Clausal Tableaux for Multimodal Logics of Belief - Semantic Scholar
Clausal Tableaux for Multimodal Logics of Belief Rajeev Gor´e1 and Linh Anh Nguyen2 1
The Australian National University and NICTA? ? ? Canberra ACT 0200, Australia [email protected] 2 Institute of Informatics, University of Warsaw ul. Banacha 2, 02-097 Warsaw, Poland [email protected]
Abstract. We develop clausal tableau calculi for seven multimodal logics variously designed for reasoning about multi-degree belief, reasoning about distributed systems of belief and for reasoning about epistemic states of agents in multi-agent systems. Our tableau calculi are sound, complete, cut-free and have the analytic superformula property, thereby giving decision procedures for all of these logics. We also use our calculi to obtain complexity results for five of these logics. The complexity of one was known and that of the seventh remains open. Keywords: modal logics for agent-based systems, theorem proving for modal logics, complexity and decidability of modal and temporal logics.
1
Introduction
Modal logics have been widely studied for reasoning about knowledge and belief, usually based on the monomodal logics S 5 and KD45 for reasoning about knowledge and belief respectively. Both logics have the axioms 4 : 2ϕ → 22ϕ and 5 : ¬2ϕ → 2¬2ϕ, which mean that knowledge and belief both satisfy positive and negative introspection. The logic S 5 also has the axiom T : 2ϕ → ϕ, which means that knowledge is veridical, while KD45 also has the axiom D : 2ϕ → ¬2¬ϕ, which means that belief is consistent. For multiple agents, the usual solution is to use a multimodal logic which has m pairs of modal operators 2i and 3i , for some fixed number m. The base multimodal logics for reasoning about knowledge and belief are typically S5(m) and KD45(m) , respectively, where each pair 2i and 3i is of type S5 and KD45, usually with no interactions between modal operators of different indices (see, e.g., [14, 8, 22, 35]). An alternative is to use S 5 and KD45 type modalities labelled with agent terms [10], again without interaction axioms. When the knowledge and beliefs of multiple agents interact then interactions between the modalities are used. One approach is to use bi-modal logics with interactions [33, 27]. Another is to extend the multimodal grammar logics of Fari˜ nas del Cerro and Penttonent [5] with (seriality) axioms D for reasoning about knowledge and belief. Recall that a grammar logic is a multimodal logic with axioms of the form [s1 ] . . . [sn ]ϕ → [t1 ] . . . [tm ]ϕ, where [si ] and [tj ] are universal (box-like) modal operators labelled by constants si and tj , respectively (see [2, 3, 6, 34, 16, 7, 12]). ???
National ICT Australia (NICTA) is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council.
For knowledge distributed over groups of agents there is the well-known multimodal logic of common knowledge and distributed knowledge among a group of agents (see [14, 8, 22]) as well as a similar logic for reasoning about mutual beliefs of agents [1]. There are also multi-modal logics of dynamic belief and knowledge for use in multi-agent systems [20, 32]. We concentrate on seven specific multimodal logics of belief, studied by Nguyen in the context of modal logic programming and modal deductive databases [28, 30]. They are introduced in Section 2.2 as: KDI4, KDI4s , KDI4s 5, KDI45 for reasoning about multi-degree belief; KD4s 5s for reasoning about distributed systems of belief; and KD45(m) , KD4Ig 5a for reasoning about epistemic states of agents in multi-agent systems. All except KD45(m) contain interaction axioms. Both KDI4 and KDI4s are grammar logics extended with axiom D. Tableau methods