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On the Complexity of Dynamic Epistemic Logic ∗ On the Complexity of Dynamic Epistemic Logic ∗ Guillaume Aucher François Schwarzentruber University of Rennes 1 - INRIA ENS Cachan - Brittany extension [email protected] [email protected] cachan.fr ABSTRACT knowledge change of multiple agents, and more generally Although Dynamic Epistemic Logic (DEL) is an influential about information change. logical framework for representing and reasoning about in- The theoretical work of the above mentioned research formation change, little is known about the computational fields has already been applied to various practical problems complexity of its associated decision problems. In fact, we stemming from telecommunication networks, World Wide only know that for public announcement logic, a fragment Web, peer to peer networks, aircraft control systems, and so of DEL, the satisfiability problem and the model-checking on. In general, in all applied contexts, the investigation of problem are respectively PSPACE-complete and in P. We the algorithmic aspects of the formalisms employed plays an contribute to fill this gap by proving that for the DEL lan- important role in determining whether and to what extent guage with event models, the model-checking problem is, they can be applied. For this reason, the algorithmic aspects surprisingly, PSPACE-complete. Also, we prove that the of DEL need to be studied. satisfiability problem is NEXPTIME-complete. In doing so, To this aim, a preliminary step consists in studying the we provide a sound and complete tableau method deciding computational properties of its main associated decision prob- the satisfiability problem. lems, namely the model checking problem and the satisfia- bility problem. Indeed, it will indirectly provide algorithmic methods to solve these decision problems and give us a hint Categories and Subject Descriptors of whether and to what extent our methods can be applied. I.2.4 [Knowledge representation formalisms and meth- However, surprisingly little is known about the computa- ods]: Modal logic; F.1.3 [Complexity measure and classes]: tional complexity of these problems. We only know that Reducibility and completeness for public announcement logic, a fragment of DEL [Plaza, 1989], the model checking problem is in P and the satisfi- General Terms ability problem is PSPACE-complete. Here, we aim to fill this gap for the full language of DEL with event models. Theory DEL is built on top of epistemic logic. An epistemic model represents how a given set of agents perceive the actual world Keywords in terms of beliefs and knowledge about this world and about Dynamic epistemic logic, computational complexity, model the other agents’ beliefs. The insight of the DEL approach checking, satisfiability is that one can describe how an event is perceived by agents in a very similar way: an agent’s perception of an event can also be described in terms of beliefs and knowledge. For 1. INTRODUCTION example, at the battle of Waterloo, when marshal Blucher¨ Research fields like distributed artificial intelligence, dis- received the message of the duke of Wellington inviting him tributed computing and game theory all deal with groups of to join the attack at dawn against Napoleon, Wellington did human or non-human agents which interact, exchange and not know at that very moment that Blucher¨ was receiving receive information. The problems they address range from his message, and Blucher¨ knew it. This is a typical example multi-agent planning and design of distributed protocols to of announcement which is not public. This led Baltag, Moss strategic decision making in groups. In order to address ap- and Solecki to introduce the notion of event model [Baltag propriately and rigorously these problems, it is necessary to et al., 1998]. The definition of an event model, denoted be able to provide formal means for representing and reason- (M,w), is very similar to the definition of an epistemic ing about such interactions and flows of information. The model. They also introduced a product update, which defines framework of Dynamic Epistemic Logic (DEL for short) is a new epistemic model representing the situation after the very well suited to this aim. Indeed, it is a logical frame- event. Then, they extended the epistemic language with work where one can represent and reason about beliefs and dynamic operators [M,w]ϕ standing for ‘ϕ holds after the M ∗An extended version of this article with full proofs can be occurrence of the event represented by ( ,w )’. found at the following url: http://hal.inria.fr/docs/00/ Using the so-called reduction axioms, it turns out that 75/95/44/PDF/RR-8164.pdf any formula with dynamic operator(s) can be translated to an equivalent epistemic formula without dynamic oper- TARK 2013, Chennai, India. ator. As a first approximation, we could be tempted to Copyright 2013 by the authors. 