Hybrid Logics and NP Graph Properties

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1 Hybrid Logics ad NP Graph Properties Fracicleber Martis Ferreira 1, Cibele Matos Freire 1, Mario R. F. Beevides 2, L. Measché Schechter 3 ad Aa Teresa Martis 1 1 Uiversidade Federal do Ceará, Departameto de Computação 2 Uiversidade Federal do Rio de Jaeiro, COPPE/Sistemas 3 Uiversidade Federal do Rio de Jaeiro, CCMN {fra, cibelemf, aa}@lia.ufc.br; mario@cos.ufrj.br; luisms@dcc.ufrj.br Jauary 15, 2011 Abstract We show that for each property of graphs G i NP there is a sequece φ 1, φ 2,... of formulas of the full hybrid logic which are satisfied exactly by the frames i G. Moreover, the size of φ is bouded by a polyomial. We also show that the same holds for each graph property i the polyomial hierarchy. These results lead to the defiitio of sytactically defied fragmets of hybrid logic whose model checkig problem is complete for each degree i the polyomial hierarchy. 1 Itroductio The use of graphs as a mathematical abstractio of objects ad structures makes it oe of the most used cocepts i computer sciece. Several typical problems i computer sciece have their iputs modeled by graphs, ad such problems commoly ivolve evaluatig some graph property. To metio a well-kow example, decidig whether a map ca be colored with a certai umber of colors is related to a similar problem o plaar graphs [HAK89, RSST97]. The applicatios of graphs i computer sciece are ot restricted to modellig the iput of problems. Graphs ca be used i the theoretical framework i which some braches of computer sciece are formalized. This is the case, for example, i distributed systems, i which the model of computatio is built o top of a graph [Bar96, Ly96]. Agai, properties of graphs ca be exploited i order to obtai results about such models of computatio. We ca use logic to express properties of structures like graphs. From the sematical stadpoit, a logic ca be regarded as a pair L = (L, =), where L, the laguage of L, is a set of elemets called formulas, ad = is a biary satisfactio relatio betwee some set of objects or structures ad formulas. Such set of structures ca be a set of relatioal structures, for example, the 1

2 class of graphs. A setece φ of L ca be used to express some property of structures, hece oe could check whether a structure A has some property by evaluatig whether A = φ holds or ot. The problem of checkig whether a give model satisfies a give formula is called model checkig. I the last few decades, modal logics have attracted the attetio of computer scietists workig with logic ad computatio [BDRV02]. Amog the reasos is the fact that modal logics ofte have iterestig computer theoretical properties, like decidability [Var96, Grä01]. This is due to a lack of expressive power i compariso with other logics such as first-order logics ad its extesios. May modal logics preset also good logical properties, like iterpolatio, defiability, ad so o. Research i modal logic icludes augmetig the expressive power of the logic usig resources as fixed-poit operators [BS07] or hybrid laguages [AtC07, ABM01]. Modal logics are particularly suitable to deal with graphs because the models of most modal logics are built up from structures called frames, which are essetially graphs. I [BS09], hybrid logics are used to express graph properties, like beig coected, hamiltoia or euleria. Several hybrid logics ad fragmets were studied to defie graph properties through the cocept of validity i a frame (see Defiitio 5 below). Some graph properties, like beig hamiltoia, require a high expressive power ad caot be expressed by a sigle setece i traditioal hybrid logics. There are, however, seteces φ which ca express such properties for frames of size. We are iterested i expressig graph properties i NP usig hybrid logics. Hybrid Modal logics have low expressive power, hece we do ot aim to associate to each graph property a sigle formula. Istead, we preset, to each graph property, a sequece of hybrid seteces φ 1, φ 2,..., such that a graph of size has the desired property iff φ is valid i the graph, regarded as a frame. I Sectio 2, we defie the hybrid logic which we will study ad defie a preex form for such logic. I Sectio 3, we show that, for ay graph property, there is a sequece of seteces φ 1, φ 2,... of the fragmet of hybrid logic with omials ad operator such that a graph of size has the property iff φ is valid i the correspodig frame. However, the size of φ obtaied is expoetial o. I Sectio 4, we show that, for graph properties i NP, ad more geerally i the polyomial hierarchy, there is such a sequece, but the size of the seteces is bouded by a polyomial o. I Sectio 5, we show how to obtai the results of the previous sectio for the fragmet of hybrid logic without the global modality E ad without omials, provided that graphs are coected. I Sectio 6, we show fragmets of hybrid logic whose model-checkig problem is complete for each degree of the polyomial hierarchy based o the results of the other sectios. This gives a alterative proof for the NP-hardess of the model-checkig problems for the fragmet FHL\ of full hybrid logic give i [tcf05]. The proofs of the mai theorems appears i Appedix A. 2

