AGM Revision in Classical Modal Logics

AGM-style revision operators are defined for several systems of classical modal logic. · · · draft of November 28, 2009 · · · Within AGM, consistency maintenance is done by classical logic. But the reliance on classical consistency presents a problem for exporting AGM revision to non-classical logics in general, and to modal logic in particular. A general technique for solving this problem is to translate a non-classical logic into classical logic, together with a specification of consistency particular to the non-classical logic, perform the operation of revision on this translation within classical logic, then translate the result back into the non-classical logic we started with (Gabbay, Rodrigues, and Russo 2008). In the case of normal modal logic, both the modal language and the semantic structure must be translated into first-order logic, and this translation for well-known normal systems will rely upon well-known frame-theoretic properties, expressed in first-order logic, of modally defined classes of Kripke frames (Goldblatt 1993). But there is a problem extending this technique to classical modal logics (Chellas 1980), because there is no (direct) correspondence between neighborhood frames and firstorder logic. This paper proposes to solve this problem by adapting Marc Pauly’s (Hansen 2003) technique of first simulating neighborhood structures by polymodal Kripke structures, then define a correspondence to first-order logic from the polymodal Kripke semantics wherein AGM revision can be defined. In modal logic the technique of simulation was first used to construct counter-examples within polymodal modal logic to export back to monomodal systems of interest (Thomason 1974; 1975). More recently the technique has been used to study the relationship between neighborhood semantics and Kripke’s relational semantics, with a particular focus on supplemented neighborhood models (Gasquet and Herzig 1996; Kracht and Wolter 1999; Hansen 2003). Supplemented neighborhood models underpin a variety of non-additive, monotone modal logics appearing Copyright c © 2009, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. in knowledge representation formalisms, including Game Logic (Parikh 1985), Concurrent Propositional Dynamic Logic (Goldblatt 1992), Alternating-time logic (Alur, Henzinger, and Kupferman 1992), Risky Knowledge (Kyburg and Teng 2001), and Coalition Logic (Pauly 2002). Nonmonotone classical modal logics have also been pressed into service, including Local Reasoning (Fagin and Halpern 1988), and the logic of Only Knowing (Humberstone 1987; Levesque 1990; Halpern and Lakemeyer 1995), which are based on the Inaccessible Worlds semantics of (Humberstone 1983). In addition to discovering properties of a logic by studying its simulation within a well-understood system, the techniques of simulation theory together with correspondence theory may be used to bring new capabilities to a classical modal logic. AGM belief revision is but one example, which is the subject of this paper. There is recent interest in supplying revision to specifically tailored classical modal logics, specifically dynamic epistemic logic (van Benthem 2007; van Eijck 2009), and to polymodal normal logics, like branching time temporal logic (Bonanno 2009). These papers have focused primarily on particular issues that arise within each logic—principally updating and common knowledge, in the case of the former, and the interaction of temporal and epistemic operators on the standard interpretion in the latter. But as we observed above, there are many interpretations of classical modal logics within knowledge representation. So, the goal of this paper is to provide a general strategy for supplying AGM revision to classical modal logic that may be adapted or extended to suit particular purposes. Classical Modal Logic To begin, we highlight the difference between neighborhood structures and standard Kripke structures. Whereas Kripke frames are characterized by a binary accessibility relation defined over a set of worlds, a neighborhood frame for the propositional modal language L∇(Φ) is a pair F = (W,N ) where a) W is a non-empty set of worlds, b) N : W 7→ ℘(℘(W )) is a neighborhood function, i.e. N (w) ⊆ ℘W , for each w ∈W . If F = (W,N ) is a neighborhood frame, Φ a countable set of propositional variables, and V : Φ 7→ ℘(W ) is a valuation on F, then M = (W,N , V ) is a neighborhood model based on F. The satisfiability conditions for non-modal propositional formulas on neighborhood models are analogous to Kripke models, but modal necessity (∇φ) and possibility ( ∇ φ) statements on neighborhood models are different. Like the normal modal logic (K) and its extensions, classical modal logics are based on the classical system (E) and the meaning of necessity statements in different classical systems is determined by the properties of neighborhood frames just as the meaning of necessity statements in different normal systems is determined by the properties of a Kripke frame. That said, there are four important classes of neighborhood models (minimal, supplemented, quasi-filters, augmented) that determine four modal systems (classical, monotone, regular, normal). The differences between these models can be reflected by the truth conditions for (∇φ). Let M = (W,N , V ) be a neighborhood model, w be a world in W , X a set of worlds, and p ∈ Φ. Then: