee first-order terms with a implicit arity restrictio) ca uiquely describe a costructor tree as a sigleto set. For istace, the INES costrait x a describes the sigleto a. As a cosequece, the etailmet propositio x a a y x y holds i INES. No similar etailmet pheomeo exists for FT.

3 Feature Costraits. The costrait system CFT [24] exteds FT by arity costraits of the form x f 1 f, sayig that the deotatio of x has subtrees exactly at the features f 1 through f. CFT subsumes Colmerauer s ratioal tree costrait system RT [6] but provides fier-graied costraits. The system EF [25] exteds CFT by feature costraits x y z, providig for first-class features. Complete axiomatizatios for FT ad CFT have bee give i [5] ad [4], respectively. The satisfiability of EF costraits is show NP-hard i [25]. The system FT sort exteds FT by allowig a partial order o labels [15]. Subsumptio Costraits. Subsumptio is a orderig o the domai of feature algebras. Subsumptio costraits have bee cosidered i the cotext of uificatio-based grammars to model coordiatio pheomea i atural laguage [9, 7, 21]. There, oe wats to express that two feature structures represetig differet parts of speech share commo properties. For example, the aalysis of programmig ad liguistics i the phrase Feature costraits for [ NP programmig] ad [ NP liguistics] should share (but might refie differetly) the iformatio commo to all ou phrases. Sice the satisfiability of subsumptio costraits is udecidable [9], Dörre proposed weak subsumptio as a decidable approximatio of subsumptio. As we show, the iformatio orderig over feature trees (as ivestigated i this paper) coicides with the weak subsumptio orderig iterpreted over (the algebra of) feature trees. Idepedet Costrait Systems. A costrait system has the fudametal idepedece property if egated cojucts are idepedet from each other, or: its costraits caot express disjuctios (we will give a formal defiitio later). Apart from the metioed tree costrait systems RT, FT, CFT [6, 1, 24, 3], costrait systems with the idepedece property iclude liear equatios over the real umbers [14], or ifiite boolea algebras with positive costraits [10]. 3 Sytax ad Sematics of FT The costrait system FT is defied by a set of costraits together with a iterpretatio over feature trees. We assume a ifiite set of variables raged over by x y z, ad a ifiite set L of labels raged over by a b. Feature Trees. A path p is a fiite sequece of labels. The empty path is deoted by ε ad the free-mooid cocateatio of paths p ad p as pp ; we have εp pε p. Give paths p ad q, p is called a prefix of p if p p p for some path p.atree domai is a o-empty prefix closed set of paths. A feature tree τ is a pair D L cosistig of a tree domai D ad partial labelig fuctio L : D L. Give a feature tree τ, we write D τ for its tree domai ad L τ for its labelig fuctio. The set of all feature trees is deoted by F. A feature tree is called fiite if its tree domai is fiite, ad ifiite otherwise. Sytax. A FT costrait ϕ is defied by the followig abstract sytax. ϕ :: x y a x x a y x y ϕ 1 ϕ 2

4 A FT costrait is a cojuctio of basic costraits which are either iclusio costraits x y, labelig costraits a x, selectio costraits x a y, or compatibility costraits x y. Compatibility costraits are eeded i our algorithm ad ca be expressed by first-order formulae over iclusio costraits (see Propositio 1). We idetify FT costraits ϕ up to associativity ad commutativity of cojuctio, i.e., we view ϕ as a multiset of iclusio, labelig, selectio, ad compatibility costraits. We write ϕ i ϕ if all cojucts i ϕ are cotaied i ϕ. The size of a costrait ϕ is defied as the umber of label ad variable occurreces i ϕ. Sematics. We ext defie the structure F over feature trees i which we iterpret FT costraits. The sigature of F cotais the biary relatio symbols ad ad for every label a a uary relatio symbol a ad a biary relatio symbol a. I F these relatio symbols are iterpreted such: τ 1 τ 2 iff D τ1 D τ2 ad L τ1 L τ2 τ 1 a τ 2 iff D τ2 p ap D τ1 ad L τ2 p b ap b L τ1 a τ iff ε a L τ τ 1 τ 2 iff L τ1 L τ2 is a partial fuctio (o D τ1 D τ2 ) Let Φ deote first-order formulae built from FT costraits with the usual first order coectives. We call Φ satisfiable (valid) if Φ is satisfiable (valid) i the structure F. We say that Φ etails Φ, writte Φ Φ, if Φ Φ is valid, ad that Φ is equivalet to Φ if Φ 1 Φ 2 is valid. We deote with V Φ the set of variables occurrig free i Φ ad with L Φ the set of labels occurrig i Φ. Propositio 1. The formulae x y ad z x z y z are equivalet i F. Proof. Let σ be a variable assigmet ito F which also is a solutio of the formula z x z y z. Sice L σ x L σ y L σ z ad L σ z is a partial fuctio, L σ x L σ y is also a partial fuctio. Hece σ is a solutio of x y. Coversely, if σ is a solutio of x y the L σ x L σ y is a partial fuctio. Thus, the pair τ de f D σ x D σ y L σ x L σ y is a feature tree ad the variable assigmet σ defied by σ z τ ad σ x σ x for x z is a solutio of x z y z. 4 Satisfiability Test We preset a set of axioms valid for FT ad the iterpret these axioms as a algorithm that solves the satisfiability problem of FT. The axioms ad the algorithm are ispired by the oes for INES costraits preseted i [17]. Table 1 cotais five axiom schemes - that we regard as sets of axioms. The uio of these sets of axioms is deoted by, i.e.,. For istace, a axiom scheme x x represets the ifiite set of axioms obtaied by istatiatio of the meta variable x. A axiom is either a costrait ϕ, a implicatio betwee costraits ϕ ϕ, or a implicatio ϕ false. Propositio 2. The structure F is a model of the axioms i.

5 x x ad x y y z x z x a x x y y a y x y x y x y ad x y y z x z ad x y y x x a x x y y a y x y a x x y b y false for a b Table 1. Satisfiability of FT Costraits. Proof. By a routie check. For illustratio, we prove the statemet for the secod rule i, amely x y y y x y. The followig implicatios hold: x y y y x y z y z y z Propositio 1 z x z y z Trasitivity x y Propositio 1 The Algorithm. The set of axioms iduces a fixed poit algorithm that, give a iput costrait ϕ, iteratively adds logical cosequeces of ϕ to ϕ. (Observe that actually oly costraits of the form x y ad x y are derived). More precisely, i every step iputs a costrait ϕ ad termiates with false or outputs a costrait ϕ ϕ. Termiatio with false takes place if there exists ϕ i ϕ such that ϕ false. Output of ϕ ϕ is possible if ϕ or there exists ϕ i ϕ with ϕ ϕ. Example 1. Icosistecy ca be due to icompatible upper bouds. Cosider: a x x z y z b y false for a b We may add x z by 1, the z x via 3, the y x with 2, ad fially termiate with false via. Example 2. We eed for derivig the usatisfiability of the costrait: a x x a x x z y z y a y b y false for a b Algorithm may add x y after several steps as show i Example 1. The it may proceed with x y via ad termiate with false via. Termiatio. The fixed poit algorithm termiates whe reflexivity of iclusio x x ( 1) is restricted to variables x V ϕ. Give a subset F of, a costrait ϕ is called F-closed if algorithm uder this restrictio ad w.r.t. the axioms i F caot proceed o ϕ. Note that false is ot F-closed sice it is ot a costrait by defiitio.

