Comparative concept similarity over Minspaces: Axiomatisation and Tableaux Calculus
We study the logic of comparative concept similarity $\CSL$ introduced by Sheremet, Tishkovsky, Wolter and Zakharyaschev to capture a form of qualitative similarity comparison. In this logic we can formulate assertions of the form " objects A are mor…
Authors: Regis Alenda (LSIS), Nicola Olivetti (LSIS), Camilla Schwind (LIF)
Comparativ e onept similarit y o v er Minspaes: Axiomatisation and T ableaux Calulus Régis Alenda 1 , Niola Oliv etti 1 , and Camilla S h wind 2 1 LSIS - UMR CNRS 6168 Domaine Univ ersitaire de Sain t-Jérme, A v en ue Esadrille Normandie-Niemen , 13397 MARSEILLE CEDEX 20 regis.alendalsis.org et niola.olivettiuniv-ezanne .fr 2 LIF - UMR CNRS 6166 Cen tre de Mathématiques et Informatique 39 rue Joliot-Curie - F-13453 Marseille Cedex13. amilla.shwindlif.univ-mrs.f r Abstrat. W e study the logi of omparativ e onept similarit y C S L in tro dued b y Sheremet, Tishk o vsky , W olter and Zakhary as hev to ap- ture a form of qualitativ e similarit y omparison. In this logi w e an form ulate assertions of the form " ob jets A are more similar to B than to C". The seman tis of this logi is dened b y strutures equipp ed with distane funtions ev aluating the similarit y degree of ob jets. W e on- sider here the partiular ase of the seman tis indued b y minsp a es , the latter b eing distane spaes where the minim um of a set of distanes alw a ys exists. It turns out that the seman tis o v er arbitrary minspaes an b e equiv alen tly sp eied in terms of preferen tial strutures, t ypial of onditional logis. W e rst giv e a diret axiomatisation of this logi o v er Minspaes. W e next dene a deision pro edure in the form of a tableaux alulus. Both the alulus and the axiomatisation tak e adv an tage of the reform ulation of the seman tis in terms of preferen tial strutures. 1 In tro dution The logis of omparativ e onept similarit y C S L ha v e b een prop osed b y Sheremet, Tishk o vsky , W olter et Zakhary as hev in [8℄ to apture a form of qualitativ e om- parison b et w een onept instanes. In these logis w e an express assertions or judgmen ts of the form: "Renault Clio is more similar to P eugeot 207 than to WW Golf". These logis ma y nd an appliation in on tology languages, whose logial base is pro vided b y Desription Logis (DL), allo wing onept denitions based on pro ximit y/similarit y measures. F or instane [8℄, the olor "Reddish " ma y b e dened as a olor whi h is more similar to a protot ypial "`Red"' than to an y other olor (in some olor mo del as R GB). The aim is to disp ose of a language in whi h logial lassiation pro vided b y standard DL is in tegrated with lassiation me hanisms based on alulation of pro ximit y measures. The latter is t ypial for instane of domains lik e bio-informatis or linguistis. In a series of pap ers [8, 10, 5, 9℄ the authors prop oses sev eral languages omprising ab- solute similarit y measures and omparativ e similarit y op erator(s). In this pap er w e onsider a logi C S L obtained b y adding to a prop ositional language just one binary mo dal onnetiv e ⇔ expressing omparativ e similarit y . In this language the ab o v e examples an b e eno ded (using a desription logi notation) b y: (1) Reddish ≡ { Red } ⇔ { Gr een, . . . , black } (2) C l io ⊑ ( P eug eot 207 ⇔ Gol f ) In a more general setting, the language migh t on tain sev eral ⇔ F eature where ea h F eatur e orresp onds to a sp ei distane funtion d F eature measuring the similarit y of ob jets with resp et to one F eatur e (size, prie, p o w er, taste, olor...). In our setting a KB ab out ars ma y ollet assertions of the form (2) and others, sa y: (3) C l io ⊑ ( Gol f ⇔ F e rr ar i 430) (4) C l io ⊑ ( P eug eot 2 07 ⇔ M aser atiQP ) together with some general axioms for lassifying ars: P eug eot 207 ⊑ C i ty car S por tLuxur y C ar ≡ M a s erati QP ⊔ F er rar i 43 0 Comparativ e similarit y assertions su h as (2)(4) migh t not neessarily b e the fruit of an ob jetiv e n umerial alulation of similarit y measures, but they ould b e determined just b y the (in tegration of ) sub jetiv e opinions of agen ts, answ er- ing, for instane, to questions lik e: "Is Clio more similar to Golf or to F errari 430?"'. In an y ase, the logi C S L allo ws one to p erform some kind of reasoning, for instane the follo wing onlusions will b e supp orted: C l io ⊑ ( P eug eot 2 07 ⇔ F err ar i 430 ) C l io ⊑ ( C ity car ⇔ S por tLuxur y C ar ) and also C l io ⊑ ( C ity car ⇔ S por tLuxur y C ar ⊓ 4 W heel s ) . The seman tis of C S L is dened in terms of distane spaes, that is to sa y strutures equipp ed b y a distane funtion d , whose prop erties ma y v ary aord- ing to the logi under onsideration. In this setting, the ev aluation of A ⇔ B an b e informally stated as follo ws: x ∈ A ⇔ B i d ( x, A ) < d ( x, B ) meaning that the ob jet x is an instane of the onept A ⇔ B (i.e. it b elongs to things that are more similar to A than to B ) if x is stritly loser to A -ob jets than to B -ob jets aording to distane funtion d , where the distane of an ob jet to a set of ob jets is dened as the inmum of the distanes to ea h ob jet in the set. In [8, 10, 5, 9℄, the authors ha v e in v estigated the logi C S L with resp et to dieren t lasses of distane mo dels, see [10℄ for a surv ey of results ab out de- idabilit y , omplexit y , expressivit y , and axiomatisation. Remark ably it is sho wn that C S L is undeidable o v er subspaes of the reals. Moreo v er C S L o v er arbi- trary distane spaes an b e seen as a fragmen t, indeed a p o w erful one (inluding for instane the logi S4 u of top ologial spaes), of a general logi for spatial reasoning omprising dieren t mo dal op erators dened b y (b ounded) quan tied distane expressions. The authors ha v e p oin ted out that in ase the distane spaes are assumed to b e minsp a es , that is spaes where the inm um of a set of distanes is atually their minimum , the logi C S L is naturally related to some onditional logis. The seman tis of the latter is often expressed in terms of preferen tial strutures, that is to sa y p ossible-w orld strutures equipp ed b y a family of strit partial (pre)-orders ≺ x indexed on ob jets/w orlds [6, 11℄. The in tended meaning of the relation y ≺ x z is namely that x is more similar to y than to z . It is not hard to see that the seman tis o v er minspaes is equiv alen t to the seman tis o v er preferen tial strutures satisfying the w ell-kno wn priniple of Limit Assumption aording to whi h the set of minimal elemen ts of a non-empt y set alw a ys exists. The minspae prop ert y en tails the restrition to spaes where the distane funtion is disrete. This requiremen t do es not seem inompatible with the pur- p ose of represen ting qualitativ e similarit y omparisons, whereas it migh t not b e reasonable for appliations of C S L to spatial reasoning. In this pap er w e on tribute to the study of C S L o v er minspaes. W e rst sho w (unsurprisingly) that the seman tis of C S L on minspaes an b e equiv a- len tly restated in terms of preferen tial mo dels satisfying some additional on- ditions, namely mo dularit y , en tering, and limit assumption. W e then giv e a dir e t axiomatization of this logi. This problem w as not onsidered in detail in [10℄. In that pap er an axiomatization of CSL o v er arbitrary distane mo dels is prop osed, but it mak es use of an additional op erator. Our axiomatisation is simpler and only emplo ys ⇔ . Next, w e dene a tableaux alulus for he king satisabilit y of C S L form ulas. Our tableaux pro edure mak es use of lab elled for- m ulas and pseudo-mo dalities indexed on w orlds x , similarly to the aluli for onditional logis dened in [2, 3℄. T ermination is assured b y suitable blo king onditions. T o the b est of our kno wledge our alulus pro vides the rst kno wn pratially-implemen table deision pro edure for C S L logi. 2 The logi of Comp ar ative Con ept