Complexity of Modal Logics
Edith Spaan
Abstract:
Complexity of Modal Logics
Edith Spaan
The actual "use" of modal logics in fields like distributed systems, computational linguistics, and program verification has given rise to new questions in the field of modal logic itself. For instance, though a logician might be satisfied by knowing that a logic is decidable, a typical "user" might want more precise information, e.g., how decidable that logic is, or, in other words, what the complexity of the provability problem for the logic is. The literature contains many results about the complexity of modal logics. However, all these results are on the complexity of specific, "fixed" logics, and, for each new logic, the complexity has to be discovered and proven anew. In this thesis, we develop a theory of the sources of complexity in modal logics--by identifying specific features that, when possessed by a logic, ensure or preclude a certain level of complexity for the logic's provability problem.
In "traditional" modal logic, there are a number of results showing decidability, completeness, and the finite model property for large classes of logics (see for instance [Fin85, Zak92]). Perhaps the easiest example of this phenomenon is Bull's theorem: all the (uncountably many) S4:3 extensions have the finite model property (and are decidable). In the next chapter we prove the following complexity analog of Bull's theorem: the satisfiability problems for all S4:3 extensions are NP-complete. This theorem is proven by a straightforward adaptation of Fine's proof of Bull's theorem [Fin72]. Its importance lies in the fact that it shows that general theorems on the complexity of modal logics exist. Looking at constructions used to prove decidability or the finite model property often leads to non-trivial upper bounds on the complexity of the satisfiability problems. These upper bound results, however, are restricted to uni-modal logics, though "useful" modal logics are usually multi-modal. Unfortunately, this situation is much more complex than the uni-modal case, and only recently has some progress been made towards proving general theorems for the multi-modal case. These theorems--called transfer theorems in [FS]--are of the following form. Suppose we have a collection of logics and we construct a new logic from these logics; we say that a property transfers if the resulting logic has this property whenever all logics in the collection have this property.
The simplest instance of this phenomenon is the following. Given two uni-modal logics with different modal operators, consider the smallest (bi-modal) logic that contains both uni-modal logics. Such a construction is called the (independent) join, also known as bimodal sum or fusion. Joins inherit many properties from their uni-modal fragments: decidability, the finite model property, and (strong) completeness all transfer [FS, KW91]. However, it is in general not the case that upper bounds transfer. A counterexample (under the assumption that NP 6= PSPACE) is given by the join of two S5 logics, since S5 satisfiability is NP-complete [Lad77], though S5 \Phi S5-satisfiability is PSPACE-complete [HM85]. On the other hand, the join does not always increase the complexity, as the satisfiability problems for K and K \Phi K are both PSPACE-complete ([Lad77, HM85]). In chapter 3, we show that under reasonable conditions upper bounds containing PSPACE transfer. In addition, we completely characterize what happens with the join of two sub-PSPACE logics.
Multiprocessor systems are often modeled in the literature by the independent join. However, if we want to make global statements about the system, we need more expressive power. One of the simplest ways to do this is by enriching the language with the universal modality [u], with the semantics: [u]OE is true iff OE is true in every world of the model. Another auxiliary modality that occurs in various guises in the literature is the reflexive transitive closure. This modality occurs, for instance, in temporal logic, as the "always" operator is the reflexive transitive closure of the "nexttime" operator, and in logics of knowledge, where "common knowledge" is defined as the reflexive transitive closure of the S5 logics that model the processors.
Chapter 4 is devoted to the complexity of these two extensions. We show that, in contrast to the join, decidability does not transfer to the enriched versions, even if we add a number of extra restrictions. In fact, the complexity of the satisfiability problem can jump from NP-complete to highly undecidable. Fortunately, this is not always the case: Halpern and Vardi's multiprocessor system with common knowledge [HV89] is (only) EXPTIME-complete. In chapter 4, we show that it is impossible to do better than that: except in some very trivial cases, adding the universal modality or the reflexive transitive closure always forces EXPTIME-hardness.
It is clear that much work remains. The last two chapters give some idea of the tremendous task that lies ahead: we will look at the complexity of various complex logics that occur in computational linguistics (chapter 5) and distributed systems (chapter 6).
Computational Linguistics
Attribute Value Structures are probably the most widely used means of representing linguistic structure in computational linguistics, and the process of unifying Attribute Value descriptions lies at the heart of many parsers. As a number of researchers have recently observed, the most common formalisms for describing AVSs are variants of languages of propositional modal logic. Furthermore, testing whether two Attribute Value descriptions unify amounts to testing for modal satisfiability. In chapter 5, which is based on joint work with Patrick Blackburn [BS], we put this observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express re-entrancy, the ability to express generalizations, and the ability to express recursive constraints.
Distributed Systems
Recent work has shown that modal logics are extremely useful in formalizing the design and analysis of distributed protocols. Halpern and Vardi [HV89] unify current formalisms for reasoning about knowledge and time, and prove the complexity for all cases corresponding to different choices of knowledge operators and different assumptions made about the distributed system. In the cases of most interest to distributed systems, the validity problems for the logics modeling these systems are highly undecidable---in fact, \Pi 11-complete. Since this is a situation we want to avoid, it is essential to determine what causes this complexity. The \Pi 11-hardness proofs of Halpern and Vardi [HV89] rely heavily upon the presence of various temporal operators, and in that paper it is conjectured that reducing the temporal expressive power decreases the complexity of the corresponding validity problem. However, in chapter 6, based on [Spa90], we show that this is not the case--even restricting the temporal operators to just the "always" operator is not enough to defy \Pi 11-completeness.
These results provide evidence that undecidability is caused not by having a fancy set of operators, but rather by the ability to express generalizations in combination with certain assumptions of the models. For example, in the logics for distributed systems mentioned above, undecidability is caused by assuming that processors never forget.