have been successively applied to modal logics to give decision procedures (see, e.g., [9, 13, 3, 35, 11, 19, 18]). There are also tableau calculi for a number of multimodal logics of knowledge and/or belief, e.g., [3, 35, 10]. But [3] concerns only grammar logics, while [35, 10] lack interaction axioms between modal operators of different indices. As far as we know, only KDI4 and KD45(m) from the seven multimodal logics (listed above) have labelled tableau calculi [3, 35]. We know of no non-labelled tableau calculi for any of these logics although Shvarts’ calculus for KD45 from [11] can be extended trivially to KD45(m) . Nor do we know of any tableau calculus for the multimodal logic of mutual belief introduced by Aldewereld et al. [1], which is similar to our KD4s 5s . Here, we develop (non-labelled) tableau calculi for all seven of the above multimodal logics of belief. Our tableau calculi are sound, complete, cut-free and have the analytic superformula property, so they are decision procedures. Our tableau calculi require clausal form and are called clausal tableau calculi. Transformation to clauses was first used for sequent calculi by Mints [23] and Hudelmaier [17] and then applied to tableau calculi by Nguyen [24]. Hudelmaier and Nguyen showed that clausal form can give better space bounds and improved decision procedures for some modal logics [17, 24]. We show that clauses can also simplify the task of developing tableau calculi for certain modal logics by giving clausal tableau calculi for the complicated logics KDI45 and KD4Ig 5a . Finally, we show that our tableaux can be used to estimate the complexity of the satisfiability problem in five of the multimodal logics we consider: NP-complete for KDI4s 5, KD4s 5s , KDI45, and PSPACE-complete for KDI4 and KDI4s . The complexity for KD45(m) is known to be PSPACE-complete [14]. We do not have an exact complexity result for KD4Ig 5a . The rest of this paper is organized as follows. In Section 2, we specify the seven multimodal logics of belief and give definitions for tableau calculi. In Section 3, we define modal clauses and consider transformation to clauses. In Section 4, we present our tableau calculi and prove their soundness. In Section 5, we prove completeness of the calculi. Section 6 deals with the complexity of the logics. Section 7 contains concluding remarks. 2
2 2.1
Preliminaries Definitions for Multimodal Logics
We consider multimodal logics with m pairs of modal operators 2i and 3i , where 1 ≤ i ≤ m. We use p and q to denote primitive propositions. Formulae of our language are recursively defined using the BNF grammar below: ϕ ::= ⊥ | p | ¬ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ | 2i ϕ | 3i ϕ The modal depth of a formula ϕ is the maximal nesting depth of modal operators occurring in ϕ. For example, the modal depth of 21 (22 p ∨ 31 q) is 2. A Kripke frame is a tuple hW, τ, R1 , . . . , Rm i, where W is a nonempty set of possible worlds, τ ∈ W is the actual world, and each Ri is a binary relation on W , called the accessibility relation for 2i and 3i . If Ri (w, u) holds, then we say that the world u is accessible from the world w via Ri . A Kripke model is a tuple hW, τ, R1 , . . . , Rm , hi, where hW, τ, R1 , . . . , Rm i is a Kripke frame and h is a function mapping worlds to sets of primitive propositions. For w ∈ W , the set of primitive propositions “true” at w is h(w). Given a Kripke model M = hW, τ, R1 , . . . , Rm , hi and a world w ∈ W , the satisfaction relation is defined as usual for the classical connectives with two extra clauses for the modalities as below: M, w 2i ϕ M, w 3i ϕ
iff iff
∀v ∈ W.Ri (w, v) implies M, v ϕ ∃v ∈ W.Ri (w, v) and M, v ϕ.
