Characterizing the NP-PSPACE Gap in the Satisfiability Problem For

Characterizing the NP-PSPACE Gap in the Satisﬁability Problem for Modal Logic∗

Joseph Y. Halpern Leandro Chaves Regoˆ † Computer Science Department Statistics Department Cornell University, U.S.A. Federal University of Pernambuco, Brazil e-mail: [email protected] e-mail: [email protected]

Abstract By way of contrast, we show that for all of the (infinitely many) modal logics L containing K5 (that is, every modal There has been a great deal of work on character- logic containing the axiom K—Kϕ ∧ K(ϕ ⇒ ψ) ⇒ Kψ— izing the complexity of the satisfiability and valid- and the negative introspection axiom, which has traditionally ity problem for modal logics. In particular, Ladner been called axiom 5), if a formula ϕ is consistent with L,then showed that the satisfiability problem for all logics |ϕ| K S4 S5 it is satisfiable in a Kripke structure of size linear in .Us- between and is PSPACE-hard, while for it ing this result and a characterization of the set of finite struc- is NP-complete. We show that it is negative intro- L K5 ¬Kp ⇒ K¬Kp tures consistent with a logic containing due to Nagle spection,theaxiom , that causes and Thomason 1985, we can show that the consistency (i.e., the gap: if we add this axiom to any modal logic L K S4 satisfiability) problem for is NP-complete. Thus, roughly between and , then the satisfiability problem speaking, adding negative introspection to any logic between becomes NP-complete. Indeed, the satisfiability K and S4 lowers the complexity from PSPACE-hard to NP- problem is NP-complete for any modal logic that complete. includes the negative introspection axiom. The fact that the consistency problem for specific modal logics containing K5 is NP-complete has been observed be- 1 Introduction fore. As we said, Ladner already proved it for S5; an easy There has been a great deal of work on characterizing the modification (see [Fagin et al., 1995]) gives the result for complexity of the satisfiability and validity problem for KD45 and K45.1 That the negative introspection axiom modal logics (see [Halpern and Moses, 1992; Ladner, 1977; plays a significant role has also been observed before; indeed, Spaan, 1993; Vardi, 1989] for some examples). In particular, Nagle 1981 shows that every formula ϕ consistent with a nor- Ladner 1977 showed that the validity (and satisfiability) prob- mal modal logic2 L containing K5 has a finite model (indeed, lem for every modal logic between K and S4 is PSPACE- a model exponential in |ϕ|) and using that, shows that the hard; and is PSPACE-complete for the modal logics K, T, provability problem for every logic L between K and S5 is and S4. He also showed that the satisfiability problem for S5 decidable; Nagle and Thomason 1985 extend Nagle’s result is NP-complete. to all logics containing K5 not just normal logics. Despite What causes the gap between NP and PSPACE here? We all this prior work and the fact that our result follows from show that, in a precise sense, it is the negative introspection a relatively straightforward combination of results of Nagle axiom: ¬Kϕ ⇒ K¬Kϕ. It easily follows from Ladner’s and Thomason and Ladner’s techniques for proving that the proof of PSPACE-hardness that for any modal logic L be- consistency problem for S5 is NP-complete, our result seems tween K and S4, there exists a family of formulas ϕn,all to be new, and is somewhat surprising (at least to us!). consistent with L such that such that |ϕn| = O(n) but the smallest Kripke structure satisfying ϕ has at least 2n states The rest of the paper is organized as follows. In the next (where |ϕ| is the length of ϕ viewed as a string of symbols). section, we review standard notions from modal logic and the key results of Nagle and Thomason 1985 that we use. In Sec- ∗ This work was supported in part by NSF under grants CTC- tion 3, we prove the main result of the paper. We discuss 0208535, ITR-0325453, and IIS-0534064, by ONR under grants related work in Section 4. N00014-00-1-03-41 and N00014-01-10-511, and by the DoD Multi- disciplinary University Research Initiative (MURI) program admin- istered by the ONR under grant N00014-01-1-0795. The second author was also supported in part by a scholarship from the Brazilian Government through the Conselho Nacional de Desenvolvimento 1Nguyen 2005 also claims the result for K5, referencing Ladner. Cient´ıfico e Tecnológico (CNPq). While the result is certainly true for K5, it is not immediate from †Most of this work was done while the author was at the Ladner’s argument. School of Electrical and Computer Engineering at Cornell Univer- 2A modal logic is normal if it satisfies the generalization rule sity, U.S.A. RN: from ϕ infer Kϕ.