6 Example 3. Our cotrol takes care of termiatio i presece of cycles like x a x. For istace, the followig costrait is -closed. x a x x y y a y x x y y x x y y x y y x I particular, ad do ot loop through the cycle x a x ifiitely ofte. This example also illustrates why the fixed poit algorithm would ot be termiatig if based i the axiom x a x x y y y a y x y. Propositio 3. If ϕ is a costrait with m variables the algorithm termiates uder the above cotrol i at most 2 m 2 steps. with iput ϕ Proof. Sice does ot itroduce ew variables, it may add at most m 2 o-disjoitess costraits x y ad m 2 iclusios x y. Propositio 4. Every -closed costrait ϕ is satisfiable over FT. Proof. See Sectio 8. Theorem 5. The satisfiability of FT costraits ca be decided i time O 3 offlie ad olie see 11 where is the costrait size. Proof. Propositio 2 shows that ϕ is usatisfiable if started with ϕ termiates with false. Propositio 4 proves that ϕ is satisfiable if started with ϕ termiates with a costrait. Sice termiates for all iput costraits uder the above cotrol (Prop. 3) this yields a effective decisio procedure. The mai idea of the complexity proof is that oe eeds at most O 2 steps (Prop. 3) each of which ca be implemeted i time O. The implemetatio ca be orgaized icremetally by exploitig that algorithm leaves the order uspecified i which the axioms are applied. Hece, we obtai that off-lie ad o-lie complexity are the same. The implemetatio details ad the complexity proof are omitted here, sice they are similar to those preseted i [17]. 5 Etailmet, Idepedece, Negatio I this sectio, we give a cubic algorithm testig etailmet ϕ ϕ betwee FT costraits ϕ ad ϕ. We the prove the idepedece property of FT. Hece we ca solve cojuctios of positive ad egative FT -costraits ϕ ϕ 1 ϕ i time O 3. A basic costrait µ is a cojuctio free costrait ϕ, i.e., give by the followig abstract sytax: µ :: x y x y a x x a y The etailmet ϕ ϕ is equivalet to the fact that the etailmet ϕ µ holds for all basic costraits µ i ϕ.

7 Next we characterize etailmet problems ϕ µ sytactically. We say that a costrait ϕ sytactically cotais µ, writte ϕ µ, if oe of the followig holds: ϕ a x if exists x such that x x a x ϕ ϕ x y if x y i ϕ or x y ϕ x y if x y i ϕ or x y ϕ x a y if exist x, y such that x a y i ϕ ad ϕ x x ϕ x x ad ϕ y y ϕ y y We say that a first-order formula Φ sytactically cotais µ, Φ µ, if Φ ϕ Φ for some ϕ ad Φ such that ϕ µ. Lemma 6. Give a -closed costrait ϕ, we ca compute a represetatio of ϕ i liear time that allows to test sytactic cotaimet ϕ µ for all µ i time O 1. Proof. Simple. It is easy to see that sytactic cotaimet is sematically correct, i.e., ϕ µ implies ϕ µ. For decidig etailmet, we have to show that our otio of sytactic cotaimet is sematically complete, i.e., if ϕ µ the ϕ µ (Propositio 13). The idea is to costruct a satisfiable extesio of ϕ (its saturatio) which sytactically ad simultaeously cotradicts all µ ot sytactically cotaied by ϕ (Lemma 12). Saturatio is defied i terms of two operators Γ 1 ad Γ 2 o costraits. The operator Γ 2 is such that Γ 2 ϕ cotradicts all µ of the form x y, x y, ad a x (i.e., o selectio costraits) which are ot sytactically cotaied i ϕ (Lemma 10). The operator Γ 1 serves for cotradictig selectio costraits. For istace, cosider ϕ x a y where ϕ x x y y. I this case, Γ 1 ϕ eforces the existece of the feature a i the deotatio of x by addig to ϕ the costrait x a v xa for a fresh variable v xa. Now Γ 2 Γ 1 ϕ is such that it cotradicts either y v xa or v xa y. (see Example 4). I this sese, Γ 1 is a preprocessor for Γ 2. Defiitio 7. Let ϕ be a costrait, v 1 ad v 2 distict fresh variables, ad l 1 ad l 2 distict labels. Furthermore, for every pair of variables x y V ϕ, ad label every label a L ϕ let l x ad l xy be fresh labels ad v xa a fresh variable. We defie Γ 1 ϕ ad Γ 2 ϕ i depedece of v 1 v 2 l x l xy v x as follows: Γ 1 ϕ ϕ x a v xa x V ϕ a L ϕ Γ 2 ϕ ϕ x l x v x y y l x y ϕ x y x y V ϕ 1 x l xy v 1 y l xy v 2 ϕ x y x y V ϕ 2 x v 1 x v 2 for all labels a : ϕ a x x V ϕ 3 Example 4. Cosider the costrait ϕ 0 x a x y x which is -closed up to trivial costraits ad which does ot etail x a y. I order to cotradict x a y we compute the -closure of Γ 1 ϕ 0 which is Γ 1 ϕ 0 x a x y x x a v xa y a v ya v ya v xa v xa x x v xa y v xa ad observe that it does ot v xa y. By defiitio of Γ 2, Γ 2 Γ 1 ϕ cotradicts v xa y. Hece, Γ 2 Γ 1 ϕ also cotradicts x a y.

