The Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19) Disjunctive Normal Form for Multi-Agent Modal Logics Based on Logical Separability Liangda Fang,1 Kewen Wang,2 Zhe Wang,2 Ximing Wen3 1Department of Computer Science, Jinan University, China 2School of Information and Communication Technology, Griffith University, Australia 3Guangdong Institute of Public Administration, Guangzhou, China [email protected], fk.wang,zhe.wangg@griffith.edu.au, [email protected] Abstract and the action preconditions. If a normal form is defined such that its entailment check, bounded conjunction and for- Modal logics are primary formalisms for multi-agent systems getting are tractable, then both progression and entailment but major reasoning tasks in such logics are intractable, which check in the normal form are tractable (Bienvenu, Fargier, impedes applications of multi-agent modal logics such as au- tomatic planning. One technique of tackling the intractabil- and Marquis 2010). As a result, in the normal form, an ef- ity is to identify a fragment called a normal form of multi- fective search algorithm is obtained. In the case of proposi- agent logics such that it is expressive but tractable for reason- tional logic, a normal form can be disjunctive normal form ing tasks such as entailment checking, bounded conjunction (DNF), conjunctive normal form (CNF) or prime implicate. transformation and forgetting. For instance, DNF of proposi- Usually, a planner based on DNF is faster than those based tional logic is tractable for these reasoning tasks. In this pa- on CNF and prime implicate (To, Pontelli, and Son 2011; per, we first introduce a notion of logical separability and then To, Son, and Pontelli 2011). This is due to the fact that, define a novel disjunctive normal form SDNF for the multi- in propositional logic, DNF possesses tractable entailment agent logic Kn, which overcomes some shortcomings of exist- check, bounded conjunction and forgetting, but CNF and ing approaches. In particular, we show that every modal for- prime implicate do not (Darwiche and Marquis 2002). mula in Kn can be equivalently casted as a formula in SDNF, major reasoning tasks tractable in propositional DNF are also Thus, in order to develop effective search algorithms to tractable in SDNF, and moreover, formulas in SDNF enjoy multi-agent epistemic planning, researchers aimed to de- the property of logical separability. To demonstrate the use- velop suitable DNF-like normal forms for multi-agent modal fulness of our approach, we apply SDNF in multi-agent epis- logics such that major reasoning tasks, such as bounded temic planning. Finally, we extend these results to three more conjunction, forgetting and entailment check, are tractable. complex multi-agent logics Dn, K45n and KD45n. Such a disjunctive normal form, named S5-DNF, is defined for modal logic S5 but only for the single-agent case (Bi- 1 Introduction envenu, Fargier, and Marquis 2010). When this DNF is ex- tended to multi-agent modal logics, both entailment and for- It is crucial for an intelligent agent system to be capable getting will not be tractable any longer. Two other normal of representing and reasoning about high-order knowledge forms, cover disjunctive normal forms (CDNFs) (ten Cate et in the multi-agent setting. A general representative frame- al. 2006) and prime implicate normal forms (PINFs) (Bi- work for these scenarios is multi-agent modal logics. How- envenu 2008), have been introduced for description logic ever, some important reasoning tasks, including satisfiabil- ALC, a syntactic variant of the multi-agent modal logic ity checking and forgetting, are intractable in such logics Kn. However, these two normal forms have some shortcom- (Halpern and Moses 1992; Bienvenu 2009). The intractabil- ings. CDNF is relatively less compact, that is, the CDNF ity results impede applications of multi-agent modal logics, representation is exponential large for some simple formu- e.g., multi-agent epistemic planning (Kominis and Geffner las. Bounded conjunction for PINF is intractable and in 2015; Huang et al. 2017). the worst case a compiled formula in PINF is double ex- Traditionally, the forward search algorithm is effective for ponentially large. In addition, Moss (2007) introduced a agent planning (Bryce, Kambhampati, and Smith 2006), in canonical form of modal formulas for Kn and Dn. While which the search is performed in the space of knowledge modal formulas can be equivalently transformed into Moss’ bases (KBs) proceeding forward from the initial KB towards canonical form, the complexity of the transformation is non- a goal KB entailing the goal formula. Two types of reasoning elementary. Hence, Moss’ canonical formulas are not a prac- tasks are essential to the search algorithm, namely, progres- tical normal