3 2 Hybrid Logic I this sectio, we preset the hybrid logic ad its fragmets which we will use. Hybrid modal logics exted classical modal logics by addig omials ad state variables to the laguage. Nomials ad state variables behave like propositioal atoms which are true i exactly oe world. Other extesios iclude the operators (bider) The allows oe to assig the curret state to a state variable. This ca be used to keep a record of the visited states. operator allows oe to evaluate a formula i the state assiged to a certai omial or state variable. Defiitio 1. The laguage of the hybrid graph logic with the bider is a hybrid laguage cosistig of a set P ROP of coutably may propositio symbols p 1, p 2,..., a set NOM of coutably may omials i 1, i 2,..., a set S of coutably may state-variables x 1, x 2,..., such that P ROP, NOM ad S are pairwise disjoit, the boolea coectives ad ad the modal i, for each omial x, for each state-variable x,, 1 ad. The laguage L F HL of the (Full) Hybrid Logic ca be defied by the followig BNF rule: α := p t α α α α 1 α t α x.α, where t is either a omial or a state variable. For each C {@,, 1, E}, we defie HL(C) to be the correspodig fragmet. I particular, we defie FHL = HL(@,, 1, E). We also use HL(C)\NOM ad HL(C)\PROP to refer to the fragmets of HL(C) without omials ad propositioal symbols respectively. The stadard boolea abbreviatios,, ad ca be used with the stadard meaig as well as the abbreviatios of the dual modal operators: φ := φ, 1 φ := 1 φ ad Aφ := E φ. Formulas of hybrid modal logics are evaluated i hybrid Kripke structures (or hybrid models). These structures are build from frames. Defiitio 2. A frame is a graph F = (W, R), where W is a o-empty set (fiite or ot) of vertices ad R is a biary relatio over W, i.e., R W W. Defiitio 3. A (hybrid) model for the hybrid logic is a pair M = (F, V), where F is a frame ad V : P ROP NOM P(W ) is a valuatio fuctio mappig propositio symbols ito subsets of W, ad mappig omials ito sigleto subsets of W, i.e, if i is a omial the V(i) = {v} for some v W. I order to deal with the state-variables, we eed to itroduce the otio of assigmets. Defiitio 4. A assigmet is a fuctio g that maps state-variables to vertices of the model, g : S W. We use the otatio g = g[v 1 /x 1,..., v /x ] to deote a assigmet such that g (x) = g(x) if x / {x 1,..., x } ad g (x i ) = v i, otherwise. The sematical otio of satisfactio is defied as follows: 3

4 Defiitio 5. Let M = (F, V) be a model. The otio of satisfactio of a formula ϕ i a model M at a vertex v with assigmet g, otatio M, g, v ϕ, ca be iductively defied as follows: M, g, v p iff v V(p); M, g, v always; M, g, v ϕ iff M, g, v ϕ; M, g, v ϕ 1 ϕ 2 iff M, g, v ϕ 1 ad M, g, v ϕ 2 ; M, g, v ϕ iff there is a w W such that vrw ad M, g, w ϕ; M, g, v 1 ϕ iff there is a w W such that wrv ad M, g, w ϕ; M, g, v i iff v V(i); M, g, i ϕ iff M, g, d i ϕ, where d i V(i); M, g, v x iff g(x) = v; M, g, x φ iff M, g, d φ, where d = g(x); M, g, v x.φ iff M, g[v/x], v φ. For each omial i, the i ϕ meas that if V(i) = {v} the ϕ is satisfied at v. If M, g, v ϕ for every vertex v, we say that ϕ is globally satisfied i the model M with assigmet g (M, g ϕ) ad if ϕ is globally satisfied i all models M ad assigmets of a frame F, we say that ϕ is valid i F (F ϕ). The ext lemma follows directly from the defiitio of satisfactio. Lemma 1. The followig equivaleces hold i HL: ( α β) x. x β), (Aα β) x β); ( α β) x. x β), (Aα β) x β); ( α β) x. x β), (Eα β) x β); ( α β) x. x β), (Eα β) x β); ( 1 α β) x. 1 x β), ( 1 α β) x. 1 x β); (( x.α) β) x.(α β), (( x.α) β) x.(α β). I the followig, we defie a preex form for formulas i FHL ad show that ay formula i FHL has a equivalet i preex form. We use this form to defie classes of formulas whose model-checkig problem is complete for the degrees of the polyomial hierarchy (see Sectio 6). Defiitio 6 (Preex Form). A formula φ i FHL is i preex form iff φ = q 1... q ψ where each q i is, 1,, 1, E, A or x., for some x, ad ψ has o ocurrece of modalities or. It follows from Lemma 1 that each formula of FHL ca be put i this preex form. Lemma 2. If φ FHL, the there is ψ FHL i preex form which is equivalet to φ. This preex form ca be stregtheed with the followig lemma: Lemma 3. x. y.φ(x, y) x.φ(x, x), if x does ot occur boud i φ. 4

