PSPACE Bounds for Rank-1 Modal Logics

PSP A CE Bounds for Rank-1 Mo dal Logics LUTZ SCHR ¨ ODER DFKI-Lab Bremen and Dept. of Comput. Sci., Universit¨ at Bremen and DIRK P A TTINSON Department of Computing, Imp erial Co llege London F or lack of gene ral algorithmic methods that apply to wide classes of logics, establishing a com- plexit y bound for a given modal logic is often a laborious t ask. The presen t w ork i s a step tow ards a gene ral theory of the complexity of m o dal logics. Our main r esult i s that all rank-1 logics enjoy a shallow model prop ert y and thus are, under m ild assumptions on the format of their axioma- tisation, in PSP A CE . This leads to a uniﬁed deriv ation of tigh t PSP ACE -bounds for a num b er of logics i ncluding K , K D , coalition l ogic, graded m o dal logic, ma jor it y logic, and probabilistic modal logic. Our generic algorithm moreov er ﬁnds tableau pro ofs that witness pleasan t pro of- theoretic prop erties including a weak subformula proper t y . This generality is made possible by a coalgebraic semantics, which con venien tly abstracts f rom the details of a given mo del class and th us allows co vering a broad range of logics in a uniform wa y . Categories and Sub ject Descriptors: F. 4.1 [ Ma thematical Logic and F ormal Languages ]: Mathematical Logic— Mo dal L o gic ; Computational L o gic ; F. 2.2. [ Analysis of Algor ithms and Problem Complexity ]: Nonnume rical Algorithms and Problems— Complexity of Pr o of Pr o c e- dur es General T erms : Algorithms, Languages, Theory Additional Key W ords and Phrases: Shallo w models, resolution, coalgebra 1. INTRODUCTION Mo dal log ics ar e attractive from a co mputational p oin t o f view, a s they often com- bine expr essiv eness with decida bilit y . F or many modal logics no t inv olving dy- namic features, satisﬁability is known to b e in PSP AC E . This is typically prov ed for o ne logic at a time, e.g. b y modiﬁcatio ns of t he w itnes s alg o rithm for the mo dal logic K [Ladner 1 977; Blackburn et al. 2 001], but also using markedly dif- ferent methods suc h as the c onstraint-based PSP ACE -alg orithm for g raded mo dal logic [T obies 2001]. V ardi [198 9] gives a ﬁrs t glimpse of a generalisable metho d, equipping v ar ious epistemic logics with a neighbourho o d frame seman tics and show- ing them to b e in NP and PSP ACE , resp ectiv e ly (with the K axiom being r espon- sible for PSP ACE -har dness; r ecen t w o rk by Halper n and R ˆ ego [20 07] s hows tha t negative in tro spection br ings the complexity back do wn to NP ). Nev ertheless , there is to date no generally applicable theorem that allows establishing PSP ACE -b ounds for la rge classes of mo dal log ics in a uniform wa y . Here, we gener alise the metho ds of [V ardi 1989 ] to o bta in PSP ACE b ounds for rank-1 mo dal log ics, i.e. logics a xiomatisable by formulas whose mo dal depth uni- formly equals one, in a systematic w ay . Although pre s en tly limited to r ank 1 , our approach covers nu merous relev ant and no n- trivial e xamples. W e recov er known PSP ACE b ounds not only for norma l mo dal logics s uc h as K a nd K D , but most notably also for a rang e of non-normal mo dal logics suc h as graded mo dal logic [Fine 2 · L. Schr¨ oder an d D. Pattinson 1972], coa lition logic [Pauly 2002], and probabilistic mo dal log ic [Larsen and Sk ou 1991; Heifetz a nd Mongin 200 1]. Mor eo ver, o ur metho ds lead to a previously un- known PSP ACE upper b ound for ma jority logic [Pacuit and Salame 2004] that was independently discov er ed b y Demri and Lugiez [2006 ] at the sa me time. These logics are far from exotic: gr aded mo dal logic plays a role e.g. in decision sup- po rt and k nowledge represe ntation [v an der Ho ek and Meyer 1992; Ohlbach and Ko ehler 19 99], a nd probabilistic moda l log ic has app eared in co nnec tio n with model chec king [La r sen and Skou 1991] and in modelling economic b ehaviour [Heifetz and Mongin 2001]. The key to such a degr ee of genera lit y is to parametrise the theory over the t ype of systems deﬁning the seman tics, using coalgebraic metho ds. Coalgebr a co n ve- nient ly abstr acts from the details of a concrete class o f mode ls as it encapsulates the precise nature of mo dels in an endofunctor on the category of sets . As sp e- ciﬁc instances , one obtains e.g. (serial) Kripke frames, (monotone) neighbo ur hoo d frames [Hansen and Kupk e 2004 ], game frames [Pauly 2002], pro babilistic transi- tion systems and automata [Rabin 196 3 ; Bartels et a l. 2 004], w eighted automata, linear automata [Carlyle and Paz 19 7 1], and mult igraphs [D’Agos tino and Visser 2002]. Despite the broa d rang e of systems covered by the co algebraic approach, a substantial b ody of concepts and non-trivial results has emerged, encompassing e.g. generic notions of bisimilarity and co inductio n [Bartels 2003 ], c o recursion [T uri and Plo tkin 1997], duality , and ultraﬁlter extensions [K upk e et al. 20 05]. On the applications side, coalg e braic mo dal logic fea tur es in a c tua l sp eciﬁcation languages such as the ob ject o rien ted sp eciﬁcation language CCSL [Rothe et al. 2001] and CoCasl [Mossakowski et al. 2006]. The coalgebr aic study of computatio na l asp ects of mo dal log ic was initiated in [Schr¨ oder 20 07], where the ﬁnite mo del pr oper ty and asso ciated NEXPTIME - bo unds were pro ved. Here, we push these results fur ther and pres e nt a shallow mo del pro perty based on coalgebr aic semantics. This leads to a gene r ic PSP ACE - algorithm for deciding satisﬁability that trav erses a shallow mo del and strips oﬀ one layer of mo dalities in every step. Alternatively , our algorithm may b e seen as computing a shallow pr oof that enjoys a n umber of pleas an t pro of-theor etic pro p- erties, including a weak subformula prop ert y (i.e. it men tions only pro positiona l combinations of subformulas of the goal). The model construction relies o n extending the axioma tis a tion of a given logic to a set of rules which is closed under rule r esolution , i.e. ev er y re s olv e nt of tw o substituted rule co nclusions can also be derived direc tly using a third rule. This pro cess typically results in an inﬁnite but recurs iv e set of rules. Reso lutio n clo sed- ness then enables us to build the shallow model using induction on the moda l dept h of form ulas. Since we are w orking with an inﬁnite set of r ules, we hav e to imp ose a second condition to ens ure that w e can decide satisﬁa bilit y: a rule set is closed under c ont r action if every substituted rule c onclusion with duplicate literals can be deriv ed using a s ubstitution instance of a second r ule in whose co nclusion a ll literals rema in distinct. The decisio n pr ocedure will run in PSP ACE if b oth c lo- sure under r e solution and clos ure under contraction can b e co n tro lled, i.e. there is a p olynomial b ound o n the size of rules that ar e applicable at every step of the deductive pro cess. This turns out to b e the case for all examples men tioned above. PSP ACE Boun ds for R ank-1 Mo dal Logics · 3 The materia l is or ganised as follows. In Section 2, w e give a brief introduction to the generic coalge braic semantics of modal logic. In Section 3, w e disc uss deduction systems for coa lgebraic mo da l logics and their prop erties, notably the (equiv alent) central notions of strict one-step completeness and reduction closedness of rule sets. Sections 4 a nd 5 a re devoted to the tableau-base d shallow mode l constr uction and the pro of-theore tic view there o f. The ensuing PSP ACE -alg orithm a nd its example applications are presented in Section 6. 2. CO ALGEBRAIC M OD AL LOGIC W e br ieﬂy recapitulate the basics of the coalgebra ic interpretation of mo dal logic. T o begin, we ﬁx the syntactic framework. A mo dal s ignatur e is just a set Λ of unary mo dal op erators (all our results generalis e stra igh tforwardly to a po ly adic setting as in [Schr¨ oder 2 005]). The signatur e Λ induces a mo dal la nguage F (Λ), with formulas φ, ψ ∈ F (Λ) deﬁned by the grammar φ ::= ⊥ | φ ∧ ψ | ¬ φ | L φ, where L rang es over Λ. Disjunctions φ ∨ ψ , tr uth ⊤ , a nd other b o o lean op erations are deﬁned as usual. The depth of a for m ula is its maximal nesting depth of mo dal op erators. W e work in the framework of c o algebr aic mo dal lo gic , introduced by Pattinson [2004], g eneralising previous results [J acobs 2000; R¨ oßiger 2000; Kurz 2001; Pattin- son 2001], where mo da l languages are interpreted ov er coalgebra s for a Set -functor: Deﬁnition 2. 1. [Rutten 2000] Let T : Set → S et b e a functor, referred to a s the signatur e functor , where Set is the category of se ts. A T -c o algebr a A = ( X , ξ ) is a pair ( X , ξ ) wher e X is a set (of states ) and ξ : X → T X is a function called the tr ansition function. A m orp hism f : A → B betw ee n T -coalgebr as A = ( X, ξ ) and B = ( Y , ζ ) is a map f : X → Y suc h that T f ξ = ζ f . W e view coalgebra s as genera lised transition systems: the transition function de- livers a structured set of successo rs a nd observ ations for a state. Mutatis m utandis, we can in fact allow T to tak e prop er classes as v alues, as we never itera te T or otherwise assume tha t T X is a set; deta ils are left implicit. This allows us to treat more examples, in particular Pauly’s coalition logic (Example 2.7.8 b elo w). Assumption 2 . 2. W e can assume w.l.o .g. that T preser v es injectiv e ma ps [Barr 1993]. F or convenience of notation, w e will in fa ct sometimes assume that T X ⊆ T Y in case X ⊆ Y . Moreover, we assume w.l.o.g . that T is non-tr ivial, i.e. T X = ∅ = ⇒ X = ∅ (other wise, T X = ∅ for all X ). Deﬁnition 2. 