Shallow Models for Non-Iterative Modal Logics

Shallo w Mo dels for Non-Iterative Mo dal Logics Lutz Sc hr¨ oder DFKI-Lab Bremen and Departmen t of Computer Science, Univ ersit¨ at Bremen Dirk P attinson Departmen t of Computing Imp e rial College London Octob er 25, 2018 Abstract The metho ds used to establish P SP A CE -b ounds for modal logics can roughly be g roup ed int o t wo cla sses: syntax driven metho ds establish that exhaus tive pro of sea rch can b e per formed in p olynomia l space where as semantic approaches directly constr uct shallow mo dels. In this pap er, we follow the latter approach and establish ge ne r ic PSP ACE - bo unds for a larg e and heterogeneous class of mo dal logic s in a coalgebraic framework. In particular , no co mplete axioma tisation o f the logic under sc r utiny is needed. This do es not only complement our ea r lier, syntactic, appro ach conceptually , but also covers a wide v ariety of new exa mples whic h are difficult to harness b y purely syntactic mea ns. Apart from re-proving known complexity b o unds for a la r ge v ariety of structurally different logics, we apply our metho d to o bta in previously unknown PSP ACE -b ounds for Elgesem’s logic of ag ency and for g r aded mo dal logic over reflexive frames. 1 In tro duction Sp ecial p urp ose mo dal logic s often com b ine exp ressivit y and d ecidabilit y , usually in a lo w complexit y class. In the absence of fixed p oint op erators, these logics are frequently decidable in P SP ACE, i.e. not dramatically worse than prop ositional logic. While lo wer PSP A CE b ound s for mo dal logics can t ypically b e obtained directly from seminal results of Ladner [20] b y em b edd ing a PSP A CE-hard logic such as K or K D , u p p er b ound s are often non -trivial to establish. In particular PSP A CE upp er b ound s for non-normal logics hav e recentl y receiv ed m uc h atten tion: • A PSP A CE u p p er b ound for graded mo dal logic [11] is obtained using a constraint set algorithm in [30]. This corrects a previously published incorrect algorithm and refutes a previous EXPTIME h ardness conjecture. • More r ecen tly , a P S P AC E upp er b ound for Presbu rger mo dal logic (which contai ns graded mo dal logic and ma jorit y logic [23]) h as b een established u sing a L adner-t yp e algo- rithm [6] . • Using a v ariant of a shallo w n eighb ou r ho o d frame construction from [31], a PS P A CE upp er b ound for coalition logic is established in [26]. • PSP A CE up p er b ound s f or CK and r elated conditional logics [3] are obtained in [22] b y a detailed pr o of-theoretic analysis of a lab elled sequen t calculus. The metho d s used to obtain these results can b e broadly group ed in to tw o classes. S ynt ac- tic approac hes presup p ose a complete tableaux or Gentze n system and establish that pr o of searc h can b e p erformed in p olynomial space. Semantics-driv en approac hes, on the other hand, d irectly construct shallo w tree mo dels. Both approac hes are in timately connected in the case of normal mo dal logics interpreted o v er Kripke frames: counte r mo dels can usu ally b e d eriv ed directly fr om searc h trees [16]. It should b e noted that this metho d is not im m edi- ately applicable in the non-normal case, wh ere the structure of mo dels often go es far b eyond mere graphs. Using coalgebraic tec hniques, we ha ve previously sho wn [29 ] that the synta ctic appr oac h uniformly generalises to a large class of mo dal logic s: starting from a one-step c omplete ax- iomatisatio n, we hav e ap p lied r esolution closur e to obtain complete tableaux systems. Generic PSP ACE-b ounds follo w if the ensuing rule s et is PSP ACE-tr actable . Here, we present a d iffer- en t, semanti c, set of metho d s to establish un iform PSP A CE b oun ds by dir ectly constructing shallo w mo dels for logics sub ject to the one- step p olysize mo del pr op erty , or a v arian t of the latter. In particular, n o axiomatisat ion of the logic itself is needed. Apart from the fact that b oth metho ds use su b stan tially d ifferen t tec hniqu es, they apply to d ifferen t classes of examples. Wh ile it is e.g. relativ ely easy to obtain a resolution closed rule set for coalitio n logic [26], pr o ving the one-step p olysize mod el prop ert y f or (the coalge braic seman tics of ) coalition logic is a non-trivial task. On the other hand, small one-step mo dels are comparativ ely easy to construct for complex mo d al logics suc h as probabilistic mo dal logic [8] or Presburger mo dal logic [6] th at are not straightforw ard ly amenable to th e synt actic approac h via resolution closure, either b ecause n o axiomatisa tion has b een giv en or b ecause the complexit y of th e axiomatisation makes th e resolution closure h ard to harn ess. Moreo ver, the p resen t semant ic approac h to PSP ACE-b ounds tak es a significant step to o v ercome an imp ortan t b arrier in the coalgebraic treatmen t of mo dal logic s. Existing d ecid- abilit y and completeness r esults [25, 5, 28, 29] are limited to r ank-1 lo gics , giv en by axioms whose mo d al nesting depth is u niformly equal to one. While this already encompasses a large class of examples (includin g all logic s men tioned so far), the seman tic mo del construction in the p resen t p ap er applies to non-iter ative lo g ics [21], i.e. logics axiomatised without nested mo dalities (rank-1 logics additionally exclude top-leve l prop ositional v ariables). Despite the seemingly min ute difference b et we en the t w o classes of logics, this generalisatio n is not only tec h nically non-trivial but also subs tan tially extends the scop e of the coalgebraic metho d . Besides the m o dal logic T , the class of non-iterativ e logics includes e.g. all conditional logics co v ered in [22] (of wh ich only 4 are r ank-1), in particular CK + M P [3], as we ll as Elgesem’s logic of agency [7, 12 ] and the graded version T n of T [11]. As in [29], we work in th e framework of c o algebr aic mo dal lo gic [25] to obtain r esults that are p arametric in the un derlying s eman tics of particular logics. While n ormal m o dal logics are usually interpreted o ver Kripk e frames, non-normal logics see a large v ariety of different seman tics, e.g. pr obabilistic systems [8 ], frames with ordered branching [6], game frames [26], or conditional frames [3]. The coalg ebraic treatmen t allo w s us to encapsu late the seman tics in the c hoice of a signatur e functor , wh ose coalgebras then pla y the role of mo dels, leading to results that are unformly applicable to a large class of different logics. Since the class of al l coalge bras for a giv en signature fun ctor can alw a ys b e completely axiomatised in rank 1 [28], in analogy to the fact th at the K -axioms are complete for the class of al l Kripke f rames, the standard coalgebraic app roac h is not d irectly applicable to non- iterativ e logics. T o o v ercome this limitation, w e in tro d u ce the new concept of interpreting mo dal logics o ve r coalgebras for c op ointe d functors , i.e. functors T equipp ed with a natural 2 transformation of type T → Id . In this setting, our main tec hnical to ol is to cut bac k mod el constru ctions from mo d al logics to the lev el of one-step lo gics whic h seman tically do not in v olv e state transitions, and then amalgamate the corr esp onding one-step mo dels in to sh allo w mo dels for the full mo dal logic, whic h ideally can b e trav ersed in p olynomial space. F or this approac h to w ork, the logic at h and n eeds to s upp ort a small mo del pr op erty for its one-step fr agmen t, the one-step p olysize mo del pr op erty (OSPMP) . Our first m ain theorem s h o ws that the OSPMP guarante es decidabilit y in p olynomial space. Crucially , th e OSPMP is muc h easier to establish than a shallo w mo del p rop erty for the logic itself. T o r epro v e e.g. Ladner’s PSP A CE u pp er b ound for K , one ju st observ es that to construct a set that in tersects n giv en sets, one n eeds at most n elements. F or the conditional logics CK , CK + ID , and CK + MP , the OSPMP is similarly easy to chec k. F or other logics, in