5 Lemma 4. If φ F HL, the there is ψ F HL without modalities or biders ad a prefix q = q 1... q where each q i is, 1,, 1, E, A or x. for some x, ad there is o cosecutive applicatio of biders i q. Based o Lemma 4, we defie the followig classes of formulas: Defiitio 7. Let EX = {, 1, E} ad UN = {, 1, A}. We recursively defie the classes of formulas σ i ad π i i preex form as: σ 0 = π 0 = {φ HL φ has o modalities}; σ i+1 = {φ HL φ = q 1... q ψ, ψ π i, q j EX { x.}, for some x}; π i+1 = {φ HL φ = q 1... q ψ, ψ σ i, q j UN { x.}, for some x}. We say that a formula is σ i (resp. π i ) if it is equivalet to a formula i σ i (resp. π i ). From Lemma 2 it follows that each formula i HL is π i or σ i for some i. 3 Properties of Graphs i HL I [BS09], it was show that there is a formula φ of FHL such that a graph of size is Hamiltoia iff it globally satisfies φ. The mai questio which uderlies this ivestigatio is whether there is a sequece of formulas (φ ) N for each graph property G i NP such that a graph G of size is i G iff G, as a frame, globally satisfies φ. Actually, we ca show that such sequece exists for each graph property. Let G = (V, E) be a graph of cardiality. Let us cosider that the set V of vertices coicides with the set {1,..., } of omials. Cosider the formula: ψ G i i j. (i,j) E (i,j) E Let G be ay property of graphs. We defie the formulas ψg = ψ G, θ i j ad φ G = θ ψg. G G, G = i,j {1,...,},i j Lemma 5. Let G be a graph of cardiality ad G a property of graphs. The G G iff G φ G. Sice there are 2 2 graphs with vertices i {1,..., }, we have that the size of φ G is O(22 ) for ay graph property G. Obviously, there is o hope for that sequece of formulas to be always computable. We ca show, however, that, for problems i the polyomial hierarchy, such sequece is recursive ad, moreover, there is a polyomial boud i the size of formulas. 5