8 Lemma 8. Let ϕ be a -closed (ad hece satisfiable) costrait. The Γ 1 ϕ is satisfiable ad its closure Γ 1 µ satisfies the followig two properties for all basic costraits µ: 1. If ϕ µ ad V µ V ϕ, the Γ 1 ϕ µ. 2. If ϕ x a y the Γ 1 ϕ y v xa or Γ 1 ϕ v xa y. Proof. The -closure Γ 1 ϕ of Γ 1 ϕ has the followig form up-to trivial costraits ad symmetry of compatibility costraits. Γ 1 ϕ Γ 1 ϕ v xa v ya Φ x y a L ϕ 4 1 z v xa exists y : Φ z y y a y y x Φ x a 4 2 v xa z exist y y : Φ x y y a y y z Φ x a 4 3 v xa z exist y y : Φ x y y a y y z Φ x a 4 4 v xa z exist y y : Φ x y y a y y z Φ x a 4 5 (For istace ote that v xa x x v xa i Γ 1 ϕ if x a x i ϕ by clauses (4.2, 4.3) ad reflexivity). All costraits i Γ 1 ϕ either belog to Γ 1 ϕ or a derived from it by axioms i. The -closedess of Γ 1 ϕ ca be proved by a somewhat tedious case distictio. The same holds for the two additioal properties of Γ 1 ϕ claimed. Lemma 9. If ϕ is -closed the Γ 2 ϕ is satisfiable. Proof. It is ot difficult to show that the costrait part of Γ 2 ϕ is -closed up to trivial costraits (x x ad x x) ad symmetric compatibility costraits. The critical bit is to check that the egated selectio costraits added i clause 1 of Γ 2 ϕ are cosistet. Let y y l x y i Γ 2 ϕ. We must show that Γ 2 ϕ y y l x y. Assume the coverse, Γ 2 ϕ y y l x y. The, by Corollary 27 i Sectio 8, there exist z ad z such that Γ 2 ϕ z y z l x z. By defiitio of Γ 2 ϕ we kow that z x. However, if Γ 2 ϕ x y ad hece (by defiitio of Γ 2 ) ϕ x y holds, clause 1 does ot apply. Thus y y l x y caot be cotaied i Γ 2 ϕ, i cotradictio to our assumptio. Lemma 10. Let ϕ be a FT -costrait ad let µbe a basic costrait of the form x y, x y, or a x (i.e., ot a selectio costrait). The Γ 2 ϕ µ if ad oly if ϕ µ. Proof. By ispectio of the defiitio of Γ 2 ϕ. Clause 1 cotradicts etailmet of x y by ϕ by forcig x to have a feature l x which y must ot have. Clause 2 cotradicts x y by forcig x ad y to have a commo feature l xy such that the subtrees of x ad y at l xy are icompatible. Clause 3 cotradicts a x for ay label by forcig x to be ulabeled (i.e., compatible with at least two trees with distict label). Defiitio 11 Saturatio. Let ϕ be a -closed costrait ad Γ 1 ϕ the -closure of Γ 1 ϕ which exists accordig to Lemma 8. The saturatio of ϕ is the formula Sat ϕ give by Sat ϕ Γ 2 Γ 1 ϕ.