particular v arious logics of quantitat iv e uncertain t y , the OSPMP can b e ob tained by sharp enin g known off-the-shelf r esults. A s a new resu lt, we establish the OSPMP for Elgesem’s logic of agency to obtain a pr eviously u nknown PSP ACE upp er b ound. As a by-prod u ct of our constru ction, we obtain NP-b ounds for the b oun d ed rank fragmen ts of all logics with the O SPMP , generalizing the corresp ond ing result for th e logics K and T from [13] to a large v ariet y of s tructurally d ifferent (non-iterativ e) logics. While the OSPMP is usually easy to establish, a weak er prop erty , the one-step p ointwise p olysize mo del pr op erty (OSPPMP) , can b e used in cases w here the OS PMP fails, pr o vided that the signature functor sup p orts a n otion of p oin t wise smallness for ov erall exp onen tial- sized one-step mo dels. This allo ws tra v ersing exp onen tially branc hing shallo w mod els in p olynomial space by dealing with the su ccessor structures of single states in a p oint wise fashion. Our second main result, which yields PSP ACE upp er b ounds for logics with the OSPPMP , is app lied to reprov e the kno wn PSP A CE b ound for P resburger mo dal logic [6] and to derive a new PS P A CE b ound for Presburger mo dal logic o v er reflexiv e frames, and hence for T n [11] (which w as so far only kno wn to b e decidable [10]). T h e latter result extends straigh tforw ardly to a d escription logic with role hierarc hies, qualified num b er r estrictions, and reflexive roles. 2 Coalgebraic Mo dal Logic W e recall th e coalgebraic interpretatio n of mo d al logic an d extend it to non-iterativ e logics using cop ointe d functors. A mo dal signatur e Λ is a set of mo dal op erators with asso ciated fi nite arit y . The signature Λ determines t wo languages: fir stly , the one-step lo g ic of Λ, whose form ulas ψ , . . . (the one- step formulas ) o v er a set V of prop ositional v ariables are defined by the grammar ψ ::= ⊥ | ψ 1 ∧ ψ 2 | ¬ ψ | L ( φ 1 , . . . , φ n ) , where L ∈ Λ is n -ary and the φ i are p rop ositional form ulas o ve r V ; and secondly , the mo dal lo gic of Λ, whose set F (Λ) of Λ - formulas ψ , . . . is defined by the grammar ψ ::= ⊥ | ψ 1 ∧ ψ 2 | ¬ ψ | L ( ψ 1 , . . . , ψ n ) . Th us, the mo dal logic of Λ is distinguish ed fr om th e one-step logic in that it adm its nested mo dalities. The b o olean op erations ∨ , → , ↔ , ⊤ are defined as usual. The r ank rank( φ ) 3 of φ ∈ F (Λ) is th e maximal nesting depth of mo dalities in φ (note ho wev er that th e notion of r ank-1 logic [28 , 29] is stricter than suggested b y this definition, as it excludes top-lev el prop ositional v ariables in axioms; the latter are allo w ed only in n on -iterativ e logics). W e denote by F n (Λ) the set of f orm ulas of rank at most n ; w e refer to the languages F n (Λ) as b ounde d-r ank fr agments . W e treat one-step logics as a tec hnical to ol in the study of mo dal logics. H o we ve r, one- step logics also app ear as logics of indep en den t inte rest in the literature [9, 14, 15]. One of the cen tral id eas of coalgebraic mo dal logic is that prop erties of the full mo dal logic, su c h as soundness, completeness, and d ecidabilit y , can b e reduced to prop erties of the muc h simpler one-step logic. This is also the sp irit of the presen t work, w hose core is a construction of p olynomially b ranc hing sh allo w mo dels for the mo dal logic assuming a small mo d el pr op ert y for the one-step logic . The semantic s of b oth the one-step logic and the mo d al logic of Λ are parametrized coalge braically by the choic e of a set functor. The standard setup of coalgebraic mo d al logic using all coalgebras for a plain set functor co v ers only r ank-1 lo gi cs , i.e. logics axiomatised one-step form ulas [28] (a t ypical example is the K -axiom  ( a → b ) →  a →  b ). Here, we impro v e on this by considering the class of coalgebras for a giv en c op ointe d set functor, whic h enables u s to co v er the more general class of non-iter ative lo g ics , axiomatised b y arbitrary form ulas without nested mo dalities (such as the T -axiom  a → a ). W e follo w a pur ely seman tic approac h and hen ce do not formally consid er axiomatisations in the present w ork (where we do men tion axioms, this is for solely explanatory purp oses). How eve r, the extended scop e of th e new framework and its relation to n on-iterativ e m o dal logics (wh ic h can b e made precise in the same wa y as for p lain fu n ctors and rank-1 logics [28]) will b ecome clear in the examples. In general, a copointe d functor ( T , ǫ ) consists of a fu n ctor T : Set → Set , wh ere S et is the category of sets, and a n atural trans formation ǫ : T → Id . F or our pr esen t purp oses, a sligh tly r estricted notion is more con v enien t: Definition 2.1. A (r estricte d) c op ointe d f u nctor S with signatur e functor S 0 : Set → Set is a subfun ctor of S 0 × Id (where ( S 0 × Id ) X = S 0 X × X ). W e sa y that S is trivial ly c op ointe d if S = S 0 × Id . An S - c o algebr a A = ( X, ξ ) consists of a set X of states and a tr ansition function ξ : X → S 0 X suc h that ( ξ ( x ) , x ) ∈ S X f or all x . Remark 2.2. The mo dal logic F (Λ) d o es not explicitly in clude p rop ositional v ariables. These ma y b e regarded as n ullary mo d al op erators in Λ; their semant ics is th en defin ed ov er coal- gebras f or S 0 × P ( V ), where V is the set of v ariables (cf. also e.g. [28]). W e omit d iscussion of prop ositional v ariables in the examples, ev en in cases like the mo dal logic of probabilit y that b ecome trivial in the absence of v ariables; our treatment extends straight forwardly to the case with v ariables in the manner j ust indicated. W e view coalgebras as generalized transition s y s tems: th e transition fun ction map s a state to a structur ed set of successors and observ ations, with the structure pr escrib ed by the sig- nature functor. Th us, the latter en capsu lates the branching t yp e of the underlyin g transition systems. Cop oin ted functors additionally imp ose lo cal frame conditions that r elate a state to the collection of its successors. Assumption 2.3. W e assu m e w.l.o.g. that S 0 preserve s injectiv e maps [2], and even S 0 X ⊆ S 0 Y in case X ⊆ Y , and th at S is non-trivial , i.e. S X = ∅ ⇒ X = ∅ . 4 Generalising earlier w ork (e.g. [17, 19]), c o algebr aic mo dal lo gi c abstractly captures the in ter- pretation of mo dal op erators as p oly adic pr edicate liftings [25, 27], Definition 2.4. An n - ary pr e dic ate lifting ( n ∈ N ) for S 0 is a natural transformation λ : Q n → Q ◦ S op 0 , where Q denotes the con tra v ariant p o we rset fu nctor Set op → S et (i.e. Q X is the p ow ers et P ( X ), and Q f ( A ) = f − 1 [ A ]), and Q n is defin ed by Q n X = ( Q X ) n . A coalge braic semant ics for Λ is formally defined as a Λ -structur e M ( over S ) consisting of a copointe d fun ctor S w ith signatur e functor S 0 and an assignmen t of an n -ary predicate lifting [ [ L ] ] for S 0 to ev ery n -ary mo dal op erator L ∈ Λ. When S is trivially cop oin ted, we will also call M a simple Λ -structur e (over S 0 ) . We fix the notation Λ , M , S , S 0 thr oughout the p ap er . The s eman tics of the mo dal language F (Λ) is then given in term s of a satisfaction relation | = C b et we en states x of S -coalgebras C = ( X , ξ ) and F (Λ)-form ulas o ver V . The relation | = C is defined inductiv ely , with the usual clauses for b o olean op erators. The clause for an n -ary mod al op erator L is x | = C L ( φ 1 , . . . , φ n ) ⇔ ξ ( x ) ∈ [ [ L ] ]([ [ φ 1 ] ] C , . . . , [ [ φ n ] ] C ) where [ [ φ ] ] C = { x ∈ X | x | = C φ } . W e drop the sub scripts C when clear fr om the con text. Our main inte rest is in the (lo cal) satisfiability pr oblem ov er M : Definition 2.5. An F (Λ)-form ula φ is satisfiable if there exist an S -coalgebra C and a state x in C suc h that x | = C φ . Dually , φ is valid if x | = C φ for all C , x . Con trastingly , the seman tics of the one-step logic is give n in terms of s atisfactio n relati ons | = 1 X,τ b et we en elements t ∈ S 0 X and one-step formulas