6 4 Traslatio I this sectio, we show that for each graph property G i the polyomial hierarchy there is a sequece (φ ) N of formulas such that a graph G of size is i G iff G φ ad such that φ is bouded from above by a polyomial o. We will use the well-kow characterizatio of problems i PH ad classes of fiite models defiable i secod-order logic (SO) from descriptive complexity theory [Imm99]. To this ed, we defie a traslatio from formulas i SO to formulas i FHL which are equivalet with respect to frames of size, for some N. Such traslatio will give us formulas whose size is bouded by a polyomial o. Moreover, the formulas obtaied by the traslatio do ot use propositioal symbols, omials or free state variables, which meas that, for these formulas, the complexity of model-checkig ad frame-checkig coicides. We use the well kow defiitios ad cocepts related to first-order logic (FO) ad secod-order logic which ca be fouded i most textbooks (see, for istace, [EFT94]). Defiitio 8 (Traslatio from FO to HL). Let φ be a first-order formula i the vocabulary S = {E, R 1,..., R m } where E is biary, a atural umber ad f a fuctio from the set of first-order variables ito {1,..., }. Let t, z 1,..., z be state variables ad for each R {R 1,..., R m } of arity h, let yj R 1,...,j h be a state variable, with j i {1,..., }, 1 i h. We defie the fuctio tr f : L S F O L F HL as: tr(x f 1 x 2 ) z zf(x1 ) f(x 2); tr(e(x f 1, x 2 )) z zf(x1 ) f(x 2); tr(r(x f 1,..., x k )) t yf(x R 1 ),...,f(x k ), for each R {R 1,..., R m }; tr(γ f θ) = tr(γ) f tr(θ); f tr( γ) f = tr(γ); f tr( xγ) f = x i=1 trf i (γ); tr( xγ) f = (γ). x i=1 trf i I the traslatio above, t is iteded to represet a state v such that, if zj R 1,...,j h is assiged to v ad z j1,..., z jh are assiged to v 1,..., v h, the (v 1,..., v h ) belogs to the iterpretatio of R. Note that if φ is a setece, the tr(φ) f = tr f (φ). Hece we write tr (φ) istead of tr(φ) f for a setece φ. Example 1. We give a example of applicatio of the traslatio above. Let φ ψ (exclusive or ) be a abbreviatio for (φ ψ) (φ ψ). Cosider the followig first-order setece: φ := x ( R(x) G(x) B(x) ) x y ( (E(x, y) x y) ( (R(x) R(y)) (G(x) G(y)) (B(x) B(y))) ). The setece above says that each elemet belogs to oe of the sets R, G ad B, each adjacet pair does ot belog to the same set, ad o elemet 6

7 belogs to more tha oe set. This setece is true iff the sets R, G ad B forms a 3-colorig of a graph with edges i E. Below we traslate φ ito a formula of hybrid logic usig the traslatio give above ad settig = 3: 3 ( tr (φ) yi t yi t yi B ) 3 [ 3 ( (@zi z zi z j ) i=1 i=1 j=1 ((@ t y R t y R j ) (@ t y G t y G j ) (@ t y B t y B j )) )]. Lemma 6. tr 3 (φ) has polyomial size i, that is, tr (φ) O( k ) for some 0 k. Proof. By iductio o φ oe ca see that tr (φ) is O( k ), where k is the quatifier rak of φ, that is, the maximum umber of ested quatifiers. Lemma 7. Let G = (V, E G ) be a graph of cardiality, R 1,..., R m relatios o V with arities r 1,..., r m, g a assigmet of state variables, β a assigmet of first-order variables, S = {E, R 1,..., R m } a vocabulary ad f a fuctio from the set of first-order variables to {1,..., } such that: (i) g assigs to each variable z i a differet elemet i V ; (ii) g(y R i 1,...,i k ) = g(t) iff (g(z i1 ),..., g(z ik )) R for each R {R 1,..., R m }; (iii) β(x) = g(z f(x) ) for each first-order variable x. If φ is a first-order formula i the vocabulary S, the (G, R 1,..., R m, β) = φ iff for all w V, (G, g, w) tr f (φ). Defiitio 9 (Traslatio from SO to FHL). Let φ = Q 1 X 1... Q l X l ψ be a SO formula where Q i {, } ad ψ is a first-order setece. We defie T (φ) = 1 y X y X1.... l y X l l y X l tr (ψ), where i = E if Q i = ad A otherwise. Example 2. Cosider the setece φ of Example 1. Let ψ be the followig secod-order setece: ψ := R G B(φ). The setece ψ above states that there are three sets R, G ad B which forms a 3-colorig of elemets i the domai of a structure. Hece, φ is satisfied i a graph with edges i E iff such graph is 3-colorable. Decidig whether a graph is 3-colorable is a NP-complete problem [Pap03]. We apply the traslatio T for = 3 below. Let ˆQ := E y R 1.E y R 2.E y R 3.E y G 1.E y G 2.E y G 3.E y B 1.E y B 2.E y B 3.. We have T 3 (ψ) := ˆQtr 3 (φ). That is, ( 3 T 3 ( (φ) := yi t yi t yi B i=1 [ ) 3 3 ( (@zi z zi z j ) i=1 j=1 ((@ t y R t y R j ) (@ t y G t y G j ) (@ t y B t y B j )) )]). 7