3. If for a subset Z ⊆ X of a coalgebra A = ( X , ξ ), ξ restricts to a map ξ Z : Z → T Z , then C = ( Z , ξ Z ) is a sub c o algbr a of A ; in this case, the inclusion Z ֒ → X is a mor phism C → A . In the same way that the signa ture functor abstra cts from a concrete cla ss of mo dels, the interpretation of mo dal op erators is encapsula ted in ter ms of predicate liftings: Deﬁnition 2. 4. A pr e dic ate lifting for a functor T is a natural transfor mation Q → Q ◦ T op , 4 · L. Schr¨ oder an d D. Pattinson where Q denotes the contrav aria n t powerset functor Set op → Set (i.e. Q ( X ) = P ( X ) is the p ow erset, a nd Q f ( B ) = f − 1 [ B ] for f : X → Y and B ∈ Q ( X )). A coalg ebraic semant ics for a modal s ignature Λ is giv en by a Λ -stru ctur e , cons is ting of a signature functor T and an assignment of a predicate lifting [ [ L ] ] for T to every mo dal op erator L ∈ Λ; by abuse of notation, we re fer to the entire Λ-s tructure just as T . Giv en a Λ- structure T , the s atisfaction r elation | = C betw een states x of a T -coa lgebra C = ( X, ξ ) and F (Λ)-for m ulas is deﬁned inductively , with the usual clauses for the b o olea n op erations. The c la use for the mo dal op erator L is x | = C Lφ ⇐ ⇒ ξ ( x ) ∈ [ [ L ] ] C ([ [ φ ] ] C ) , where [ [ φ ] ] C = { x ∈ X | x | = C φ } . W e drop the s ubscripts C when these a re clear from the co n text. W e o ccasionally make use of the fact that the logic F (Λ) is ade quate for T - coalgebr as [Pattinson 2004]: Proposition 2.5. If f : A → B is a morphism of T -c o algebr as, then x | = A φ iﬀ f ( x ) | = B φ for al l states x in A and al l F (Λ) -formulas φ . Our main interest here is in the lo cal satisﬁability pr oblem : Deﬁnition 2. 6. An F (Λ)-for m ula φ is satisﬁable (ov er T ) if there e x ist a T - coalgebr a A = ( X, ξ ) and a state x in X such that x | = A φ . Dually , φ is valid if x | = A φ fo r a ll T -coa lgebras A = ( X, ξ ) and a ll x ∈ X . Example 2.7. [Pattinson 2004; C ˆ ırstea and Pattinson 2007; Sc hr ¨ oder 2 0 07] W e illustrate how the coalgebr aic a pproach subsumes a larg e class of mo dal logics . This includes not only lo gics with a standard Kripke se ma n tics, but in particular also non-normal mo dal logics whose semantics is deﬁned ov er structures that diﬀer substantially from classical Kripke frames. (1) Mo dal lo gic K : The signature Λ K of the mo dal logic K consis ts of a single mo dal operato r ✷ . Let P b e the co v a r ian t pow er set functor. Then P -coalgebr as are graphs, tho ugh t of as transition systems or indeed Kr ipk e fr ames. A Λ K -structure ov er P is deﬁned by [ [ ✷ ] ] X ( A ) = { B ∈ P ( X ) | B ⊆ A } ; this induces pre cisely the standar d Kripke semantics of mo dal log ic (note that no restrictions are imp o sed o n frames). (2) Mo dal lo gic K D : K D is obtained from K by a dding the ax iom ¬ ✷ ⊥ , i.e. by restricting the seman tics to serial Kripke frames ( X , R ), c haracterize d by the condition that for every state x , there exists a s tate y such that xRy . Thus, the signature Λ K D of the normal modal logic K D is the same as that of K , and a Λ K D - structure is deﬁned in the same wa y as for K , but over the non-empty p o werset functor P ∗ deﬁned by P ∗ ( X ) = { A ∈ P ( X ) | A 6 = ∅} . (3) Mo dal lo gic E : The signa ture Λ E of the mo dal logic E , the smallest classi- cal mo dal logic [Chellas 19 80], has a s ingle mo dal op erator ✷ ; the pro of sys tem o f E comprises, b esides prop ositional rea soning, only repla c e men t of equiv alents (i.e. PSP ACE Boun ds for R ank-1 Mo dal Logics · 5 the rule a ↔ b/ ✷ a → ✷ b ). The standar d neighbour hoo d se man tics o f E is coalge- braically captured b y a Λ E -structure over the neighb ourho o d functor N = Q ◦ Q op (comp osition of the contrav aria n t pow erset functor with itself ); coalgebra s for this functor are neigh bo urhoo d fra mes. The mo dal op erator ✷ is interpreted over N b y [ [ ✷ ] ] X ( A ) = { α ∈ N ( X ) | A ∈ α } . (4) Mo dal lo gic M : The mo dal logic M , the smallest monotonic mo dal logic [Chellas 19 8 0], is o btained from the mo dal logic E by adding the mono- tonicity rule a → b/ ✷ a → ✷ b . The neighbourho o d semantics of M is captured coalgebr aically ana lo gously to the previous exa mple as a structure over the s ub- functor Up P of N assig ning to a set X the set of upw a rds clo sed subsets of Q X . Coalgebr a s for Up P a re mo no tone neighbourho o d frames [Hansen and Kupke 2 004]. (5) Gr ade d mo dal lo gic [Fine 1972]: The mo dal signature of gr ade d mo dal lo gic (GML) is Λ GML = { ✸ k | k ∈ N } ; the intended reading of ✸ k φ is ‘ φ ho lds in more than k successor states’. The semantics of GML is origina lly deﬁned by count- ing succe ssor states in Kripke frames. This semantics fails to be coalgebr a ic, as the naturality condition for the asso ciated predicate liftings fails. How ever, one may deﬁne a coalgebra ic semantics whic h is equiv alent for purpo ses o f satisﬁabil- it y [Sc hr ¨ oder 2007], as fo llo ws. The ﬁnite multiset (or b ag ) functor B maps a set X to the set of maps B : X → N with ﬁnite supp ort, the in tuition b eing tha t B is a multiset co n taining x ∈ X with multiplicit y B ( x ). W e extend B to P ( X ) by putting B ( A ) = P x ∈ A B ( x ). The action on mo rphisms f : X → Y is then given b y B f : B X → B Y , B 7→ λy . B ( f − 1 [ { y } ]). Coa lgebras for B are dir ected gr a phs with N -weigh ted edges, often referr ed to as multigr aphs [D’Agostino and Visser 2002]. The graded mo dal op erator ✸ k is intepreted ov er B b y [ [ ✸ k ] ] X ( A ) = { B : X → N ∈ B ( X ) | B ( A ) > k } . Thu s, x  ✸ k φ for a state x in a B -coalgebra iﬀ φ holds fo r mor e than k s uc c e ssor states of x , taking int o a c coun t multiplicities. The dual o p erator s ¬ ✸ k ¬ are denoted  k , i.e.  k φ reads ‘ φ fails in at most k successor states ’. Note that  k is monotone, but fails to be normal unless k = 0. A non-monotone v ariation of GML a rises when negative multiplicities a r e admitted. (6) Majority lo gic [Pacuit and Sala me 2 004]: Gr a ded moda l logic is extended to majority lo gic by a dding a we ak majority op er ator W , read ‘in at least half of the successor states , it is the case that . . . ’. The str ucture for GML ov er the m ultiset functor B describ ed in the previous example is extended to W by putting [ [ W ] ] X ( A ) = { B : X → N ∈ B ( X ) | B ( A ) ≥ B ( X − A ) } . The dual o perato r M = ¬ W ¬ captur e s strict ma jor it y ‘in mor e than half of the successor states, it is the ca se that’. (7) Pr ob abilistic mo dal lo gic [Larsen and Skou 1991 ; Heifetz a nd Mongin 2001]: The moda l signature Λ PML of pr ob abilistic mo dal lo gic (P ML) co mprises opera - tors L p , p ∈ [0 , 1] ∩ Q , to b e re ad ‘in the next step, it is with pro babilit y at least p the case that. . . ’. W e deﬁne a Λ PML -structure ov er the ﬁn it e distribution func- tor D ω which maps a s et X to the set of probability distributions o n X with ﬁnite 6 · L. Schr¨ oder an d D. Pattinson suppo rt. Coalgebr as fo r D ω are probabilistic transition systems (also called pr ob- abilistic typ e sp ac es [Heifetz a nd Mongin 200 1]) with ﬁnite branching degr ee. O ur deﬁnition contrasts with that of [Heifetz and Mo ng in 200 1], where there is no re- striction o n the branching degree, but since PML ha s the ﬁnite mo del prop ert y (cf. lo c. cit.), this has no b earing on sa tisﬁabilit y . The interpretation o f L p ov er D ω is deﬁned by [ [ L p ] ] ( A ) = { P ∈ D ω X | P A ≥ p } . PML is non-normal ( L p ( a ∨ b ) → L p a ∨ L p b is not v alid for p > 0 ). (8) Co alition lo gic [Pauly 2002]: Let N = { 1 , . . . , n } b e a ﬁxed set o f agents . Subsets of N are calle d c o alitions . The signature Λ Co al of coalition lo gic consists of mo da l op erators [ C ], where C r anges over coalitions, read ‘coalition C has a collab orative strategy to ensure that . . . ’. A coalg ebraic sema n tics for coalition logic is ba sed on the class -v alued signature functor T deﬁned by T X = { ( S 1 , . . . , S n , f ) | ∅ 6 = S i ∈ Set , f : Q i ∈ N S i → X } . The elemen ts of T X ar e understo o d as str ate gic games with set X of states, i.e . tuples consisting o f nonempty sets S i of st ra t e gies for all agents i , and an ou t c ome function ( Q S i ) → X . A T -coa lgebra is a game fr ame [Pauly 2 0 02]. W e denote the set Q i ∈ C S i by S C , and for σ C ∈ S C , σ ¯ C ∈ S ¯ C , where ¯ C = N − C , ( σ C , σ ¯ C ) denotes the obvious elemen t of Q i ∈ N S i . A Λ Co al -structure ov er T is then deﬁned by [ [[ C ]] ] X ( A ) = { ( S 1 , . . . , S n , f ) ∈ T X | ∃ σ C ∈ S C . ∀ σ ¯ C ∈ S ¯ C . f ( σ C , σ ¯ C ) ∈ A } . All the ab o ve ex a mples can b e ca nonically extended to sys tems that pro cess inputs fr om a s et I by passing fro m the s ignature functor T to o ne of the functors T I or T ( I × ) and suitably indexing the mo dal op erator s. W e r efer to [C ˆ ırstea and Pattinson 2007 ] for a detaile d acco un t of the induced lo gics. R emark 2.8. In the mo dal grammar g iven ab o ve, atomic prop ositional sy mbo ls are delib erately not included. This is for the sa k e of b oth generality , a s some mo dal logics suc h as Henness y-Milner logic do not include such atomic propo s itions, and economy o f presentation, as a set U of atomic pr opositio nal symbols ma y be int egrated in the basic fra mework as follows. Given a modal signature Λ and a Λ-structure T , we deﬁne a structure for the mo dal signa ture Λ U = Λ ∪ U ov er the functor T U deﬁned by T U X = T X × P ( U ): mo dal op erators from Λ are interpreted by taking the preimage o f their in ter pretation o ver T under the pro jection T U → T , and a prop ositional symbol a ∈ U is int erpreted by putting [ [ a ] ] X ( A ) = { ( t, B ) ∈ T X × P ( U ) | a ∈ B } . Since [ [ a ] ] is independent of its ar gumen t, the mo dal op erator a