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 au- thor 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 Tecnol´ogico (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 .