8 Lemma 8. Let G = (V, E G ) be a graph of cardiality, R 1,..., R m relatios o V with arities r 1,..., r m, g a assigmet of state variables, β a assigmet of first-order variables, S = {E, R 1,..., R m } a vocabulary ad f a fuctio from the set of first-order variables to {1,..., } such that: (i) g assigs to each variable z i a differet elemet i V ; (ii) g(y R i 1,...,i k ) = g(t) iff (g(z i1 ),..., g(z ik )) R for each R {R 1,..., R m }; (iii) β(x) = g(z f(x) ) for each first-order variable x. If φ = Q 1 X 1... Q l X l ψ is a secod-order formula i the symbol set S, the (G, R 1,..., R m, β) = φ iff for all w V, (G, g, w) T (φ). We have the followig: Theorem 1. Let φ be a secod-order setece ad G a graph of cardiality. The G = φ iff G t.e z E zi z j T (φ). A well kow result of descriptive complexity is the correspodece betwee the polyomial hierarchy ad the alteratio hierarchy of secod-order logic (with respect to fiite models) [Imm99, Fag74]. There are several ways to defie this hierarchy, for example usig alteratig Turig machies [Imm99]. I this paper we assume the defiitio preseted i [Pap03], which uses Turig machies with oracles to defie PH. A Turig machie with a oracle is a machie that has the special ability of guessig some specific questios. Whe a Turig machie has a oracle for a decisio problem B, durig its executio it ca ask for the oracle if some istace of problem B is positive or egative. This is doe i costat time, regardless the size of the istace. We use the otatio M B to defie a Turig machie M with a oracle for a problem B. I a similar way, we defie C B, where C ad B are complexity classes, as the class of problems solved by a Turig machie i C with a oracle i B. Defiitio 10. Cosider the followig sequece of complexity classes. First, p 0 = Σp 0 = Πp 0 = PTIME ad, for all i 0, 1. p i+1 = PΣp i 2. Σ p i+1 = NPΣp i 3. Π p i+1 = conpσp i. We defie the Polyomial Time Hierarchy as the class PH = Σ p i. I particular we have: Theorem 2 ([Imm99]). Let G be a graph property i the polyomial hierarchy. The there is a secod-order setece φ i the laguage of graphs such that G G iff G = φ. The followig is the mai theorem of this sectio: i 0 8

9 Theorem 3. Let G be a graph property i the polyomial hierarchy. The there is a set of seteces Φ = {φ 1, φ 2,...} of FHL, such that: (1) G G iff G Φ iff G φ G, ad (2) φ m is O( k ) for some costat k depedig oly o G. Corollary 1. If φ SO, the existecial fragmet of SO, the T is i HL(@,, E)\{, P ROP }, that is, the fragmet of HL(@,, E) without propositioal symbols ad the patter. 5 Coected Frames with Loops Let FHL\{E, NOM} be the fragmet of full hybrid logic without the modality E ad without omials. Its is ot difficult to show that: Lemma 9. Frame validity ad model (global) satisfactio for seteces from FHL\{E, N OM} are ivariat uder disjoit uio. Thus a aalogous to Theorem 3 does ot hold for FHL\{E, NOM}. Corollary 2. There are graph properties i PTIME for which there is o set Φ = {φ 1, φ2,...} from seteces i FHL\{E, NOM} which satisfies coditios (1) ad (2) from Theorem 3 above. Proof. Coectivity is oe such a property. However, Theorem 3 still hold if we restrict ourselves to coected, frames with loops. Cosider the followig traslatio from SO to FHL\{E, N OM}: Defiitio 11. Let φ = Q 1 X 1... Q l X l ψ be a secod-order formula where Q i {, } ad ψ is a first-order setece. We defie ˆT (φ) = 1 y X y X l y X l l y X l tr (ψ), where i = ( 1 ) if Q i = ad ( 1 ) otherwise. Lemma 10. Let G = (V, E G ) be a coected graph with loops o each vertex, R 1,..., R m relatios o V with arities r 1,..., r m, g a assigmet of state variables, β a assigmet of first-order variables, S = {E, R 1,..., R m } a vocabulary ad f a fuctio from the set of first-order variables to {1,..., } such that: (i) g assigs to each variable z i a differet elemet i V ; (ii) g(yi R 1,...,i k ) = g(t) iff (g(z i1 ),..., g(z ik )) R for each R {R 1,..., R m }; (iii) β(x) = g(z f(x) ) for each first-order variable x. If φ = Q 1 X 1... Q l X l ψ is a secod-order formula i the symbol set S, the (G, R 1,..., R m, β) = φ iff for all w V, (G, g, w) ˆT (φ). Proof. Aalogous to the proof of Lemma 8 9