can b e written as just the pr opositio nal symbol a (without an argument formula). In a framework with po ly a dic mo dal op erator s [Sc hr ¨ o de r 2005], prop ositional consta n ts corr espond to nullary mo dalities. Some of the logics ab o ve indeed r e quir e pro positiona l s ym b ols lest they co llapse into trivia lit y . This holds in those cases where T 1 (for 1 a singleton set) is a singleton, e.g. probabilistic mo dal logic, co alition logic, and the mo dal logic KD . W e nev ertheles s gener ally co n tinue to omit the treatmen t of prop ositional symbols in the sequel, since the addition of prop ositional s ym b ols as indicated abov e PSP ACE Boun ds for R ank-1 Mo dal Logics · 7 has no b earing on the rule sets forming the core of our method, and the mo del construction is ent irely analogo us. 3. PROOF SYSTEM S FOR COALG EBRAIC MODAL LOGIC Our decision pro cedure for rank-1 lo g ics relies o n a co mplete axiomatisatio n in a certain format. Deduction for mo dal lo g ics with co algebraic semantics has b een considered in [Pattinson 2003; C ˆ ırstea and Pattinson 20 07; Kupke et al. 2 005; Schr¨ oder 2007 ]. It ha s b een shown that ev e ry mo dal logic ov er coalgebr as can be axiomatised in rank 1 using either rank-1 axioms or rules leading from ra nk 0 to rank 1 [Schr¨ oder 2007], esse n tially b ecause functors, a s oppo sed to comonads, only enco de the one-step b ehaviour o f systems. Here, w e fo cus on r ules. The crucial ingredients for the shallow model c o nstruction and the ensuing PSP ACE algo rithm are novel notions of r esolut ion closur e a nd strict one-step c ompleteness of rule sets. F or the remainder of the pap er, we ﬁx a mo dal signature Λ and a Λ-structure T . W e reca ll a few basic notio ns from prop ositional logic, a s well a s notation for coal- gebraic mo dal log ic in tr o duced in [Pattinson 200 3; C ˆ ır stea and Pattinson 20 07]: Deﬁnition 3. 1. W e denote the set of pr opositio nal form ulas ov er a set V (con- sisting e.g. of prop ositional v a riables or modal formulas) b y Prop ( V ). Here, we regar d ¬ and ∧ as the ba sic connectives, with all other co nnec tives deﬁned in the standard w ay . F or φ, ψ ∈ Prop ( V ), w e sa y that φ pr op ositional ly entails ψ a nd write φ ⊢ PL ψ if φ → ψ is a pr opos itional tautology . Similar ly , Φ ⊆ Prop ( V ) prop osition- ally entails ψ (Φ ⊢ PL ψ ) if there e xist φ 1 , . . . , φ n ∈ Φ s uc h that φ 1 ∧ · · · ∧ φ n ⊢ PL ψ . A liter al ov er V is either an ele men t o f V or the negatio n of such an element. W e use the meta-v aria ble ǫ (po ssibly indexed) to denote either nothing or ¬ , s o that a literal over V has the gener al for m ǫa , a ∈ V . A clause is a ﬁnite (p ossibly empty) disjunction of literals, which then takes the form W n i =1 ǫ i a i with a 1 , . . . , a n ∈ V . Similarly , a c onjunctive clause is a ﬁnite conjunction of literals. A (conjunctive) clause is c ontra cte d if all its litera ls are distinct. Th e set of all clauses ov e r V is denoted by Cl ( V ). Althoug h we regar d clauses as formu las rather than sets of literals, we shall sometimes use terminolog y such as ‘a literal is contained in a clause’ or ‘a claus e contains another’, with the obvious meaning. W e denote b y Up ( V ) the s e t { La | L ∈ Λ , a ∈ V } . If V consists of pro positiona l v ar iables, then we hav e the us ual notions o f v a lua- tion and substitution: A valuation is just a map κ : V → {⊤ , ⊥} assigning bo olean truth v alues to v ar iables; for φ ∈ Prop ( V ), we write κ | = φ if κ is a satisfying v aluation for φ . More generally , given a set X , a P ( X ) -valuation for V is a map V → P ( X ). F or φ ∈ Prop ( V ), a P ( X )-v alua tio n τ induces in the obvious w ay a subset [ [ φ ] ] τ of X ; we wr ite X , τ | = φ if [ [ φ ] ] τ = X . Using the structure T for Λ, we int erpret ψ ∈ Prop ( Up ( V )) as a subset [ [ ψ ] ] τ o f T X by putting [ [ Lφ ] ] τ = [ [ L ] ][ [ φ ] ] τ , and we w r ite T X , τ | = ψ if [ [ ψ ] ] τ = T X . Moreover, given a set Z , a Z -su bst itution for V is a map σ : V → Z ; for a formula φ ov er V (e.g. φ ∈ Prop ( Up ( Prop ( V ) ))), we denote the result o f p erforming the substitution σ o n φ by φσ and refer to φσ as a Z -instanc e of φ . Lemma 3. 2 . F or φ, ψ ∈ Cl ( V ) , φ ⊢ PL ψ iﬀ either φ is c ontaine d in ψ or ψ is a tautolo gy ( i.e. c ontains b oth a and ¬ a for some a ∈ V ). 8 · L. Schr¨ oder an d D. Pattinson Deﬁnition 3. 3. A (one-step) rule R over a set V of prop o sitional v aria bles is a rule φ/ψ , where φ ∈ Prop ( V ) and ψ ∈ Cl ( Up ( V )). W e silently iden tify rules mo dulo α -equiv a le nc e . The r ule R is sound if, whenever φσ is v alid for an F (Λ)- substitution σ , then ψ σ is v alid. Moreov er , R is one-step sound if T X, τ | = ψ for each set X a nd each P ( X )-v aluatio n τ suc h that X , τ | = φ . Our hitherto informal use o f the term r ank- 1 lo gic formally means axiomatisable by one-step ru les . The term rank- 1 logic ha s b een used in the litera ture [Pattinson 2003; C ˆ ırstea and Pattinson 2007 ; Kupke et al. 200 5 ; Schr¨ oder 2007 ] to describ e logics axiomatisa ble b y ra n k-1 axioms , i.e. prop ositional com bina tions of formu- las Lφ where L is a mo dal o perato r a nd φ is pur ely pr opositio nal (in the notatio n int ro duced ab ov e, formulas from Prop ( Up ( Prop ( V )))). This class of axioms includes e.g. the K axiom ✷ ( a → b ) → ( ✷ a → ✷ b ), but excludes axioms co ntaining nested mo dalities or top-level pro positiona l v a r iables such as the axioms 4 and T , resp ec- tively . It has b een shown in [Schr¨ oder 2007 ] that one-step rule s and ra nk-1 a xioms determine the same class o f logic s. R emark 3.4. W e can alwa y s a ssume that every prop ositional v ariable a a ppear- ing in the premise φ of a one-step rule app ears also in the conclusion: other wise, we can eliminate a b y passing fro m φ to φ [ ⊤ /a ] ∨ φ [ ⊥ /a ]. Proposition 3.5. [Schr¨ oder 2007] Every one-step sound ru le is sound. The conv er s e holds under additional a ssumptions [Schr¨ oder 20 05]; note howev er that the obviously sound r ule ⊥ / ⊥ is one-step sound iﬀ T ∅ = ∅ (as is the ca se e.g. for P ML). A given set R of one- s tep sound rules induces a pro of sys tem fo r F (Λ) as follows. Deﬁnition 3. 6. Let R C denote the set of rules obtained b y extending R with the c ongruenc e rule ( C ) a ↔ b La → Lb for every L ∈ Λ . (This rule o f course implies a rule where → is replaced b y ↔ , which how ever does no t ﬁt the format for one-step rules.) The s et of pr ovable formulas is the s ma llest set closed under prop ositional e ntailment a nd the rules in R C , with prop ositional v ariables instantiated to fo r m ulas in F (Λ). W e say that a form ula φ is c onsist en t if ¬ φ is not prov able. It is easy to see that this pro of s ystem is so und. Completeness requires ‘enough’ rules in the following sense. Deﬁnition 3. 7. The set R is (strictly) one-step c omplete if, w hene ver T X , τ | = χ for a set X , χ ∈ Cl ( Up ( V )), a nd a P ( X )-v aluation τ , then χ is (strictly) pr ovable ov er X , τ , i.e. prop ositionally entailed by clause s (a cla use) ψ σ where φ/ψ ∈ R C (Deﬁnition 3 .6) and σ is a Prop ( V )-substitution (a V -substitution) such that X, τ | = φσ . Strict o ne-step completeness is one of cr ucial notions in this work. Its distinc- tive feature is that strict prov a bilit y larg ely disp enses with pro p ositional rea soning by restricting instantiations to propos itional v ar iables, and b y replacing gener a l PSP ACE Boun ds for R ank-1 Mo dal Logics · 9 prop ositional entailmen t by the r ather trivial concept of pro p ositional en tailment betw een single clauses (cf. Lemma 3 .2 ). This plays a cent ral role in the shallow mo del cons tr uction presented in Sec tio n 4. R emark 3.8. It is shown in [Schr¨ oder 200 7] that the set of all one-step sound rules is alwa ys str ictly one-s tep complete and that the pro of system induced by a one-step complete s et of rules is we akly c omplete , i.e. prov es all v alid form ulas. In the further tr eatmen t, we need a further technical co nditio n. Deﬁnition 3. 9. A one-step rule φ/ψ ov er V is inje ctive if every v a riable in V o ccurs at most once in ψ . Assumption 3 . 10. W e ass ume for the remainder of the pape r that the given rules in R ar e inje ct ive . This restriction will be satisﬁed b y the naturally arising rule sets in our examples; it can always be forc e d b y introducing new prop ositional v ariables and adding premis es stating the equiv ale nce to the o riginal v a riables (e.g. a rule ⊤ / ( ✷ a → ✸ a ) can b e replaced by ( a ↔ b ) / ( ✷ a → ✸ b )). Strictly one - step complete se ts of rules are g e nerally more complica ted tha n one- step complete sets of rules or axioms [Pattinson 2003 ; Schr¨ oder 20 07]. In our terminology , pa rt of the eﬀort of [V ar di 1989] and [P a uly 20 02] is devoted to ﬁnding strictly o ne-step complete sets of rules . W e now dev elo p a systematic pro cedure fo r turning one-step complete rule sets into strictly one-step co mplete o nes. F or the following, recall that g iv en clauses φ a nd ψ con taining literals a a nd ¬ a , respe c tiv ely , a r esolvent of φ and ψ ( at a ) is obtained b y removing a and ¬ a from the clause φ ∨ ψ . A set Φ o f clauses is called r esolution close d if, for φ, ψ ∈ Φ, all resolven ts of φ and ψ are prop ositionally entailed by some clause in Φ. This is genera lised to rules as follows: Deﬁnition 3. 