Similarity C S L The language L C S L of C S L is generated from a set of prop ositional v ariables V i b y ordinary prop ositional onnetiv es plus ⇔ : A, B ::= V i | ¬ A | A ⊓ B | A ⇔ B . The seman tis of C S L in tro dued in [8℄ mak es use of distan e sp a es in order to represen t the similarit y degree b et w een ob jets. A distane spae is a pair ( ∆, d ) where ∆ is a non-empt y set, and d : ∆ × ∆ → R ≥ 0 is a distan e funtion satisfying the follo wing ondition: (ID) ∀ x, y ∈ ∆, d ( x, y ) = 0 i x = y T w o further prop erties are usually onsidered: symmetry and triangle inequalit y . W e briey disuss them b elo w. The distane b et w een an ob jet w and a non-empt y subset X of ∆ is dened b y d ( w, X ) = inf { d ( w , x ) | x ∈ X } . If X = ∅ , then d ( w, X ) = ∞ . If for ev ery ob jet w and for ev ery (non-empt y) subset X w e ha v e the follo wing prop ert y (MIN) inf { d ( w, x ) | x ∈ X } = min { d ( w , x ) | x ∈ X } , w e will sa y that ( ∆, d ) is a minsp a e . W e next dene C S L -distane mo dels as Kripk e mo dels based on distane spaes: Denition 1 ( C S L -distane mo del). A C S L -distan e mo del is a triple M = ( ∆, d, . M ) wher e: ∆ is a non-empty set of ob jets . d is a distan e on ∆ M (so that ( ∆, d ) is a distan e sp a e). . M : V p → 2 ∆ is the ev aluation funtion whih assigns to e ah pr op ositional variable V i a set V M i ⊆ ∆ . W e further stipulate: ⊥ M = ∅ ( ¬ C ) M = ∆ − C M ( C ⊓ D ) M = C M ∩ D M ( C ⇔ D ) M = w ∈ ∆ d ( w, C M ) < d ( w , D M ) . If ( ∆, d ) is a minsp a e, M is al le d a C S L -distane minspae mo del (or simply a minspae mo del ). W e say that a formula A is v alid in a mo del M if A M = ∆ . W e say that a formula A is v alid if A is valid in every C S L -distan e mo del. As men tioned ab o v e, the distane funtion migh t b e required to satisfy the further onditions of symmetry ( S Y M ) ( d ( x, y ) = d ( y , x ) ) and triangular in- equalit y ( T R ) ( d ( x, z ) ≤ d ( x, y ) + d ( y , z ) ). It turns out that C S L annot distin- guish b et w een minspae mo dels whi h satisfy ( T R ) from mo dels whi h do not. In on trast [8℄, C S L has enough expressiv e p o w er in order to distinguish b et w een symmetri and non-symmetri minspae mo dels. As a rst step, w e onen trate here on the general non-symmetri ase, lea ving the in teresting symmetri ase to further resear h. C S L is a logi of pure qualitativ e omparisons. This motiv ates an alternativ e seman tis where the distane funtion is replaed b y a family of omparisons relations, one for ea h ob jet. W e all this seman tis pr efer ential seman tis, similarly to the seman tis of onditional logis [7, 6℄. Preferen tial strutures are equipp ed b y a family of strit pre-orders. W e ma y in terpret this relations as expressing a omparativ e similarit y b et w een ob jets. F or three ob jets, x ≺ w y states that w is more similar to x than to y . The preferen tial seman tis in itself is more general than distane mo del se- man tis. Ho w ev er, if w e assume the additional onditions of the denition 2, it turns out that these t w o are equiv alen t (theorem 4). Denition 2. W e wil l say that a pr efer ential r elation ≺ w over ∆ : (i) is mo dular i ∀ x, y , z ∈ ∆ , ( x ≺ w y ) → ( z ≺ w y ∨ x ≺ w z ) . (ii) is en tered i ∀ x ∈ ∆ , x = w ∨ w ≺ w x . (iii) satises the Limit Assumption i ∀ X ⊆ ∆ , X 6 = ∅ → min ≺ w ( X ) 6 = ∅ . 3 wher e min ≺ w ( X ) = { y ∈ X | ∀ z ∈ ∆ ( z ≺ w y → z / ∈ X ) } . 3 W e note that the Limit Assumption implies that the preferen tial relation is asym- metri. On the other hand, on a nite set, asymmetry implies Limit Assumption. Mo dularit y and asymmetry imply that this relation is also transitiv e and irreexiv e. Mo dularit y is strongly related to the fat that the preferen tial relations repre- sen ts distane omparisons. This is the k ey prop ert y to enfore the equiv alene with distane mo dels. Cen tering states that w is the unique minimal elemen t for its preferen tial relation ≺ w , and an b e seen as the preferen tial oun terpart of (ID) . The Limit Assumption states that ea h non-empt y set has at least one minimal elemen t wrt. a preferen tial relation (i.e it do es not on tain an innitely desending hain), and orresp onds to (MIN) . Denition 3 ( C S L -preferen tial mo del). A C S L -pr efer ential mo del is a triple M = ( ∆, ( ≺ w ) w ∈ ∆ , . M ) wher e: ∆ M is a non-empty set of ob jets (or p ossible w orlds ). ( ≺ w ) w ∈ ∆ is a family of preferen tial relation , e ah one b eing mo dular, en- ter e d, and satisfying the limit assumption . . M is the evaluation funtion dene d as in denition 1, ex ept for ⇔ : ( A ⇔ B ) M = w ∈ ∆ ∃ x ∈ A M suh that ∀ y ∈ B M , x ≺ w y V alidity is dene d as in denition 1. W e no w sho w the equiv alene b et w een preferen tial mo dels and distane minspae mo dels. W e sa y that a C S L -preferen tial mo del I and a C S L -distane minspae mo del J are e quivalent i they are based on the same set ∆ , and for all form ulas A ∈ L C S L , A I = A J . Theorem 4 (Equiv alene b et w een C S L -preferen tial mo dels and C S L - distane mo dels). 1. F or e ah C S L -distan e minsp a e mo del, ther e is an e quivalent C S L -pr efer ential mo del. 2. F or e ah C S L -pr efer ential mo del, ther e is an e quivalent C S L -distan e minsp a e mo del. Pr o of. 1. ([8℄): giv en I = ( ∆ I , d, . I ) a C S L -distane minspae mo del, just de- ne a preferen tial mo del J b y stipulating x ≺ w y i d ( w, x ) < d ( w, y ) , and for all prop ositional v ariable V i , V J i = V I i . It is to he k that ≺ w is mo dular, en tered, and satises the limit assumption, and that I and J are equiv alen t. 2. Sine the relation ≺ w is mo dular, w e an assume that there exists a r anking funtion r w : ∆ → R ≥ 0 su h that x ≺ w y i r w ( x ) < r w ( y ) . Therefore, giv en a C S L -preferen tial mo del J = ( ∆ J , ( ≺ w ) w ∈ ∆ J , . J ) , w e an dene a C S L - distane minspae mo del I = ( ∆ J , d, . J ) , where the distane funtion d is dened as follo w: if w = x then d ( w, x ) = 0 , and d ( w, x ) = r w ( x ) otherwise. W e an easily he k that (i) I is a minspae b eause of the limit assumption, and that (ii) I and J are equiv alen t; this is pro v ed b y indution on the omplexit y of form ulas. W e ha v e men tioned the relation with onditional logis. These logis, orig- inally in tro dued b y Lewis and Stalnak er [6, 11℄, on tain a onnetiv e A > B whose reading is appro ximativ ely "`if A w ere true then B w ould also b e true"' 4 . The idea is that a w orld/state x v eries A > B if B holds in all states y that are most similar to x that is: x ∈ A > B M i min ≺ x ( A M ) ⊆ B M The t w o onnetiv es ⇔ are in terdenable as sho wn in [8℄: A > B ≡ ( A ⇔ ( A ∧ ¬ B )) ∨ ¬ ( A ⇔ ⊥ ) A ⇔ B ≡ (( A ∨ B ) > A ) ∧ ( A > ¬ B ) ∧ ¬ ( A > ⊥ ) By means of this equiv alene, an (indiret) axiomatization of ⇔ an b e obtained: just tak e an axiomatization of the suitable onditional logi (w ell kno wn) and add the denition ab o v e. On the other hand an axiomatisation of C S L o v er ar- bitrary distane mo dels is presen ted in [10℄, ho w ev er it mak es use of an extended language, as w e ommen t b elo w. Moreo v er, the ase of minspaes has not b een studied in details. Our axiomatisation is on tained in g. 1. The axioms (1) and (1) ¬ ( A ⇔ B ) ⊔ ¬ ( B ⇔ A ) (2) ( A ⇔ B ) → ( A ⇔ C ) ⊔ ( C ⇔ B ) (3) A ⊓ ¬ B → ( A ⇔ B ) (4) ( A ⇔ B ) → ¬ B (5) ( A ⇔ B ) ⊓ ( A ⇔ C ) → ( A ⇔ ( B ⊔ C )) (6) ( A ⇔ ⊥ ) → ¬ ( ¬ ( A ⇔ ⊥ ) ⇔ ⊥ ) ( M on ) ⊢ ( A → B ) ⊢ ( A ⇔ C ) → ( B ⇔ C ) ( T aut ) Classial tautologies and rules. Fig. 1. CSMS axioms. (2) apture resp etiv ely the asymmetry and mo dularit y of the preferene rela- tions, whereas (3) and (4) eno de en tering and the minspae prop ert y . By (5) , w e obtain that ⇔ distributes o v er disjuntion on the seond argumen t, sine the opp osite diretion is deriv able. The axiom (6) is similar to axiom (33) of the ax- iomatization in [10℄, it sa ys that the mo dalit y ♦ A ≡ A ⇔ ⊥ has the prop erties of S5. Finally , the rule ( M on ) states the monotoniit y of ⇔ in the rst argumen t, a dual rule stating the an ti-monotoniit y in the seond argumen t is deriv able as w ell. The axiomatisation of C S L pro vided in [10℄ for arbitrary distane spaes mak es use of the op erator ◦ R A that, referring to preferen tial mo dels, selets 4 T o this regard, in alternativ e to the onept/subset in terpretation men tioned so far, the form ula A ⇔ B ma y p erhaps b e read as "` A is (stritly) more plausible than B "'. This in terpretation ma y in tuitiv ely explain the relation with the onditional op erator. elemen ts x for whi h min ≺ x ( A ) is non-empt y . As observ ed in [10℄, an axioma- tization of C S L o v er minspaes an then b e obtained b y just adding the axiom ◦ R A ↔ ( A ⇔ ⊥ ) . Ho w ev er our axiomatization is signian tly simpler (almost one half of the axioms). W e an sho w that our axiomatization is sound and omplete with resp et to the preferen tial seman tis, whene wrt minspae mo dels (b y theorem 4). Theorem 5. A formula is derivable in CSMS i it is valid in every C S L - pr efer ential mo del. The follo wing theorems and inferene rule are deriv able from the axioms: T1 A → ( A ⇔ ⊥ ) b y (3) T2 ¬ ( A ⇔ A ) b y (1) T3 ¬ ( A ⇔ ⊤ ) b y (2) T4 (( A ⇔ ⊥ ) ⇔ ⊥ ) → ( A ⇔ ⊥ ) b y T1 and (6) T5 ( A ⇔ B ) ⊓ ( B ⇔ C ) → ( A ⇔ C ) b y (1) and (4) T6 ∀ n > 0 , ⊢ ( A ⇔ B 1 ) ⊓ . . . ⊓ ( A ⇔ B n ) → ( A ⇔ ( B 1 ⊔ . . . ⊔ B n )) b y indution o v er n and (5) T7 ∀ n > 0 , ⊢ ( A ⇔ B 1 ) ⊓ . . . ⊓ ( A ⇔ B n ) → (( A ⊓ ¬ B 1 ⊓ . . . ⊓ ¬ B n ) ⇔ ( B 1 ⊔ . . . ⊔ B n )) R1 If ⊢ ( A → B ) then ⊢ ( C ⇔ B ) → ( C ⇔ A ) b y (1) and RM Theorem (T1) orresp onds to the T -axiom A → ♦ A ; axiom (6) is the S5 axiom (Eulidean) ♦ A → ♦ A . Hene making use of (2) and (6) w e an deriv e the S4 axiom (T4). W e an sho w that our axiomatisation is sound and omplete with resp et to the preferen tial seman tis in tro dued ab o v e. Theorem 6 (Soundness of CSMS ). If a formula is derivable in CSMS , then it is CSMS -valid. Theorem 7 (Completeness of CSMS ). If a formula is CSMS − valid, then it is derivable in CSMS . Soundness is straigh tforw ard. W e sho w that ev ery axiom is C S L -v alid. The ompleteness is sho wn b y the onstrution of a anonial mo del. W e dene onsisten t and maximal onsisten t form ula sets in the usual w a y: Denition 8. A set of formulas Γ is al le d inonsisten t with r esp e t to CSMS i ther e is a nite subset of Γ , { A 1 , . . . A n } suh that ⊢ CSMS ¬ A 1 ⊔ ¬ A 2 ⊔ . . . ¬ A n . Γ is al le d onsisten t if Γ is not in onsistent. If an (in) onsistent Γ ontains only one formula A , we say that A is (in) onsistent. A set of formulas Γ is al le d maximal onsisten t i it is onsistent and if for any formula A not in Γ , Γ ∪ { A } is in onsistent. W e will use prop erties of maximal onsisten t sets, the pro ofs of whi h an b e found in most textb o oks of logi. In partiular: Lemma 9. Every onsistent set of formulas is ontaine d in a maximal onsis- tent set of formulas. Lemma 10. L et w b e a maximal onsistent set of formulas and A , B formulas in L C S L . Then w has the fol lowing pr op erties: 1. If ⊢ CSMS A → B and A ∈ w , then B ∈ w 2. If fr om A ∈ w we infer B ∈ w , then A → B ∈ w . 3. A ⊓ B ∈ w i A ∈ w and B ∈ w 4. A 6∈ w i ¬ A ∈ w Let U b e the set of all maximal onsisten t sets. Denition 11. L et x, y b e maximal onsistent formula sets and A, B b e L C S L - formulas. W e dene 1 R ( x, y ) i ∀ A ∈ L C S L if A ∈ y then ( A ⇔ ⊥ ) ∈ x 2 w A = {¬ B | ( A ⇔ B ) ∈ w } Pr op erty 12. R is an equiv alene relation. Pr o of. 1 R is reexiv e b y T 1 2 R is transitiv e b y T 4 3 R is symmetri b y axiom (6) F or x ∈ U , w e note ˜ x the equiv alene lass of x with resp et to R . The follo wing prop erties hold for w A : Lemma 13. 1. If { A } is onsistent, then w A ∪ { A } is onsistent. 2. ( A ⇔ B ) ∈ w i ∀ x if w A ⊆ x then ¬ B ∈ x . 3. w A ⊆ w 4. If { A } is onsistent and ( A ⇔ ⊥ ) ∈ w then ∃ z ∈ ∆ suh that A ∈ z and w A ⊆ z Pr o of. 1. Supp ose that w A ∪{ A } is in onsistent. Then ther e ar e formulas ¬ B 1 , . . . ¬ B n ∈ w A suh that ⊢ CSMS B 1 ⊔ · · · ⊔ B n ⊔ ¬ A . W e an then derive (i) ⊢ CSMS A → B 1 ⊔ · · · ⊔ B n (ii) ⊢ CSMS ( A ⇔ ( B 1 ⊔ · · · ⊔ B n )) → ( A ⇔ A ) fr om (i) by R1 (iii) ⊢ CSMS ¬ (( A ⇔ B 1 ) ⊓ · · · ⊓ ( A ⇔ B n )) (fr om (ii) by T2 and axiom (5) Contr adition with the onsisteny of w sin e al l ( A ⇔ B i ) ∈ w . 2. ⇒ imme diately by denition of w A ⇐ W e rst show that w A ∪ { B } is in onsistent. Supp ose that this is not the ase. Then ther e is z ∈ U and w A ∪ { B } ⊆ z . Hen e ¬ B ∈ z , fr om the pr e ondition. But B ∈ z , ontr aditing the onsisteny of z . Sin e w A ∪ { B } is in onsistent, ther e ar e formulas ¬ B 1 , . . . ¬ B n ∈ w A and ⊢ CSMS ¬ B 1 ⊓ · · · ⊓ ¬ B n → ¬ B . W e an then derive (i) ⊢ CSMS B → B 1 ⊔ · · · ⊔ B n (ii) ⊢ CSMS ( A ⇔ ( B 1 ⊔ · · · ⊔ B n ) → ( A ⇔ B ) fr om (i) by R1 (iii) ⊢ CSMS (( A ⇔ B 1 ) ⊓ ( A ⇔ B 1 ) · · · ⊓ ( A ⇔ B n )) → ( A ⇔ B ) fr om (ii) by (T6) Sin e ( A ⇔ B i ) ∈ w , we onlude ( A ⇔ B ) ∈ w 3. imme diate by axiom (2). 4. By 1, we have that w A ∪ { A } is onsistent, hen e is is ontaine d in a maximal onsistent formula set z ∈ U by lemma 9. W e show then that w A is ontaine d in a set x ∈ ∆ W e show that for al l x ∈ U , if w A ⊆ x and ( A ⇔ ⊥ ) ∈ w , then R ( w , x ) . W e have ∀ C ∈ w , ( C ⇔ ⊥ ) ∈ w , b e ause of the r eexivity of R . By axiom (4) we have ( A ⇔ ⊥ ) → ( A ⇔ ¬ ( C ⇔ ⊥ )) ⊔ ( ¬ ( C ⇔ ⊥ ) ⇔ ⊥ ) . By axiom (6), we obtain ( ¬ ( C ⇔ ⊥ ) ⇔ ⊥ ) 6∈ w , sin e ( C ⇔ ⊥ ) ∈ w . Hen e ( A ⇔ ¬ ( C ⇔ ⊥ )) ∈ w . This entails ( C ⇔ ⊥ ) ∈ w A and ther efor e ( C ⇔ ⊥ ) ∈ x . This me ans that we have R ( w , x ) . Hen e we have x ∈ ∆ and w A ⊆ x . W e no w are in a p osition to dene the anonial mo del. Denition 14 (Canonial Mo del). Sin e C is not derivable in CSMS , ¬ C is onsistent, and so ther e is a maximal onsistent set z ∈ U suh that ¬ C ∈ z . W e dene the anoni al mo del M C = ( ∆, ( ≺ w ) w ∈ ∆ , . M C ) as fol lows: ∆ = ˜ z . x ≺ w y i ther e exists a formula B ∈ y suh that for al l formulas A ∈ x , ( A ⇔ B ) ∈ w . V M C i = { x ∈ ∆ | V i ∈ x } , for al l pr op ositional variables V i . F or A ∈ L C S L , w e dene the set of ob jets in ∆ on taining A b y k A k = { z | A ∈ z ∩ ∆ } . Lemma 15. F or e ah w ∈ ∆ , ≺ w is enter e d and mo dular. Pr o of. 1 ≺ w is enter e d. L et b e x 6 = y . Then ther e is (i) B ∈ y and B 6∈ x , i.e. ¬ B ∈ x . L et b e any L C S L -formula A with A ∈ x . Then A ⊓ ¬ B ∈ x fr om whih fol lows that (ii) ( A ⇔ B ) ∈ x by axiom (3). F r om (i) and (ii) we obtain x ≺ x y . 2 ≺ w is mo dular. L et b e x ≺ w y and supp ose ther e is u ∈ ∆ suh that x 6≺ w u and u 6≺ w y . W e get then: (i) ∃ B ∈ y suh that ∀ A ∈ x, ( A ⇔ B ) ∈ w (ii) ∀ C ∈ u ∃ A ′ ∈ x suh that ¬ ( A ′ ⇔ C ) ∈ w and (iii) ∀ B ′ ∈ y ∃ C ′ ∈ u suh that ¬ ( C ′ ⇔ B ′ ) ∈ w Then we have ( A ′ ⇔ B ) ∈ w fr om (1), ¬ ( A ′ ⇔ C ′ ) ∈ w fr om (ii) and ¬ ( C ′ ⇔ B ) ∈ w fr om (iii). By axiom (1) and (ii), we get ( C ⇔ A ′ ) ∈ w and by tr ansitivity (T5) and (i), we obtain ( C ′ ⇔ B ) ∈ w whih ontr adits the onsisteny of w . Subsequen tly , w e sho w a w eak v arian t of the limit assumption for sets of ob jets satisfying a form ula. Lemma 16. If k A k 6 = ∅ then min ≺ w ( k A k ) 6 = ∅ . Pr o of. L et b e k A k 6 = ∅ . Then w A ∪ { A } is onsistent by lemma 13.1. Ther efor e ther e is a maximal onsistent set z ∈ U suh that w A ∪ { A } ⊆ z . W e show that z ∈ min ≺ w ( k A k ) . Supp ose for the ontr ary that ther e is y ∈ A M c and y ≺ w z . By denition of anoni al mo del, this me ans that ∃ C ∈ z suh that ∀ B ∈ y , ( B ⇔ C ) ∈ w . Sin e A ∈ y , we have ( A ⇔ C ) ∈ w , hen e ¬ C ∈ w A , whih entails ¬ C ∈ z ontr aditing the onsisteny of z. It is not har d to se e that a formula A is valid wrt. the we ak variant of the limit assumption i it is valid wrt. the str ong variant. Lemma 17. k A k = A M c Pr o of. The pr o of is by indution on the onstrution of