10 Theorem 4. Let φ be a secod-order setece ad G a coected graph of cardiality with loops. The G = φ iff G t.( 1 ) z ( 1 ) zi z j ˆT (φ). Proof. Aalogous to the proof of Theorem 1 Theorem 5. Let G be a property of coected graphs with loops i the polyomial hierarchy. There is a set of seteces Φ = {φ 1, φ 2,...} of the fragmet FHL\{E, NOM}, such that: (1)for all coected graphs G with loops, G G iff G Φ iff G φ G, ad (2) φ m is O( k ) for some costat k depedig oly o G. 6 Polyomial Hierarchy I [tcf05], it is proved that the model checkig for the FHL\ fragmet is NP-complete. The traslatio give i Sectio 4 above ca be used to produce hybrid formulas of polyomial size usig formulas of secod-order logic. This leads to a alterative proof that the model checkig problem for the fragmet FHL\ is hard for NP, sice there is a polyomial reductio for ay istace of a NP problem to the model checkig of FHL\. Theorem 6 ([tcf05]). The model checkig problem for σ 1 FHL\ is NP-hard. Actually, for each degree of the polyomial hierarchy, there is a sytactically defied fragmet of HL whose model checkig problem is hard. Theorem 7. The model checkig problem for σ i (resp. π i ) is Σ p i -hard (resp. Π p i -hard). Also, the model checkig for σ i ad π i are i Σ p i ad Πp i, respectively. Theorem 8. The model checkig problem for σ i (resp. π i ) is i Σ p i (resp. Πp i ). Corollary 3. Let Φ = {φ 1, φ 2,...} be such that each φ i ca be costructed i time polyomial o i ad each φ i is i π j (resp. σ j ). The the graph property G defied as: G G iff G φ G is i Π p j (resp. Σp j ). From Theorems 7 ad 8 we have: Corollary 4. The model checkig problem for σ i (resp. (resp. Π i -complete). π i ) is Σ p i -complete Corollary 5. The frame checkig problem for seteces i σ i \{P ROP, NOM} (resp. π i \{P ROP, NOM}) is Σ p i -complete (resp. Πi -complete). 10

11 Refereces [ABM01] [AtC07] [Bar96] C. Areces, P. Blackbur, ad M. Marx. Hybrid logics: Characterizatio, iterpolatio ad complexity. Joural of Symbolic Logic, 66(3): , C. Areces ad B. te Cate. Hybrid logics. I P. Blackbur, J. va Bethem, ad F. Wolter, editors, Hadbook of modal logic, volume 3, pages Elsevier Sciece Ltd, V.C. Barbosa. A itroductio to distributed algorithms. The MIT Press, [BDRV02] P. Blackbur, M. De Rijke, ad Y. Veema. Modal logic. Cambridge Uiv Pr, [BS07] [BS09] J. Bradfield ad C. Stirlig. Modal mu-calculi. Hadbook of Modal Logic, 3: , M.R.F. Beevides ad L.M. Schechter. Usig modal logics to express ad check global graph properties. Logic Joural of IGPL, 17(5):559, [EFT94] H.D. Ebbighaus, J. Flum, ad W. Thomas. Mathematical logic. Spriger, [Fag74] Roald Fagi. Geeralized first-order spectra ad polyomial-time recogizable sets. I R.Karp, editor, Complexity of Computatio, volume 7 of SIAM-AMS Proceedigs, pages AMS, [Grä01] Erich Grädel. Why are modal logics so robustly decidable? I G. Pau, G. Rozeberg, ad A. Salomaa, editors, Curret Treds i Theoretical Computer Sciece. Eterig the 21st Cetury, pages World Scietific, [HAK89] W. Hake, K. Appel, ad J. Koch. Every plaar map is four colorable, volume 98 of Cotemporary Mathematics. America Mathematical Society, [Imm99] N. Immerma. Descriptive complexity. Spriger Verlag, [Ly96] N.A. Lych. Distributed algorithms. Morga Kaufma, [Pap03] C.H. Papadimitriou. Computatioal complexity. Joh Wiley ad Sos Ltd., [RSST97] Neil Robertso, Daiel Saders, Paul Seymour, ad Robi Thomas. The four-colour theorem. Joural of Combiatorial Theory, Series B, 70(1):2 44,