We say that ϕ is satisfied at w in M if M, w ϕ. We say that ϕ is satisfied in M , and write M ϕ, and call M a model of ϕ, if M, τ ϕ. If we allow all Kripke models (with no restrictions on the accessibility relations) then we obtain a multimodal logic which has a standard Hilbert-style axiomatisation denoted by K(m) . Other normal multimodal logics are obtained by adding certain axioms to K(m) . Suppose a normal first-order definable modal logic L is characterised by a class C of Kripke frames. The first order formulae describing C are L-frame restrictions, and any frame in C is an L-frame. A model M is an L-model if its underlying frame is an L-frame. We say that ϕ is L-satisfiable if there exists an L-model of ϕ, i.e. an L-model satisfying ϕ. A formula ϕ is said to be L-valid and called an L-tautology if ϕ is true in every L-model. 2.2
Multimodal Logics of Belief
To reflect properties of belief, one can extend K(m) with some of the axioms: Name (D) (I) (4) (4s ) (5) (5s )
Schema 2i ϕ → ¬2i ¬ϕ 2i ϕ → 2j ϕ if i > j 2 i ϕ → 2i 2i ϕ 2 i ϕ → 2j 2i ϕ ¬2i ϕ → 2i ¬2i ϕ ¬2i ϕ → 2j ¬2i ϕ
Meaning belief is consistent subscript indicates degree of belief belief satisfies positive introspection belief satisfies strong positive introspection belief satisfies negative introspection belief satisfies strong negative introspection 3
The following systems are intended for reasoning about multi-degree belief: KDI4s = K(m) + (D) + (I) + (4s ) KDI4 = K(m) + (D) + (I) + (4) KDI4s 5 = K(m) + (D) + (I) + (4s ) + (5) KDI45 = K(m) + (D) + (I) + (4) + (5) In the above systems, the axiom (I) gives 2i ϕ the meaning “ϕ is believed up to degree i”, and 3i ϕ can be read as “it is possible weakly at degree i that ϕ”. The axioms (5) are controversial as they are quite strong. For this reason, we consider also KDI4 and KDI4s . Note that the axiom (5s ) is derivable in KDI4s 5. Although the logic KDI4 belongs to the class of logics considered by Debart et al. [4] (for multimodal logic programming) and the axioms I and 4s are possible axioms for grammar logics, the reading of modal indices as degrees of belief has not been considered by other authors. Our logics of multi-degree belief are similar to graded modal logics [22], where grades constrain the number of worlds accessible from the current world, but our degrees of belief are symbolic rather than numeric. For multi-agent systems, we use subscripts on 2 and 3 to denote agents and assume that 2i ϕ stands for “agent i believes that ϕ is true” and 3i ϕ stands for “ϕ is considered possible by agent i”. For distributed systems of belief we can use the logic system KD4s 5s = K(m) + (D) + (4s ) + (5s ) In this system, all agents have full access to the belief bases of other agents: they are united as “friends”. Our logic KD4s 5s has features similar to the logic S5P(m) introduced by Meyer and van der Hoek [21], but S5P(m) is intended for formalising preference and the operators Pi of [21] need not be serial. In another kind of multi-agent system, agents are “opponents” who play against each other, and each agent may want to simulate the epistemic states of the others. To write a program for one agent, we may need to use modal operators of the other agents. A suitable logic for this problem is: KD45(m) = K(m) + (D) + (4) + (5) As mentioned earlier, the logic KD45(m) has been intensively studied. We use a subscript in KD45(m) to distinguish the logic from the monomodal logic KD45, while