formulas. F or atomi V i it fol lows fr om the mo del denition. F or lassi al formulas the pr o of is standar d. " ⇒ ": L et b e F ∈ w , F = ( A ⇔ B ) . w A is onsistent by lemma 13, 3. L et b e x ∈ ∆ , suh that w A ∈ x (13, 4). By axiom (4), we have (( A ⇔ C ) ⊔ ( C ⇔ B ) ∈ w for any C ∈ x . By lemma 13, 2, we have then ( A ⇔ C ) 6∈ w , hen e ¬ ( A ⇔ C ) ∈ w fr om whih we get ( C ⇔ B ) ∈ w ∀ C ∈ x . By the denition of ≺ w we have then if for al l y ∈ ∆ , if B ∈ y then x ≺ w y . This me ans that w ∈ ( A ⇔ B ) M C " ⇐ ": L et b e w ∈ ( A ⇔ B ) M C . Then A M C 6 = ∅ . By indution hyp othesis , A ∈ x for al l x ∈ A M C . Then A is onsistent and sin e ( w, x ) ∈ R , ( A ⇔ ⊥ ) ∈ w . By lemma 10.1, and axiom (4), we have ( i )( A ⇔ B ) ⊔ ( B ⇔ ⊥ ) ∈ w By lemma 13.4, ther e is z ∈ ∆ suh that w A ⊆ z . W e onsider two ases: (a) B M C = ∅ . If B is in onsistent, we have trivial ly ( A ⇔ B ) ∈ w , by ( A ⇔ ⊥ ) ∈ w and (RM), R1. If B is not in onsistent, we observe that ( ii ) ¬ ( B ⇔ ⊥ ) ∈ w if not, by lemma 13.4, ther e would b e z ′ ∈ ∆ with B ∈ z ′ , i.e. z ′ ∈ B by indution hyp othesis whih is imp ossible sin e B M C = ∅ . W e onlude ( A ⇔ B ) ∈ w by (i). (b) B M C 6 = ∅ , sin e B is onsistent, ther e is z ′ ∈ ∆ suh that w B ⊆ z ′ and B ∈ z ′ . By w ∈ ( A ⇔ B ) M C we have z ≺ w z ′ ( z is minimal for ≺ w in k A k ). This me ans that ther e is C ∈ z ′ suh that for al l D ∈ z , ( D ⇔ C ) ∈ w . Sin e A ∈ z , we have ( A ⇔ C ) ∈ w fr om whih we obtain ( A ⇔ B ) ⊔ ( B ⇔ C ) ∈ w . But we have ¬ ( B ⇔ C ) ∈ w by lemma 13.2, fr om whih we obtain ( A ⇔ B ) ∈ w . By virtue of theorem 4, w e obtain: Corollary 18. CSMS is sound and omplete wrt. the CSMS -distan e min- mo dels. 3 A T ableaux Calulus In this setion, w e presen t a tableau alulus for C S L , this alulus pro vides a deision pro edure for this logi. W e iden tify a tableau with a set of sets of form ulas Γ 1 , . . . , Γ n . Ea h Γ i is alled a table au set 5 . Our alulus will mak e use of lab els to represen t ob jets of the domain. Let us onsider form ulas ( A ⇔ B ) and ¬ ( A ⇔ B ) under preferen tial seman tis. W e ha v e: w ∈ ( A ⇔ B ) M i ∃ x ( x ∈ A M ∧ ∀ z ( z ∈ B M → x ≺ w z )) In minspae mo dels, the righ t part is equiv alen t to: w ∈ ( A ⇔ B ) M i ∃ u ∈ A M and ∀ y ( y ∈ B M → ∃ x ( x ∈ A M ∧ x ≺ w y )) W e no w in tro due a pseudo-mo dalit y w indexed on ob jets: x ∈ ( w A ) M i ∀ y ( y ≺ w x → y ∈ A M ) Its meaning is that x ∈ ( w A ) M i A holds in all w orlds preferred to x with resp et to ≺ w . Observ e that w e ha v e then the equiv alene: Claim 1. w ∈ ( A ⇔ B ) M i A M 6 = ∅ and ∀ y ( y / ∈ B M or y ∈ ( ¬ w ¬ A ) M ) . This equiv alene will b e used to deomp ose ⇔ -form ulas in an analyti w a y . The tableau rules mak e also use of a univ ersal mo dalit y (and its negation). The language of tableaux omprises the follo wing kind of form ulas: x : A, x : ( ¬ ) ¬ A, x : ( ¬ ) y ¬ A, x < y z , where x, y , z are lab els and A is a C S L -form ula. The meaning of x : A is the ob vious one: x ∈ A M . The reading of the rules is the follo wing: w e apply a rule Γ [ E 1 , . . . , E k ] Γ 1 | . . . | Γ n to a tableau set Γ if ea h form ula E k is in Γ . W e then replae Γ with an y tableau set Γ 1 , . . . , Γ n . As usual, w e let Γ, A stand for for Γ ∪ { A } , where A is a tableau form ula. The tableaux rules are sho wn in gure Figure 2. Let us ommen t on the rules whi h are not immediately ob vious. The rule for (T ⇔ ) eno des diretly the seman tis b y virtue of laim 1. Ho w ev er in the negativ e ase the rule is split in t w o: if x satises ¬ ( A ⇔ B ) , either A is empt y , or there m ust b e an y ∈ B su h that there is no z ≺ x y satisfying A ; if x satises B then x itself fullls this ondition, i.e. w e ould tak e y = x , sine x is ≺ x -minimal (b y en tering). On the other hand, if x do es not satises B , then x annot satisfy A either (otherwise x w ould satisfy A ⇔ B ) and there m ust b e an y as desrib ed ab o v e. This ase analysis with resp et to x is p erformed b y the (F1 ⇔ ) rule, whereas the reation y for the latter ase is p erformed b y (F2 ⇔ ). W e ha v e a similar situation for the (F x ) rule: let z satisfy ¬ x ¬ A , 5 A tableau set orresp onds to a br anh in a tableau-as-tree represen tation. ( T ⊓ ) Γ [ x : A ⊓ B ] Γ, x : A, x : B ( F ⊓ ) Γ [ x : ¬ ( A ⊓ B )] Γ, x : ¬ A | Γ, x : ¬ B ( N E G ) Γ [ x : ¬¬ A ] Γ, x : A ( F 1 ⇔ ) Γ [ x : ¬ ( A ⇔ B )] Γ, x : ¬ A | Γ , x : B | Γ, x : ¬ A, x : ¬ B ( T ⇔ ) ( ∗ ) Γ [ x : A ⇔ B ] Γ, x : ¬ ¬ A, y : ¬ B | Γ, y : B, y : ¬ x ¬ A ( F 2 ⇔ )( ∗∗ ) Γ [ x : ¬ ( A ⇔ B ) , x : ¬ A, x : ¬ B ] Γ, y : B, y : x ¬ A ( F 1 x ) Γ [ z : ¬ x ¬ A ] Γ, x : ¬ A | Γ, x : A ( T x )( ∗ ) Γ [ z : x ¬ A, y < x z ] Γ, y : ¬ A, y : x ¬ A ( F 2 x )( ∗∗ ) Γ [ z : ¬ x ¬ A, x : ¬ A ] Γ, y < x z , y : A, y : x ¬ A ( T )( ∗ ) Γ [ x : ¬ A ] Γ, y : ¬ A, y : ¬ A ( F )( ∗∗ ) Γ [ x : ¬ ¬ A ] Γ, y : A ( M od )( ∗ ) Γ [ z < x u ] Γ, z < x y | Γ , y < x u ( C ent ) ( ∗ ∗ ∗ ) Γ Γ, x < x y | Γ [ x/y ] (*) y is a lab el o urring in Γ . (**) y is a new lab el not o urring in Γ . (***) x and y are t w o distint lab els o urring in Γ . Fig. 2. T ableau rules for C S L . then there m ust b e an y ≺ x z satisfying A ; but if x satises A w e an tak e x = y , sine x ≺ x z (b y en tering). If x do es not satisfy A then w e m ust reate a suitable y and this is the task of the (F2 x ) rule. Observ e that the rule do es not simply reate a y ≺ x z satisfying A but it reates a minimal one. The rule is similar to the (F ) rule in mo dal logi GL (Gö del-Löb mo dal logi of arithmeti pro v abilit y) [1℄ and it is enfored b y the Limit Assumption. This form ulation of the rules for (F ⇔ ) and for (F x ) prev en ts the unneessary reation of new ob jets whenev er the existene of the ob jets required b y the rules is assured b y en tering. The rule ( C ent ) is of a sp eial kind: it has no premises (ie. it an alw a ys b e applied) and generates t w o tableau sets: one with Γ ∪ { x < x y } , where x and y are t w o distint lab els o urring in Γ ), and one where w e replae x b y y in Γ , i.e. where w e iden tify the t w o lab els. Denition 19 (Closed set, losed tableau). A table au set Γ is losed if one of the thr e e fol lowing onditions hold: (i) x : A ∈ Γ and x : ¬ A ∈ Γ , for any formula A , or x : ⊥ ∈ Γ . (ii) y < x z and z < x y ar e in Γ . (iii) x : ¬ x A ∈ Γ . A C S L -table au is lose d if every table au set is lose d. In order to pro v e soundness and ompleteness of the tableaux rules, w e in- tro due the notion of satisabilit y of a tableau set b y a mo del. Giv en a tableau set Γ , w e denote b y Lab Γ the set of lab els o urring in Γ . { x : A, x : ¬ B , x : ¬ ( A ⇔ A ) } { x : A, x : ¬ B , x : ¬ ( A ⇔ A ) x : ¬ A } ( F 1 ⇔ ) { x : A, x : ¬ B , x : ¬ ( A ⇔ A ) x : ¬ A x : ¬ A } losed b y def 6-(i) ( T ) { x : A, x : ¬ B , x : ¬ ( A ⇔ A ) x : B } losed b y def 6-(i) ( F 1 ⇔ ) { x : A, x : ¬ B , x : ¬ ( A ⇔ A ) x : ¬ A } losed b y def 6-(i) ( F 1 ⇔ ) Fig. 3. An example of tableau: pro v abilit y of A ⊓ ¬ B → ( A ⇔ B ) . Denition 20 ( C S L -mapping, satisable tableau set). L et M = ( ∆ M , ( ≺ w ) w ∈ ∆ M , . M ) b e a pr efer ential mo del, and Γ a table au set. A C S L -mapping fr om Γ to M is a funtion f : Lab Γ − → ∆ M satisfying the fol lowing ondition: for every y < x z ∈ Γ, we have f ( y ) ≺ f ( x ) f ( z ) in M . Given a table au set Γ , a C S L -pr efer ential mo del M , and a C S L -mapping f fr om Γ to M , we say that Γ is satisable under f in M if x : A ∈ Γ implies f ( x ) ∈ A M . A table au set Γ is satisable if it is satisable in some C S L -pr efer ential mo del M under some C S L -mapping f . A C S L -table au is satisable if at le ast one of its sets is satisable. W e an sho w that our tableau alulus is sound and omplete with resp et to the preferen tial seman tis, whene with resp et to minspae mo dels (b y theorem 4). Theorem 21 (Soundness of the alulus). A formula A ∈ L C S L is satis- able wrt. pr efer ential semantis then any table au starte