11. A set R of one-step rules is r esolution close d if it sa tisﬁes the following req uiremen t. Let R 1 , R 2 ∈ R , where R 1 = φ 1 /ψ 1 and R 2 = φ 2 /ψ 2 . W e can assume that R 1 and R 2 hav e disjoin t s ets V 1 , V 2 of prop ositional v a riables. Let La be in ψ 1 , and let ¬ Lb be in ψ 2 for some L ∈ Λ, s o that we have a r e s olv e nt ¯ ψ of ψ 1 and ψ 2 [ a/b ] at L a ; by Assumption 3 .10, ¯ ψ is a clause over Up ( V ) where V = V 1 ∪ V 2 − { a, b } . Then R C is required to co ntain a rule R = φ/ψ s uc h φ 1 ∧ φ 2 [ a/b ] ⊢ PL φσ and ψ σ ⊢ PL ¯ ψ for some V -substitution σ ; in this case, R is called a r esolvent o f R 1 and R 2 . Resolution closure will play a cen tral role in the following development , a s it for ms the syntactic counterpart of strict one-step completeness. R emark 3.12. O ne can construc t reso lution closed sets by iterated addition of missing reso lv ents. Here, an obvious c hoice for a r esolven t φ/ψ of φ 1 /ψ 1 and φ 2 /ψ 2 as ab o ve is to take ψ a s the r esolven t ¯ ψ of ψ 1 and ψ 2 , and φ as φ 1 ∧ φ 2 [ a/b ], with a eliminated according to Remark 3 .4 as a is not contained in ψ by Assumption 3.1 0. It is clea r that φ 1 ∧ φ 2 [ a/b ] / ¯ ψ is one-step sound if R 1 and R 2 are one-step sound. R emark 3.13. No te that o ur approach is diﬀerent to existing resolution-bas ed approaches to decision pr ocedur es for mo dal logic (e.g . [De Nivelle et al. 2000]), which rely on translating mo dal log ic in to ﬁrst-order log ic . 10 · L. Schr¨ oder and D . Pattinson Lemma 3. 1 4. L et ψ ∈ Cl ( V ) , and let ∅ 6 = Φ ⊆ Cl ( V ) b e re solution close d. Then Φ ⊢ PL ψ iﬀ φ ⊢ PL ψ for some φ ∈ Φ . Proof. The ‘if ’ dir ection is clear. ‘Only if ’: W.l.o.g. ψ is not a tautolo gy . W e can assume that V is ﬁnite and then pr ove the contrapo sition of the claim by induction o ver the s iz e of V . T hus assume, recalling Lemma 3.2, that Φ does not contain a sub clause of ψ . Pick a cla use χ ∈ Φ that con tains a minimal num b er o f literals not in ψ (this num b er is no n- zero); w.l.o.g. χ co n tains a p ositiv e literal a such that a is not in ψ . Remov e all cla uses containing a from Φ, and remove ¬ a from the remaining clauses and from ψ , obtaining a new set Φ ′ of clauses and a new clause ψ ′ , respectively . Then Φ ′ is resolutio n closed and do es not contain a sub c lause o f ψ ′ (otherwise there exists a cla us e ρ ∈ Φ whose only literal not in ψ is ¬ a , and resolv ing ρ with χ yields a clause in Φ with less litera ls not in ψ than χ , contradiction). By induction we th us have a v aluation τ ′ for V − { a } satisfying Φ ′ but not ψ ′ . W e extend τ ′ to a v aluation τ for V by putting τ ( a ) = ⊤ ; then τ satisﬁes Φ but not ψ . Lemma 3. 1 5. R is r esolution close d iﬀ R C is r esolution close d. Proof. The ‘if ’ direction is trivial. The ‘o nly if ’ dir e ction follows from the fact that every rule R is a r esolven t of R a nd a n y congruence r ule, since rules a r e injectiv e (Assumption 3.10). Theorem 3. 16. L et R b e one-step c omplete. Then R is strictly one-step c om- plete iﬀ R is r esolution close d. Proof. ‘If ’: Let X be a set, let τ b e a P ( X )-v alua tion, and let χ ∈ Cl ( Up ( V )) such that T X , τ | = χ ; w.l.o.g. χ is not a tautolo gy . By one-s tep completeness, χ is prop ositionally entailed by the (non- empt y) set of cla uses Ψ = { ψ σ | φ/ψ ∈ R C , σ a Prop ( V )-substitution , X, τ | = φσ } . The set Ψ is r esolut ion close d: for i = 1 , 2, let φ i /ψ i ∈ R C be a rule ov er W i (with W 1 , W 2 disjoint) , let σ i be a Prop ( V )-substitution such that X , τ | = φ i σ i , and let ψ 1 σ 1 and ψ 2 σ 2 contain liter als Lρ a nd ¬ Lρ , resp ectively . Th us , ψ 1 and ψ 2 contain literals La and ¬ Lb , r espectively , where σ 1 ( a ) = σ 2 ( b ) = ρ ; le t ¯ ψ be the resolven t of ψ 1 , ψ 2 [ a/b ] at La , a clause ov er W = W 1 ∪ W 2 − { a , b } . Then the resolven t o f ψ 1 σ 1 , ψ 2 σ 2 at Lρ is ¯ ψ σ , wher e σ acts like σ 1 on W 1 − { a } and like σ 2 on W 2 − { b } . By re s olution closedness of R C (Lemma 3.1 5), we hav e φ/ψ ∈ R C and a W -substitution θ such that φ 1 ∧ φ 2 [ a/b ] ⊢ PL φθ and ψ θ ⊢ PL ¯ ψ . Then X , τ | = φθ σ , so that ψ θσ ∈ Ψ, a nd ψ θ σ ⊢ PL ¯ ψ σ as req uired. By Lemma 3.14, it now follows that ψ σ ⊢ PL χ for some clause ψ σ in Ψ, where by Lemma 3.2 necess arily σ ( v ) ∈ V for every v ariable v in ψ . ‘Only if ’: Let φ 1 /ψ 2 , φ 2 /ψ 2 ∈ R b e rules over disjoint sets V 1 , V 2 of v ar iables, where ψ 1 contains La and ψ 2 contains ¬ Lb . L e t ¯ ψ denote the reso lvent of ψ 1 , ψ 2 [ a/b ] at La , a clause over V = V 1 ∪ V 2 − { a, b } . Let X b e the set of satisfying v a luations for φ 1 ∧ φ 2 [ a/b ], and deﬁne the P ( X )-v alua tion τ by τ ( a ) = { κ ∈ X | κ ( a ) = ⊤ } . Then X , τ | = φ 1 ∧ φ 2 [ a/b ] and hence T X , τ | = ¯ ψ by one- step soundness of R . By strict one-step co mpleteness, it follows that there exists a rule φ/ψ ∈ R C and a V -substitution σ such that X , τ | = φσ and ψ σ ⊢ PL ¯ ψ . By constructio n of X , τ , we may conclude fr om X , τ | = φσ tha t φ 1 ∧ φ 2 [ a/b ] ⊢ PL φσ as re q uired. PSP ACE Boun ds for R ank-1 Mo dal Logics · 11 In summary , stric tly one-step complete rule sets can be constructed by resolving the rules of a o ne - step complete axiomatisa tion ag a inst each other. Below, we give examples of str ictly one-s tep complete systems obtained in this way . In order to simplify the presentation for the ca s e of graded mo dal logic and probabilistic mo dal logic, we use the follo wing nota tion. If φ i is a formula, r i ∈ Z for a ll i ∈ I , and k ∈ Z , we abbrevia te X i ∈ I r i φ i ≥ k ≡ ^ J ⊆ I r ( J ) 0, and ¬ ✸ 0 φ for n = 0 . These axio ms may b e derived from the system o f one-s tep rules ( RG 1) a → b ✸ n +1 a → ✸ n b ( A 1) c → a ∨ b ✸ n 1 + n 2 c → ✸ n 1 a ∨ ✸ n 2 b ( A 2) a ∨ b → c a ∧ b → d ✸ n 1 a ∧ ✸ n 2 b → ✸ n 1 + n 2 +1 c ∨ ✸ 0 d ( RN ) ¬ a ¬ ✸ 0 a (( G 1) and ( N ) are easily derived fro m ( RG 1) and ( R N ), resp ectiv e ly ; ( G 2) follows by ( A 1) taking n 2 = 0; and ( G 3) may b e derived using ( A 1) and ( A 2)). All these rules are subsumed by the rule schema ( G ) P n i =1 a i ≤ P m j =1 b j V n i =1 ✸ k i a i → W m j =1 ✸ l j b j , where n , m ≥ 0, sub ject to the side condition P n i =1 ( k i + 1) ≥ 1 + P m j =1 l j (whic h ent ails that n and m cannot bo th b e 0 ). O ne - step soundness of ( G ) follows from one- step soundness of the rule sy s tem for ma jority lo gic pr oved in the next example. By the preceding considerations , ( G ) is weakly complete, a nd hence one-step complete by Prop osition 5.3. R esolution close dness of ( G ) : T ake tw o instances of ( G ), one denoted lik e in the general form of the rule and one with all entities primed ( a ′ i etc.), with the reso lution taking place w.l.o.g. b y matc hing ✸ k ′ 1 a ′ 1 ≡ ✸ l 1 b 1 . The conclusion of the aris ing resolven t is n ^ i =1 ✸ k i a i ∧ m ′ ^ i =2 ✸ k ′ i a ′ i → m _ i =2 ✸ l j b j ∨ m ′ _ j =1 ✸ l ′ j b ′ j . Since a ′ 1 ≡ b 1 , the premises P n i =1 a i ≤ P m j =1 b j and P n ′ i =1 a ′ i ≤ P m ′ j =1 b ′ j imply n X i =1 a i + n ′ X i =2 a ′ i ≤ m X j =2 b j + m ′ X j =1 b ′ j , and since k 1 = l ′ 1 , t he side conditions P n i =1 ( k i + 1) ≥ 1 + P m j =1 l j and P n ′ i =1 ( k ′ i + 1) ≥ 14 · L. Schr¨ oder and D . Pattinson 1 + P m ′ i =1 l ′ j imply n X i =1 ( k i + 1) + n ′ X i =2 ( k ′ i + 1) ≥ 1 + m X j =2 l j + m ′ X j =1 l ′ j , so that we arrive again a t a n instance of ( G ). (7) Majority lo gic: in [Pacuit and Salame 2004], the extensio n o f the axiomati- zation of graded mo dal logic with the axioms ( M 1) M a ∧ M b → ✸ 0 ( a ∧ b ) ( M 2) M a ∧  0 ( a → b ) → M b ( M 3) W a ∧ W b ∧ ✸ n ( ¬ a ∧ ¬ b ) → ✸ n ( a ∧ b ) ( M 4) W a ∧ M b ∧ ✸ n ( ¬ a ∧ ¬ b ) → ✸ n +1 ( a ∧ b ) is proved to b e weakly complete for ma jority logic including prop ositional symbols. These a xioms ar e der iv able from the set of rules ( RM 1 ) a ∨ b W a ∨ W b ( RM 2 ) a → b ∨ c W a → W b ∨ ✸ 0 c ( RM 3 ) ¬ ( a ∧ c ) ¬ ( b ∧ c ) a ∧ b → d W a ∧ W b ∧ ✸ n c → ✸ n d ( RM 4 ) ¬ a ∧ b a → c ∨ d b → c W a ∧ ✸ n b → W c ∨ ✸ n +1 d (( M 2), ( M 3) and ( M 4) follow directly from ( R M 2), ( RM 3) and ( RM 4 ), resp ec- tively; ( RM 1) proves M a → W a , whence ( M 1) is obtained from ( RM 2 )). These rules and r ule ( G ) for GML are subsumed by the rule schema ( M u ) P n i =1 a i + P v r =1 c r + u ≤ P m j =1 b j + P w s =1 d s V n i =1 ✸ k i a i ∧ V v r =1 W c r → W m j =1 ✸ l j b j ∨ W w s =1 W d s ( u ∈ Z ) with side co nditio ns P n i =1 ( k i + 1) − P m j =1 l j − 1+ w − max( u, 0) ≥ 0 and v − w + 2 u ≥ 0 (take u = 1 for ( R M 1), u = 0 for ( R M 2), ( R M 4), and ( G ), and u = − 1 for ( RM 3 )). Resolution close dnes s is chec ked a nalogously as for graded mo dal log ic, cov ering the tw o cases of resolution a t literals ✸ n a and W a , r espectively; in b oth cases, an instance of M u 1 + u 2 can be taken as a r e s olv e n t o f an instance of M u 1 and an insta nce of M u 2 . One-step soundness of ( M u ) : Let τ b e a P ( X )-v aluatio n such that X, τ | = P n i =1 a i + P v r =1 c r + u ≤ P m j =1 b j + P w s =1 d s . Let B ∈ B ( X ). Using Lemma 3.17, we obtain by summation over x ∈ X n X i =1 B ( σ ( a i )) + v X r =1 B ( σ ( c r )) + uB ( X ) ≤ m X j =1 B ( σ ( b j )) + w X s =1 B ( σ ( d s )) . Now put p = ⌈ B ( X ) / 2 ⌉ (with ⌈ x ⌉ = min { z ∈ Z | z ≥ x } ) so that B satisﬁes W a iﬀ B ( τ ( a )) ≥ p . T o esta blis h that B is in the interpretation of the co nclusion o f M u , PSP ACE Boun ds for R ank-1 Mo dal Logics · 15 it suﬃces to prov e n X i =1 ( k i + 1) + v p + uB ( X ) ≥ m X j =1 l j + w ( p − 1) + 1 . By the s ide conditions, this inequa lit y is equiv alent to − 2 up + uB ( X ) + max( u, 0) ≥ 0 , which is easily established by distinguishing the cas es B ( X ) = 2 p a nd B ( X ) = 2 p − 1. (8) Pr ob abilistic mo dal lo gic: By refor m ulating the o ne-step co mplete s et of ax- ioms for probabilis tic moda l logic given by C ˆ ır stea a nd P a ttinson [2007] as one-step rules and subs e quen tly apply ing r e solution, one o btains the rules ( P u ) P n i =1 a i + u ≤ P m j =1 b j V n i =1 L p i a i → W