there is not such a need for the other considered multimodal logics. To capture common belief of a group of agents, one can extend the logic KD45(m) with modal operators for groups of agents and some additional axioms. Suppose that there are n agents and m = 2n − 1. Let g be an one-to-one function that maps every natural number less than or equal to m to a non-empty subset of {1, . . . , n}. Suppose that an index 1 ≤ i ≤ m stands for the group of agents whose indices form the set g(i). We can adopt the axioms (D), (4), and additionally (Ig ) : 2i ϕ → 2j ϕ if g(i) ⊃ g(j), so i indicates a group that contains the group identified by j, and (5a ) : ¬2i ϕ → 2i ¬2i ϕ if g(i) is a singleton, so that i stands for an agent. Thus, for reasoning about belief and common belief, we can use: KD4Ig 5a = K(m) + (D) + (4) + (Ig ) + (5a ) 4
Here we want to catch the most important properties of belief and common belief, and the aim is not to give an exact formulation of belief or common belief. This logic is therefore different in nature from the well-known multimodal logics of common knowledge with fixed-points. It also differs from the multimodal logic of mutual belief introduced by Aldewereld et al. [1]. Our modal operator of common belief satisfies positive introspection, while the operator of mutual belief introduced in [1] lacks this property. On the other hand, the latter operator has some properties that the former does not have. The given axioms correspond to the following frame restrictions: Axiom (D) (I) (Ig ) (4) (4s ) (5) (5s ) (5a )
2.3
Corresponding Condition ∀u ∃v Ri (u, v) Rj ⊆ Ri if i > j Rj ⊆ Ri if g(i) ⊇ g(j) ∀u, v, w (Ri (u, v) ∧ Ri (v, w) → Ri (u, w)) ∀u, v, w (Rj (u, v) ∧ Ri (v, w) → Ri (u, w)) ∀u, v, w (Ri (u, v) ∧ Ri (u, w) → Ri (w, v)) ∀u, v, w (Rj (u, v) ∧ Ri (u, w) → Ri (v, w)) as for (5) if g(i) is a singleton
Definitions for Tableau Calculi
As in our previous works on tableau calculi [11, 27, 12], our tableau formulation traces their roots to Hintikka via [31]. A tableau rule σ consists of a numerator N above the line and a (finite) list of denominators D1 , D2 , . . . , Dk (below the line) separated by vertical bars. The numerator is a finite formula set, and so is each denominator. As we shall see later, each rule is read downwards as “if the numerator is L-satisfiable, then so is one of the denominators”. The numerator of each tableau rule contains one or more distinguished formulae called the principal formulae. A tableau calculus CL for a logic L is a finite set of tableau rules. A CL-tableau for X is a tree with root X whose nodes carry finite formula sets obtained from their parent nodes by instantiating a tableau rule with the proviso that if a child s carries a set Z and Z has already appeared on the branch from the root to s then s is an end node. Let ∆ be a set of tableau rules. We say that Y is obtainable from X by applications of rules from ∆ if there exists a tableau for X which uses only rules from ∆ and has a node that carries Y . A branch in a tableau is closed if its end node carries only ⊥. A tableau is closed if every one of its branches is closed. A tableau is open if it is not closed. A finite formula set X is CL-consistent if every CL-tableau for X is open. If there is a closed CL-tableau for X then X is CL-inconsistent. A tableau calculus CL is sound if for all finite formula sets X, if X is L-satisfiable then X is CL-consistent. It is complete if for all finite formula sets X, if X is CLconsistent then X is L-satisfiable. Let σ be a rule of CL. We say that σ is sound w.r.t. L if for every instance σ 0 of σ, if the numerator of σ 0 is L-satisfiable then so is one of the denominators of σ 0 . Any CL containing only rules sound w.r.t. L is sound. 5