IJCAI-07 2306 2 Modal Logic: A Brief Review Modal logics are typically characterized by axiom systems. Consider the following axioms and inference rules, all of We briefly review basic modal logic, introducing the notation which have been well-studied in the literature [Blackburn et used in the statement and proof of our result. The syntax of al., 2001; Chellas, 1980; Fagin et al., 1995]. (We use the tra- the modal logic is as follows: formulas are formed by start- Φ={p,q,...} ditional names for the axioms and rules of inference here.) ing with a set of primitive propositions, and These are actually axiom schemes and inference schemes;we then closing off under conjunction (∧), negation (¬), and the K consider all instances of these schemes. modal operator K. Call the resulting language L1 (Φ).(We often omit the Φ if it is clear from context or does not play a Prop. All tautologies of propositional calculus ϕ∨ψ ϕ ⇒ ψ significant role.) As usual, we define and as ab- K. (Kϕ ∧ K(ϕ ⇒ ψ)) ⇒ Kψ (Distribution Axiom) breviations of ¬(¬ϕ ∧¬ψ) and ¬ϕ ∨ ψ, respectively. The in- tended interpretation of Kϕ varies depending on the context. T. Kϕ ⇒ ϕ (Knowledge Axiom) It typically has been interpreted as knowledge, as belief, or as 4. Kϕ ⇒ KKϕ (Positive Introspection Axiom) necessity. Under the epistemic interpretation, Kϕ is read as “the agent knows ϕ”; under the necessity interpretation, Kϕ 5. ¬Kϕ ⇒ K¬Kϕ (Negative Introspection Axiom) ϕ can be read “ is necessarily true”. D. ¬K(false) (Consistency Axiom) The standard approach to giving semantics to formulas in K ϕ ϕ ⇒ ψ ψ L1 (Φ) is by means of Kripke structures. A tuple F =(S, K) MP. From and infer (Modus Ponens) is a (Kripke) frame if S is a set of states, and K is a binary RN. From ϕ infer Kϕ (Knowledge Generalization) relation on S.Asituation is a pair (F, s),whereF =(S, K) is a frame and s ∈ S. A tuple M =(S, K,π) is a Kripke The standard modal logics are characterized by some sub- structure (over Φ) if (S, K) is a frame and π : S × Φ → set of the axioms above. All are taken to include Prop, MP, {true, false} is an interpretation (on S) that determines and RN; they are then named by the other axioms. For exam- K5 which primitive propositions are true at each state. Intu- ple, consists of all the formulas that are provable using itively, (s, t) ∈Kif, in state s, state t is considered pos- Prop, K, 5, MP, and RN; we can similarly define other sys- KD45 KT5 KT sible (by the agent, if we are thinking of K as representing tems such as or . has traditionally been T KT4 S4 KT45 an agent’s knowledge or belief). For convenience, we define called ; has traditionally been called ;and S5 K(s)={t :(s, t) ∈K}. has traditionally been called . L Depending on the desired interpretation of the formula For the purposes of this paper, we take a modal logic Kϕ, a number of conditions may be imposed on the binary to be any collection of formulas that contains all instances relation K. K is reflexive if for all