ov er V , w here X is a set and τ is a P ( X ) -valuation for V , i.e. a map τ : V → P ( X ). Th e v aluation τ canonically in duces an in terpretation of [ [ φ ] ] 0 τ ⊆ X of p rop ositional formulas φ ov er V . W e write X , τ | = 0 φ if [ [ φ ] ] 0 τ = X . The relation | = 1 X,τ is th en defined by th e u sual clauses for b o olean op erators, and t | = 1 X,τ L ( φ 1 , . . . , φ n ) ⇔ t ∈ [ [ L ] ]([ [ φ 1 ] ] 0 τ , . . . , [ [ φ n ] ] 0 τ ) . Note in particular that the seman tics of the one-step logic d o es not in v olv e a notion of s tate transition. Definition 2.6. A one-step mo del ( X , τ , t, x ) ov er V consists of a set X , a P ( X )-v aluation τ for V , t ∈ S 0 X , and x ∈ X suc h that ( t, x ) ∈ S X . The latter condition is v acuous if S is trivially cop oin ted, in which case we omit the mentio n of x . F or a one-step formula ψ o ver V , ( X , τ , t, x ) is a one-step mo del of ψ if t | = 1 X,τ ψ . W e recall some basic notation: Definition 2.7. W e denote the set of prop ositional formulas o v er a set Z , generated by the basic connectiv es ¬ and ∧ , b y Prop ( Z ). W e use v ariables ǫ etc. to denote either nothing or ¬ . Thus, a liter al o v er Z is a formula of the form ǫa , with a ∈ Z . A (c onjunctive) clause is a fi nite, p ossib ly empt y , disjunction (conjunction) of literals. W e denote b y Λ( Z ) the set { L ( a 1 , . . . , a n ) | L ∈ Λ n -ary , a 1 , . . . , a n ∈ Z } . 5 In the ab o v e notation, the set of one-step form ulas o v er V is Prop (Λ( Prop ( V ))). The one-step logic may alternativ ely b e pr esen ted in terms of pairs of formulas separating out the low er prop ositional lay er: Definition 2.8. A one-step p air ( φ, ψ ) o ve r V consists of f orm ulas ψ ∈ Prop (Λ( V )) and φ ∈ Prop ( V ). A one-step mo d el ( X , τ , t, x ) is a one-step mo del of ( φ, ψ ) if X, τ | = 0 φ and t | = 1 X,τ ψ . In analogy to the equiv alence b et w een axioms and one-step rules describ ed in [28], one-step pairs and one-step form ulas ma y replace eac h other for p urp oses of satisfiabilit y: Lemma 2.9. F or ev e ry one-step p air ( φ, ψ ) over V with φ satisfiable, ther e e xists a Prop ( V ) - substitution σ such that the one-step formula ψ σ is e qu ivalent to ( φ, ψ ) in the sense that if t | = 1 X,τ ψ σ then t | = 1 X,σ τ ( φ, ψ ) , and if t | = 1 X,τ ( φ, ψ ) then t | = 1 X,τ ψ σ (and σ τ = τ ). H er e, σ τ denotes the P ( X ) -valuation taking a to [ [ σ ( a )] ] 0 τ . Conversely, we have, for ψ ∈ Prop (Λ( Prop ( V ))) , an e q u ivalent one-step p air ( φ, ψ 1 ) over V ∪ W , wher e ψ de c omp oses as ψ ≡ ψ 1 σ , with ψ 1 ∈ Prop (Λ( W )) , σ a Prop ( V ) - substitution, and V ∩ W = ∅ , and wher e φ is the c onjunction of the formulas a ↔ σ ( a ) , a ∈ W . Her e, r estricting valuations to V induc es a bije ction b etwe en one-step mo dels of ( φ, ψ 1 ) and one-step mo dels of ψ . Pr o of. The second claim is clear. The fi rst claim is pro v ed as follo ws. As in [28], let κ b e a satisfying truth v aluation for φ and put σ ( a ) = a ∧ φ if κ ( a ) = ⊥ , and σ ( a ) = φ → a otherwise. Then φσ and the formulas φ → ( a ↔ σ ( a )), for a ∈ V , are tautologies [28]. Both directions of the first claim no w follo w straigh tforw ardly . The coalge braic approac h subs umes many interesting mo dal logics, includin g e.g. graded and probabilistic mo dal logics and coalition logic [29]. Belo w, we presen t the most basic examples, the mod al logics K and T , as well as v arious conditional logics and logics of quantita tiv e uncertain t y . The treatmen t of Elgesem’s mo dal logic of agency is deferr ed to Sect. 4. Example 2.10. 1. M o dal lo gic K : Let Λ = {  } , with  a unary mo dal oper ator. W e define a simp le Λ-structure o v er the co v arian t p ow erset fu nctor P (i.e. P X is p o we rset, an d P f ( A ) = f [ A ]) by pu tting [ [  ] ] X ( A ) = { B ∈ P X | B ⊆ A } . Naturality of [ [  ] ] is j ust the equiv alence f [ B ] ⊆ A ⇐ ⇒ B ⊆ f − 1 [ A ]. P -coalge bras are Kripke frames, and P -mo dels are Kripke mo dels. The mo dal logic of Λ is precisely the mo dal logic K , equipp ed with its standard K ripke seman tics. C on trastingly , a one-step formula ov er V is a prop ositional com bination of ato ms of the form  φ , wh ere φ ∈ Prop ( V ). F or A ∈ P X , we hav e A | = 1 X,τ  φ iff A ⊆ [ [ φ ] ] 0 τ . One easily c hec ks that the one-step logic is NP-complete, while the mo d al logic K is PSP A CE-complete [20]. 2. M o dal lo gic T : The logic T has the same synt ax as K . Its coalgebraic seman tics is a structure o v er the cop oin ted functor R with signatur e functor P , given by RX = { ( A, x ) ∈ P X × X | x ∈ A } . Th us, R -coalgebras are reflexive Kripke frames. The interpretation of  is defined as f or K . (Axiomatica lly , T is determined by the non-iterativ e axiom  a → a .) 6 3. Conditional lo gic CK : The signature of conditional logic has a single binary mod al op- erator ⇒ , written in infix n otation. F orm ulas φ ⇒ ψ are read as non-monotonic conditionals. The seman tics of the conditional logic CK [3] is giv en by a s im p le stru cture o v er the functor Cf given by Cf ( X ) = ( Q X → P X ), with → denoting function space and Q contra v arian t p o werset, cf. Definition 2.4. Cf -coalgebras are c onditiona l fr ames [3]. The op erator ⇒ is in terpreted ov er Cf by [ [ ⇒ ] ] X ( A, B ) = { f : Q X → P X | f ( A ) ⊆ B } . 4. Conditional lo g ic CK + ID : The cond itional logic CK + ID [3] extends CK with the r ank-1 axiom a ⇒ a , referred to as ID . The s eman tics of CK + ID is mo delled by restricting the structure for CK to the subfun ctor Cf ID of Cf defin ed b y Cf ID ( X ) = { f ∈ Cf ( X ) | ∀ A ∈ Q X. f ( A ) ⊆ A } . 5. Conditional L o gic CK + MP : T he logic CK + MP [3] extends CK with the non-iterativ e axiom ( MP ) ( a ⇒ b ) → ( a → b ) . (This axiom is un desirable in default logics, but r easonable in r elev ance logics.) S eman tically , this amounts to passing fr om the functor Cf to the cop oin ted fun ctor Cf MP with signature functor Cf , d efi ned b y Cf MP ( X ) = { ( f , x ) ∈ Cf ( X ) × X | ∀ A ⊆ X . x ∈ A ⇒ x ∈ f ( A ) } . 6. M o dal lo gic s of quantitative unc ertainty: The mo d al signature of likeliho o d has n -ary mo dal op erators P n i =1 a i l ( ) ≥ b for a 1 , . . . , a 1 , b ∈ Q . The terms l ( φ ) are called like liho o ds . The in terpretation of lik eliho o ds v aries. E.g. the semantic s of the mo dal lo gic of pr ob abil- ity [8] is mo delled coalgebraically b y a structure ov er the (finite) distribu tion functor D ω , where D ω X is the set of finitely supp orted pr ob ab ility distr ibutions on X , and D ω f acts as image measure form ation. Coalge bras for D ω are pr obabilistic trans ition systems (i.e. Mark o v c hains). Lik eliho o ds are int erpr eted as probabilities; i.e. [ [ P n i =1 a i · l ( ) ≥ b ] ] X ( A 1 , . . . , A n ) = { P ∈ D ω X | P n i =1 a i P ( A i ) ≥ b } . Alternativ ely , lik eliho o ds ma y b e interpreted as upp er pr ob abilities [14], i.e. the fun ctor D ω is replaced by P ◦ D ω , and in the ab o v e definition, P ( A i ) is replaced b y P ∗ ( A i ), wh ere for P ∈ P D ω X , P ∗ ( A ) = sup { P A | P ∈ P } . Th is setting describ es s itu ations w here agen ts are unsu re ab out the actual probabilit y distribu tion. F urther alternativ e notions of like liho o d include Demps ter-S h afer b elief f u nctions and Dub ois-Prade p ossibilit y measures [15]. An extension of the mo dal signature of likel iho o d is the mo d al signature of exp e ctation [15], where in s tead of likel iho o ds one more generally considers exp ectations e ( P n k j =1 b ij φ ij ). Here, linear com binations of f orm ulas represent gambles , i.e. real-v alued outcome functions, where the pa yo ff of φ is the charact eristic fu nction of φ . T he exact defin ition of exp ectation dep ends on the un derlying notion of lik eliho o d. One-step logics of qu an titativ e uncertaint y are often considered to b e of ind ep endent in - terest. E .g. the one-step logic of probabilit y , i.e. a logic w ithout nesting of lik eliho o ds that talks only ab out a sin gle probab ility distribution, is int ro d uced indep endent ly [9] and only later extended to a full mo d al logic [8]. In fact, log ics of exp ectation [15] an d the logic of upp er probabilit y [14] so far app ear in the literature only as one-step logic s; the corresp ond ing mo dal logics are of in terest as natural v ariations of the mo dal logic of p robabilit y . 