9 Lemma 12. Let ϕ be a -closed costrait For all µ such that V µ V ϕ, ϕ µ implies Sat ϕ µ. Proof. Let Γ 1 ϕ the -closure of Γ 1 ϕ such that Sat ϕ Γ 2 Γ 1 ϕ. If ϕ µ the Γ 1 ϕ µ by Lemma 8.1. If µ is ot a selectio costrait, the Γ 2 Γ 1 ϕ µ by Lemma 10. Otherwise, let µ x a y. Hece, Γ 1 ϕ v xa y or Γ 1 ϕ y v xa by Lemma 8.2. By Lemma 10, either Γ 2 Γ 1 ϕ v xa y or Γ 2 Γ 1 ϕ y v xa holds. I both cases, Γ 2 Γ 1 ϕ µ follows. Propositio 13. The otios of etailmet ad of sytactic cotaimet coicide for basic costraits: If ϕ is -closed ad µ a basic costrait the ϕ µ iff ϕ µ. Proof. We assume ϕ µ ad show ϕ µ. (The coverse is correctess of sytactic cotaimet.) If V µ V ϕ the µ is of the form x x or x x such that ϕ µ. Otherwise, V µ V ϕ. If ϕ µ, the Sat ϕ µ sice Sat ϕ cotais ϕ. Moreover, Sat ϕ is satisfiable (Lemmas 8 ad 9) such that Sat ϕ µ. Hece, ϕ µ by Lemma 12. Theorem 14 Etailmet. Etailmet problems of the form ϕ cubic time. ϕ ca be tested i Proof. Let be the size of ϕ ϕ. To decide ϕ ϕ, first test whether ϕ is satisfiable. By Theorem 5 this ca be doe by computig the -closure ϕ of ϕ i time O 3. If this test fails the the etailmet test is trivial. Otherwise, from Lemma 12 we obtai ϕ µ if ϕ µ, ad hece that ϕ ϕ iff ϕ µ for all µ i ϕ. There are O such µ ad ϕ is of size O 2, hece, by Lemma 6, this is decidable i time O. The overall complexity sums up to O 3. has the idepedece prop- Theorem 15 Idepedece. The costrait system FT erty; i.e., for every 1 ad costraits ϕ ϕ 1 ϕ : if ϕ i 1 ϕ i the ϕ ϕ i for some i 1 Proof. Assume ϕ i 1 ϕ i. If ϕ is usatisfiable we are doe. Also, if ϕ ϕ i is osatisfiable for some j, the ϕ i 1 ϕ i iff ϕ i 1 i j ϕ i is. Now let ϕ ad ϕ ϕ i be satisfiable for all i ad let ϕ be -closed (wlog. by Prop. 2). If there exists i with ϕ µ for all µ sytactically cotaied by ϕ i, the ϕ ϕ i ad we are doe. Otherwise, for all i there exists µ i such that ϕ µ i. Lemma 12 yields Sat ϕ i 1 ϕ i. Sice Sat ϕ is satisfiable (Lemma 8) ad etails ϕ, this cotradicts our assumptio that ϕ i 1 ϕ i. Corollary 16 Negatio. The satisfiability of cojuctios of positive ad egative FT costraits ϕ ϕ 1 ϕ k ca be tested i time O 3 where is the size of the give cojuctio. Proof. If ϕ is o-satisfiable the ϕ i 1 ϕ i is trivially o-satisfiable. By Propositio 5, satisfiability of ϕ is decidable i time O 3. Now assume ϕ to be satisfiable. By the Idepedece Theorem 15, ϕ i 1 ϕ i is osatisfiable if ad oly if ϕ ϕ i for some i. By Lemma 12 this is equivalet to the existece of i such that for all µ if ϕ i µ the ϕ µ. Overall, there are O 2 cadidates µ to be tested for sytactic cotaimet ad O possible ϕ i. By Lemma 6, ϕ µ ca be tested i time O 1 such that the total complexity sums up to time O 3.