3
Modal Clauses
A classical literal is a formula of the form p or ¬p, where p is a primitive proposition. We use a, b, c to denote classical literals, and to denote a possibly empty sequence of universal (box-like) modal operators, which is also called a universal modality. We write [ϕ1 , . . . , ϕk ] to denote (ϕ1 ∨ . . . ∨ ϕk ) with the emphasis that the order of disjuncts is not important. A (modal) clause is a formula of the form [a1 , . . . , ak ], [a, 2i b], [a, 3i b], or 3i b, where is called the modal context. Proposition 1. Every formula set can be transformed in quadratic time to a set of clauses which is equisatisfiable in any normal modal logic to the first set. Proof. Repeatedly replace a complex formula by a new primitive proposition which is “defined” by that formula (see, e.g., [23, 17, 24]). For example, (ϕ ∨ 3i ψ), where ψ is not a primitive proposition, is replaced by (ϕ ∨ 3i p) and 2i (¬p ∨ ψ), where p is a fresh primitive proposition (defining ψ via the second added clause). We need to do at most n such replacements, where n is the sum of the lengths of formulae belonging to the input set. Hence the transformation can be done in quadratic time. We now define L-clauses using a case-distinction on L as: For L ∈ {KDI4, KDI4s , KD4Ig 5a }: every clause is an L-clause; For L ∈ {KD4s 5s , KDI4s 5}: an L-clause is a clause with modal depth 1 or 0; For KD45(m) : a KD45(m) -clause is a clause ϕ whose modal context has rightmost modal operator index l, and contains no subsequence of the form 2i 2i , and if ϕ is of the form [a, 2j b], [a, 3j b] or 3j b then j 6= l; For KDI45: a KDI45-clause is a clause ϕ whose modal context has rightmost modal operator index l, and contains no subsequence of the form 2i 2j with i ≤ j, and if ϕ is of the form [a, 2k b], [a, 3k b] or 3k b, then k < l. For convenience, if L ∈ {KDI45, KD4Ig 5a } then we also call a formula an Lclause if it is of the form [ϕ, ψ], where ϕ and ψ are of the form 2i a, 3i a or a. Proposition 2. Every formula set can be transformed in quadratic time to a set of L-clauses preserving L-satisfiability, where L is one of our seven logics. Proof. First, transform the formula set to a set of clauses preserving L-satisfiability, as in the proof of Proposition 1. For L ∈ {KDI4, KDI4s , KD4Ig 5a } we are done. Now observe that if ∇i is either 2i or 3i then in: KD4s 5s and KDI4s 5: we have ∇i ∇0j ϕ ≡ ∇0j ϕ KD45(m) : we have ∇i ∇0i ϕ ≡ ∇0i ϕ KDI45: if i ≤ j then ∇i ∇0j ϕ ≡ ∇0j ϕ. Also observe that, if 2i ∇j ϕ ≡ 3i ∇j ϕ ≡ ∇j ϕ in L for a fixed i and a fixed ∇j , then 2i (a ∨ ∇j b) ≡ 2i a ∨ ∇j b in L. Using these equivalences and the technique of replacing a complex formula by a new primitive proposition which is “defined” by that formula, we can transform every set of clauses to a set of L-clauses preserving L-satisfiability, for L ∈ {KD4s 5s , KDI4s 5, KD45(m) , KDI45}. It is easily seen that the complexity of the whole transformation is quadratic, as is the size of the resulting set of clauses. 6
(⊥)
X; p; ¬p ⊥
(∨)
X; [ϕ1 , . . . , ϕk ] X; ϕ1 | . . . | X; ϕk
X X; 3i >
(D)
(3L )
X; 3i a a ; trans(L, i, X)
(I5a )
X; 2i 2j ϕ (i > j) X; 2i 2j ϕ; 2j ϕ