d by x : A is op en. The pro of of the soundness is standard: w e sho w that rule appliation pre- serv es satisabilit y . Pr o of (Soundness of the T able au System). Let Γ b e a tableau set satisable in a C S L -mo del M = h ∆, ( ≺ w ) w ∈ ∆ , . M i under a C S L -mapping f . W e pro v e that if w e apply one of the tableau rules to Γ , at least one of the new tableau sets generated b y it is satisable. Moreo v er, a satisable set annot b e losed. The ases of ( T ⊓ ) , ( F ⊓ ) , ( T x ) , ( T ) and ( F ) are easy and left to the reader. Soundness of ( F 1 x ) is trivial (it's a ut-lik e rule), as ( M od ) and ( C ent ) whi h ame from the mo dularit y and en tering prop ert y of the pref- eren tial relation ≺ w . ( T ⇔ ) . Let x : ( A ⇔ B ) ∈ Γ . F or an y lab el y ∈ L ab Γ , the appliation of this rule to Γ will generate t w o tableau sets: Γ 1 = Γ ∪ { y : ¬ ¬ A, y : ¬ B } Γ 2 = Γ ∪ { y : B , y : ¬ x ¬ A } . where y ∈ Lab Γ . As Γ is satisable in M under f , w e ha v e that f ( x ) ∈ ( A ⇔ B ) M . By laim 1, for all y ∈ Lab Γ w e ha v e f ( y ) ∈ ( ¬ ¬ A ) M (as A M annot b e empt y) and f ( y ) ∈ ( B → ¬ x ¬ A ) M . W e then ha v e t w o ases: • either f ( y ) ∈ ( ¬ B ) M , and then Γ 1 is satisable. • or f ( y ) ∈ B M and then f ( y ) ∈ ( ¬ x ¬ A ) M . Γ 2 is then satisable. ( F 1 ⇔ ) . Let x : ¬ ( A ⇔ B ) ∈ Γ . The rule will generate three tableau sets: Γ 1 = Γ ∪ { x : ¬ A } Γ 2 = Γ ∪ { x : B } Γ 3 = Γ ∪ { x : ¬ A, x : ¬ B } As Γ is satisable in M under f , w e ha v e f ( x ) ∈ ¬ ( A ⇔ B ) M . By laim 1, w e ha v e that f ( x ) ∈ ( ¬ A ) M or ∃ y ( y ∈ B M and y ∈ ( x ¬ A ) M ) . If f ( x ) ∈ ( ¬ A ) M , then Γ 1 is satisable. If not, w e ha v e t w o ases: • either f ( x ) ∈ B M , and th us Γ 2 is satisable. • either f ( x ) ∈ ( ¬ B ) M , and then there is some ob jet y ′ ∈ ∆ su h that y ′ ∈ B M and y ′ ∈ ( x ¬ A ) M , so that y ′ 6 = f ( x ) . Sine the re- lation ≺ x satises the en tering prop ert y , w e ha v e x ≺ x y ′ . And from y ′ ∈ ( x ¬ A ) M , w e an dedue that f ( x ) ∈ ( ¬ A ) M , th us making Γ 3 satisable. ( F 2 ⇔ ) . This rule will generate the follo wing set: Γ 1 = Γ ∪ { y : x ¬ A, y : B } where y / ∈ Lab Γ . As sho wn in the pro of for ( F 1 ⇔ ) , sine x : ¬ ( A ⇔ B ) , x : ¬ A, x : ¬ B b elong to Γ and Γ satisable in M b y f , w e ha v e that there exists some y ′ ∈ ∆ su h that y ′ ∈ ( B ⊓ x ¬ A ) M . W e onstrut a C S L - mapping f ′ b y taking ∀ u 6 = y , f ′ ( u ) = f ( u ) ; and f ′ ( y ) = y ′ . It's then easy to sho w that f ′ is a C S L -mapping, and that Γ 1 is satisable in M under f ′ . ( F 2 x ) . Let z : ¬ x ¬ A, x : ¬ A . This rule will generate the follo wing tableau set: Γ 1 = Γ ∪ { y < x z , y : A, y : x ¬ A } . where y / ∈ Lab Γ . Sine Γ is satisable in M under f , w e ha v e that f ( z ) ∈ ( ¬ x ¬ A ) M . Therefore w e ha v e that ∃ y ′ ∈ ∆ su h that y ′ ∈ A M and y ′ ≺ x f ( z ) . W e then let y ′′ ∈ min ≺ x { y ′ | y ′ ∈ ∆ ∧ y ′ ∈ A M ∧ y ′ ≺ x f ( z ) } . As ≺ x satises the limit assumption, y ′′ exists, and it's easy to see that y ′′ ∈ ( x ¬ A ) M (if not it w ould not b e minimal). As y / ∈ Lab Γ , w e dene a new C S L -mapping f ′ b y taking ∀ u 6 = y , f ′ ( u ) = f ( u ) , and f ( y ) = y ′′ . It's then easy to he k that f ′ is indeed a C S L -mapping, and that Γ 1 is satisable in M under f ′ . Finally , w e sho w that if Γ is satisable, then Γ is op en. Supp ose it is not. Then if Γ is losed b y def. 19-(i) or (ii), w e immediately nd a on tradition. Supp ose that Γ is losed b y ondition (iii), then x : ¬ x ¬ A in Γ for some form ula x : ¬ x ¬ A . Sine Γ is satisable, w e w ould ha v e that there is y ∈ ∆ su h that y ≺ f ( x ) f ( x ) , obtaining a on tradition (b y en tering and asymmetry). In order to sho w ompleteness, w e need the follo wing denition: Denition 22 (Saturated tableau set). W e say that a table au set Γ is sat- ur ate d if: ( T ⊓ ) If x : A ⊓ B ∈ Γ then x : A ∈ Γ and x : B ∈ Γ . ( F ⊓ ) If x : ¬ ( A ⊔ B ) ∈ Γ then x : ¬ A ∈ Γ or x : ¬ B ∈ Γ . ( N E G ) If x : ¬¬ A ∈ Γ then x : A ∈ Γ . ( T ⇔ ) If x : ( A ⇔ B ) ∈ Γ then for al l y ∈ L ab Γ , either y : ¬ B ∈ Γ and ¬ ¬ A ∈ Γ , or y : B and y : ¬ x ¬ A ar e in Γ . ( F ⇔ ) If x : ¬ ( A ⇔ B ) ∈ Γ then either (i) x : ¬ A ∈ Γ , or (ii) x : B ∈ Γ , or (iii) x : ¬ A and x : ¬ B ar e in Γ and ther e exists y ∈ Lab Γ suh that y : B and y : x ¬ A ar e in Γ . ( T x ) If z : x ¬ A ∈ Γ and y < x z ∈ Γ , then y : ¬ A and y : x ¬ A ar e in Γ . ( F x ) If z : ¬ x ¬ A ∈ Γ , then either (i) x : A ∈ Γ , or (ii) x : ¬ A ∈ Γ and ther e exists y ∈ Lab Γ suh that y < x z , y : A and y : x ¬ A ar e in Γ . ( T ) If x : ¬ A ∈ Γ , then for al l y ∈ Lab Γ , y : ¬ A and y : ¬ A ar e in Γ . ( F ) If x : ¬ ¬ A ∈ Γ , then ther e is y ∈ Lab Γ suh as y : A ∈ Γ . ( C ent ) F or al l x, y ∈ Lab Γ suh that x 6 = y , x < x y is ∈ Γ . ( M od ) If y < x z ∈ Γ , then for al l lab els u ∈ W Γ , either u < x z ∈ Γ , or y < x u ∈ Γ . W e say that Γ is satur ate d wrt. a rule R if Γ satises the orr esp onding satur a- tion ondition for R of ab ove denition. The follo wing lemma sho ws that the preferene relations < x satises the Limit Assumption for an op en tableau set. Lemma 23. L et Γ b e an op en table au set ontaining only a nite numb er of p ositive ⇔ -formulas x : A 0 ⇔ B 0 , x : A 1 ⇔ B 1 , x : A 2 ⇔ B 2 , . . . , x : A n − 1 ⇔ B n − 1 . Then Γ do es not ontain any innite des ending hain of lab els y 1 < x y 0 , y 2 < x y 1 , . . . , y i +1 < x y i , . . . . Pr o of. By absurdit y , let Γ on tain a desending hain of lab els . . . , y i +1 < x y i < x . . . < x y 1 < x y 0 . This hain ma y only b e generated b y suessiv e appliations of ( T ⇔ ) and ( F 2 x ) to form ulas x : A i ⇔ B i for 0 ≤ i < n . Γ then on tains the follo wing form ulas for 0 ≤ i < n : y i : ¬ x ¬ A i , y i +1 < x y i , y i +1 : A i , y i +1 : x ¬ A i . Here ( T ⇔ ) has b een applied to ev ery form ula x : A i ⇔ B i one and with parameter y i previously (and newly) generated b y ( F 2 x ) from y i − 1 : ¬ x ¬ A i − 1 . The only w a y to mak e the hain longer is b y applying ( T ⇔ ) a seond time to one of the p ositiv e ⇔ -form ulas lab elled x on Γ . Let this form ula b e x : A k ⇔ B k where 0 ≤ k < n . Then Γ on tains further y n +1 : A k (together with y n : ¬ x ¬ A k , y n +1 < x y n , y n +1 : x ¬ A k ). By the mo dularit y rule, w e get y n +1 < x y k +1 . Moreo v er, Γ on tains also y k +1 : x ¬ A k , from whi h w e obtain b y ( T x ) y n +1 : ¬ A k so that Γ is losed. Theorem 24. If Γ is an op en and satur ate d table au set, then Γ is satisable. (Pro of ) : Giv en an op en tableau set Γ , w e dene a anonial mo del M Γ = h ∆, ( ≺ w ) w ∈ ∆ , . M Γ i as follo ws: ∆ = Lab Γ and y ≺ x z i y < y z ∈ Γ . F or all prop ositional v ariables V i ∈ V p , V M Γ i = { x | x : V i ∈ Γ } M Γ is indeed a C S L -mo del, as ea h preferen tial relation is en tered, mo dular, and satises the limit assumption. The rst t w o ame from the rules ( C ent ) and ( M od ) , and w e ha v e the latter b y lemma 23. W e no w sho w that Γ is satisable in M Γ under the trivial iden tit y mapping, i.e for all form ula C ∈ L C S L : (i) if x : C ∈ Γ , then x ∈ C M Γ . (ii) if x : ¬ C ∈ Γ , then x ∈ ( ¬ C ) M Γ . Pr o of. W e reason b y indution on the omplexit y cp ( C ) of a form ula C , where w e supp ose that cp (( ¬ ) ¬ A ) , cp (( ¬ ) x ¬ A ) < cp ( A ⇔ B ) . if C = V i , C ∈ V p , x ′ ∈ C M Γ b y the denition of M Γ . if C is a lassial form ula, the pro of is standard. if C = ( A ⇔ B ) : sine Γ is saturated, for ev ery y ∈ Lab Γ w e ha v e either x : ¬ ¬ A ∈ Γ and y : ¬ B ∈ Γ , or y : B ∈ Γ and y : ¬ x ¬ A ∈ Γ . By indution h yp othesis , in the rst ase w e get A M Γ 6 = ∅ and y / ∈ B M Γ , and in the seond w e ha v e that y ∈ ( ¬ x ¬ A ) M Γ whi h also en tails A M Γ 6 = ∅ . Th us, b y laim 1, w e ha v e x ∈ ( A ⇔ B ) M Γ . if C = ¬ ( A ⇔ B ) : b y the saturation onditions, w e ha v e 3 ases. (a) x : ¬ A ∈ Γ . By appliation of the rule ( T ) , for all lab el y , y : ¬ A ∈ Γ . By our indution h yp othesis, A M Γ = ∅ , and so x ∈ ( ¬ ( A ⇔ B )) M Γ . (b) x : B ∈ Γ . By indution h yp othesis , x ∈ B M Γ , and so x ∈ ( ¬ ( A ⇔ B )) M Γ b y axiom (4). () x : ¬ A, x : ¬ B are in Γ , and there is a lab el y su h that y : B , y : x ¬ A are in Γ . By indution h yp othesis, w e ha v e y ∈ B M Γ and y ∈ ( x ¬ A ) M Γ , so that b y laim 1, w e ha v e x ∈ ( ¬ ( A ⇔ B )) M Γ if C = y ¬ A , b y saturation w e ha v e: for all z , if z < x ∈ Γ then z : ¬ A ∈ Γ and z : y ¬ A ∈ Γ . Then b y indution h yp othesis , w e ha v e that for all z , if z ≺ y x then z ∈ ( ¬ A ) M Γ whi h means that x ∈ ( y ¬ A ) M Γ if C = ¬ y ¬ A , b y saturation w e ha v e either y : A ∈ Γ or y : ¬ A ∈ Γ . In the rst ase, sine ≺ y satises en tering w e ha v e y ≺ y x , and b y indution h yp othesis , y ∈ A M Γ . Th us x ∈ ( ¬ y ¬ A ) M Γ . In the seond ase, b y saturation there is z ∈ Lab Γ su h that z < y x ∈ Γ , z : A ∈ Γ . By indution h yp othesis and the denition of ≺ y , w e onlude that x ∈ ( ¬ y ¬ A ) M Γ . 