m j =1 L q j b j , where m, n ≥ 0, m + n ≥ 1, and u ∈ Z , sub ject to the side co ndition P n i =1 p i + u ≥ P m j =1 q j and P n i =1 p i + u > 0 if m = 0 . One-step c ompleteness o f ( P u ): The rule schema is one-step complete, a s it sub- sumes the following a xiomatisation that has bee n shown to b e one-s tep co mplete in lo c.cit.: (0) L 0 a ( ⊤ ) a L p a ( > 1) ¬ a ∨ ¬ b ¬ L p a ∨ ¬ L q b ( p + q > 1) (1) a ∨ b L p a ∨ L q b ( p + q = 1) ( 1 ) P r i =1 c i = P s j =1 ¯ d j V r i =1 L u i c i ∧ V s j =2 L (1 − v j ) d j → L v 1 d 1 , where ¯ d 1 = d 1 and ¯ d j = ¬ d j for j ≥ 2, and rule ( 1 ) is sub ject to the s ide condition s X j =1 v j = r X i =1 u i . These rules ar e subsumed by the rule schema ( P u ), a s follows. Rule (0): take m = 1 , n = 0, u = 0, q 1 = 0 . Rule ( ⊤ ): take m = 1, n = 0, u = 1. Rule > 1: take n = 2, m = 0, u = − 1 . Rule (1): take n = 0, m = 2, u = − 1. Rule ( 1 ): take m = 1, n = r + s − 1, u = 1 − s , and insta n tiate b i to c i for i = 1 , . . . , r , b i to d i − r +1 for i = r + 1 , . . . , r + s − 1, a 1 to d 1 , q i to u i for i = 1 , . . . , r , q i to 1 − v i − r +1 for i = r + 1 , . . . , r + s − 1, and p 1 to v 1 . One-step soundness : Analog ously to the previous exa mple, us ing a dditionally that one a lw ays has P ( X ) = 1. R esolution close dness: Analog ously as for gra ded mo dal lo gic; as a r e s olv e nt of an instance o f P u 1 and a n instance of P u 2 , one c an take an instance of P u 1 + u 2 . 16 · L. Schr¨ oder and D . Pattinson 4. THE SHALLOW MODE L CONSTRUCTION W e now present the announced generic shallow model construction, which is ba sed on strictly o ne-step complete axioma tisations. The co ns truction ge ne r alises res ults from [V ardi 198 9 ] (where the use of axiomatisatio ns is implicit in certain lemmas). Deﬁnition 4. 1. The set MA ( φ ) of (top level) mo dal atoms of a formula φ is deﬁned r e c ursiv ely b y MA ( φ ∧ ψ ) = MA ( φ ) ∪ MA ( ψ ), MA ( ¬ φ ) = MA ( φ ), and MA ( Lρ ) = { Lρ } . (Note that φ ∈ Pro p ( MA ( φ )).) A pseudovaluation is a conjunc- tive clause H over Up ( F (Λ)), repres en ted as a set of literals (i.e. pseudov alua tio ns are identiﬁed mo dulo contraction and reo rdering of literals, whic h do es not aﬀect the set MA ( H ) of mo dal atoms). A pseudov aluation is c onsistent if it is consisten t as an F (Λ)-formula. W e say that H is a pseudovaluation for φ if MA ( H ) ⊆ MA ( φ ) a nd H ⊢ PL φ . If φ/ψ is a rule in R C and σ is a substitution such t hat ψ σ ∈ Cl ( MA ( H )) and H ⊢ PL ¬ ψ σ , then the negated instance ¬ φσ o f the premise φ is a de mand of H . This generalises the no tion of demand [Blackburn et al. 2001, Deﬁnition 6.43] to a co algebraic setting. Note that by the dual of Lemma 3.2, all demands of a pseudov aluation H ar e con tained in H when regar ded as s ets of literals, unles s H is prop ositionally inconstent (i.e. c o n tains b oth L ρ and ¬ Lρ for some mo dal ato m Lρ ). Lemma 4. 2 . Every c onsistent formula has a c onsistent pseudovaluation. Proof. If φ is consistent, then one of th e conjunctiv e clauses from its disjunctive normal form (DNF) is co nsisten t and hence is a consisten t pseudov a luation for φ . Lemma 4. 3 . Every demand of a c onsistent pseudovaluation is c onsistent. Proof. By co n tra position: Let H be a pseudov aluation, a nd let φ/ ψ b e a rule in R C such that ψ σ ∈ Cl ( MA ( H )) and H ⊢ PL ¬ ψ σ . If the demand ¬ φσ is incon- sistent, then φσ is prov able; he nce , ψ σ is pr o v able using φ/ψ , a nd consequently H is inco nsisten t. Deﬁnition 4. 4. A supp orting Kripke fr ame o f a T -coalge br a ( X , ξ ) is a Kripke frame ( X , K ) (consisting of a se t X and a transition rela tion K ⊆ X × X ) such that for e ac h x ∈ X , ξ ( x ) ∈ T { y | x K y } ⊆ T X . Lemma and Definition 4.5. If a c o algebr a C = ( X, ξ ) is e quipp e d with a sup- p orting Kripke fr ame ( X , K ) , then for every state x ∈ X , the s et X x of states r e achable fr om x in ( X , K ) is the c arrier of a sub c o algebr a C x = ( X x , ξ x ) of C , the submo del gene r ated b y x . Note that by Prop osition 2.5 , y | = C x φ iﬀ y | = C φ fo r y ∈ X x . Deﬁnition 4. 6. A shal low table au is a Kr ipke frame ( X , K ) with a distinguished r o ot H 0 ∈ X such tha t X is a s et of pseudov a lua tions, ev er y state is reachable from H 0 , for all H , G ∈ X , H K G = ⇒ G is a pseudov aluation for a demand of H , and for every dema nd φ of H ∈ X ther e exis ts a pseudov aluation G ∈ X for φ s uc h that H K G . Given a formula φ , a shal low table au for φ is a sha llo w tableau whose ro ot is a pseudov aluatio n for φ . PSP ACE Boun ds for R ank-1 Mo dal Logics · 17 A shal low table au mo del is a T - c o algebra C = ( X , ξ ) which has a supp orting Kripke frame ( X , K ) such that ( X , K ) is a shallow tableau and the trut h lemma H ⊢ PL χ = ⇒ H | = C χ for a ll F (Λ)-formulas χ holds fo r a ll H ∈ X (hence in pa rticular H | = C χ if H is a pseudov aluatio n for χ ). A shallow tablea u is almost a dag , ex cept tha t in the presence of the r ule ⊥ / ⊥ (cf. Section 3) the pseudov aluation ⊤ is a pseudov aluation for one of its own demands. Explicitly: Proposition 4.7. A shal low table au ( X , K ) with r o ot H 0 is, u p t o a p ossible lo op at the st at e ⊤ , a dag of depth at most the depth of H 0 , and the br anching de gr e e at H ∈ X is exp onential ly b ounde d in | H | . Proof. The ﬁrs t cla im follows from the fa c t the the depth of all dema nds of a pseudov aluation H is strictly less than the depth of H . T o prov e the b ound on branching, note tha t pseudov aluations for demands of H are co njunctiv e clauses ov er the set of s ubfor m ulas of H . Lemma 4. 8 . If a formula φ has a pseudovaluation H 0 such that al l demands of H 0 ar e c onsistent, then ther e exists a shal low table au for φ . (By Lemmas 4.2 and 4 .3, the conditions of the ab ov e lemma hold in pa r ticular if φ is sa tisﬁable.) Proof. Let Z consist of H 0 and all cons is ten t ps eudo v alua tions, and for H , G ∈ Z put H ¯ K G iﬀ G is a pseudov aluation for a demand of H . Let ( X, K ) b e the subframe of ( Z, ¯ K ) g enerated b y H 0 (i.e. X is the s e t of states reachable from H 0 in ( Z, ¯ K ), and K = ¯ K ∩ ( X × X )). By the as sumption on H 0 and Lemmas 4.2 and 4.3 , ( X, K ) is a shallow tableau for φ . Theorem 4. 9. If R is strictly one-step c omplete, t hen every shal low table au is a su pp orting Kripke fr ame of a shal low table au mo del. Proof. Let ( X , K ) b e a shallow tableau; we ha ve to construct a shallow tableau mo del C = ( X , ξ ) for which ( X , K ) is a supp orting Kripke fr ame. T o beg in, note that to ensure the truth lemma, it suﬃces that C is c oher ent in the se ns e that for H ∈ X and Y = { G | H K G } , H ⊢ PL Lρ ⇐ ⇒ ξ ( H ) ∈ [ [ L ] ] Y { G ∈ Y | G | = C G ρ } fo r a ll Lρ ∈ MA ( H ) (cf. Lemma and Deﬁnition 4.5) : note that { G ∈ Y | G | = C G ρ } = [ [ ρ ] ] C ∩ Y , so that by naturality of predicate liftings, coherence implies that H ⊢ PL Lρ ⇐ ⇒ H | = C Lρ fo r a ll Lρ ∈ MA ( H ) . The extension to prop ositional co nsequences of H is then straightforward (noting that for Lρ ∈ MA ( H ), either H ⊢ PL Lρ o r H ⊢ PL ¬ Lρ ). W e construct a coher e n t coa lg ebra structure ξ by induction o ver the depth of pseudov aluations . Th us, let H ∈ X , put Y = { G | H K G } , and a ssume that ξ is already constructed for all pseudov aluations of smaller depth in X , in particular for all states G rea c hable from H in ( X , K ). Thus, the submodel C G generated b y such 18 · L. Schr¨ oder and D . Pattinson a state G is alrea dy deﬁned, and c oherence at G is unaﬀected b y the constr uction of ξ ( H ). W e hav e to pr ove tha t there exists ξ ( H ) ∈ T Y ⊆ T X sa tisfying the co he r ence condition. Ass ume the contrary . Let V b e the set of prop ositional v aria bles b ρ , where Lρ ∈ MA ( H ) for so me L . Let θ ∈ Cl ( Up ( V )) consist of the litera ls ¬ Lb ρ for Lρ ∈ H and L b ρ for ¬ Lρ ∈ H . By as s umption, T Y , τ Y | = θ , where τ Y is the P ( Y )-v aluation taking b ρ to { G ∈ Y | G | = C G ρ } . By s trict one-step completeness, it follows that ψ η ⊢ PL θ for a rule φ/ψ in R C and a V - substitution η suc h that Y , τ Y | = φη . By constructio n of θ , H ⊢ PL ¬ θσ a nd hence H ⊢ PL ¬ ψ ησ . Thus, ¬ φη σ is a demand for H , and hence there exists in Y a pseudov aluation G for ¬ φη σ . By the tr uth lemma for G , G | = C G ¬ φη σ , in co n tradictio n to Y , τ Y | = φη . Corollar y 4. 10. If R is strictly one-step c omplete, then t he fol lowing ar e e quivalent for an F (Λ) -formula φ . ( 1 ) φ is satisﬁable. ( 2 ) φ is c onsistent. ( 3 ) φ has a pseudovaluation H such that al l demands of H ar e c onsistent. ( 4 ) φ has a pseudovaluation H such that al l demands of H ar e satisﬁable. ( 5 ) Ther e ex ists a shal low table au for φ . ( 6 ) φ is satisﬁable at the r o ot of a shal low t able au mo del. Proof. (1) = ⇒ (2): By soundness. (2) = ⇒ ( 3): By Lemmas 4.2 and 4.3 (3) = ⇒ ( 5): By Lemma 4.8. (5) = ⇒ ( 6): By Theorem 4.9. (6) = ⇒ ( 1): T rivia l. (3) ⇐ ⇒ (4): By the equiv alence (1 ) ⇐ ⇒ (2) a lready established. The ab o ve implies in pa rticular that the proof system is we akly c omplete , i.e. prov es all v a lid formulas; this reproves a result of [Pattinson 2003]. By Remark 3.8, we obtain mo reov er that coalg ebraic moda l log ic has the shallow mo del pr oper ty: Corollar y 4. 