7 Con v en tion 2.11. W e assu me that Λ is equipp ed with a size measure, thus in ducing a size measure for F (Λ). F or one-step f orm ulas φ o ve r V , we assum e w.l.o.g. that | V | ≤ size( φ ). F or finite X , we assu me giv en a representat ion of elemen ts ( t, x ) ∈ S X as str ings of s ize size( t, x ) o v er some fi nite alphab et. W e do not r equire that all elemen ts of S X are representa ble, nor that all strin gs denote element s of S X . When S is trivially cop oin ted, we represent only t ∈ S 0 ( X ). W e require that in clusions S X ⊆ S Y induced according to Assump tion 2.3 by inclusions X ⊆ Y int o a finite b ase set Y p reserv e represen table elemen ts and in crease their size by at m ost log | Y | . W e mak e these issues explicit for the ab o ve examples: Example 2.12. 1. M o dal lo gics K and T : F or X finite, elemen ts of P X are rep r esen ted as lists of elements of X . 2. Conditional lo gics CK and CK + ID : F or X finite, elemen ts of Q X → P X are represente d as p artial maps f 0 : Q X ⇀ P X ; s u c h an f 0 represent s the total map f that extends f 0 b y f ( A ) = ∅ wh en f 0 ( A ) is u n defined. (The use of partial maps av oids exp onential blo wup.) 3. Conditional lo gic CK + MP : F or X fin ite, a pair ( f 0 , x ) consisting of a partial map f 0 : Q X ⇀ P X and x ∈ X represen ts the pair ( f , x ), wh ere f : Q X → P X extends f 0 b y f ( A ) = A ∩ { x } in case f 0 ( A ) is u n defined. 4. M o dal lo gic s of q u antitative u nc ertainty: Su itable compact rep resen tations are d escrib ed in [9, 14, 15]. 3 P olynomiall y br anc hing shallo w mo dels W e no w turn to the announced construction of p olynomially b ranc hing shallo w mo d els for mo dal logics whose one-step logic has a small m o del prop erty; this construction leads to a PSP ACE decision pro cedure. Definition 3.1. W e sa y that M has the one-step p olysize mo del pr op erty (OSPMP) if there exist p olynomials p and q suc h that, w h enev er a one-step pair ( φ, ψ ) o v er V has a one-step mo del ( X , τ , t, x ), then it has a one-step mo d el ( Y , κ, s, y ) suc h that | Y | ≤ p ( | ψ | ), ( s, y ) is represent able with size( s, y ) ≤ q ( | ψ | ), and y ∈ κ ( a ) iff x ∈ τ ( a ) f or all a ∈ V . In analogy to the tr an s ition b et w een rules and axioms describ ed in [28], one-step pairs are in terc hangeable with one-step formulas. In particular, w e ha v e Prop osition 3.2. The Λ -structur e M has the OSPMP iff ther e exist p olynomials p , q such that, whenever a one-step formula ψ over V ha s a one-step mo del ( X , τ , t, x ) , then it has a one-step mo del ( Y , κ, s, y ) such that y ∈ κ ( a ) iff x ∈ τ ( a ) for al l a ∈ V , | Y | ≤ p ( | ψ 1 | ) , and ( s, y ) is r epr esentable with s ize( s, y ) ≤ q ( | ψ 1 | ) , wher e ψ ≡ ψ 1 σ with ψ 1 ∈ Prop (Λ( W )) and σ a Prop ( V ) -substitution. Pr o of. Only if: Let ( X , τ , t, x ) b e a one-step mo del of a one-step form ula ψ ov er V . By Lemma 2.9, ψ is equiv alen t to a one-step pair of the form ( φ, ψ 1 ), with ψ 1 as in the statemen t. By the O SPMP , ( φ, ψ 1 ) h as a one-step mo del ( Y , κ, s, y ) suc h that | Y | ≤ p ( | ψ 1 | ), size( s, y ) ≤ q ( | ψ 1 | ), and y ∈ κ ( a ) iff x ∈ τ ( a ) for all a ∈ V ; by Lemma 2.9, this mo d el give s r ise to a one-step mo del of ψ with the comp onents Y , s , y unchanged. 8 If: Let ( X , τ , t, x ) b e a on e-step mo del of a one-step p air ( φ, ψ ) ov er V . By Lemma 2.9, ( φ, ψ ) is equiv alen t to a one-step f ormula of the form ψ σ , w here σ is a Prop ( V )-sub stitution. By assumption, ψ σ has a one-step mo del ( Y , κ, s, y ) such that | Y | ≤ p ( | ψ | ), size( s, y ) ≤ q ( | ψ | ), and y ∈ κ ( a ) iff x ∈ τ ( a ) for all a ∈ V . By Lemma 2.9 , this mo del giv es rise to a one-step mo del of ( φ, ψ ) with the comp onents Y , s , y unc hanged. Both formulat ions of the OSPMP easily r ed uce to the case that ψ is a conjun ctiv e clause. Remark 3.3. It is sho wn in [28] that the one-step logic alw a ys h as an exp onent ial-size m o del prop erty: a one-step form ula ψ ov er V h as a one-step mo d el iff it h as a on e-step mo del with carrier set P ( V ). W e are now ready to prov e the shallo w mo del theorem. Definition 3.4. A supp orting K ripke fr ame of an S -coalgebra ( X, ξ ) is a K r ipk e frame ( X , R ) suc h that f or eac h x ∈ X , ξ ( x ) ∈ S 0 { y | xR y } ⊆ S 0 X (equiv alen tly ( ξ ( x ) , x ) ∈ S { y | xRy } ). A state x ∈ X is a lo op if xRx . Theorem 3.5 (Shallo w mo del pr op erty) . If M has the OSPM P , then F (Λ) has the p oly- nomially branc hing sh allo w mo del prop ert y : Ther e exist p olynomials p , q such that every satisfiable F (Λ) -formula ψ is satisfiable at the r o ot of an S -c o algebr a ( X , ξ ) which has a sup- p orting Kripke fr ame ( X , R ) such that r emoving al l lo ops fr om ( X, R ) yields a tr e e of depth at most rank( ψ ) and br anching de gr e e at most p ( | ψ | ) , and ( ξ ( x ) , x ) ∈ S { y | xR y } is r epr esentable with size( ξ ( x ) , x ) ≤ q ( | ψ | ) . Definition 3.6. F or x ∈ X and a P ( X )-v aluation τ , w e pu t Th τ ( x ) ≡ V x ∈ τ ( a ) a ∧ V x / ∈ τ ( a ) ¬ a. Pr o of of The or em 3.5. Indu ction o v er the rank of ψ . I f rank( ψ ) = 0, then ψ ev aluates to ⊤ and hence is satisfied in a singleton S -coalgebra ( X, ξ ), w hic h exists by Ass u mption 2.3. No w let rank( ψ ) = n + 1. Let z 0 b e a s tate in an S -coalgebra ( Z, ζ ) suc h that z 0 | = ( Z,ζ ) ψ . Let MSub ( ψ ) denote the set of subf ormulas of ψ o ccuring in ψ w ith in the scop e of a mo dal op erator, let V b e th e set of v ariables a ρ , in dexed ov er ρ ∈ M Sub ( ψ ), and let σ denote the substitution taking a ρ to ρ for all ρ . Let ¯ ψ b e th e conju nction of all literals ǫL ( a ρ 1 , . . . , a ρ n ) suc h that L ( ρ 1 , . . . , ρ n ) is a sub form ula of ψ and z 0 | = ( Z,ζ ) ǫL ( ρ 1 , . . . , ρ n ). (Recall that ǫ denotes either nothing or n egation.) Moreo v er, let φ denote the p r op ositional theory of σ , i.e. the conjun ction of all clauses χ o v er V such that χσ is L -v alid. Then ( Z , κ, ζ ( z 0 ) , z 0 ) is a one-step mo del of ( φ, ¯ ψ ), where κ ( a ) = [ [ σ ( a )] ] ( Z,ζ ) . By the OSPMP , it follo ws th at ( φ, ¯ ψ ) has a one-step mo d el ( Y , τ , t, x 0 ) of p olynomial size in | ¯ ψ | s uc h that for all ρ ∈ MSub ( ψ 1 ), x 0 ∈ τ ( a ρ ) iff z 0 ∈ κ ( ρ ), w h ic h in turn is equiv alen t to z 0 | = ( Z,ζ ) ρ . F rom this mod el, we n o w construct a sh allo w mo del ( X , ξ ) for ψ . T o b egin, note th at Th τ ( y ) σ is L -satisfiable for eve ry y ∈ Y . F or supp ose not; then ¬ Th τ ( y ) σ is L -v alid, h ence ¬ Th τ ( y ) is a conjun ct of φ . Th u s, Y , τ | = 0 ¬ Th τ ( y ), in con tradiction to the fact that y ∈ [ [ Th τ ( y )] ] τ by constru ction. By ind uction, w e thus ha v e, for eve ry y ∈ Y , a shallo w mo d el ( X y , ξ y ) of Th τ ( y ) σ , where we ma y assume y ∈ X y and y | = ( X y ,ξ y ) Th τ ( y ) σ , with depth at most rank( Th τ ( y ) σ ) = n . W e tak e ( X , ξ ) as the disjoint un ion of the ( X y , ξ y ) o v er y ∈ Y − { x 0 } , extended by the state x 0 , for wh ic h w e p ut ξ ( x 0 ) = t ∈ S 0 Y ⊆ S 0 X . 