4 T ermination of the T ableau Calulus The alulus presen ted ab o v e an lead to non-terminating omputations due to the in terpla y b et w een the rules whi h generate new lab els (the dynami rules ( F 2 ⇔ ) , ( F ) and ( F x ) ) and the stati rule ( T ⇔ ) whi h generates for- m ula ¬ x A to whi h ( F x ) ma y again b e applied. Our alulus an b e made terminating b y dening a systemati pro edure for applying the rules and b y in tro duing appropriate blo king onditions. The systemati pro edure simply presrib es to apply stati rules as far as p ossible b efore applying dynami rules. T o prev en t the generation of an innite tableau set, w e put some restritions on the rule's appliations. The restritions on all rules exept ( F 2 ⇔ ) and ( F 2 x ) are easy and prev en t redundan t appliations of the rules. W e all the restritions on ( F 2 ⇔ ) and ( F 2 x ) blo king onditions in analogy with standard onditions for getting termination in mo dal and desription logis tableaux; they prev en t the generation of innitely man y lab els b y p erforming a kind of lo op- he king. T o this aim, w e rst dene a total ordering ⊏ on the lab els of a tableau set su h that x ⊏ y for all lab els x that are already in the tableau when y is in tro dued. If x ⊏ y , w e will sa y that x is older than y . W e dene Box + Γ,x ,y as the set of p ositiv e b o xed form ulas indexed b y x lab elled b y y whi h are in Γ : Box + Γ,x ,y = { x ¬ A | y : x ¬ A ∈ Γ } and Π Γ ( x ) as the set of non b o xed form ulas lab elled b y x : Π Γ ( x ) = { A | A ∈ L C S L and x : A ∈ Γ } . Denition 25. (Stati and dynami rules) W e al l dynami the fol lowing rules: ( F 2 ⇔ ) , ( F 2 x ) and ( F ) . W e al l stati al l the other rules. (Rules restritions) 1. Do not apply a stati rule to Γ if at le ast one of the onse quen es is alr e ady in it. 2. Do not apply the rule ( F 2 ⇔ ) to a x : ¬ ( A ⇔ B ) , x : ¬ A, x : ¬ B (a) if ther e exists some lab el y in Γ suh that y : B and y : x ¬ A ar e in Γ . (b) if ther e exists some lab el u suh that u ⊏ x and Π Γ ( x ) ⊆ Π Γ ( u ) . 3. Do not apply the rule ( F 2 x ) to a z : ¬ x ¬ A, x : ¬ A (a) if ther e exists some lab el y in Γ suh that y < x z , y : A and y : x ¬ A ar e in Γ . (b) if ther e exists some lab el u in Γ suh that u ⊏ x and Π Γ ( x ) ⊆ Π Γ ( u ) . () if ther e exists some lab el v in Γ suh that v ⊏ z and v : ¬ x ¬ A ∈ Γ and Box + Γ,x ,z ⊆ Box + Γ,x ,v . 4. Do not apply the rule ( F ) to a x : ¬ ¬ A in Γ if ther e exists some lab el y suh that y : A is in Γ . (Systemati pro edure) (1) Apply stati rules as far as p ossible. (2) Apply a (non blo ke d) dynami rule to some formula lab el le d x only if no dynami rule is appli able to a formula lab el le d y , suh that y ⊏ x . W e pro v e that a tableau initialized with a C S L -form ula alw a ys terminates pro vided it is expanded aording to Denition 25. Theorem 26. L et Γ b e obtaine d fr om { x : A } , wher e A is a C S L -formula, by applying an arbitr ary se quen e of rules r esp e ting denition 25. Then Γ is nite. Pr o of. Supp ose b y absurdit y that Γ is not nite. Sine the stati rules (and also the ( F ) rule) ma y only add a nite of n um b er of form ulas for ea h lab el, Γ m ust on tain an innite n um b er of lab els generated b y the dynami rules, either (F2 ⇔ ) or (F2 ) (or b oth). Let Γ on tain innitely man y lab els in tro dued b y (F2 ⇔ ). Sine the n um- b er of negativ e ⇔ form ulas is nite, there m ust b e one form ula, sa y ¬ ( B ⇔ C ) , su h that for an innite sequene of lab els x 1 , . . . , x i , . . . , x i : ¬ ( B ⇔ C ) ∈ Γ . By blo king ondition (2b) w e then ha v e that for ev ery i , Π Γ ( x i ) 6⊆ Π Γ ( x 1 ) , . . . , Π Γ ( x i ) 6⊆ Π Γ ( x i − 1 ) . But this is imp ossible sine ea h Π Γ ( x i ) is nite (namely b ounded b y O ( | A | ) ) and the rules are non-dereasing wrt. Π Γ ( x i ) (an appliation of a rule an nev er remo v e form ulas from Π Γ ( x i ) ). Let no w Γ on tain innitely man y lab els in tro dued b y (F2 ). That is to sa y , Γ on tains x i : ¬ y i ¬ B for innitely man y x i and y i . If all y i are distint, Γ m ust on tain in partiular innitely man y form ulas x : ¬ y i for a xed x . The reason is that x i : ¬ y i ¬ B ma y only b e in tro dued b y applying (T ⇔ ), th us there m ust b e innitely man y y i : B ⇔ C ∈ Γ . By the systemati pro edure, the rule (T ⇔ ) has b een applied to a lab el x for ev ery y i : B ⇔ C ∈ Γ generating x : ¬ y i ¬ B for all y i . But then w e an nd a on tradition with resp et to blo king ondition (3b) as in the previous ase, sine for ea h i w e w ould ha v e Π Γ ( y i ) 6⊆ Π Γ ( y 1 ) , . . . , Π Γ ( y i ) 6⊆ Π Γ ( y i − 1 ) . W e an onlude that Γ annot on tain x i : ¬ y i ¬ B , for innitely man y distint y i and distint x i . W e are left with the ase Γ on tains x i : ¬ y ¬ B for a xed y and innitely man y x i . In this ase, b y blo king ondition (3), w e ha v e that for ea h i , Box + Γ,y , x i 6⊆ Box + Γ,y , x 1 , . . . , Box + Γ,y , x i 6⊆ Box + Γ,x i − 1 . But again this is imp ossible giv en the fat that ea h Box + Γ,y , x i is nite (b ounded b y O ( | A | ) ) and that the rules are non- dereasing wrt. the sets Box + Γ . T o pro v e ompleteness, w e will onsider tableau sets saturated under blo k- ing. A tableau set Γ is saturated under blo king i (a) it is build aording to Denition 25 (b) No further rules an b e applied to it. It is easy to see that if Γ is saturated under blo king, it satises all the saturation onditions in Denition 22 exept p ossibly for onditions ( F ⇔ ) .(iii) and ( F x ) .(ii). By the termination theorem, w e get that an y tableau set generated from an initial set on taining just a C S L form ula, will b e either losed or saturated under blo king in a nite n um b er of steps. W e no w sho w that an op en tableau set saturated under blo king an b e extended to an op en saturated tableau set, that is satisfying all onditions of denition 22. By means of theorem 21 w e obtain the ompleteness of the termi- nating pro edure. Theorem 27. If Γ is satur ate d and op en under blo king, then ther e exists an op en and satur ate d set Γ ∗ suh that for al l A ∈ L C S L , if x : A ∈ Γ then A ∈ Γ ∗ . Let Γ b e an op en and saturated set under blo king. W e will onstrut the set Γ ∗ from Γ in three steps. First, w e onsider form ulas z : ¬ x ¬ A whi h are blo k ed b y ondition 3 (and not b y 3b). W e onstrut a set Γ 1 from Γ whi h satises the saturation ondition ( F x ) wrt. these form ulas. Step 1. F or ea h form ula z : ¬ x ¬ A ∈ Γ for whi h ondition ( F x ) is not fullled and that is blo k ed only b y ondition 3, w e onsider the oldest lab el u that blo ks the form ula. Therefore, the form ula u : ¬ x ¬ A is in Γ and it is not blo k ed b y ondition 3 6 . Sine z : ¬ x ¬ A is not blo k ed b y ondition 3b, u : ¬ x ¬ A is not blo k ed for this ondition either, and th us the rule ( F 2 x ) has b een applied to it. Hene there exists a lab el y su h that y : A, y : x ¬ A and y < x u are in Γ . W e then add y < x z to Γ . W e all Γ 1 the resulting set. Claim 2. (I) Γ 1 is satur ate d, ex ept for ( M od ) and the formulas x : ¬ ( A ⇔ B ) and z : ¬ x ¬ A r esp e tively blo ke d by ondition 2b and 3b. (II) It is op en. The step 2 will no w build a set Γ 2 saturated wrt. ( M od ) from Γ 1 . Step 2. F or ea h y < x z ∈ Γ , if Box + Γ,x ,z ⊂ Box + Γ,x ,y , then for ea h z 0 su h that Box + Γ,x ,z 0 = Box + Γ,x ,z w e add y < x z 0 to Γ 1 . W e all Γ 2 the resulting set. Claim 3. (I) Γ 2 is satur ate d ex ept for the formulas x : ¬ ( A ⇔ B ) and z : ¬ x ¬ A r esp e tively blo ke d by ondition 2b and 3b. (II) It is op en. W e will no w onsider the form ulas blo k ed b y onditions 2b and 3b, and nally build a set Γ 3 saturated wrt. all rules from Γ 2 . Step 3. F or ea h lab el x su h that there is a form ula x : ¬ ( A ⇔ B ) ∈ Γ or z : ¬ x ¬ A ∈ Γ resp etiv ely blo k ed b y ondition 2b or 3b, w e let u b e the oldest lab el whi h aused the blo king. W e then onstrut the set Γ 3 b y the follo wing pro edure: 1. w e remo v e from Γ 2 ea h relation < x , and all form ulas v : ¬ x ¬ A and v : x ¬ A ( v ∈ La b Γ ). 