11 Shallow model proper ty. Every satisﬁable F (Λ) - formula φ is satisﬁable in a shallo w mo del , i. e. in a T -c o algebr a that has a supp orting Kripke fr ame ( X , K ) which has ﬁnal state x ⊤ , i.e. x ⊤ K x implies x = x ⊤ , and which, u p to a p ossible lo op at x ⊤ , is a dag of depth at most the depth of φ and of s ize at most 3 n , wher e n is the numb er of subformulas of φ . Proof. All that rema ins to b e chec ked is the b ound on the size: every state in a sha llo w tableau is a s et r epresent ing a conjunctive clause over subfor m ulas of φ , in w hich a given subformula may o ccur a s a pos itiv e literal, as a negative litera l, or not at a ll. 5. SHALLOW PROOFS The satisﬁability criterio n of Corolla ry 4.10 can b e rephra sed in terms of a shal- low pr o of pr op erty . This prop erty can b e prov ed semantically b y dualising Coro l- lary 4.1 0, as done in the pro of of Cor ollary 5.1 b elow. Alternatively , the shallow pro of prop e r t y can b e established purely syn tactically , without any re ference to PSP ACE Boun ds for R ank-1 Mo dal Logics · 19 mo dels; we present such an argument in the pro of of The o rem 5.2 b elow. The shal- low model construction presented in the pr evious section is how ever o f independent int erest. Corollar y 5. 1 S hallow Proof P r oper ty . L et R b e strictly one-step c om- plete. Then an F (Λ) -formula φ is pr ovable iﬀ for e ach clause ρ in the c onjun ctive normal form (CNF) of φ , ther e ex ists a rule χ/ψ ∈ R C and a substitut ion σ such that ψ σ ⊢ PL ρ and χσ is pr ovable. Proof. The ‘if ’ dire ction is triv ia l; we prov e ‘only if ’. Dualizing the implica tion (3) = ⇒ (2) in Coro llary 4.10 yields if φ is prov able then each pseudov aluatio n H fo r ¬ φ has a demand χ such that ¬ χ is prov able. Now let ρ b e a clause in the CNF of φ . Then ¬ ρ is a conjunctive clause in the DNF of ¬ φ , in par ticular a pse udo v aluatio n for ¬ φ . By the ab o ve condition, there exists a rule χ/ψ ∈ R C and a substitution σ such that ¬ ρ ⊢ PL ¬ ψ σ , hence ψ σ ⊢ PL ρ , and χσ is pr o v able. In a purely syntactic formulation of the shallow pr oof pro perty , we ha ve to replace strict completeness by closedness under resolution. The statement th us ta k es the following form. Theorem 5. 2 Shallow Proof Prope r ty. L et R b e r esolution close d. Then an F (Λ) -formula φ is pr ovable u nder R iﬀ for e ach clause ρ in the CNF of φ , ther e exists a rule χ/ψ ∈ R C and a substitution σ such t hat ψ σ ⊢ PL ρ and χσ is pr ovable. (This reprov es Coro lla ry 5.1 , a s strict one-step completeness implies r esolution closedness by Theorem 3.16.) Proof. Again, ‘if ’ is trivial, and we prov e ‘only if ’. Let φ b e prov able, and let ρ be a clause in the CNF of φ . Then ρ is prov able. By deﬁnition of the pro of system, ρ is prop ositionally entailed by the set of clauses Φ = { ψ σ | χ/ψ ∈ R C , χσ prov able } . One s ho ws ana logously as in the ‘if ’ directio n o f the pr oof o f Theorem 3 .1 6 that Ψ is r esolution closed. By Lemma 3.14 , there exists ψ σ in Φ such that ψ σ ⊢ PL ρ . W e hop e that both pro ofs of the shallow pr o of prop ert y provide a ha ndle for g en- eralizations to lo gics outside rank 1. One a pplication of the shallow pro of pr operty is Proposition 5.3. L et Λ c ontain an inﬁnite s et U of pr op ositional symb ols, mo d- el le d as in R emark 2.8 over a functor T U of the form T U X = T X × P ( U ) . Then the pr o of system induc e d by R is we akly c omplete iﬀ R is one-step c omplete. Proof. W.l.o.g. R is resolution closed (one ca n close under r esolution, thereby aﬀecting neither completeness nor one-step completeness). The ‘if ’ direction is known (cf. Remark 3.8). T o pr o ve the ‘o nly if ’ direction, let ψ ∈ Cl ( Up ( V )), let X be a set, and let τ be a P ( X )-v a luation such that T U X , τ | = ψ . Since U is inﬁnite and V may be a ssumed to b e ﬁnite, we ca n ass ume w.l.o.g. that V ⊆ U . Let φ denote 20 · L. Schr¨ oder and D . Pattinson the prop ositional theory of τ , i.e. the conjunction of all contracted clauses χ over V such that X , τ | = χ . Then o ne chec k s as in the pro of of Theorem 17 in [Sc hr¨ oder 2007] that the rule φ/ψ is one-step sound. By Le mma 16 in [Schr¨ oder 20 07], there exists a Prop ( V )-substitution σ such that φσ and φ → ( a ↔ σ ( a )) (for each a ∈ V ) are pro positiona l tautolog ies. Since V ⊆ U , we ca n r egard φσ as an F (Λ)-formula. As such, φσ is v alid. By soundness of φ/ψ , it follows that ψ σ , again re g arded as an F (Λ)-formula, is v alid, hence prov able b y weak completeness. B y the shallow pro of prop ert y (Theor em 5.2), there exist a rule χ/ρ o ver W and a Prop ( V )-substitution θ such that χθ is prov able and ρθ ⊢ PL ψ σ . By Lemma 3.2 and Assumption 3 .10, it follows that there exists a V -substitution κ suc h that σ ( κ ( b )) = θ ( b ) for all b ∈ W and ρκ ⊢ PL ψ . It r emains to prove that X , τ | = χκ . F ro m X , τ | = φ and the construction of σ , we obtain X , τ | = a ↔ σ ( a ) for all a ∈ V and hence X , τ | = κ ( b ) ↔ θ ( b ) for all b ∈ W , so that the goal follows from X , τ | = χθ . R emark 5.4. In the a bov e result, the ass umption that Λ cont ains enough prop o- sitional symbols is essential. E.g. in cases like coalition logic or probabilistic mo dal logic where the logic collapses into tr ivialit y without pro p ositional symbols, the empt y set o f rules is complete, but not one-s tep complete. The proof-theor etic con ten t of Theorem 5.2 goes beyond the mer e fact that pro ofs are shallow. The theorem a sserts that if the rule system is resolution clos e d, then prop ositional rea soning can a lways be limited to deco mposing a formula into the clauses of its CNF a nd pr opos itional entailmen t (i.e. by Lemma 3.2 ess en tially containmen t) b et ween clauses. Mor eo ver, shallow proofs witness a we ak su bfo rmula pr op erty : every prov able fo r m ula ha s a pro of that mentions only prop ositional combinations of subformulas. F or mally: Theorem 5. 5 Weak subf o rmula pr o per ty. Supp ose that R is r esolution close d and φ is de rivable u n der R . Then ther e ex ists a pr o of of φ t hat mentions only pr op ositional c ombinations of subformulas of φ . Proof. Assume that φ is deriv able under R and ρ is a clause o f the CNF o f φ ; w.l.o.g. ρ is not a tautology . By The o rem 5.2 we ﬁnd a rule χ/ψ ∈ R and a substitution σ s uc h that ψ σ ⊢ PL ρ and χσ is prov able under R ; by Lemma 3.2, ρ contains ψ σ , hence w e can assume w.l.o.g. that σ ma ps pro p ositional v ariables to subformulas of ρ . As χ is a purely pro positiona l form ula, the substituted premise χσ is a prop ositional combinations of subfor mulas of ρ , hence also of φ . The claim now follows inductively . As a consequence, it is immediate that F (Λ) is a conserv ative extensio n of any sublanguage F (Λ 0 ) induced by a sub-signatur e Λ 0 ⊆ Λ: Corollar y 5. 6 Con s er v a tivity. S upp ose R is r esolution close d, Λ 0 ⊆ Λ is a sub-signatur e and R 0 c onsists of those φ/ψ ∈ R that ment ion only mo dal op er ators in Λ 0 . Then a formula φ ∈ F (Λ 0 ) is R -derivable iﬀ it is R 0 -derivable. In p articular, i f R is we akly c omplete for F (Λ) , t hen R 0 is we akly c omplete for F (Λ 0 ) . Example 5.7. F ro m completeness of the rules ( M u ) for ma jority logic (Exam- PSP ACE Boun ds for R ank-1 Mo dal Logics · 21 ple 3.18.7 ), we obtain that the r ules ( W u ) P v r =1 c r + u ≤ P w s =1 d s V v r =1 W c r → W w s =1 W d s ( u ∈ Z ) with side conditio ns w − 1 − max( u, 0) ≥ 0 and v − w + 2 u ≥ 0 fo r m a complete axiomatisa tio n of the ma jority opera to r W alone. (Pauly [200 5] considers a similar language, but w itho ut nesting of mo dal op erators in formulas.) 6. A GE NERIC PSP A CE ALGORITHM W e will now exploit the shallow mo del r esult (Cor ollary 4.10 ) to desig n a decision pro cedure for sa tisﬁabilit y in the s pirit of [V ardi 1989 ]. This requires one more preparato ry s tep: s ince res olution clo sed rule sets are in general inﬁnite, we must ensure that we never need to instantiate a r ule in such a wa y that the conclusion contains the s ame literal twice; otherwise, determining the demands of a given pseudov aluation (Deﬁnition 4.1) might req uire chec king inﬁnitely many r ules. This is for mally captured as follows. Deﬁnition 6. 1. An instance φσ /ψ σ of a rule φ/ψ is c ontr acte d if the clause ψ σ is contracted (Deﬁnition 3.1). In this case, if H is a pseudov alua tio n (Deﬁnition 4.1) such that ψ σ ∈ Cl ( MA ( H )) and H ⊢ PL ¬ ψ σ , the demand ¬ φσ of H is called an essential demand . W e say that a s et R o f rules is close d un der c ontra ct io n if for every V -instanc e φσ /ψ σ of a rule φ/ψ over V in R , ther e exists a contracted V - instance φ ′ σ ′ /ψ ′ σ ′ of a rule φ ′ /ψ ′ ∈ R such that ψ ′ σ ′ prop ositionally en tails ψ σ and φσ pr opos itionally ent ails φ ′ σ ′ . I.e. a rule set is clo s ed under contraction if ev ery ins ta nce of a rule that duplicates literals in the conclusion can b e replaced by a contracted instance of a diﬀer en t rule. Not all the rule sets discusse d in Example 3.18 satisfy this prop erty , but they can easily b e closed under contraction: just add a r ule φ ′ /ψ ′ for every rule φ/ψ over V in R a nd every V -substitution σ , where φ ′ is some suitably chosen pro positiona l equiv alent of φσ and ψ ′ is obtained from ψ σ b y removing duplicate litera ls. It is clear that the new rules remain one-s tep sound. Note that extending the rule set trivially pres erv es strict one-step completeness , so that there is no need to close the extended rule set under resolution aga in. F o r conv enience, we in tr oduce further notation for prop ositional formulas: if r ∈ Z − { 0 } and φ is a for m ula, then we put sgn ( r ) φ = ( φ r > 0 ¬ φ r < 0 . Example 6.2. (1) The strictly one-step complete r ule sets of Examples 3.18.1– 5 ( E , M , K , K D , and c o alition logic) are easily seen to b e closed under con tr action, essentially b ecause in all relev a n t rule schemas, the premise is a clause of the same general fo rmat as the conclusion. (2) Gr ade d mo dal lo gic: The rule schema ( G ) of Ex ample 3.1 8.6 fa ils to b e closed under co n tractio n, as duplica ting litera ls in the conclusion subs ta n tially a ﬀects bo th the pr emise and the side condition. W e can close ( G ) under cont raction as described 22 · L. Schr¨ oder and D . Pattinson ab o ve; this results in the rule schema ( G ′ ) P n i =1 r i a i ≥ 0 W n i =1 sgn ( r i ) ✸ k i a i , where n ≥ 1 and r 1 , . . . , r n ∈ Z − { 0 } , sub ject to the side co ndition P r i < 0 r i ( k i + 1) ≥ 1 + P r i > 0 r i k i . (3) Majority lo gic: Similarly , closing the rule schem a ( M m ) for ma jority logic under co n tra c tion yields the rule s c hema ( M ′ m ) m ≤ P n i =1 r i a i + P v j =1 s j b j W sgn ( r i ) ✸ k i a i ∨ W sgn ( s j ) W b j ( r i , s j ∈ Z − { 0 } , m ∈ Z ) with side conditio ns P r i < 0 r i ( k i + 1 ) − ( P r i > 0 r i k i ) − 1 + P s j > 0 s j − ma x( m, 0) ≥ 0 and 2 m − P s j ≥ 0. (4) Pr ob abilistic mo dal lo gic: The rule sc hema ( P k ) of Ex a mple 3.18 .8 fails to be clos ed under co n tractio n. Closure under contraction as de s cribed ab ov e leads to the rule s c hema ( P ′ k ) P n i =1 r i a i ≥ k W 1 ≤ i ≤ n sgn ( r i ) L p i a i where n ≥ 1 and r 1 , . . . , r n ∈ Z − { 0 } , sub ject to the side condition P n i =1 r i p i ≤ k , and if ∀ i . r i < 0 then P n i =1 r i p i < k . The crucial pr operty of contraction clo sed rule sets is Lemma 6. 