9 W e hav e to v erify that x 0 | = ( X,ξ ) ψ . W e will pr ov e the stron ger statemen t x 0 | = ( X,ξ ) ¯ ψ σ , i.e. t | = 1 X,θ ¯ ψ , (1) where θ ( a ρ ) = [ [ ρ ] ] ( X,ξ ) for ρ ∈ MSub ( ψ ). By ind uction o ve r χ and naturalit y of pr edicate liftings, y | = ( X,ξ ) χ iff y | = ( X y ,ξ y ) χ for y ∈ Y − { x 0 } and for ev ery form ula χ . In p articular, y | = ( X,ξ ) Th τ ( y ) σ for all y ∈ Y − { x 0 } , i.e. y | = ( X,ξ ) ρ ⇐ ⇒ y ∈ τ ( a ρ ) (2) for all ρ ∈ M Sub ( ψ ). W e prov e b y induction o v er ρ ∈ MSub ( ψ ) that x 0 | = ( X,ξ ) ρ ⇐ ⇒ x 0 ∈ τ ( a ρ ) , (3) whic h in connection with (2) yields [ [ ρ ] ] ( X,ξ ) ∩ Y = τ ( a ρ ) . (4) The steps for b o olean op erations are straigh tforw ard. F or L ( ρ 1 , . . . , ρ n ) ∈ MSub ( ψ ), we hav e x 0 | = ( X,ξ ) L ( ρ 1 , . . . , ρ n ) ⇐ ⇒ t ∈ [ [ L ] ] Y ([ [ ρ i ] ] ( X,ξ ) ∩ Y ) i =1 ,...,n = [ [ L ] ] Y ( τ ( a ρ 1 ) , . . . , τ ( a ρ n )) ⇐ ⇒ t | = 1 ( Y ,τ ) L ( a ρ 1 , . . . , a ρ n ) , using n aturalit y of [ [ L ] ] in the first step and the in ductiv e h yp othesis in the shap e of (4) in the subsequent equ alit y . Sin ce t | = ( Y ,τ ) ¯ ψ , the last statemen t is equiv alen t to z 0 | = ( Z,ζ ) L ( ρ 1 , . . . , ρ n ). By the d efinition of κ , th is is equiv alen t to z 0 ∈ κ ( a L ( ρ 1 ,...,ρ n ) ), wh ic h in turn is equiv alent to x 0 ∈ τ ( a L ( ρ 1 ,...,ρ n ) ) b y constru ction of ( Y , τ , t, x 0 ). By (4) and n aturalit y of pred icate liftings, our remaining goal (1) reduces to t | = 1 Y , τ ¯ ψ , whic h holds by construction. Finally , we hav e to establish that th e o v erall branc hing degree of the m o del is p olyno- mial in | ψ | . The mo del is recursiv ely constructed fr om p olynomial-size one-step mo dels for pairs whose second comp onen ts are conjunctiv e clauses o v er atoms L ( a ρ 1 , . . . , a ρ n ), where L ( ρ 1 , . . . , ρ n ) is a subformula of ψ . S uc h conjunctiv e clauses are of at most quadr atic size in | ψ | (ev en O ( | ψ | log | ψ | ) if subformulas of ψ are represente d b y p oint ers into ψ ); this pro ve s the claim. Remark 3.7. While it is to b e exp ected that the constr u ction of p olynomially branching mo dels dep ends on a condition lik e the OSPMP , it do es not s eem to b e the case that the precise formulat ion of th is condition is implicit in the literature (not ev en for the trivially cop oin ted case). Note in particular that the p olynomial b oun d dep end s only on the second comp onent of a one-step pair. This is crucial, as th e first comp onent of the one-step pair constructed in th e ab o v e pr o of ma y b e of exp onential size. Wh en w e sa y in the in tro duction that the OSPMP can b e obtained from off-the-shelf results (e.g. [9 , 14, 15]), we refer to p olynomial-size mo del theorems in which the p olynomial b ou n d dep ends, in the notation of Prop osition 3.2, on | ψ | , w hic h may b e exp onenti ally larger than | ψ 1 | ; t ypically , on ly an insp ection of the giv en pro ofs sh o ws that the b ound can b e sharp ened to b e p olynomial in | ψ 1 | . The pro of of T heorem 3.5 leads to th e follo w ing nondeterministic decision pro cedu re. 10 Algorithm 3.8. (Decide satisfiabilit y of an F (Λ)-formula ψ ) Let M ha ve the OSPMP , and let p , q b e p olynomial b ounds as in Defin ition 3.1. 1. I f r ank( ψ ) = 0, terminate successfully if ψ ev aluates to ⊤ , else uns u ccessfully . Other w ise: 2. T ak e V and σ as in the p ro of of Theorem 3.5, and guess a conjunctive clause ¯ ψ o v er Λ( V ) con taining for eac h subformula L ( ρ 1 , . . . , ρ n ) of ψ either L ( a ρ 1 , . . . , a ρ n ) or ¬ L ( a ρ 1 , . . . , a ρ n ) suc h that ¯ ψ σ p rop ositionally enta ils ψ . 3. Gu ess a P ( Y )-v aluation τ f or V and ( t, x ) ∈ S Y with size( t, x ) ≤ q ( | ¯ ψ | ), wh ere Y = { 1 , . . . , p ( | ¯ ψ | ) } , su c h that t | = 1 Y , τ ¯ ψ . 4. F or eac h y ∈ Y , chec k recursivel y that Th τ ( y ) σ is satisfiable. Since th e rank decreases with eac h recur siv e call, th e ab o v e algorithm can b e imp lemen ted in p olynomial sp ace, pro vided that Step 3 can b e p erformed in p olynomial space. Definition 3.9. T he one-step mo del che cking pr oblem is to c hec k, giv en a string s , a fi nite set X , A 1 , . . . , A n ⊆ X , and L ∈ Λ n -ary , wh ether s r epresen ts some ( t, x ) ∈ S X and wh ether t ∈ [ [ L ] ] X ( A 1 , . . . , A n ). This prop erty and the ab o v e algorithm lead to a PSP A CE b ound for th e mo d al logic. More- o v er, for b ounded-rank fr agmen ts, the p olynomially branching sh allo w mo del p rop erty b e- comes a p olynomial size mo d el p r op erty , thus leading to an NP up p er b ound: Corollary 3.10. L et M have the OSPMP. 1. If one-step mo del che cking is in PSP ACE, then the satisfiability pr oblem of F (Λ) is in PSP ACE. 2. If one-step mo del che cking is in P , then the satisfiability pr oblem of F n (Λ) is in N P for every n ∈ N . Pr o of. 1: By Algorithm 3.8. 2: L et p and q b e p olynomial b oun ds on the br anc hing degree of su pp orting Kripk e f rames and on the size of successor structur es ξ ( x ) as guarante ed by Theorem 3.5. Let n ∈ N . Then b y Theorem 3.5, ev ery satisfiable formula ψ ∈ F n (Λ) is satisfiable in a mo del ( X, ξ ) suc h that | X | ≤ P n i =0 p ( | ψ | ) i =: N and size( ξ ( x )) ≤ log ( N ) q ( | ψ | ) for all x ∈ X , where the second inequalit y relies also on Conv ention 2.11 . Thus, the en tire representati on s ize of th e m o del ( X, ξ ) is b oun ded b y M := N log ( N ) q ( | ψ | ), which is p olynomial in | ψ | . T h us, the follo wing non-deterministic algorithm d ecides satisfiabilit y of ψ in p olynomial time: 1. Gu ess a mo d el ( X , ξ ) of size at most M 2. C hec k that ( X, ξ ) is an S -coalgebra. 3. C hec k that [ [ ψ ] ] ( X,ξ ) 6 = ∅ . The second step can b e p erformed in p olynomial time b ecause one-step mo del c hec king is in P . The third step can b e p erformed in p olynomial time b y recursiv ely computing extensions [ [ φ ] ] ( X,ξ ) , again b ecause one-step mo d el c hec king is in P . This generalises results f or the mo d al logics K and T established in [13]. 11 Example 3.11. 1. M o dal lo gic s K and T : O ne-step m o del c hec king for K and T amoun ts to c hec king a subset inclusion and, in the case of T , add itionally an elementhoo d ; this is clearly in P . T o verify the OSPMP for K , let ( X , τ , A ) b e a one-step mo del of a one-step pair ( φ, ψ ) o v er V ; w.l.o.g. ψ is a conjunctiv e clause o ver atoms  a , where a ∈ V . F or ¬  a in ψ , there exists x a ∈ A such that x a / ∈ τ ( a ). T aking Y to b e the set of th ese x a , w e obtain a p olynomial-size one-step mo del ( Y , τ Y , Y ) of ( φ, ψ ), wh ere τ Y ( a ) = τ ( a ) ∩ Y f or all a . The construction for T is the same, except that the p oin t x of the original one-step mo d el ( X, τ , A, x ) is retained in th e carrier set Y , and b ecomes the p oin t of the small m o del. By Corollary 3.10, this r ep ro v es Ladner’s PSP A CE upp er b ound s for K and T [20], as well as Halp ern’s NP up p er b ounds for b oun d ed-rank fr agmen ts [13]. 2. Conditional lo gic: It is easy to see that one-step mo del chec king for CK , CK + ID , and CK + MP is in P . (In p articular, deciding whether a give n string represen ts an element of Cf ID ( X ) just amounts to c hec king subset inclusions. Moreo v er, deciding whether ( f , x ) ∈ Cf MP ( X ), i.e. whether x ∈ A implies x ∈ f ( A ), can b e d one in p olynomial time thanks to the c hoice of d efault v alue for f ; cf. E xample 2.12.2.) T o pro ve th at CK has the O SPMP , let ( X , τ , f ) b e a one-step mo del of a one-step pair ( φ, ψ ), wh ere w.l.o.g . ψ is a conjunctiv e clause V n i =1 ǫ i ( a i ⇒ b i ). I f τ ( a i ) 6 = τ ( a j ), fix an elemen t y ij in the symmetric difference of τ ( a i ) and τ ( a j ). Moreo v er, if ǫ i is negation, fix z i ∈ f ( τ ( a i )) \ τ ( b i ). Let Y b e th e set of all y ij and all z i . Let τ Y b e th e P ( Y )-v aluation defined by τ Y ( v ) = τ ( v ) ∩ Y , and let f Y ∈ Cf ( Y ) b e represen ted by the p artial map taking τ Y ( a i ) to f ( τ ( a i )) ∩ Y for all i (this is well-defined by constru ction of Y ). Then ( Y , τ Y , f Y ) is a one-step mo del of ( φ, ψ ). T he cardinalit y of Y is quadr adic in ψ , and the represen tation size of f Y is p olynomial. Thanks to the c hoice of default v alue, this constru ction of p olynomial-size one-step mo dels w orks also for CK + ID . The construction for CK + MP is almost iden tical, except that the p oin t x of ( X , τ , f , x ) is retained in the small one-step m o del ( Y , τ Y , f Y , x ); here, ( f Y , x ) ∈ Cf MP ( Y ) due to the different c hoice of d efault v alue. W e th us obtain that CK , CK + ID , and CK + M P are in PSP ACE (hence PSP ACE-co mplete, as these logics conta in K and — in the case of CK + M P — T , resp ectiv ely , as s ublogics). T his has previously b een pr o v ed u s ing