s ∈ S, (s, s) ∈K;itis of Prop and is closed under modus ponens (MP) and substitu- ϕ L p transitive if for all s, t, u ∈ S,if(s, t) ∈Kand (t, u) ∈K, tion, so that if is a formula in and is a primitive proposi- ϕ[p/ψ] ∈ L ϕ[p/ψ] then (s, u) ∈K;itisserial if for all s ∈ S there exists t ∈ S tion, then ,where is the result of replacing p ϕ ψ such that (s, t) ∈K;itisEuclidean iff for all s, t, u ∈ S,if all instances of in by . A logic is normal if it contains (s, t) ∈Kand (s, u) ∈Kthen (t, u) ∈K. We use the super- all instances of the axiom K and is closed under the inference scripts r, e, t and s to indicate that the K relation is restricted rule RN. In terms of this notation, Ladner 1977 showed that L K S4 to being reflexive, Euclidean, transitive, and serial, respec- if is a normal modal logic between and (since we tively. Thus, for example, Srt is the class of all situations are identifying a modal logic with a set of formulas here, that K ⊆ L ⊆ S4 ϕ ∈ L where the K relation is reflexive and transitive. just means that ), then determining if We write (M,s) |= ϕ if ϕ is true at state s in the Kripke is PSPACE-hard. (Of course, if we think of a modal logic as M being characterized by an axiom system, then ϕ ∈ L iff ϕ is structure . The truth relation is defined inductively as fol- L ϕ lows: provable from the axioms characterizing .) We say that is consistent with L if ¬ϕ/∈ L. Since consistency is just the (M,s) |= p,forp ∈ Φ,ifπ(s, p)=true dual of provability, it follows from Ladner’s result that testing (M,s) |= ¬ϕ if (M,s) |= ϕ consistency is PSPACE-hard for every normal logic between K and S4. Ladner’s proof actually shows more: the proof (M,s) |= ϕ ∧ ψ (M,s) |= ϕ (M,s) |= ψ if and holds without change for non-normal logics, and it shows that (M,s) |= Kϕ if (M,t) |= ϕ for all t ∈K(s) some formulas consistent with logics between K and S4 are satisfiable only in large models. More precisely, it shows the Aformulaϕ is said to be satisfiable in Kripke structure following: M if there exists s ∈ S such that (M,s) |= ϕ; ϕ is said to be valid in M, written M |= ϕ,if(M,s) |= ϕ for all Theorem 2.1: [Ladner, 1977] s ∈ S N . A formula is satisfiable (resp., valid) in a class of (a) Checking consistency is PSPACE complete for every Kripke structures if it is satisfiable in some Kripke structure logic between K and S4 (even non-normal logics). in N (resp., valid in all Kripke structures in N ). There are analogous definitions for situations. A Kripke structure M = (b) For every logic L between K and S4, there exists a fam- L (S, K,π) is based on a frame F =(S , K ) if S = S and ily of formulas ϕn , n =1, 2, 3,..., such that (i) for all L K = K ϕ (F, s) n, ϕn is consistent with L, (ii) there exists a constant d . The formula is valid in situation , written L (F, s) |= ϕ F =(S, K) s ∈ S (M,s) |= ϕ such that |ϕn |≤dn, (iii) the smallest Kripke structure ,where and ,if for n all Kripke structure M based on F . that satisfies ϕ has at least 2 states.