3 . If R is close d un der c ontr action, t hen al l the demands of a pseu- dovaluation ar e satisﬁable iﬀ al l its essential demands ar e satisﬁable. Proof. The ‘only if ’ directio n is tr ivial. W e prov e ‘if ’: Let R be closed un- der contraction. Then als o R C is closed under contraction, since instances o f the congruence rule nev er co n tain duplicate literals. Thus, every demand of a pseu- dov aluation H is pro positiona lly e n tailed by an essential demand. Thu s we can ex tend Coro llary 4.10 as follows. Corollar y 6. 4. If R is strictly one-step c omplete and close d under c ontr action, then an F (Λ) -formula φ is satisﬁable iﬀ φ has a pseudovaluation H such that al l essential demands of H ar e satisﬁable. In the alg orithm suggested by Cor ollary 6.4, we will enco de demands , which are themselves to o large to be passed aro und directly , by the r ules that induce them. Here, we need to repre sen t rules by suitable c o des , i.e. s trings over so me alpha bet, since a naive direct representation of rules would in par ticular hav e to deal with rule pr emises of p otent ially e xponential size. Deﬁnition 6. 5. W e say that a rule R ∈ R matches a clause ρ ≡ W n i =1 ǫ i L i φ i if the conclusio n o f R is of the for m W n i =1 ǫ i L i a i . In this case, let σ ( R, ρ ) denote the arising substitution [ φ i /a i ] i =1 ,...,n . Two rules matc hing the same clause are e quivalent if their premises a re prop ositionally equiv alent; equiv alence classes [ R ] are called R -matchings . The co de of R is also a c o de for [ R ]. PSP ACE Boun ds for R ank-1 Mo dal Logics · 23 W e ﬁx so me size measures for the r epresent ation of formulas and rules : Deﬁnition 6. 6. The s iz e size ( a ) of a n integer a is ⌈ log 2 ( | a | + 1 ) ⌉ , wher e ⌈ r ⌉ = min { z ∈ Z | z ≥ r } as usual. The size size ( p ) of a r ational num b er p = a/b , with a, b rela tiv ely prime, is 1 + size ( a ) + size ( b ). The size | φ | of a fo r m ula φ ov er V is deﬁned by counting 1 for each prop ositional v a riable, bo olean op erator, or mo dal op erator, and additionally the size of ea c h index of a mo dal op erator. (In the examples, indices a re either num b ers, with sizes as ab ov e, o r subsets o f { 1 , . . . , n } , assumed to b e of size n .) Assumption 6 . 7. W e assume a reasonable enco ding o f mo dal for m ulas in which bo olean oper ators take up cons ta n t space and moda l op erators take up space ac- cording to a g iv en co ding of Λ; we assume that this co ding is in NP (i.e. it is decidable in N P whether a given co de is a v alid co de for a mo dal oper ator in Λ). Graded or probabilistic mo dal oper ators a r e assumed to b e co ded in bina ry , with sizes according to Deﬁnition 6.6. Example 6.8. F or the r ules of Examples 3.1 8 and 6 .2, we just tak e t he par a meters of a rule as its co de in the obvious wa y . E.g. the co de of an instance o f ( P ′ k ) as displayed in Example 6.2 .4 consists of n , k , the r i , and the p i . T he size of the co de is determined b y the sizes of these n um ber s plus separa ting letters, say , P (1 + si ze ( a i )) + P (1 + siz e ( p i )) + size ( n ) + size ( k ) + 1. No te that not all such co des repre s en t instances of ( P ′ k ). The following decision pro cedure o n a n alternating T uring machine generalise s the PSP ACE algor ithms in [V a r di 19 8 9], g iven a strictly one-s tep complete and con- traction closed rule set R . Algo rithm 6. 9. (Decide satisﬁability of φ ∈ F (Λ)) (1) (Existential) Guess a pr o positiona lly consistent pseudov aluation H for φ . (2) (Univ ersal) Cho ose a contracted cla us e ⊥ 6 = ρ ov er MA ( H ) suc h that H ⊢ PL ¬ ρ . (3) (Univ ersal) Cho ose an R C -matching [ R ] of ρ . (4) (Existential) Guess a cla use γ from the CNF of the premise o f R . (5) Recursively chec k that ¬ γ σ ( R, ρ ) is satisﬁa ble. The algorithm succeeds if all p ossible c hoices at steps marked universal lead to successful termination, and fo r all steps marked existential , there ex ists a choice leading to successful termination. Concerning Step 1 , note that the only wa y for a pseudov aluation to b e prop o sitionally incons isten t is to contain b oth L ρ and ¬ Lρ for s ome mo dal ato m L ρ . W e emphasise that in Step 3 , it suﬃces to guess o ne co de for each matching. Proposition 6.10. Algorithm 6.9 suc c e e ds iﬀ the input formula φ is satisﬁable. Proof. Induction over the depth n of φ . If n = 0, then the prop ositional for- m ula φ will ev aluate to either ⊤ or ⊥ , as it do es not contain any prop ositional v ariables; moreov er, the only candidate for a pseudov aluation for φ is the empt y conjunctive clause ⊤ . Thus, the algorithm terminates unsuccessfully in the existen- tial step (1) iﬀ φ ev aluates to ⊥ , since ⊤ is a pseudov alua tion for φ iﬀ φ ev aluates 24 · L. Schr¨ oder and D . Pattinson to ⊤ . Otherwise, the algorithm terminates successfully in the universal step (2 ), since the o nly clause ρ over MA ( ⊤ ) = ∅ s uc h that ⊤ ⊢ PL ¬ ρ is ⊥ . F o r n > 0, cor- rectness of the algorithm follows from Corollary 6.4 and the inductive hypothesis: the essential demands o f φ are the neg ated premises ¬ φ σ ( φ/ψ, ρ ) fo r R C -matchings [ φ/ψ ] o f contracted clauses ρ as in the alg orithm, and such a demand is satisﬁable iﬀ the neg ation of one of the clauses in the CNF of φσ ( φ/ψ, ρ ) is satisﬁa ble. R emark 6.11. In Step 1 of Algorithm 6.9, it suﬃces to co ns ider the co njunctiv e clauses in some DNF of φ rather than all pseudov aluations. A canonical, if not necessarily the most eﬀective choice for such a DNF is to take all pseudov aluations H for φ such that MA ( H ) = MA ( φ ) (rather t han only MA ( H ) ⊆ MA ( φ )); in a co ncrete implemen tation, a heuristic pro cedure for determining some DNF eﬀectiv e ly may be preferable. Note that due to the no n-deterministic nature of the algorithm, the ab ov e prop o- sition do es not imply decidability of F (Λ). This follows only if the algor ithm resp ects suitable r e source b ounds. W e are in terested in cases wher e the algor ithm runs in p olynomial time. The crucial re quiremen t for this is that Steps 3 and 4 can b e per formed in p olynomial time, i.e. by suitable nondeterministic poly nomial- time multiv a lue d functions (NPMV) [Bo ok et al. 1 984]. W e recall that a function f : Σ ∗ → P (∆ ∗ ), where Σ and ∆ a re alphab ets, is NP MV iﬀ (NPMV1) there exists a p olynomial p such that | y | ≤ p ( | x | ) for a ll y ∈ f ( x ), where | · | denotes size, and (NPMV2) the graph { ( x, y ) | y ∈ f ( x ) } of f is in NP . This motiv ates the following conditions: Deﬁnition 6. 12. A set R o f r ules is called PSP ACE -tr actable if there exists a po lynomial p such that a ll R -matchings of a con tracted cla use ρ ov er F (Λ) hav e some co de of size at most p ( | ρ | ) (recall that ma tc hings are equiv a lence clas s es of rules and thus may have s e v era l code s ), and it can be decided in N P (1) whether a given co de is the co de of some rule in R ; (2) whether a rule matc hes a given contracted clause; and (3) whether a cla us e b elongs to the CNF o f the premise of a given rule. Theorem 6. 