12 [tcf05] [Var96] Balder te Cate ad Massimo Fraceschet. O the complexity of hybrid logics with biders. I L. Og, editor, Proceedigs of Computer Sciece Logic 2005, volume 3634 of Lecture Notes i Computer Sciece, pages Spriger Verlag, M.Y. Vardi. Why is modal logic so robustly decidable. I Neil Immerma ad Phokio G. Kolaitis, editors, Descriptive complexity ad fiite models, volume 31 of Series i Discrete Mathematics ad Theoretical Computer Sciece, pages America Mathematical Society, A Proofs of Mai Theorems Proof of Lemma 7. We proceed by iductio o φ. φ is atomic: I this case φ = x y, φ = E(x, y) or φ = R i (x 1,..., x ). If φ = x y, the (G, R 1,..., R m, β) = φ iff β(x) = β(y) iff, by (iii), g(z f(x) ) = g(z f(y) ) iff (G, g, zf(x) z f(y) = tr f (φ). If φ = E(x, y), the (G, R 1,..., R m, β) = φ iff (β(x), β(y)) E G iff, by (iii), (g(z f(x) ), g(z f(y) ) E G iff (G, g, zf(x1 ) z f(x 2 ) = tr f (φ). If φ = R i (x 1,..., x h ), the (G, R 1,..., R m, β) = φ iff (β(x 1 ),..., β(x )) R i iff, by (iii), (g(z f(x1)),..., g(z f(x))) R i iff, by (ii), g(y R f(x 1),...,f(x k ) ) = g(t) iff (G, g, t y R f(x 1),...,f(x k ) = trf (φ). φ = γ θ or φ = γ: These cases follow directly from the defiitio of tr f ad the iductive hypothesis. φ = xγ: I this case, (G, R 1,..., R m, β) = φ iff there is a v V such that (G, R 1,..., R m, β x v ) = γ. By (i), there is a j be such that v = z j. Hece we have β x v (y) = g(z f x j (y) ) for each first-order variable y. By iductive hypothesis we have, (G, R 1,..., R m, β x v ) = γ iff (G, g, w) x trf j (γ) iff (G, g, w) (γ) = tr(φ). f x i=1 trf i φ = xγ: I this case, (G, R 1,..., R m, β) = φ iff, for each v V, (G, R 1,..., R m, β x v ) = γ. By (i), for each v V there is a j such that v = z j. Hece we have β x v (y) = g(z f x j (y) ) for each first-order variable y. By iductive hypothesis we have, for each v V, (G, R 1,..., R m, β x v ) = γ iff, for each j {1,..., }, (G, g, w) tr f x j tr(φ). f (γ) iff (G, g, w) i=1 trf x i (γ) = Proof of Lemma 8. Let v V such that v g(t). For each X 1 o V, let = g(t) if (g(z i1 ),..., g(z ih )) ad v otherwise. The, for each X 1 o V v X1 i 1,...,i h 12

13 there is a assigmet g X 1 defied as g X 1 = g yx 1 1,...,1... yx 1 i 1,...,i h... y X 1,..., v X 1 1,...,1... vx 1 i 1,...,i h... v X. 1,..., O the cotrary, give a assigmet g we ca fid X 1 such that g = g X1. The assigmet g X1 ca be described as: g(t), if s = y X 1 i 1,...,i h ad (i 1,..., i h ) X 1 ; g X1 (s) = v, for some v g(t), if s = y X1 i 1,...,i h ad (i 1,..., i h ) X 1 ; g(s), otherwise; It follows that g X 1 ad X 1 satisfies (i) ad (iii) ad (ii ) g X1 (y R i 1,...,i k ) = g X1 (t) iff (g X1 (z i1 ),..., g X1 (z ik )) R for each R {R 1,..., R m, X 1 }. Now, we proceed by iductio o the size l of the prefix Q 1 X 1... Q l X l. If l = 0, the φ is first-order ad the result follows immediately from Lemma 7. Suppose that l > 0. If φ = X 1... Q l X l ψ, the (G, R 1,..., R m, β) = φ iff there is X 1 V r1 such that (G, R 1,..., R m, X 1, β) = Q 2 X 2... Q l X l ψ iff, by the iductive hypothesis, there is g X 1 such that iff iff (G, g X 1, w) T (Q 2 X 2... Q l X l ψ) (G, g yx 1 1,...,1... yx 1 i 1,...,i h... y,..., X1 v X 1 1,...,1... vx 1 i 1,...,i h... v X, w) T (Q 1 2 X 2... Q l X l ψ),..., (G, g, w) E y X1.... E 1 yx1.t (Q 2 X 2... Q l X l ψ) = T (φ). If φ = X 1... Q l X l ψ, the iff, for all X 1 V r 1, (G, R 1,..., R m, β) = φ (G, R 1,..., R m, X 1, β) = Q 2 X 2... Q l X l ψ iff, by the iductive hypothesis, for all g X 1, iff, for all v X 1 1,...,1... vx 1 i 1,...,i h... v X1,...,, (G, g X 1, w) T (Q 2 X 2... Q l X l ψ),..., (G, g yx1 1,...,1... yx1 i 1,...,i h... y X 1 v X 1 1,...,1... vx 1 i 1,...,i h... v X 1,...,, w) T (Q 2 X 2... Q l X l ψ) iff (G, g, w) A y X A 1 yx 1.T (Q 2 X 2... Q l X l ψ) = T (φ). 13