IJCAI-07 2307 There is a well-known correspondence between properties That is, if ϕ is satisfiable at all, it is satisfiable in a situation of the K relation and axioms: reflexivity corresponds to T, with a number of states that is linear in |ϕ|. transitivity corresponds to 4, the Euclidean property corre- One more observation made by Nagle and Thomason will sponds to 5, and the serial property corresponds to D. This be important in the sequel. correspondence is made precise in the following well-known theorem (see, for example, [Fagin et al., 1995]). Definition 2.4 : A p-morphism (short for pseudo- epimorphism) from situation ((S, K),s) to situation Theorem 2.2: Let C be a (possibly empty) subset of {T, 4, 5, ((S, K),s) is a function f : S → S such that } C {r, t, e, s} D and let be the corresponding subset of .Then • f(s)=s; {Prop, K, MP, RN}∪Cis a sound and complete axiomatiza- K C 3 • (s ,s ) ∈K (f(s ),f(s )) ∈K tion of the language L1 (Φ) with respect to S (Φ). if 1 2 ,then 1 2 ; L • (f(s ),s ) ∈K s ∈ S Given a modal logic L,letS consist of all situations if 1 3 , then there exists some 2 such (s ,s ) ∈K f(s )=s (F, s) such that ϕ ∈ L implies that (F, s) |= ϕ. An imme- that 1 2 and 2 3. diate consequence of Theorem 2.2 is that Se, the situations where the K relation is Euclidean, is a subset of SK5. Nagle and Thomason 1985 provide a useful semantic char- This notion of p-morphism of situations is a variant of stan- acterization of all logics that contain K5. We review the rele- dard notions of p-morphism of frames and structures [Black- ] vant details here. Consider all the finite situations ((S, K),s) burn et al., 2001 . It is well known that if there is a p- such that either morphism from one structure to another, then the two structures satisfy the same formulas. An analogous result, which 1. S is the disjoint union of S1, S2,and{s} and K = is proved in the full paper, holds for situations. ({s}×S1) ∪ ((S1 ∪ S2) × (S1 ∪ S2)),whereS2 = ∅ if (F ,s) S1 = ∅;or Theorem 2.5: If there is a p-morphism from situation to (F, s), then for every modal logic L,if(F ,s) ∈SL then K = S × S L 2. . (F, s) ∈S . Using (a slight variant of) Nagle and Thomason’s nota- S m ≥ 1 n ≥ 0 (m, n)=(0, 0) Now consider a partial order on pairs of numbers, so that tion, let m,n, with and or , (m, n) ≤ (m,n) iff m ≤ m and n ≤ n. Nagle and denote all situations of the first type where |S1| = m and Thomason observed that if (F, s) ∈Sm,n, (F ,s) ∈Sm,n , |S2| = n,andletSm,−1 denote all situations of the second (1, −1) ≤ (m, n) ≤ (m,n) |S| = m and , then there is an obvi- type where . (The reason for taking -1 to be the ous p-morphism from (F ,s) to (F, s):ifF =(S, K), second subscript for situations of the second type will be- S = S1 ∪ S2, F =(S , K ), S = S1 ∪ S2 (where Si and come clearer below.) It is immediate that all situations in Si for i =1, 2 are as in the definition of Sm,n), then define Sm,n for fixed m and n are isomorphic—they differ only in f : S → S so that f(s )=s, f maps S1 onto S1, and, if the names assigned to states. Thus, the same formulas are S2 = ∅,thenf maps S2 onto S2;otherwise,f maps S2 to valid in any two situations in Sm,n. Moreover, it is easy to S1 arbitrarily. The following result (which motivates the sub- check that the K relation in each of the situations above in script −1 in Sm,−1) is immediate from this observation and Euclidean, so each of these situations is in SK5.Itiswell K5 Theorem 2.5. known that the situations in Sm,−1 are all in S and the sit- KD45 Sm,−1 ∪Sm,0 S S5 Theorem 2.6: If (F, s) ∈Sm,n, (F ,s) ∈Sm,n , and uations in are all in . In fact, (resp., KD45) is sound and complete with respect to the situations (1, −1) ≤ (m, n) ≤ (m ,n), then for every modal logic L (F ,s) ∈TL (F, s) ∈TL in Sm,−1 (resp., Sm,−1 ∪Sm,0). Nagle and Thomason show ,if then . L that much more is true. Let T =(∪{Sm,n : m ≥ 1,n ≥ −1 or (m, n)=(0, 0)}) ∩SL. 3 The Main Results Theorem 2.3: [Nagle and Thomason, 1985] For