2. F or all lab el z ∈ Lab Γ su h that z 6 = x , w e add x < x z . 3. F or all lab els z , v ∈ Lab Γ su h that z 6 = x , if v < u z ∈ Γ 2 , then w e add v < x z . 4. F or ea h v : u ¬ A ∈ Γ , if A ∈ Π Γ ( x ) w e then add v : x ¬ A . 5. F or ea h v : ¬ u ¬ A ∈ Γ su h that v 6 = x , w e add v : ¬ u ¬ A 6. F or ea h form ula A ∈ Π Γ ( x ) , w e add x : x A . Claim 4. (I) Γ 3 is satur ate d wrt. al l rules. (II) It is op en. W e then let Γ ∗ = Γ 3 . It is easy to see that for all form ulas A ∈ L C S L , if x : A ∈ Γ then x : A ∈ Γ ∗ , as none of these form ulas are remo v ed b y the onstrution of Γ ∗ . W e no w pro v e the preeden t laims. 6 If it w as, let v older than u the lab el whi h auses the blo king. Then v will also blo k u : ¬ x ¬ A , on tradition, as u is b y h yp othesis the oldest lab el blo king z : ¬ x ¬ A . Claim 5. 1. If y < x z ∈ Γ , then Box + Γ,x ,z ⊆ Box + Γ,x ,y . 2. If y < x z is in Γ 1 , then Box + Γ,x ,z ⊆ Box + Γ,x ,y . 3. If z : ¬ x ¬ A is blo ke d by ondition 3 and if u is the oldest lab el (a or ding to ⊏ ) blo king it, then u : ¬ x ¬ A is not blo ke d by ondition 3. 4. If z : ¬ x ¬ A and x : ¬ ( A ⇔ B ) ar e blo ke d by ondition 3b or 2b, and if u is the oldest lab el blo king it, then u annot b e blo ke d by ondition 3b nor 2b. Pr o of. 1. T rivial, sine Γ is saturated wrt. ( T x ) . 2. If y < x z ∈ Γ , w e are in the preeden t ase. If not, then y < x z w as added b y step 1. Th us there is a form ula z : ¬ x ¬ A blo k ed b y ondition 3. Let u b e the oldest lab el blo king it. W e then ha v e, b y denition of blo king ondition 3, Box + Γ,x ,z ⊆ Box + Γ,x ,u . W e also ha v e, b y denition of step 1, that y < x u ∈ Γ . Th us, as sho wn in part 1 of this lemma, w e ha v e Box + Γ,x ,u ⊆ Box + Γ,x ,y . W e an no w onlude that Box + Γ,x ,z ⊆ Box + Γ,x ,y . 3. Supp ose that u is the oldest lab el blo king z : ¬ x ¬ A , and that u : ¬ x ¬ A is blo k ed b y v . By denition of the blo king ondition, w e ha v e that v ⊏ u ⊏ z , and Box + Γ,x ,z ⊆ Box + Γ,x ,u ⊆ Box + Γ,x ,v . Then w e ha v e that v also blo ks z : ¬ x ¬ A : on tradition, as u should b e, b y h yp othesis, the oldest lab el blo king this form ula. 4. Supp ose that u blo ks z : ¬ x ¬ A or x : ¬ ( A ⇔ B ) b y ondition 3b or 2b. Then w e ha v e u ⊏ x and Π ( x ) ⊆ Π ( u ) . No w supp ose that there is a form ula u : ¬ ( A ⇔ B ) or w : ¬ u ¬ A ( w ∈ Lab Γ ) blo k ed b y ondition 2b or 3b b y a lab el v . Then v ⊏ u and Π ( u ) ⊆ Π ( v ) . Sine w e ha v e v ⊏ x and Π ( x ) ⊆ Π ( v ) , w e w ould ha v e that v also blo ks x : on tradition, as u is b y h yp othesis the oldest lab el blo king it. Pr o of (Claim 2-(I)). As Γ is saturated under blo king, and Γ ⊆ Γ 1 , w e only ha v e to he k that the form ulas z : ¬ x ¬ A whi h w ere blo k ed b y ondition 3 (and not b y 3b) satisfy the saturation ondition ( F x ) . As w e add a preferen tial relation y < x z , w e also need to he k the saturation wrt. to ( T x ) . ( F 2 x ) : Let z : ¬ x ¬ A blo k ed b y ondition 3b (and only b y this ondition). By onstrution, there is a lab el y su h that y : A, y : ¬ A and y < x z are in Γ 1 , so the form ula z : ¬ x ¬ A satises the saturation ondition. ( T x ) : Let z : x ¬ C and v < x z in Γ 1 . W e ha v e t w o ases: (1) v < x z is already in Γ , and as Γ is saturated wrt. ( T x ) , the saturation ondition holds in Γ 1 . (2) v < x z w as not in Γ , and so that v < x z w as added b y onstrution of Γ 1 . By laim 5-2, w e obtain Box + Γ,x ,z ⊆ Box + Γ,x ,v and the pro of is trivial. Pr o of (Claim 2-(II)). Sine Γ 1 is obtained b y adding only preferen tial form ulas to Γ , and Γ is op en, w e only ha v e to he k losure ondition (ii). Supp ose that Γ 1 is losed b y denition 19-(ii): y < x z and z < x y are in Γ 1 . Then w e ha v e 3 ases: (1) y < x z and z < x y are in Γ : on tradition, Γ is op en. (2) z < x y is in Γ , but y < x z is not. Therefore y < x z has b een added b y onstrution of Γ 1 , and so b y laim 5-2 w e ha v e Box + Γ,x ,z ⊆ Box + Γ,x ,y . Sine z < x y is in Γ w e ha v e b y laim 5-1 Box + Γ,x ,y ⊆ Box + Γ,x ,z (*). Th us w e ha v e: Box + Γ,x ,z = Box + Γ,x ,y . By denition of Γ 1 , w e also ha v e that y : x ¬ A is in Γ . By the inlusion (*), w e ha v e that z : x ¬ A m ust also b e in Γ . But z : ¬ x ¬ A ∈ Γ : on tradition, Γ is op en. The ase where y < x z is in Γ but z < x y is not is symmetri. (3) Neither y < x z nor z < x y are in Γ . Both form ulas has b een added b y onstrution of Γ 1 . Th us there are some y : ¬ x ¬ A and z : ¬ x ¬ B in Γ blo k ed b y ondition 3. By onstrution of Γ 1 , if y < x z and z < x y w ere added, then w e m ust ha v e y : x ¬ B and z : x ¬ A in Γ . Using laim 5-2, if y < x z and z < x y are in Γ 1 , then w e m ust ha v e Box + Γ,x ,z ⊆ Box + Γ,x ,y and Box + Γ,x ,y ⊆ Box + Γ,x ,z . Th us Box + Γ,x ,z = Box + Γ,x ,y . Th us w e ha v e z : x ¬ B and y : x ¬ A also in Γ : on tradition, Γ is op en. Pr o of (Claim 3-(I)). W e ha v e to he k saturation wrt. ( M od ) and ( T x ) . With regards to ( M od ) , let u < x z ∈ Γ 2 . W e ha v e sev eral ases: (1) y < x z ∈ Γ : trivial, Γ b eing saturated wrt. ( M od ) . (2) y < x z ∈ Γ 1 but not in Γ , y < x z m ust ha v e b een added b y onstrution of Γ 1 . Let u ∈ Lab Γ , w e ha v e t w o ases: (2a) either u < x z or y < x u are in Γ 1 : the saturation ondition is then satised. (2b) neither u < x z nor y < x u are in Γ 1 . By onstrution of Γ 1 , w e ha v e that there is z : ¬ x ¬ A ∈ Γ whi h is blo k ed b y v : ¬ x ¬ A b y ondition 2b, and w e ha v e that y : x ¬ A and y < x v are in Γ . W e also ha v e Box + Γ,x ,z ⊆ Box + Γ,x ,y (laim 5-2). Sine x ¬ A ∈ Bo x + Γ,x ,y but not in Box + Γ,x ,z ( Γ 1 w ould b e losed), w e ha v e that Box + Γ,x ,z ⊂ Box + Γ,x ,y . As y < x v is in Γ , ( M od ) has b een applied to it with u so either y < x u or u < x v are in Γ . The rst ase annot o ur b y our h yp othesis, so w e ha v e that u < x v is in Γ . ( M od ) had also b een applied to it with z , so either u < x z is in Γ , or z < x v is in Γ . The rst ase b eing not p ossible b y h yp othesis, w e ha v e z < x v ∈ Γ , and so Box + Γ,x ,v ⊆ Box + Γ,x ,z (b y laim 5-1). As v blo ks z b y ondition 3, w e also ha v e Box + Γ,x ,z ⊆ Box + Γ,x ,v (b y denition of blo king ondition). So Box + Γ,x ,v = Box + Γ,x ,z , and b y denition of Γ 2 , as u < x v ∈ Γ , u < x z is in Γ 2 . (3) y < x z ∈ Γ 2 but not in Γ 1 . So y < x z has b een added b y onstrution of Γ 2 . W e then ha v e that there is some y < x v in Γ su h that Box + Γ,x ,v = Box + Γ,x ,z . Then for all u ∈ Lab Γ , w e ha v e t w o ases: (3a) Either u < x z or y < x u are in Γ 1 : this ase is easy . (3b) Neither u < x z nor y < x u are in Γ 1 . As y < x v ∈ Γ , ( M od ) had b een applied to it with u : so either y < x u or u < x v are in Γ . The rst ase is imp ossible b y h yp othesis, so u < x v is in Γ . As Box + Γ,x ,v = Box + Γ,x ,z and b y onstrution of Γ 2 , u < x z is then in Γ 2 . As step 2 add some preferen tial relations y < x z , w e ha v e to he k the saturation wrt. ( T x ) . By denition of Step 2, if y < x z w as added in Γ 2 , w e ha v e Box + Γ,x ,z ⊆ Box + Γ,x ,y , and th us the saturation ondition easily follo ws. Pr o of (Claim 3-(II)). The ase of the losures onditions (i) and (iii) are trivial (as Γ 1 is op en, and as step 2 only adds preferen tial form ulas). W e no w onsider the ase of the losure ondition (ii). Supp ose that y < x z and z < x y are in Γ 2 . Then w e ha v e sev eral ases: (1) b oth form ulas are in Γ 1 : on tradition with the fat that Γ 1 is op en. (2) z < x y ∈ Γ 1 but y < x z is not. Then z < x y ha v e b een added b y on- strution of Γ 2 . So there is a lab el v su h that y < x v ∈ Γ , Box + Γ,x ,v = Box + Γ,x ,z , and Box + Γ,x ,v ⊂ Box + Γ,x ,y . As z < x y ∈ Γ 1 , w e also ha v e that Box + Γ,x ,y ⊆ Box + Γ,x ,z (laim laim 5-2), and so Box + Γ,x ,y ⊆ Box + Γ,x ,v , whi h leads to a on tradition. (3) neither z < x y nor y < x z are in Γ 1 . Both form ulas ha v e b een added b y onstrution of Γ 2 . So there are some lab els v and w su h that z < x v and y < x w are in Γ . F urthermore, w e ha v e Box + Γ,x ,v ⊂ Box + Γ,x ,z , Box + Γ,x ,w ⊂ Box + Γ,x ,y , and Box + Γ,x ,v = Box + Γ,x ,y Box + Γ,x ,w = Box + Γ,x ,z . So w e an onlude that Box + Γ,x ,y ⊂ Box + Γ,x ,z and Box + Γ,x ,z ⊂ Box + Γ,x ,y : w e get a on tradition. Pr o of (Claim 4-(I)). ( T ⊓ ) , ( N ⊓ ) and ( N E G ) : trivial, as Γ 2 w as saturated with resp et to those rules, and onsidering the fat that for all A ∈ L C S L , if x : A ∈ Γ 2 then x : A ∈ Γ 3 . ( F 1 ⇔ ) : trivial, as Γ 2 is saturated with resp et to this rule and the form ulas added b y it are not remo v ed in the onstrution of Γ 3 . ( T ⇔ ) : if x : ( A ⇔ B ) is in Γ 3 , then it m ust b e in Γ 2 . If x is not blo k ed, it's easy , as Γ 2 is saturated wrt. this rule, so either x : ¬ ¬ A, y : ¬ B or y : B , y : ¬ x ¬ A m ust b e in Γ 3 . Otherwise, if x is blo k ed b y ondition 2b or 3b, let z b e the oldest lab el blo king it. F or ea h lab el y in Γ 3 w e ha v e t w o ases: either y : ¬ B ∈ Γ 3 or y : B ∈ Γ 3 . The rst ase is easy , y : ¬ B m ust ha v e b een in Γ 2 , and b y saturation, y : ¬ ¬ A to o. As this form ula annot b e remo v ed b et w een Γ 2 and Γ 3 , w e ha v e the saturation. In the seond ase, y : B m ust ha v e b een in Γ 2 . W e ha v e, as x is blo k ed b y z , Π Γ ( x ) ⊆ Π Γ ( z ) , and so z : ( A ⇔ B ) is in Γ 2 . Sine Γ 2 is saturated wrt. ( T ⇔ ) and y : B ∈ Γ 2 , and as z is not blo k ed b y ondition 2b or 3b (b y laim 5-4), w e ha v e y : ¬ z ¬ A ∈ Γ 2 . By denition of Γ 3 , w e then ha v e y : ¬ x ¬ A ∈ Γ 3 . So Γ 3 is saturated wrt. to ( T ⇔ ) . ( F 2 ⇔ ) : Let x b e blo k ed b y ondition 2b or 3b (the ase where x is non blo k ed is trivial), and let z b e the oldest lab el blo king it. As Π Γ ( x ) ⊆ Π Γ ( z ) , z : ¬ ( A ⇔ B ) , z : ¬ A, z : ¬ B m ust b e in Γ 2 . As z is not blo k ed (b y laim 5-4), ( F 2 ⇔ ) m ust ha v e b een applied to it. So there exists a lab el u su h that u : B , u : z ¬ A are in Γ 2 . By onstrution of Γ 3 , w e ha v e that u : x ¬ A is in Γ 3 , making it saturated wrt. ( F 2 ⇔ ) . ( F 1 x ) : if x is not blo k ed b y ondition 2b or 3b, it is trivial. Otherwise, let v b e the oldest lab el blo king x . As z : ¬ x ¬ A is in Γ 3 , z : ¬ v ¬ A m ust b e in Γ 2 (b y onstrution of Γ 3 ). As v is not blo k ed (b y laim 5-4), ( F 1 x ) m ust ha v e b een applied to z : ¬ v ¬ A from whi h w e obtain the onlusion. ( T x ) : if x is not blo k ed b y 2b or 3b, the pro of is easy . Otherwise, let v b e the oldest lab el blo king it. As z : x ¬ A and y < x z are in Γ 3 , z : v ¬ A and y < v z are in Γ 2 (and note that x : ¬ A m ust b e in Γ 2 to o). As v is not blo k ed (b y laim 5-4), the rule ( T x ) ha v e b een applied to these form ulas, and so y : ¬ A and y : v ¬ A are in Γ 2 , and so in Γ 3 . ( F 2 x ) : if x is not blo k ed b y onditions 2b or 3b, the pro of is easy . Oth- erwise, let v b e the oldest lab el blo king it. As z : ¬ x ¬ A and x : ¬ A are in Γ 3 , z : ¬ v ¬ A and x : ¬ A m ust b e in Γ 2 . Moreo v er, as v is not blo k ed (b y laim 5-4), the rule ( F 2 x ) has b een applied to these form ulas. So there exists a lab el u su h that u < v z , u : A and u : v ¬ A are in Γ 2 . And so, b y denition of Γ 3 , u < x z , u : A and u : x ¬ A are also in Γ 3 . ( T ) and ( F ) : trivial. ( M od ) : if x is not blo k ed b y onditions 2b or 3b, the relation < x w as already saturated in Γ 2 , and not mo died in Γ 3 . If x is blo k ed (b y ondition 2b or 3b), let v b e the oldest lab el blo king it. As v is not blo k ed, the relation < v is saturated for ( M od ) in Γ 2 . Let z < x u ∈ Γ 3 . Note that, b y denition of Γ 3 , u 6 = x . If z = x , for all lab els y 6 = x ha v e x < x y b y onstrution of γ 3 . If z 6 = x , then for all y w e ha v e t w o ases: (a) y = x : then w e ha v e x < x u b y onstrution of Γ 3 . (b) y 6 = x : then z < v u m ust ha v e b een in Γ 2 . As Γ 2 is saturated and v not blo k ed, either y < v u or z < v y are in Γ 2 , and so either y < x u or u < x z are in Γ 3 b y onstrution. By denition of step 3, if y = x or z = x , at least one of these form ula is not in Γ 3 , on traditing our h yp othesis. ( C ent ) : easy , either b y saturation of Γ 2 if x is not blo k ed b y 2b or 3b, or b y onstrution of Γ 3 in the other ase. Pr o of (Claim 4-(II)). None of the losure onditions ould o ur in Γ 3 : Supp ose that x : C and x : ¬ C are in Γ 3 . If C ∈ L C S L , then x : C and x : ¬ C m ust b e in Γ 2 : on tradition b eause Γ 2 is op en. If C = z ¬ A , then t w o ases: (a) z is not blo k ed b y ondition 2b or 3b. Then x : z ¬ A and z : ¬ z ¬ A are in Γ 2 whi h leads to a on tradition as Γ 2 is op en. (b) z is blo k ed b y ondition 2b or 3b. Let v b e the oldest lab el blo king it. Then, b y onstrution of Γ 3 , z : v ¬ A and z : ¬ v ¬ A are in Γ 2 : on tradi- tion as Γ 2 is op en. Supp ose that y < x z and z < x y are in Γ 3 . If x is not blo k ed b y ondition 2b or 3b, b oth form ulas are in Γ 2 , whi h leads to a on tradition. If x is blo k ed b y ondition 2b or 3b, let v b e the oldest lab el blo king it. Supp ose that y 6 = x 6 = y . Then, b y onstrution of Γ 3 , z < v y and y < v z are in Γ 2 : on tradition. Supp ose that x : ¬ x ¬ A . This form ula annot ha v e b een added b y step 3 (b y denition of this step), so it m ust ha v e b een in Γ 2 (and then x m ust b e not blo k ed b y ondition 2b or 3b): on tradition, as Γ 2 is op en. The tableaux pro edure desrib ed in this setion giv es a deision pro edure for C S L . T o estimate its omplexit y , let the length of A , the initial form ula, b e n . It is not hard to see that an y tableau set saturated under blo king ma y on tain at most O (2 n ) lab els. As matter of fat b y the blo king onditions no more than O (2 n ) lab els an b e in tro dued b y dynami rules F 2 ⇔ and F 2 x . Th us a saturated set under blo king will on tain most O (2 n ) tableau form ulas. W e an hene devise a non deterministi pro edure that guesses an op en tableau set in O (2 n ) steps. This sho ws that our tableau alulus giv es a NEXPTIME deision pro edure for C S L . In ligh t of the results on tained in [8℄ our pro edure is not optimal, sine it is sho wn that this logi is EXPTIME omplete. W e will study p ossible optimization (based for instane on a hing te hniques) in subsequen t w ork. 5 Conlusion In this pap er, w e ha v e studied the logi C S L o v er minspaes, and w e ha v e ob- tained t w o main results: rst w e ha v e pro vided a diret, sound and omplete axiomatisation of this logi. F urthermore, w e ha v e dened a tableau alulus, whi h giv es a deision pro edure for this logi. In [4℄, a tableau algorithm is prop osed to handle logis for metri spaes omprising distane quan tiers of the form ∃
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