a detailed analysis of a lab elled sequent calculus [22] (the metho d of [22] yields an explicit p olynomial b ound on space usage whic h is n ot matched by the generic algorithm). The NP u pp er b ound for b oun ded-rank fr agmen ts of CK , CK + ID , and CK + MP arising from Corollary 3.10.2 is, to our kno wledge, n ew. 3. M o dal lo gics of quantitative unc ertainty: Polynomial size mo del p r op erties for one-step logics hav e b een prov ed for the logic of probabilit y [9], the logic of up p er pr obabilit y [14], and v arious logics of exp ectation [15]. As indicated in Remark 3.7, the p olynomial b oun d s are stated in the cited work as dep end ing on the size of th e entire one-step form ula ψ ; ho w ev er, insp ection of the giv en pro ofs sh o ws that the p olynomial b ound in fact dep ends on ly on the n umb er of lik eliho o ds or exp ectations in ψ , resp ective ly , and on the representat ion size of the largest co efficien t. By Pr op osition 3.2, it follo ws that th e resp ectiv e logics hav e the OS PMP . Suitable complexit y estimates for one-step m o del c hec king are also found in the cited w ork. By the ab o ve resu lts, it follo ws that th e resp ectiv e mo dal logics of quantita tiv e uncertaint y are in PS P ACE (hence PS P ACE-c omplete, as one can embed K D b y mapp ing ♦ to l ( ) > 0), and in NP wh en the mo dal nesting depth is b ounded . F or the mo dal logic of probabilit y , a pro of of the PSP ACE upp er b ound is sk etc hed in [8]. Th e PSP A CE up p er b oun ds for the remaining cases (e.g. the mo dal logic of up p er probabilit y and th e v arious mod al logics of 12 exp ectation) seem to b e new, if only f or the reason that only the one-step v ersions of these logics app ear in the literature. Similarly , all NP up p er b oun ds for b ounded -rank fr agmen ts are, to our knowle dge, new. Moreo v er, the upp er b ounds extend easily to m o dal logics of uncertain t y w ith non-iterativ e axioms, e.g. an axiom a → l ( a ) ≥ p which states that the present state remains stationary with lik eliho o d at least p . 4 Extended Example: Elgesem’s mo dal logic of agency There hav e b een numerous approac hes to captur ing the notion of agen ts bringing ab out certain states of affairs, one of the most recen t ones b eing Elgesem’s mo dal logic of agency ([7] and r eferen ces therein, [12]). Mo dal logics of agency pla y a role e.g. in plannin g and task assignmen t in multi-ag ent systems (cf. e.g. [4, 18]). Elgesem defi n es a logic with t w o mo dalities E and C (in general indexed ov er agen ts; all results b elo w easily generalise to the multi- agen t case), read ‘the agen t brin gs ab out’ and ‘the agen t is capable of r ealising’, resp ectiv ely . T he seman tics is giv en by a class of conditional frames ( X, f : X → Q X → P X ) (Example 2.10.3), called sele ction fu nction mo dels in this con text. Th e clauses for the mo dal op erators are x | = E φ iff x ∈ f ( w )([ [ φ ] ]) and x | = C φ iff f ( w )([ [ φ ] ]) 6 = ∅ . The relev ant class of selection fu nction mo dels ( X, f ) is defined b y the cond itions (E1) f ( x )( X ) = ∅ (E2) f ( x )( A ) ∩ f ( x )( B ) ⊆ f ( x )( A ∩ B ) (E3) f ( x )( A ) ⊆ A . It is shown in [7 , 12] that the logic of agency is completely axiomatised by ¬ C ⊤ , ¬ C ⊥ , E a ∧ E b → E ( a ∧ b ), E a → a , and E a → C a . Nota bly , the agent is incapable of realising what is logically necessary ( ¬ C ⊤ ), i.e. the notion of realising a state of affairs enta ils actual attributabilit y (this axiom is w eak er than previous form ulations using avoidability ; cf. the bab y fo o d example in [7]). Monotonicit y is not imp osed. The axiom ¬ C ⊥ is due to [12]. Most of the information in selection f u nction mo dels (motiv ated b y philosophical consid- erations in [7]) is irrelev an t for th e seman tics of E and C : one only needs to kno w w h ether f ( x )( A ) is n on-empt y , and whether it conta ins x . Moreo v er, the selection function seman- tics f ails to b e coalgebraic, as the naturalit y condition fails for the (generalised) predicate lifting implicit in the clause for E . Both problems are easily remedied by mo v in g to the follo w ing coalgebraic seman tics: pu t 3 = {⊥ , ∗ , ⊤} (to rep r esen t the cases f ( x )( A ) = ∅ , x / ∈ f ( x )( A ) 6 = ∅ , and x ∈ f ( x )( A ), resp ectiv ely), and tak e as signature fu nctor the 3 -value d neighb orho o d functor N 3 giv en by N 3 ( X ) = Q ( X ) → 3 (with Q ( X ) denoting con tra v ari- an t p ow ers et). W e defin e the cop oint ed fu nctor A as the subfunctor of N 3 × Id s uc h that ( f , x ) ∈ A ( X ) iff for all A, B ⊆ X , (E1 ′ ) f ( X ) = ⊥ (E2 ′ ) f ( A ) ∧ f ( B ) ≤ f ( A ∩ B ) (E3a ′ ) f ( ∅ ) = ⊥ (E3b ′ ) f ( A ) = ⊤ = ⇒ x ∈ A 13 where ∧ and ≤ refer to the ordering ⊥ < ∗ < ⊤ . W e d efine a str ucture o v er A for the mo d al logic of agency by [ [ E ] ] X A = { f : Q → 3 | f ( A ) = ⊤} [ [ C ] ] X A = { f : Q → 3 | f ( A ) 6 = ⊥} . Prop osition 4.1. A formula of the mo dal lo gic of agency is satisfiable in a sele ction function mo del iff it is satisfiable over A . Pr o of. ‘Only if:’ Giv en a selectio n fun ction mo d el ( X, f ), define an N 3 -coalg ebra ( X, ˜ f ) by ˜ f ( x )( A ) =      ⊤ if x ∈ f ( x ) ( A ) ∗ if x / ∈ f ( x )( A ) 6 = ∅ ⊥ if f ( x )( A ) = ∅ . It is clear th at ( X , ˜ f ) is an A -coalgebra and that x ∈ X satisfies the same formulas in ( X, ˜ f ) as in ( X , f ). ’If ’: Let ( X, f ) b e an A -coalg ebra. W e can assume that | [ [ φ ] ] ( X,f ) | 6 = 1 for all formulas φ (otherwise, form the copro d uct of ( X , f ) with itself, so that eac h state has a t win s atisfying the same formulas). W e define a selection function mo d el ( X , ¯ f ) b y ¯ f ( x )( A ) =      A if f ( x )( A ) = ⊤ A − { x } if f ( x )( A ) = ∗ ∅ if f ( x )( A ) = ⊥ . It is clear that ( X, ¯ f ) satisfies E1–E3. One sho ws by induction ov er the f ormula str ucture that x ∈ X satsifies the same formulas in ( X , ¯ f ) as in ( X, f ), with the only non-trivial p oin t b eing that in th e step f or the mo dal op erator C , one has to n ote that, b y the ab ov e assu mption, [ [ φ ] ] ( X,f ) − { x } 6 = ∅ w henev er f ( x )([ [ φ ] ] ( X,f ) ) = ∗ . T o a v oid exp onen tial explosion, we repr esen t elemen ts of N 3 ( X ), for X fi n ite, using partial maps f 0 : Q ( X ) ⇀ 3. T o enforce (E2 ′ ), we let suc h an f 0 represent th e map f : Q ( X ) → 3 that maps B ⊆ X to the maxim um of V n i =1 f 0 ( A i ) , tak en o v er all sets A 1 , . . . , A n ⊆ X suc h that T A i = B and f 0 ( A i ) is defin ed for all i ; when no su c h sets exist, the maxim um is understo o d to b e ⊥ . Lemma 4.2. L e t f 0 and f b e as ab ove. 1. Whenever f 0 ( A ) is define d, then f 0 ( A ) ≤ f ( A ) . 2. L et b ∈ 3 . Then f ( A ) ≥ b iff T { B ⊆ X | A ⊆ B , f 0 ( B ) ≥ b define d } = A. 3. The p air ( f , x ) satisfies (E1 ′ ) i ff f 0 ( X ) is either undefine d or e quals ⊥ . 4. The p air ( f , x ) satisfies (E2 ′ ). 5. The p air ( f , x ) satisfies (E3a ′ ) iff T { A ⊆ X | f 0 ( A ) > ⊥ define d } 6 = ∅ . 6. The p air ( f , x ) satisfies (E3b ′ ) iff whenever f 0 ( A ) = ⊤ is define d, then x ∈ A . Pr o of. 1.: T rivial. 2.: ‘If ’ is trivial. ‘Only if ’: b y assu mption, A = T n i =1 B i for some B i suc h that f 0 ( B i ) ≥ b is defined for all i ; the claim follo w s immediately . 14 3.: ‘Only if ’ is imm ediate b y 1., and ‘if ’ holds b ecause X = T A i implies A i = X for all i . 4.: By construction. 5.: Immediate b y 2. 