19 use these reduction axioms to reduce both the model check- 1,2 2 ing problem and the satisfiability problem of DEL to the Õ ×Ö × ÕÔ ×Ö ÕÔ model checking problem and the satisfiability problem of w : p ÐÑ(w, w1):pÒÓ ÐÑ(w, w1,w2):pÒÓ epistemic logic, because optimal algorithmic methods al- ready exist for these related problems. However, the re- 2 1 2 duction algorithm induced by the reduction axioms is expo- z z ¬ nential in the size of the input formula. Therefore, for the u : Yp p 1,2 p 2 satisfiability problem, we only obtain an algorithm which is 2 1 in EXPSPACE (because the satisfiability problem of epis- 1 z temic logic is PSPACE-complete), and for the model check- ¬pY p 1,2 ing problem, we only obtain an algorithm which is in EX- PTIME (because the model checking problem of epistemic 2 1 logic is in P). These algorithms are not optimal because, as ¬p we shall see, there exists an algorithm solving the satisfiabil- Y ity problem which is in NEXPTIME⊆ EXPSPACE and also an algorithm solving the model checking problem which is in 1 PSPACE⊆ EXPTIME. Our algorithm for solving the satis- M fiability problem is based on a sound and complete tableau Figure 1: Pointed epistemic models ( ,w) (left), M⊗M M⊗M ⊗ method which does not resort to the reduction axioms. (( 1), (w, w1)) (center)and( 1 M The paper is organized as follows. In Section 2, we re- 2, (w, w1,w2)) (right) call the core of the DEL framework and the different vari- ants of languages with event models which have been in- troduced in the literature. In Section 3, we prove that is an abbreviation for p ∧¬p,and is an abbreviation for the model checking problem of DEL is PSPACE-complete, ¬⊥. The formula Ba ϕ is an abbreviation of ¬Ba¬ϕ.The and in Section 4 we prove that the satisfiability problem size of a formula ϕ ∈LEL is defined by induction as follows: is NEXPTIME-complete. In Section 5, we discuss related |p| =1;|¬ϕ| =1+|ϕ|; |ϕ∧ψ| =1+|ϕ|+|ψ|; |Baϕ| =1+|ϕ|. works and whether our results still hold when we extend the expressiveness of the language with common belief and ‘star’ Intuitively, the formula Baϕ reads as ‘agent a believes that iteration operators. We conclude in Section 6. ϕ holds in the current situation’. Definition 3 (Truth conditions). 2. DYNAMIC EPISTEMIC LOGIC Given an epistemic model M =(W, R, V )andaformula Following the methodology of DEL, we split the exposi- ϕ ∈LEL, we define inductively the satisfaction relation |=⊆ tion of the DEL logical framework into three subsections. In W ×LEL as follows: for all w ∈ W , Section 2.1, we recall the syntax and semantics of the epis- M,w |= p iff w ∈ V (p) temic language. In Section 2.2, we define event models, and M,w |= ϕ ∧ ψ iff M,w |= ϕ and M,w |= ψ in Section 2.3, we define the product update. In Section 2.4, M,w |= ¬ϕ iff not M,w |= ϕ we recall the different languages that have been introduced M,w |= Baϕ iff for all v ∈ Ra(w), we have M,v |= ϕ in the DEL literature and we introduce our language LDEL. We write M|= ϕ when for all w ∈M, it holds that M,w |= 2.1 Epistemic language ϕ. Also, we write |= ϕ, and we say that ϕ is valid,whenfor In the rest of the paper, ATM is a countable set of atomic all epistemic model M,itholdsthatM|= ϕ. Dually, we propositions and AGT is a finite set of agents. say that ϕ is satisfiable when ¬ϕ is not valid. M A (pointed) epistemic model ( ,w) represents how the Example 1. Our running example is inspired by the co- actual world represented by w is perceived by the agents. ordinated attack problem from the distributed systems folk- Intuitively, in this definition, vRau means that in world v lore [Fagin et al., 1995]. Our set of atomic propositions is agent a considers that world u might be the actual world. ATM = {p} andoursetofagentsisAGT= {1, 2}. Agent Definition 1 (Epistemic model). 1 is the duke of Wellington and agent 2 is marshal Blucher;¨ An epistemic model is a tuple M =(W, R, V ) where W p stands for ‘Wellington wants to attack at dawn’. The ini- is a non-empty set of possible worlds, R maps each agent tial situation is represented in Figure 1 by the pointed epis- W temic model (M,w)=({w, u},R1 = {(w, w), (u, u)},R2 = a ∈ AGT to a relation Ra ⊆ W × W and V : ATM → 2 { } { } is a function called a valuation. We abusively write w ∈M (w, w), (w, u) ,V(p)= w ).Inthispointedepistemic M | ∧ for w ∈ W and we say that (M,w)isapointed epistemic model, it holds that ,w = p B1p: Wellington ‘knows’ that he wants to attack at dawn.
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