10 6 Weak Subsumptio Costraits We ext itroduce weak subsumptio costraits that are used i computatioal liguistics [7]. We show that their satisfiability problem is subsumed by the oe for FT. Sytax. We assume give a set C of costats c ad a set D of features d. We cosider the set of labels L C D. A weak subsumptio costrait η is a FT costrait of the followig form. η :: c x x d y x y x y η η Note that compatibility costraits do ot occur i [7]. We add them here to simplify our compariso. Sematics. We iterpret weak subsumptio costraits over the whole class of feature algebras with the iduced weak subsumptio orderig, which we will defie below. A feature algebra A over C ad D cosists of a set A that is called the domai of A, a uary relatio c A o A for every costat c C, ad a biary relatio d A o A for every feature d D, which satisfy the followig properties for all α α α A, costats c c 1 c 2 C, ad features d D: 1 if α d A α ad α d A α the α α 2 if c 1 α A ad c 2 α A the c 1 c 2 I the literature [22, 7] a slightly differet otio of feature algebras with costats has bee cosidered. We will give a formal compariso betwee the two otios at the ed of the preset sectio. Propositio 17. The structure F over L is a feature algebra over C ad D. Proof. The above properties follow from the axioms i iformatio orderig i FT (x y y x x y). ad the atisymmetry of the Give a feature algebra A, we defie the weak subsumptio orderig A as follows. A simulatio for A is a biary relatio Δ o the domai of A that satisfies the followig properties for all elemets α 1, α 2, α 1, α 2 of A s domai: 1 if α 1 Δα 2 c 1 α 1 A ad c 2 α 2 A the c 1 c 2 2 if α 1 Δα 2 α 1 d A α 1 ad α 2 d A α 2 the α 1 Δα 2 The weak subsumptio orderig A of A is the greatest simulatio relatio for A. The weak subsumptio relatio o A iduces a compatibility relatio A : α 1 A α 2 iff exists α such that α 1 A α ad α 2 A α A feature algebra A iduces a structure with the same sigature as F, i which is iterpreted as weak subsumptio orderig A, as A, c as c A, ad d as d A. Propositio 18 Dörre [8]. The structure F coicides with the structure iduced by the feature algebra defied by F.

11 Proof. It is sufficiet to prove that the weak subsumptio relatio of the feature algebra defied by F coicides with the iformatio orderig o F. The proof i the case for feature algebras with costats ca be foud i [8] o page 24 (Satz 6 ad Satz 7). There the algebra of feature trees has bee called algebra of path fuctios. A direct proof (additioal 5 lies) is omitted for lack of space. Theorem 19. A weak subsumptio costrait η is satisfiable over F is satisfiable over the structure iduced by some feature algebra A. if ad oly if η Proof. If η is satisfiable the it is satisfiable over the structure iduced by the feature algebra defied by F. Coversely, every structure iduced by a feature algebra is a model of the axioms i. Thus, if η is satisfiable over oe such structure the it is equivalet to a -closed costrait (ad ot false) ad hece satisfiable over F. Alterative Notios of Feature Algebras. I the literature [22, 7] a restricted otio of feature algebra has bee cosidered that we call feature algebra with costats i the sequel. The focus o feature algebras with costats leads to a restricted satisfiability problem. This shows that the preseted results properly exted the results i [7]. A feature algebra with costats is a feature algebra with the additioal property that if c α A the ot α d A α (1) I order to hadle the ew property we cosider the followig mappig of weak subsumptio costraits over C ad D to weak subsumptio costraits over C ad D where is a ew costat ot cotaied i C. c x y x y c y x d y x d y x y x y x y x y η η η η Propositio 20. A costrait η is satisfiable i some feature algebra if ad oly if η is satisfiable i some feature algebra with costats. Proof. If η is satisfiable over a feature algebra A with costats C ad features D the η is satisfiable over the feature algebra F with labels C D. Give a solutio σ of η over A a solutio σ of η over F ca be defied as follows: D σ x p exists α i domai of A: σ x p A α ad p D L σ x p c exists α i domai of A : σ x p A α ad c α A Coversely, let η be satisfiable i a feature algebra A. The η is satisfiable i F by Theorem 19. We cosider the followig feature algebra with costats F ad show that η is satisfiable over F. The costats ad features of F are C ad D, respectively. The domai of F cotais all feature trees τ without labeled iteral odes where a labeled iteral ode of τ is a path p such that p D τ, exists c with p c L τ, but ot exists d with pd D τ. The selectio ad labelig relatios of F are those of FT restricted to trees without iteral labels. Obviously, F satisfies all three axioms of a feature algebra with costats. Now let σ be a A-solutio of η. The the variable assigmet σ mappig x o σ x as give below is a F -solutio of η. D σ x D σ x p exists a L : p a L σ x L σ x p a p a L σ x

12 7 Expressiveess We show that FT is strictly more expressive tha FT but that FT caot express a arity costrait. A FT costrait η is of the form x y, a x, x a y, or η η, ad a arity costrait of the form x a 1 a. A arity costrait x a 1 a holds if x deotes a tree with subtrees at exactly a 1 through a. Propositio 21. There is o FT the empty feature tree, i.e., if a costrait which expresses that a variable x deotes b the there is o costrait equivalet to x y z x y x z a y b z Proof. If ϕ were such a FT costrait, the ϕ as well as its fiite -closure would etail x y for all variables y. This cotradicts Propositio 13 for all those y such that y V ϕ ad x y (because if ϕ x y the x y or x y V ϕ ). Such a variable y exists sice V ϕ is fiite. Lemma 22. Let η be a FT costrait. The η x y if ad oly if η y x. Proof. The FT costrait η is equivalet to the FT costrait ϕ obtaied from η by replacig all equalities x y by iequalities x y y x. Hece, x y i ϕ iff y x i ϕ, ad sice algorithm preserves this ivariat it also holds for the -closure of ϕ. The claim follows from Propositio 13. Propositio 23. If x y the there is o FT costrait η equivalet to x y. Proof. This follows immediately from Lemma 22 ad Propositio Completeess of the Satisfiability Test Propositio 4. Every -closed costrait ϕ is satisfiable over FT. The proof is based o the otio of path reachability ad covers the rest of the sectio. We proceed as follows. We first defie path reachability, the give two Lemmas, ad fially compose the proof of Propositio 4 from these Lemmas. For all paths p ad costrait ϕ, we defie a biary relatio ϕ p, where x ϕ p y reads as y is reachable from x over path p i ϕ : x ϕ ε y if y x i ϕ x ϕ a y if x a y i ϕ x ϕ pq y if x ϕ p z ad z ϕ q y Defie relatioships x ϕ p a meaig that a ca be reached from x over path p i ϕ : x ϕ p a if x ϕ p y ad a y i ϕ