13 Sp ace Complexity. L et R b e strictly one-s t ep c omplete, close d un der c ont r action, and PSP ACE -tr actable. Then the satisﬁability pr oblem for F (Λ) is in PSP ACE . Proof. Since R is PSP ACE -tractable, so is R C , as s uming rea sonable co des for the congruence rules (e.g. consisting of the repres en tation o f the relev ant mo dal op erator; c f. Assumption 6 .7). Thus, the functions ma pping a cla use ρ to the set of its R C -matchings and a r ule to the se t o f cla uses o ccurring in the CNF of its premise, resp ectiv ely , ar e NPMV: in the for mer case, the p o lynomial b ound requir ed b y condition (NPMV1) is ensured by the deﬁnition of PSP ACE -tra ctabilit y , as w e only need to pr oduce one co de for each matching, and in the latter ca se, the p olynomial bo und holds univ er sally , a s clauses are of polynomia l size. Conditio n (NPMV2) is ensured explicitly by Deﬁnition 6.12 and Assumption 6.7 (whic h implies that PSP ACE Boun ds for R ank-1 Mo dal Logics · 25 the s et of formulas is in NP ). Therefore, Steps 3 a nd 4 in Algo rithm 6.9 can b e per formed in p olynomial time. Steps 1 a nd 2 ha ve p olynomial run time without sp eciﬁc assumptions, as a pseudov aluation H for φ is represented as a set o f literals and must by deﬁnition satisfy MA ( H ) ⊆ MA ( φ ), and the con tracted clause ρ chosen in Step 2 is constructed as a non-rep etitiv e list of liter als who se nega tions b elong to H . Since the depth of r e cursion is b ounded b y the depth of φ , it follows that the algorithm r uns in APTIME = PSP ACE [Chandra e t a l. 1981 ]. R emark 6.14. A more careful analy sis of Algorithm 6 .9 reveals that it suﬃces for the de c is ion problems in Deﬁnition 6.12 to be in P H , the poly nomial time hierarch y . In our examples, how ever, the co mplexit y is in fact P rather than N P . W e exp ect that this situation is t y pic a l, with the crucia l condition for PSP ACE -tracta bilit y being the poly nomial b ound on R -matchings. W e are not aware o f any natur al examples of intractable r ule sets (contrived examples ar e easy to co nstruct, e.g. by impo sing computationa lly har d side conditions). R emark 6.15. Alg orithm 6.9 can b e dualised to yield a pro of-search pro cedure that determines whether φ ∈ L (Λ) is R -deriv able, thus implementing the shallow pro of prop erty (Corollar y 5.1/Theorem 5.2). Note that the dualisa tion entails that the roles o f existential and universal steps are interc hanged. In the treatment of gra de d and prop ositional mo dal logic, the p olynomial bound on rule co des follows ra ther directly from size estima tes in integer linear pro- gramming, as follows. F ollowing usua l practice, we take the s ize | W | of a r a- tional ineq ualit y W ≡ ( P n i =1 u i x i op u 0 ), op ∈ { <, ≤ , > , ≥} and u i ∈ Q , to be 1 + n + P n i =0 size ( u i ). W e recall that for n ∈ Z , sgn ( n ) = − 1 if n < 0 , sgn ( n ) = 1 if n > 0, a nd sgn ( n ) = 0 if n = 0. Lemma 6. 1 6. F or every r ational line ar ine quality W and every solution r 0 , . . . , r n ∈ Z of W , ther e exists a solution s 0 , . . . , s n ∈ Z of W such t hat sgn ( s i ) = sgn ( r i ) for al l i , t he pr op ositional formulas P n i =1 s i a i ≥ s 0 and P n i =1 r i a i ≥ r 0 (cf. Se ction 3) ar e e quivalent, and size ( s i ) ≤ 18 | W | 4 for al l i . Proof. Let V = { a 1 , . . . , a n } , and let x 0 , . . . , x n be the v ariables in W . W e note that a pro p ositional formula P n i =1 s i a i ≥ s 0 is equiv alent to φ ≡ P n i =1 r i a i ≥ a 0 iﬀ for all v aluations ν : V → { 0 , 1 } , one has P n i =1 s i ν ( a i ) ≥ s 0 if and only if P n i =1 r i ν ( a i ) ≥ r 0 , r e ad as in teg er linea r inequalities. Thus, let I denote the system o f inequalities consis ting o f W and a dditio na l inqualities F i and E ν , where i = 1 , . . . , n , ν range s ov er v aluations V → { 0 , 1 } , F i =      x i ≥ 1 if r i ≥ 1 x i = 0 if r i = 0 x i ≤ − 1 if r i ≤ − 1 (where the middle ca se actually cor respo nds to tw o inequa lities), and E ν = ( P n i =1 x i ν ( a i ) ≥ x 0 if P n i =1 r i ν ( a i ) ≥ r 0 P n i =1 x i ν ( a i ) < x 0 if P n i =1 r i ν ( a i ) < r 0 . Then the claim trans lates into the statement that I has a solution of p olynomially bo unded size in | W | . 26 · L. Schr¨ oder and D . Pattinson It follows from [Schrijv er 1986, Cor ollary 17.1b] that I has a solutio n whose size is b ounded by 6 c ( n + 1) 3 , where c is the facet complexity of the s ystem, i.e. the size of the lar gest inequa lit y in I . As the co ﬃcien ts o f the inequalities E ν and F i are of size at most 1, we have c ≤ | W | + 2( n + 1 ). Since moreover | W | ≥ n + 1, I thus has a so lution of size at most 18 | W | 4 . W e no w illustrate how Theor em 6.1 3 allows us to establish PSP ACE b ounds for many mo dal logics in a uniform wa y . Example 6.17. Co nditions (1) a nd (2) of Deﬁnition 6.1 2 are immediate for all the rule sets o f E xample 3 .18 — the decision problems in question inv o lv e no more than c hecking co mputationally ha rmless side conditions in the case o f Condition (1) (disjoint ness and containmen t of ﬁnite sets, linear inequalities), and comparing clauses of p olynomial (in fact, linear) size in the case of Condition (2). Moreover, Condition (3) is immediate in thos e cases where the premises of rules ar e ju st single clauses. This leav es only GML and P ML; but the expa nsion of P i ∈ I r i a i ≥ k to a pr o positiona l form ula is already in CNF, a nd chec king w he ther a giv en clause belo ngs to this CNF is clear ly in P . It remains to establish the p olynomial b ound on the matchings. F or GML and PML, this is gua ran teed precisely b y Lemma 6.16 . In all o ther cases, e very con- tracted c la use ρ matches at mo st one r ule, who se co de ha s size linea r in the s iz e of ρ . W e thus have obtained PSP ACE -tra ctabilit y a nd hence decidability in PSP ACE for a ll logics in Example 3.18 . The log ics E and M ar e of lesser interest here, b eing actually in NP [V ardi 19 8 9]. W e brieﬂy comment on the a lgorithms and b ounds for the o ther cases . (1) F o r the mo dal log ics K and K D (Examples 3 .1 8.3 and 4 ), Algor ithm 6.9 is essentially the witness algorithm [Ladner 1977 ; V ar di 1989; Bla ckburn et al. 2001]. Both lo gics ar e PSP ACE -har d [L a dner 197 7]. (2) F o r coalition logic (Ex a mple 3.18.5), we arrive, due to minor diﬀerences o f the rule s ets, at a slight v ariant of Pauly’s PSP ACE -algo rithm [P auly 2 002]. (3) F o r gra ded mo dal logic, we obtain a new algorithm which co nﬁrms the known PSP ACE upp er bo und [T obies 2001]. One might cla im that the new algorithm is not only nicely embedded in to a uniﬁed framework, but a lso conceptually simpler than the constraint-based algor ithm of [T o bies 2 001] (which co rrects a similar but incorrect a lgorithm previo usly g iv en elsewhere, and r e futes a previous EXPTIME hardness co njecture). Graded mo dal logic is PSP ACE -ha rd, as it extends K . (4) F o r pro babilistic mo dal lo gic, we obtain a new a lgorithm which conﬁr ms the PSP ACE upp e r b o und tha t follows from the cor respo nding b ound for the more expr essiv e (mo dal) logic of proba bilit y , a pro of of which is sketched in [F agin and Halp ern 199 4]. The bound is tight, as PML contains the PSP ACE -complete logic K D as a fragment (embedded b y mapping ✷ to L 1 ). In compariso n to the algorithm in lo c. cit. , our algorithm has additional pro of theoretic cont ent a s dis- cussed in Sec tio n 5 . Under the corr espondence outlined in Remark 6 .15, it ﬁnds pro ofs which rema in within PML rather than po ssibly diverting v ia a mor e expres- sive logic. PSP ACE Boun ds for R ank-1 Mo dal Logics · 27 (5) Our PSP ACE upp er b ound for ma jor it y logic, which a ppeare d for the ﬁrst time in the co nference presentation of [Schr¨ oder and Pattinson 20 06], tied in a pri- ority race with [Demri and Lugiez 2006], where a PSP ACE upper b ound w as proved for the mor e expressive Pr esburger moda l logic using a diﬀerent type of alg orithm. The same rema rks conce rning pro of-theore tic con ten t apply as for probabilistic mo dal logic. 7. CONCLUSION Generalising results b y V ardi [1 989], we hav e shown tha t coalgebra ic mo dal logic has the shallow mo del prop erty , and we hav e presented a g e ne r ic PSP A CE algor ithm for s atisﬁabilit y base d on depth-ﬁr st explor ation of sha llo w mo dels. W e have thus —repro duced the witness algorithm for K and K D [Blackburn et a l. 2001 ] —obtained a slig h t v ar ian t o f the known PSP ACE algor ithm for co alition logic [Pauly 2002] —obtained a new PSP AC E a lgorithm fo r gr aded mo dal logic, recov ering the known PSP ACE bound [T obies 20 01] —obtained a new PSP ACE algorithm for proba bilistic moda l logic [Larsen and Skou 1991; Heifetz a nd Mongin 2 0 01], recovering a PSP ACE upp e r b ound whic h follows from results sketc hed in [F agin and Ha lp ern 19 94]. —obtained, simultaneously with [Demri and L ug iez 20 06], a new PSP ACE upp er bo und for ma jority lo gic [Pacuit and Salame 200 4]. In all these cas e s, the PSP ACE upp er bound is tight. Our algo rithm may alter- natively b e view ed as traversing a shallow pr oof that witnesses a weak subformula prop ert y . The crucial prereq uisite for the gener ic algorithm is an axioma tisation by so- called one-step rules (go ing fro m rank 0 to ra nk 1) ob eying tw o clos edness conditions: closedness under res olution and under contraction, i.e. remov al of duplicate litera ls. In the e xamples, it ha s not only turned out that it is fea s ible to keep this closure pro cess under control, but also that the axio ma tisations obtained have pleasingly compact presentations — typically , one ends up with a single rule s chema. It has b een s ho wn that every mo dal log ic can b e equipp e d with a c anonical coalgebr aic semantics, provided it is a xiomatisable in r ank 1 and satisﬁes the con- gruence rule [Sc hr¨ oder and Pattinson 2 007b]. This means in par ticular that our shallow mo del constructio n applies to every such mo dal log ic when eq uipp ed with the canonica l semantics. Moreov er, the PSP ACE -alg o rithm presented her e can b e made mo dula r w.r .t. hetero geneous combination of systems and mo dal log ics using m ulti-sorted coalgebra [Sc hr¨ oder and P attinson 2007a ]. The extension of the theo ry beyond r ank 1 is the sub ject o f future rese a rc h, a s is the treatment of simple ﬁxed po in t oper ators, pos sibly using automa ta theore tic methods [V a r di 1996; V enema 2006] or pseudomo dels [Emer son and Halp ern 198 5]. A further p oint of interest is to inv estiga te the connection b et ween our notio n of re s olution clo sure a nd classica l pro of-theoretic issues such as cut elimination a nd interpola tion. 28 · L. Schr¨ oder and D . Pattinson AC KNOWLE DGMENTS The author s wish to thank Alexander Kurz for useful discussions and the Depart- men t of Computer Science at the Universit y o f Bremen for funding a visit o f the second author. REFERENCES Barr, M. 1993. T erminal coalgebras in well-founded set theory . The or. Comput. 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