every logic We can now state our key technical result. L K5 L containing , is sound and complete with respect to Theorem 3.1: If L is a modal logic containing K5 and ¬ϕ/∈ T L the situations in . L, then there exist m, n such that m + n<|ϕ|, a situation L The key result of this paper shows that if a formula ϕ is (F, s) ∈S ∩Sm,n, and structure M based on F such that consistent with a logic L containing K5, then there exists (M,s) |= ϕ. m, n M =(S, K,π) s ∈ S ,aKripkestructure , and a state Proof: By Theorem 2.3, if ¬ϕ/∈ L, there is a situation ((S, K),s) ∈S S ⊆TL m + n<|ϕ| L such that m,n, m,n ,and . (F ,s0) ∈T such that (F ,s0) |= ¬ϕ. Thus, there exists a Kripke structure M based on F such that (M ,s0) |= ϕ. 3We remark that soundness and completeness is usually stated C Suppose that F ∈Sm,n .Ifm + n < |ϕ|, we are done, so with respect to the appropriate class M of structures, rather than C suppose that m + n ≥|ϕ|. Note that this means m ≥ 1. the class S of situations. However, the same proof applies without S C S C We now construct a a situation (F, s) ∈Sm.n such that change to show completeness with respect to , and using (1, −1) ≤ (m, n) ≤ (m,n) m + n<|ϕ| (M,s) |= ϕ allows us to relate this result to our later results. While for normal , ,and logics it suffices to consider only validity with respect to structures, for some Kripke structure based on F . This gives the desired for non-normal logics, we need to consider validity with respect to result. The construction of M is similar in spirit to Ladner’s situations. 1977 proof of the analogous result for the case of S5.

IJCAI-07 2308 L Let C1 be the set of subformulas of ϕ of the form Kψ such Say that (m, n) is a maximal index of T if m ≥ 1, L L that (M ,s0) |= ¬Kψ,andletC2 be the set of subformulas Sm,n ⊆T , and it is not the case that Sm,n ⊆T for some of ϕ of the form Kψ such that KKψ is a subformula of ϕ and (m,n) with (m, n) < (m,n). It is easy to see that T L can (M ,s0) |= ¬KKψ ∧ Kψ. (We remark that it is not hard to have at most ﬁnitely many maximal indices. Indeed, if (m, n) show that if K is either reﬂexive or transitive, then C2 = ∅.) is a maximal index, then there can be at most m + n − 1 max- Suppose that M =(S, K,π). For each formula imal indices, for if (m ,n) is another maximal index, then C1 m

The idea for showing that the consistency problem for a 4 Discussion and Related Work logic L that contains K5 is NP-complete is straightforward. We have shown all that, in a precise sense, adding the negative Given a formula ϕ that we want to show is consistent with L, introspection axiom pushes the complexity of a logic between we simply guess a frame F =(S, K), structure M based on K and S4 down from PSPACE-hard to NP-complete. This is F ,andstates ∈ S such that (F, s) ∈Sm,n with m+n<|ϕ|, not the only attempt to pin down the NP-PSPACE gap and to L and verify that (M,s) |= ϕ and Sm,n ⊆T . Verifying understand the effect of the negative introspection axiom. We that (M,s) |= ϕ is the model-checking problem.Itiswell discuss some of the related work here. known that this can be done in time polynomial in the num- A number of results showing that large classes of logics ber of states of M, which in this case is linear in |ϕ|.Soit have an NP-complete satisfiability problem have been proved remains to show that, given a logic L containing K5, check- L 4Here we have implicitly assumed that checking whether in- ing whether Sm,n ⊆T can be done efficiently. This follows equalities such as (m, n) ≤ (m ,n ) hold can be done in one time from observations made by Nagle and Thomason 1985 show- L step. If we assume instead that it requires time logarithmic in the T Sm,n L ing that that, although may include for infinitely inputs, then checking if Sm,n ⊆T requires time logarithmic in L ∗ ∗ many pairs (m ,n), T has a finite representation that makes m + n, since we can take all of m ,n ,m1,...mk,n1,...,nk to L it easy to check whether Sm.n ⊆T . be constants.