form for epistemic planning. sion and entailment check. The progression updates KBs ac- logic sepa- cording to the action effects while the entailment check is In this paper, we first formulate the notion of rability needed to decide if the current KB entails the goal formula in multi-agent modal logic, which was originally in- vestigated for first-order logic (Levesque 1998). Informally, Copyright c 2019, Association for the Advancement of Artificial a conjunction φ of formulas is logically separable w.r.t. a Intelligence (www.aaai.org). All rights reserved. form of reasoning if the reasoning for φ can be reduced to 2817 the same reasoning for the conjuncts of φ. For example, the P (φ) for the set of variables appearing in φ. A formula φ is formula φ = (p ! q) ^ (q ! r) is not logically separable smaller than , if jφj < j j. since it logically implies a conjunct p ! r that is not de- The notions of propositional literals, terms (TE), clauses rived by any single conjunct of φ. By conjoining φ with the (CL), disjunctive normal forms (DNF) and conjunctive nor- implicit conjunct, the new formula becomes logically sepa- mal forms (CNF) are defined as usual. rable. A modal literal is a formula of the form iφ or ♦iφ, where Based on the logical separability, we introduce a disjunc- φ is a modal formula. A modal literal is positive (resp. neg- tive normal form, called separability-based DNF (or SDNF) ative), if it is of the form iφ (resp. ♦iφ). A formula is ba- for multi-agent modal logics Kn, Dn, K45n and KD45n. sic, if it is a propositional formula or modal literal. A modal Similarly, separability-based CNF (or SCNF) can also be term (resp. clause) is a conjunction (resp. disjunction) of defined. For these two normal forms and the two existing basic formulas. For a modal term (resp. clause) φ, we use normal forms for multi-agent modal logics, we investigate Prop(φ) for the set of propositional components (i.e., maxi- the expressiveness, succinctness, queries and transforma- mal propositional subformulas) that are conjuncts (resp. dis- tions (Darwiche and Marquis 2002). Thus we obtain an al- juncts) of φ, Bi(φ) for the set of formulas such that i most complete knowledge compilation map for multi-agent is a conjunct (resp. disjunct) of φ, and Di(φ) for the set of modal logics. formulas such that ♦i is a conjunct (resp. disjunct) of The main contributions of this paper contain: φ. For example, consider the modal term φ = p ^ :q ^ 1. We formulate the concept of logical separability for i:p^iq ^♦i(:p_q)^♦ip. Thus, Prop(φ) = fp ^ :qg, modal terms and formalise some desirable properties for Bi(φ) = f:p; qg, and Di(φ) = f:p _ q; pg. them. Thanks to the notion of logical separability, we are Definition 2.1. A Kripke model M is a tuple hS;R;Vi where able to define separability-based DNF and CNF, two novel normal forms for Kn. • S is a non-empty set of possible worlds; 2. For the two new normal forms as well as CDNF • R = fRi j i 2 Ag where Ri is a binary relation on S; and PINF, we investigate the expressiveness, succinctness, • V is a function assigning to each s 2 S a subset of P . queries and transformations. To the best of our knowledge, we are the first to construct this map for multi-agent modal A pointed Kripke model is a pair (M; s), where M is a Kripke model and s is a world of M, called the actual world. logics. Interestingly, SDNFL0 possesses all properties that propositional DNF has, e.g., polytime satisfiability, bounded For convenience, we assume that Kripke models are conjunction, forgetting and so on. In this sense, SDNF is (M; s) L pointed. Given a Kripke model and an -formula a proper generalisation of propositional DNF for the multi- φ, the satisfaction relation M; s j= φ is defined as usual and agent modal logic Kn. in this case, we say that (M; s) is a model of φ. A modal 3. As a case study, we illustrate the application of our re- formula φ is satisfiable if it has a model; φ entails , written sults in multi-agent epistemic planning. φ j= , if every model of φ is also a model of ; φ and 4. We extend the results of knowledge compilation for are equivalent, written φ ≡ , if φ j= and j= φ. 0 Kn to Dn without any modification on normal forms, and Throughout this paper, we use L and L for fragments of 0 to K45n and KD45n under the condition that no consecutive L L L , and 0 and 0 for propositional fragments. We also modalities of the same agent appear in the given formula. assume that every propositional term and clause has a poly- 0 nomial representation in L0 and L0. All propositional frag- 2 The multi-agent modal logic Kn ments considered in (Darwiche and Marquis 2002) conform with this assumption except for full DNF, that is, if each of In this section, we first recall the syntax and semantics of its variables appears exactly once in every term.