14 Proof of Theorem 1. iff (G, g, w) t.e z E z. (G, g t w, w) E z E z. iff there are v 1,..., v V such that (G, g tz 1... z wv 1... v, w) iff there are v 1... v V such that iff, by Lemma 8, G φ. (G, g tz 1... z wv 1... v, w) T zi z j T zi z j T zi z j T (φ) Proof of Theorem 3. Let ψ be a secod-order formula expressig G. Let θ = A z A zi z j A zi z. 1 i The setece θ says that there are at most vertices i the frame. We defie φ as: φ = θ t.e z E zi z j T (ψ). Let G G. Let g be ay assigmet of state variables ad w be ay poit i G. If G θ, the G φ. It follows that G φ for each G. Hece, G Φ iff G φ G. Let G =. The (G, g, w) φ iff (G, g, w) t.e z E z. iff, by Theorem 1, G zi z j T (ψ) Proof of Theorem 5. Let ψ be a secod-order formula expressig G. Let Q be the prefix ( 1 ) z ( 1 ) z. ad let 14

15 θ = zi z j ( 1 ) z. ψ = t.( 1 ) z ( 1 ) z. 1 zi z, zi z j T (ψ). Let φ = θ ψ. The remaiig of the proof is similar to the proof of Theorem 2. Proof of Theorem 6. Let G be a NP-complete graph property. Let φ be a SO setece which express G. By Fagi s Theorem [Fag74], such setece exists. Let G be a graph. Let T G (φ) as defied i Defiitio 9. It is easy to see that T G (φ) ca be costructed from φ i time polyomial i G. Now, (G, T G (φ)) is a istace of the model checkig for FHL\. By Theorem 3, the model checker returs true for (G, T G (φ)) iff G G. Hece, the model checkig for FHL\ is hard for NP. Proof of Theorem 7. Aalogous to the proof of Theorem 6 above, sice for each graph property i Σ p i (resp. Πp i ) ca be expressed by a SO setece i Σ1 i (resp. Π 1 i ), ad T (φ) ca always be costructed i time polyomial i, for a fixed φ SO. Proof of Theorem 8. We proceed by iductio o i. I [tcf05], it is show that the model checkig for FHL\, which cotais σ 1, is i NP. It follows that π 1 is i co-np. Now, let qφ be a setece i σ i+1, where φ is i π i. For the sake of simplicity, we discosider the modalities 1 ad E, but the proof is aalogous. It follows that q has the form q = k 1 x 1. k 2 x k m x m. k m+1. Let M be a fiite model, g be a assigmet of state variables ad w a poit i W. By iductive hypothesis, suppose that the model checkig problem for π i is i Π p i. We ca use o-determiistic Turig machie to existetially guess values v j for x j amog the poits i W which are reachable i j i=1 k i steps from w, with respect to the accessibility relatio R, i polyomial o-determiistic time, ad we ca existetially guess poits w reachable i m+1 i=1 k i steps from w i polyomial o-determiistic time also. Fially, we ca use a oracle for the model checkig of π i with the iput (M, g v 1... v m x 1... x m, w ), φ). By iductive hypothesis, such a oracle is i Π p i. As the existetial guesses iitially performed ca be made i (existetial) o-determiistic polyomial time, the model checkig for σ i+1 is i Σ p i+1. The proof is aalogous for the model checkig of π i+1. 15