13 For example, if ϕ is the costrait x y a y x a u x b z z a x b z the the followig reachability propositios hold: y ϕ ε x, x ϕ b z, x ϕ ba y, x ϕ ba x, etc., as well as x ϕ ε a, x ϕ b b, x ϕ ba a, etc. Defiitio 24 Path Cosistecy. We call a costrait ϕ path cosistet if the followig two coditios hold for all x, y, p, a, ad b. 1. If x ϕ p a, x x, ad x ϕ p b the a b. 2. If x ϕ p a, x y, ad y ϕ p b the a b. Lemma 25. Every - -closed ad path cosistet costrait is satisfiable. Proof. Let ϕ be - -closed ad path cosistet. We defie the variable assigmet ito feature trees as follows: ϕ D ϕ x p x ϕ p y ad L ϕ x p a x ϕ p a The path cosistecy of ϕ coditio 1 implies that L ϕ x is a partial fuctio. Thus ϕ x is a feature tree. We ow verify that ϕ is a solutio of ϕ. Let x y i ϕ. For all x, if y ϕ p x the x ϕ p x by the defiitio of path reachability. Thus, D ϕ y D ϕ x. For all a if y ϕ p a the x ϕ p a by the defiitio of path reachability. Thus, L ϕ y L ϕ x, i.e., ϕ y ϕ x. Cosider x a y i ϕ. We have to prove for all p, z, ad b the equivaleces x ϕ ap z iff y ϕ p z ad x ϕ ap b iff y ϕ p b The first equivalece is equivalet to D ϕ y p ap D ϕ x ad the secod oe to L ϕ y p b ap b L ϕ x. We start provig the first equivalece. If y ϕ p z the x ϕ ap z sice x a y i ϕ. Suppose x ϕ ap z. By defiitio of path reachability there exists x ad y such that x ϕ ε x x a y y ϕ p z The -closedess of ϕ ad x ϕ ε x imply x x i ϕ. The -closedess esures y y i ϕ such that y ϕ p z holds. We ow prove the secod equivalece above. If x ϕ ap b the there exists z such that x ϕ ap z ad b z. The first equivalece implies y ϕ p z ad thus y ϕ p b. The coverse is simple. Let a x i ϕ. Reflexivity ( 1-closedess) implies x x i ϕ. Thus x ϕ ε a such that ε a L x. Let x y i ϕ. We have to show that the set L ϕ x L ϕ y is partial fuctio. If p a L ϕ x ad p b L ϕ y the x ϕ p a ad y ϕ p b. The path cosistecy of ϕ coditio 2 implies a b.

14 Lemma 26. Every -closed costrait is path cosistet. Proof. Let ϕ be -closed. Coditio 1 of Defiitio 24 follows from coditio 2 of Defiitio 24 ad 1-closedess. The proof of coditio 2 is by iductio o paths p. We assume x, y, a, ad b such that x ϕ p a, x y i ϕ, ad x ϕ p b. If p ε, the there exist m 0, x 1 x, y 1 y m such that: x x 1 x 1 x a x i ϕ y y 1 y m 1 y m b y m i ϕ -closedess implies that x y m i ϕ ( 2 yields x y 1 i ϕ,, x y m i ϕ. Therefore y m x i ϕ by 3-closedess, ad hece y m x 1 i ϕ,, y m x i ϕ by 2- closedess.) Hece, -closedess implies a b. I the case p a q, the there exists there exist x, y, x,ỹ with: x ϕ ε x x a x i ϕ x ϕ p a y ϕ ε y y a ỹ i ϕ ỹ ϕ p b Sice x x i ϕ we have x y i ϕ by -closedess (as above). Thus, -closedess implies x ỹ i ϕ such that a b holds by iductio hypothesis. Proof of Propositio 4. If ϕ is thus satisfiable by Lemma 25. -closed the ϕ is path cosistet by Lemma 26 ad Corollary 27. Let ϕ be a -closed costrait. The ϕ y x a y if ad oly if there are variables x ad y such that ϕ x a y ad ϕ x x. Proof. Assume ϕ x a y x x. The it holds for the miimal solutio ϕ of a -closed costrait that a L ϕ y. Hece ϕ y x a y.. Ackowledgmets. We would like to thak Joche Dörre, Gert Smolka, ad Ralf Treie for discussios o the topic of this paper. We would also like to ackowledge may helpful remarks of the referees. The research reported i this paper has bee supported by the the Esprit Workig Group CCL II (EP 22457) the SFB 378 at the Uiversität des Saarlades. Refereces 1. H. Aït-Kaci ad A. Podelski. Etailmet ad Disetailmet of Order-Sorted Feature Costraits. I A. Vorokov, editor, 4 th Iteratioal Coferece o Logic Programmig ad Automated Reasoig, LNAI 698, pp Spriger, H. Aït-Kaci ad A. Podelski. Towards a Meaig of Life. The Joural of Logic Programmig, 16(3 ad 4): , July, Aug H. Aït-Kaci, A. Podelski, ad G. Smolka. A feature-based costrait system for logic programmig with etailmet. Theoretical Computer Sciece, 122(1 2): , Ja

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