6.: ‘On ly if ’ holds b y 1., and ‘if ’ h olds b ecause f ( B ) = ⊤ implies that B = T A i for sets A i suc h that f 0 ( A i ) = ⊤ for all i . By Lemma 4.2, it is imm ediate that one-step mo del c hec king is in P . T o pro v e the OS PMP , let ( X, τ , f : Q ( X ) → 3 , x ) b e a one-step mo del of a one-step pair ( φ, ψ ) o ve r V . By Remark 3.3, w e can assume that X is finite. Let the set Y ⊆ X consist of • the element x ; • an element y ab ∈ τ ( a ) \ τ ( b ) for eac h pair ( a, b ) ∈ V 2 suc h that τ ( a ) 6⊆ τ ( b ); • an elemen t z a ∈ T { τ ( b ) | b ∈ V , τ ( a ) ⊆ τ ( b ) , f ( τ ( b )) > f ( τ ( a )) } \ τ ( a ) f or eac h a ∈ V ( z a exists by (E2 ′ )); and • an element w 0 ∈ T { τ ( b ) | f ( τ ( b )) > ⊥} ( w 0 exists by (E2 ′ ) and (E3a ′ )). Put τ Y ( a ) = τ ( a ) ∩ Y for a ∈ V , and let f Y b e represented b y the partial map f 0 taking τ Y ( a ) to f ( τ ( a )) ( f 0 is w ell-defined b y constru ction of Y ). Then Y and ( f Y , x ) are of p olynomial size in ψ , and ( Y , τ Y ) | = φ . By Lemma 4.2, ( f Y , x ) is in A ( X ), with the criterion f or (E3a ′ ) satisfied due to w 0 ∈ Y . By Lemma 4.2.2, the z a ∈ Y ensu r e that f Y ( τ Y ( a )) = f ( τ ( a )) f or all a ∈ V , so that f Y | = 1 ( Y ,τ Y ) ψ . By Corollary 3.10, we obtain that the mo dal lo gi c of agency is in P SP ACE , and that b ound ing th e mo dal n esting depth br ings the complexit y do wn to NP. Both results (and even decidabilit y) s eem to b e new. In the ligh t of the pr evious observ ation that the agglomeration axiom E a ∧ E b → E ( a ∧ b ) tends to cause PSP ACE-hardness [31], we conjecture that th e PSP ACE up p er b ound is tigh t. 5 Exp onent ial Branc hing In cases where the OSPMP fails, it ma y still b e p ossible to obtain a PSP A CE upp er b oun d b y tra v ersing an exp onentiall y branc hing shallo w mo del (b y Remark 3.3, branching is neve r w orse th an exp onential) . The crucial pr erequisite is that exp onenti al-size one-step mo dels can b e tra ve rsed p oint wise, accumulating d uring the tra v ersal a p olynomial amount of infor- mation that suffices for one-step mo del c hec king. This requires add itional assu m ptions on the signature fun ctor S 0 : Definition 5.1. W e sa y that S 0 is p ointwise b ounde d w.r.t. a set C if for all sets X , there exists an injection S 0 X ֒ → ( X → C ) (i.e. | S 0 X | ≤ | X → C | ). W e then identify S 0 X with a subset of X → C . (Note that the ab o v e do es not require that λX. X → C is fu n ctorial.) Recal l f rom [27] that the signature functor S 0 admits a sep ar ating set of unary predicate liftings (separation is a necessary condition for the generalised Hennessy-Milner prop ert y) iff the family of maps S 0 f : S 0 X → S 0 2, ind exed ov er all maps f : X → 2 = {⊥ , ⊤} , is jointly inj ectiv e for eac h set X . Typica lly , functors S 0 satisfying this condition satisfy the stronger requir emen t that already the f amily of maps ( S 0 1 { x } : S 0 X → S 0 2) x ∈ X is join tly injectiv e, where 1 A denotes the c haracteristic function of A ⊆ X , so that S 0 is p oint wise b ounded w.r.t. S 0 2; often, eve n a quotien t of S 0 2 will suffice. Of the signature fun ctors men tioned in E x amp le 2.10, P and D ω are p oint wise b oun d ed (w.r.t. 2 and [0 , 1], resp ectiv ely), while Cf and P ◦ D ω fail to b e s o. 15 F urther examples of p oint wise b ound ed fu nctors includ e th e game frame functor app earing in the semantics of coalit ion logic [29] and the m ultiset fu nctor in tro du ced b elo w. Assume from now on that S 0 is p oin t wise b ou n ded w.r.t. C , with a giv en represent ation of elemen ts of C (Conv en tion 2.11 is no longer needed). F or t : X → C , we define maxsize( t ) = max x ∈ X size( t ( x )) , and put maxsize( t, x ) = m axsize( t ) for x ∈ X . Definition 5.2. W e sa y that M has th e one-step p ointwise p olysize mo del pr op e rty (OSPPMP) if there exists a p olynomial p su c h that, wh en ev er a one-step pair ( φ, ψ ) o v er V has a one-step mo del ( X , τ , t, x ), then it has a one-step mo del ( Y , κ, s, y ) s u c h that | Y | ≤ 2 | V | , maxsize( s ) ≤ p ( | ψ | ), and y ∈ κ ( a ) iff x ∈ τ ( a ) for all a ∈ V ; suc h a mo del is calle d p ointwise p olysize . By Remark 3.3, the actual con ten t of the OSP P MP is the p olynomial b ound on maxsize( s ). The OSPPMP holds for all p oin t wise b ounded functors menti oned so far, trivially so in cases where C is finite. W e ha v e a v ariant of Theorem 3.5, prov ed enti rely analogously , which states that under the OSPPM P , every satisfiable formula ψ is satisfie d in a shal low mo del ( X , ξ ) with br anching de gr e e at most 2 | ψ | and m axsize( ξ ( x )) p olynomial ly b ounde d in | ψ | . F or the ensuing algorithmic treatment, w e need a refined notion of one-step mo del chec king: Definition 5.3. The p ointwise one-step mo del che cking pr oblem is to chec k, giv en a m ap t : X → C , x ∈ X , a P ( X )-v aluation τ for V , Y ⊆ X , and a conjunctiv e clause ψ o v er Λ( V ), whether ( t, x ) ∈ S Y ⊆ ( X → C ) × X and t | = 1 Y , τ Y ψ , where τ Y ( a ) = τ ( a ) ∩ Y for a ∈ V . W e sa y th at this problem is PSP ACE-tr actable if it is decidable on a n on-deterministic T uring mac hine with input tap e that u ses space p olynomial in maxsize( t ) and accesses eac h input sym b ol at most once. Theorem 5.4. If M has the OSPPMP and p ointwise one-step mo del che c k ing is P SP ACE- tr actable, then the satisfiability pr oblem of F (Λ) is in P SP ACE. Pr o of. Let M b e a d ecision pro cedure for p oin t wise one-step mo del c hec king as required in the definition of PSP A CE-tractabilit y (Defn. 5.3). Let p b e a p olynomial witnessing the OS PPMP as in Definition 5.2. Then the follo wing n on-deterministic algorithm decides satisfiabilit y of F (Λ)-formulas: Algorithm 5.5. 1. If rank( ψ ) = 0, terminate su ccessfully if ψ ev aluates to ⊤ , else unsuc- cessfully . Otherwise: 2. T ak e V and σ as in the p ro of of Theorem 3.5, and guess a conjunctive clause ¯ ψ o v er Λ( V ) con taining for eac h subformula L ( ρ 1 , . . . , ρ n ) of ψ either L ( a ρ 1 , . . . , a ρ n ) or ¬ L ( a ρ 1 , . . . , a ρ n ) suc h that ¯ ψ σ p rop ositionally enta ils ψ . 3. C all M w ith arguments X, τ , Y , t as in Definition 5.3 to c hec k that t ∈ T Y and t | = 1 ( Y ,τ Y ) ¯ ψ , with τ Y as in Definition 5.3. Here, X = 2 V , τ ( a ) = { B ∈ X | a ∈ B } , Y = { B ∈ X | ^ a ρ ∈ B ρ ∧ ^ a ρ / ∈ B ¬ ρ satisfiable } is calculated r ecursiv ely , and t ∈ ( X → C ) with maxsize( t ) ≤ p ( | φ | ) is guessed. 16 It remains to see that the ab o v e algorithm can b e implement ed in p olynomial space although the in put to M in Step 3 is of o v erall exp onent ial size. T his is ac hiev ed by replacing r ead op erations on the input tap e in M b y calls to a p r o cedure passed by the caller, whic h p ro du ces the k -th inp ut symb ol on demand, and then calling th e mo dified p oint wise mo del c hec k er M ′ with such a p ro cedure instead of th e full argument. By the assu m ption that M accesses eac h sym b ol on the input tap e at m ost once, there is no need to k eep the symbols represent ing the guessed v alue t in memory after they ha v e b een passed to M ′ . Therefore, only p olynomial space ov erhead is generated by the inpu t to M ′ (the inp ut pr o cedure dep ends on φ and hence has represen tation size O ( | φ | )); the space usage of M ′ itself is p olynomial in maxsize( t ) and therefore in | φ | . Example 5.6. Theorem 5.4 applies e.g. to the mo dal logics K and T , as well as to prob- abilistic mo dal logic; how ever, as all these logics in fact enjoy the O SPMP , the metho d of Sect. 3 is preferable in these cases. A more in teresting applicatio n is giv en b y graded mo d al logic [11], or more generally Presbu rger mo dal logic [6]. In its single-age nt v ersion, Presburger mo dal logi c has n -ary mo dal operators P n i =1 a i #( ) ∼ b , w here b and the a i are in tegers and ∼ ∈ { <, >, = } ∪ {≡ k | k ∈ N } . A coalge braic seman tics f or this logic, equiv alen t for p urp oses of satisfiabilit y to the ord ered tree semantics giv en in [6], is defin ed o v er the fi nite multiset functor B , w hic h m aps a set X to the set of maps B : X → N w ith fin ite supp ort, th e intuition b eing that B is a m ultiset con taining x ∈ X with multiplicit y B ( x ). F or A ⊆ X , put B ( A ) = P x ∈ A B ( x ). B -coalgebras are graphs with N -w eigh ted edges. Th e ab o v e m o dalities are interpreted b y [ [ P n i =1 a i #( ) ∼ b ] ] X ( A 1 , . . . , A n ) = { B ∈ B ( X ) | P n i =1 a i B ( A i ) ∼ b } , with > , <, = interpreted as exp ected, and ≡ k as equalit y mo du lo k . This logic extends graded mo dal logic, w hose op erators ♦ k no w b ecome #( ) > k . Of course, B is p oint wise b ound ed w.r.t. N . It f ollo ws easily fr om estimates on solution sizes of inte ger linear