IJCAI-07 2309 recently. For example, Litak and Wolter 2005 show that All the results for this paper are for single-agent logics. the satisfiability for all finitely axiomatizatble tense logics Halpern and Moses 1992 showed that the consistency prob- of linear time is NP-complete, and Bezhanishvili and Hod- lem for a logic with two modal operators K1 and K2, each kinson 2004 show that every normal modal logic that prop- of which satisfies the S5 axioms, is PSPACE-complete. In- 2 2 erly extends S5 (where S5 is the modal logic that con- deed, it is easy to see that if Ki satisfies the axioms of Li for tains two modal operators K1 and K2, each of which satis- some normal modal logic Li containing K5, then the consis- fies the axioms and rules of inference of S5 as well as the tency problem for the logic that includes K1 and K2 must be axiom K1K2p ⇔ K2K1p) has a satisfiability problem that PSPACE-hard. This actually follows immediately from Lad- is NP-complete. Perhaps the most closely related result is ner’s 1977 result; then it is easy to see that K1K2,viewedas that of Hemaspaandra 1993, who showed that the consis- a single operator, satisfies the axioms of K. We conjecture tency problem for any normal logic containing S4.3 is also that this result continues to hold even for non-normal logics. NP-complete. S4.3 is the logic that results from adding the We have shown that somewhat similar results hold when following axiom, known in the literature as D1, to S4: we add awareness to the logic (in the spirit of Fagin and [ D1. K(Kϕ ⇒ ψ) ∨ K(Kψ ⇒ ϕ) Halpern 1988), but allow awareness of unawareness Halpern and Rêgo, 2006]. In the single-agent case, if the K operator D1 is characterized by the connectedness property: it is satisfies the axioms K, 5, and some (possibly empty) subset valid in a situation ((S, K),s) if for all s1,s2,s3 ∈ S,if of {T,4}, then the validity problem for the logic is decidable; (s1,s2) ∈Kand (s1,s3) ∈K, then either (s2,s3) ∈Kor on other hand, if K does not satisfy 5, then the validity prob- (s3,s2) ∈K. Note that connectedness is somewhat weaker lem for the logic is undecidable. With at least two agents, the than the Euclidean property; the latter would require that both validity problem is undecidable no matter which subset of ax- (s2,s3) and (s3,s2) be in K. ioms K satisfies. We conjecture that, more generally, if the K As it stands, our result is incomparable to Hemspaandra’s. operator satisfies the axioms of any logic L containing K5, To make the relationship clearer, we can restate her result as the logic of awareness of unawareness is decidable, while if saying that the consistency property for any normal logic that K satisfies the axioms of any logic between K and S4,the contains K and the axioms T, 4, and D1 is NP-complete. We logic is undecidable. do not require either 4 or T for our result. However, although All these results strongly suggest that there is something the Euclidean property does not imply either transitivity or about the Euclidean property (or, equivalently, the negative reflexivity, it does imply secondary reflexivity and secondary introspection axiom) that simplifies things. However, they do K transitivity.Thatis,if satisfies the Euclidean property, then not quite make precise exactly what that something is. More s ,s ,s ,s (s ,s ) ∈K (s ,s ) ∈K for all states 1 2 3 4,if 1 2 ,then 2 2 generally, it may be worth understanding more deeply what (s ,s ) (s ,s ) ∈K (s ,s ) ∈K and if 2 3 and 3 4 ,then 2 4 ; roughly is about properties of the K relation that makes things easy or s speaking, reflexivity and transitivity hold for all states 2 in hard. We leave this problem for future work. the range of K. Secondary reflexivity and secondary transitivity can be captured by the following two axioms: Acknowledgments T. K(Kϕ ⇒ ϕ) K(Kϕ ⇒ KKϕ) We thank the IJCAI reviewers for useful comments and for 4 . catching an error in the previous version of the paper. Both T and 4 follow from 5, and thus both are sound in any K5 logic that extends . Clearly T and 4 also both hold in any References logic that extends S4.3, since S4.3 contains T, 4, and the inference rule RN. We conjecture that the consistency property [Allen, 2005] M. Allen. Complexity results for logics of lo- for every logic that extends K and includes the axioms T, cal reasoning and inconsistent belief. In Theoretical As- 4,andD1isNP-complete. If this result were true, it would pects of Rationality and Knowledge: Proc. Tenth Confer- generalize both our result and Spaan’s result. ence (TARK 2005), pages 92–108. 2005. Vardi 1989 used a difference approach to understand the [Bezhanishvili and Hodkinson, 2004] N. Bezhanishvili and semantics, rather than relational semantics. This allowed I. M. Hodkinson. All normal extensions of S5-squared him to consider logics that do not satisfy the K axiom. He are finitely axiomatizable. Studia Logica, 78(3):443–457, showed that some of these logics have a consistency prob- 2004. lem that is NP-complete (for example, the minimal normal logic, which characterized by Prop, MP, and RN), while oth- [Blackburn et al., 2001] P. Blackburn, M. de Rijke, and ers are PSPACE-hard. In particular, he showed that adding Y. Ve n e m a . Modal Logic. Cambridge Tracts in Theo- the axiom Kϕ ∧ Kψ ⇒ K(ϕ ∧ ψ) (which is valid in K) retical Computer Science, No. 53. Cambridge University to Prop, MP, and RN gives a logic that is PSPACE-hard. He Press, Cambridge, U.K., 2001. then conjectured that this ability to “combine” information is [Chellas, 1980] B. F. Chellas. Modal Logic. Cambridge Uni- what leads to PSPACE-hardness. However, this conjecture versity Press, Cambridge, U.K., 1980. has been shown to be false. There are logics that lack this axiom and, nevertheless, the consistency problem for these [Fagin and Halpern, 1988] R. Fagin and J. Y. Halpern. Be- logics is PSPACE-complete (see [Allen, 2005] for a recent lief, awareness, and limited reasoning. Artificial Intelli- discussion and pointers to the relevant literature). gence, 34:39–76, 1988.

IJCAI-07 2310 [Fagin et al., 1995] R. Fagin, J. Y. Halpern, Y. Moses, and M. Y. Vardi. Reasoning About Knowledge. MIT Press, Cambridge, Mass., 1995. A slightly revised paperback version was published in 2003. [Halpern and Moses, 1992] J. Y. Halpern and Y. Moses. A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54:319–379, 1992. [Halpern and Rêgo, 2006] J. Y. Halpern and L. C. Rêgo. Reasoning about knowledge of unawareness. In Principles of Knowledge Representation and Reasoning: Proc. Tenth International Conference (KR ’06), pages 6–13. 2006. Full version available at arxiv.org/cs.LO/0603020. [Ladner, 1977] R. E. Ladner. The computational complexity of provability in systems of modal propositional logic. SIAM Journal on Computing, 6(3):467–480, 1977. [Litak and Wolter, 2005] T. Litak and F. Wolter. All finitely axiomatizable tense logics of linear time flows are coNP- complete. Studia Logica, 81(2):153–165, 2005. [Nagle and Thomason, 1985] M. C. Nagle and S. K. Thoma- son. The extensions of the modal logic K5. Journal of Symbolic Logic, 50(1):102–109, 1985. [Nagle, 1981] M. C. Nagle. The decidability of normal K5 logics. Journal of Symbolic Logic, 46(2):319–328, 1981. [Nguyen, 2005] L. A. Nguyen. On the complexity of frag- ments of modal logics. In R. Schmidt, I. Pratt-Hartman, M. Reynolds, and H. Wansing, editors, Advances in Modal Logic, Vol. 5, pages 249–268. 2005. [Spaan, 1993] E. Spaan. Complexity of modal logics.PhD thesis, University of Amsterdam, 1993. [Vardi, 1989] M. Y. Vardi. On the complexity of epistemic reasoning. In Proc. 4th IEEE Symposium on Logic in Com- puter Science, pages 243–252, 1989.

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