equalities [24] that Presbur ger mo d al logic has the OSPPMP [6]. Moreo v er, giv en a conju nctiv e clause ψ o v er Λ( V ), a P ( X )-v aluation τ , and B ∈ B ( X ), one can c hec k whether B | = 1 X,τ ψ by tr av ersin g X and computing the B ( τ ( a )) by successiv e summation; it is th us easy to see that p oin t wise one-step mo del c hec king is PSP A CE-tractable. It follo ws that Pr esbur ger mo dal lo gic is in P SP ACE . While this is prov ed already in [6], using essen tially the same t yp e of algorithm 1 , our metho d extends straigh tforwa rd ly to extensions of P resburger mo dal logic by certain frame conditions suc h as r eflexivit y (mo delled b y the cop oin ted functor S X = { ( B , x ) ∈ B X × X | B ( x ) > 0 } ) or e.g. the condition that at least half of all transitions from a giv en state are lo ops (mo delled by th e cop oint ed functor S X = { ( B , x ) ∈ B X × X | B ( x ) ≥ B ( X − { x }} ). In particular, this implies th at gr ade d mo dal lo gic over r eflexive fr ames (i.e. the logic T n of [11]) is in PSP ACE , to our kno wledge a n ew result. The logic T n can b e seen as a description logic with qualified num b er r estrictions on a single r efl exiv e r ole. O ur argumen ts extend straigh tforw ardly to sho w that a descrip tion log ic w ith r ole hierarc hies, reflexiv e roles, and qualified n umber restrictions has concept satisfiabilit y o v er the empt y T - b o x in PSP A CE. As reflexivity of a role R is expressed b y the role in clus ion id ( ⊤ ) ⊆ R , where id ( ⊤ ) denotes the iden tit y r ole, this logic is a fragmen t of AL C H Q ( id ) [1]. 1 The claim th at a (rank- 1) logic furth er extended by regularit y constrain ts is still in PSP ACE is retracted in the full version of [6] as b eing based on p ossibly erroneous th ird-party results. 17 6 Conclusion W e ha v e form ulated t w o lo cal seman tic conditions that guaran tee PSP ACE u pp er b ound s for the satisfiabilit y problem of mo dal logics in a coalgebraic fr amew ork: the OSPMP (one- step p olysize mo del prop er ty) and its p oin t wise v arian t, the O S PPMP , whic h is weak er bu t relies on additional assu mptions on the coalgebraic semantics. Both conditions allo w a direct construction of shallo w m o dels and their tra v ersal in p olynomial space. Th is complemen ts earlier w ork [29] where syn tactic criteria hav e b een used — in particular, b oth the OSPMP and the OSPPMP can b e applied even wh en no complete axiomatisation of the logic at hand is kn o wn. Sev eral instantia tions of our results to logics studied in the literature w itn ess b oth their generalit y and their usefulness: Apart fr om re-proving kno wn PSP ACE upp er b ound s for the normal mo dal logics K and T as w ell as for the conditional logics CK , CK + ID , and CK + M P , w e hav e • giv en a systematic accoun t of tight PSP ACE up p er b ounds in mo dal logics of quanti tativ e uncertain t y that establishes new complexit y b ounds in some cases; • obtained a new PSP ACE up p er b ound for E lgesem’s mo dal logic of agency and for graded (ev en Presburger [6]) mo d al logic o v er reflexive frames [11], an d more generally for an exten- sion of the descrip tion logic ALC H Q with r efl exiv e roles [1]; • established (to our kno wledge: new) tigh t NP upp er b ound s for b ounded-rank f ragmen ts of the conditional logic s CK , CK + ID , and CK + MP . Ongoing wo rk fo cusses on th e extension of ou r results to iter ative mo d al logics, defined by frame conditions of higher rank, which ho wev er — in particular outside the r ealm of Kripke seman tics — exhibit a tendency to w ards higher complexity or ev en un decidabilit y (ind eed, it seems to b e th e case that all k n o wn iterativ e P SP ACE-co mplete mo dal logics are norm al). References [1] F. Baader, D. Calv anese, D. L. McGuinness, D. Nardi, and P . F. Pat el-Sc hneider, editors. The Description L o gic H andb o ok . Cam brid ge Universit y Press, 2003. [2] M. Barr. T erminal coalgebras in we ll-founded set theory . The or et. Comput. Sci. , 114:29 9– 315, 1993. [3] B. Chellas. Mo dal L o gic . Cambridge Univ. Press, 1980. [4] L. Cholvy , C. Garion, and C. Saurel. Abilit y in a m ulti-agen t con text: A mo d el in the situation calculus. In Computationa l L o gic in Multi-A gent Systems, CLIMA 2005 , v olume 3900 of LNCS , pages 23–36. Sp ringer, 2006. [5] C. C ˆ ırstea and D. Pattinson. Mo du lar constru ction of m o dal logics. In Concurr ency The ory , volume 3170 of LNCS , p ages 258–2 75. Sp ringer, 2004. [6] S. Demri an d D. Lu giez. Presb urger mo dal logic is on ly PS P A CE-complete. In Automate d R e asoning, IJCAR 06 , vo lume 4130 of LNAI , pages 541–55 6. Sp ringer, 2006. [7] D. Elgesem. The mo dal logi c of agency . Nor dic J. Philos. L o gic , 2:1–4 6, 1997. [8] R. F agin and J. Y. Halp ern . Reasoning ab out kno wledge and probabilit y . J. ACM , 41:340 –367, 1994. 18 [9] R. F agin, J. Y. Halp ern, an d N. Megiddo. A logic for reasoning ab out probabilities. Inform. Comput. , 87:78–1 28, 1990. [10] M. F attorosi-Barnaba and F. De Caro. Graded mo d alities I. Stud. L o g. , 44:19 7–221, 1985. [11] K. Fin e. In so m an y p ossible wo rlds. Notr e D ame J. F ormal L o gic , 13:516–52 0, 1972. [12] G. Go v ernatori and A. Rotolo. O n th e axiomatisation of Elgesem’s logic of agency and abilit y . J. P hilos. L o gic , 34:40 3–431, 2005. [13] J. Halp ern. Th e effect of b oundin g the n umb er of pr imitiv e p rop ositions and the depth of nesting on the complexit y of m o dal logic. Art ificial Intel ligenc e , 75:361 –372, 1995. [14] J. Halp ern and R. Pucella. A logic for reasoning ab out u pp er probabilities. J. Artificial Intel ligenc e R e s. , 17:57–81 , 2002. [15] J. Halp ern and R. Pu cella. Reasoning ab out exp ectatio n. In Unc ertainty in Artificial Intel ligenc e , UA I 02 , p ages 207–2 15. Morgan Kau f man, 2002. [16] J. Y. Halp ern and Y. O. Moses. A guide to completeness and complexit y for mo dal logics of kno wledge an d b elief. A rtificial Intel ligenc e , 54:31 9–379, 1992. [17] B. Jacobs. T o w ards a dualit y result in the mod al logic of coalge bras. In Co algebr aic Metho ds in Computer Scienc e, CM CS 2000 , v olume 33 of ENTCS . Elsevier, 2000. [18] A. Jones and X. P aren t. Conv entional s ignalling acts and con v ersation. In A dvanc es in A gent Communic ation , v olume 2922 of LNAI , pages 1–17. S pringer, 2004. [19] A. Kurz. Sp ecifying coalgebras with mo dal logic. The or et. Comput. Sci . , 260:11 9–138, 2001. [20] R. Ladner. The compu tational complexit y of prov abilit y in s y s tems of mo dal prop osi- tional logic. SIAM J. Comput. , 6:467–4 80, 1977. [21] D. Lewis. In tensional logics without iterativ e axioms. J. P hilos. L o gi c , 3:457–46 6, 1975. [22] N. Olive tti, G. L. Pozza to, and C . Sc h wind . A sequent calculus and a theorem pr o v er for standard conditional logics. ACM T r ans. Comput. L o gic . T o app ear. [23] E. Pa cuit and S. Salame. Ma jorit y logic. In Principles of Know le dge R epr esentation and R e asoning, KR 04 , pages 598–605. AAAI Press, 2004. [24] C. H. P apadimitriou. On the complexit y of integer programming. J. ACM , 28:765–7 68, 1981. [25] D. P attinson. Coalgebraic mo d al logic: Soundn ess, completeness and decidabilit y of lo cal consequence. The or et. Comput. Sci. , 309:177 –193, 2003. [26] M. Pa uly . A mo d al logic for coalitional p ow er in games. J . L o gi c Comput. , 12:149– 166, 2002. [27] L. Schr¨ oder. Expressivit y of coalgebraic mo dal logic: the limits and b ey ond. The or et. Comput. Sci. In press. 19 [28] L. Sc hr¨ oder. A finite mod el construction f or coalg ebraic mo dal logic. J. L o g. Algebr. Pr o g. , 73:97– 110, 2007. [29] L. Sc hr¨ oder and D. P attinson. PSP ACE reasoning for rank -1 mo d al logics. In L o gic in Computer Scienc e, LICS 06 , pages 231–240. IEE E , 2006. Extended version to app ear in ACM T r ans. Comput. L o g. [30] S. T obies. PSP A CE reasoning for graded mod al logics. J. L o gi c Comput. , 11:8 5–106, 2001. [31] M. V ard i. On th e complexit y of epistemic reasoning. In L o gic in Computer Scienc e , LICS 89 , pages 243–251 . IE E E, 1989. 20