(I5c )
X; 2i [a, 2j b] (i > j) X; 2i [a, 2j b]; [2j a, 2j b]
(I5b )
X; 2i 3j b (i > j) X; 2i 3j b; 3j b
(I5d )
X; 2i [a, 3j b] (i > j) X; 2i [a, 3j b]; [2j a, 3j b]
(Ig )
X; 2i ϕ (‡) X; 2i ϕ; 2j ϕ
(5aa )
X; 2i 2i ϕ (†) X; 2i ϕ
(5da )
X; 2i [a, 2i b] (†) X; [2i a, 2i b]
(5ba )
(Ig0 )
X; 2j [a, 2i b] (‡) X; 2j [a, 2i b]; 2j [a, 2j b]
X; 2i 3i b (†) X; 2i 3i b; 3i b
(‡) is g(i) ⊃ g(j)
(5ca )
X; 3i b (†) X; 2i 3i b; 3i b
(5ea )
X; 2i [a, 3i b] (†) X; [2i a, 3i b]
(†) is g(i) must be a singleton
The rule (3L ) is transitional. All other rules are static. Fig. 1. Clausal Tableau Rules
4
Clausal Tableau Systems
In this section, L is one of our seven logics: i.e. KDI4, KDI4s , KDI4s 5, KD4s 5s , KD45(m) , KDI45, or KD4Ig 5a . In clausal tableau calculi, numerators of rules are sets of clauses, so for an L-clausal tableau calculus, we require that numerators of rules are sets of L-clauses. As in [11], we categorise each rule either as a static rule or as a transitional rule: see Figure 1. The intuition behind this sorting is that in the static rules, the numerator and denominator represent the same world (in the same model), whereas in the transitional rules, the numerator and denominator represent different worlds (in the same model). When constructing an L-model graph, if ϕ belongs to a world w and we want to connect w to u via Ri , then certain logic-specific formulae are added to u . For the moment we denote such formulae by trans(L, i, ϕ), and define them precisely later. For a set of L-clauses X, define [ trans(L, i, X) = {trans(L, i, ϕ) | ϕ ∈ X}. For L ∈ {KDI4, KDI4s , KDI4s 5, KD4s 5s , KD45(m) }, define the tableau calculus CL using the rules from Figure 1 as CL = {(⊥), (∨), (D)} ∪ {(3L )}. For L = KDI45, let the tableau calculus be CKDI45 = {(⊥), (∨), (D), (I5a ), (I5b ), (I5c ), (I5d )} ∪ {(3L )}. For the rules (I5a ), (I5b ), (I5c ), (I5d ), if the numerator is a set of KDI45-clauses, then the condition i > j is always true. 7
For L = KD4Ig 5a , let the tableau calculus CKD4Ig 5a be CKD4Ig 5a = {(⊥), (∨), (D), (Ig ), (Ig0 ), (5aa ), (5ba ), (5ca ), (5da ), (5ea )} ∪ {(3L )}. We now define the function trans(L, i, ϕ) for six of our seven logics. trans(KDI4, i, ϕ) = trans(KDI4s , i, ϕ) = trans(KDI4s 5, i, ϕ) = trans(KD4s 5s , i, ϕ) = trans(KD45(m) , i, ϕ) = trans(KD4Ig 5a , i, ϕ) =
{ψ, 2j ψ | ϕ = 2j ψ and j ≥ i} {ϕ | ϕ = 2j ψ} ∪ {ψ | ϕ = 2j ψ and j ≥ i} {ψ | ϕ = 2j ψ and j ≥ i} {ψ | ϕ = 2i ψ} {ψ | ϕ = 2i ψ} {ψ, 2j ψ | ϕ = 2j ψ and g(j) ⊇ g(i)}.
Note that the sets of rules for CKD4s 5s and CKD45(m) are the same, but the difference lies in the definitions of L-clauses. Also note that trans(KD4Ig 5a , i, ϕ) has similarities to trans(KDI4, i, ϕ). Only trans(KDI45, i, ϕ) remains, and the intuition for specifying trans(KDI45, i, ϕ) is that the skeleton of a desired KDI45-model is a tree such that each of its branches is a sequence of edges created via accessibility relations Ri1 , . . . , Rik with i1 > . . . > ik . Recall that a KDI45-clause is a clause ϕ whose modal context has rightmost modal operator index l, and contains no subsequence of the form 2i 2j with i ≤ j, and if ϕ is of the form [a, 2k b], [a, 3k b] or 3k b, then k < l. Consider the following cases where mmi(ψ) is the maximal modal index occurring in ψ: trans(KDI45, i, ϕ) = {ψ, 2i−1 ψ} {2i−1 ψ} {[a, ψ], [2i−1 a, ψ]} {[a, ψ], [2i−1 a, ψ]} {3i−1 b} ∅
if ϕ = 2j ψ, j ≥ i, and mmi(ψ) < i − 1 if ϕ = 2j 2i−1 ψ and j ≥ i if ϕ = 2j [a, ψ], j ≥ i, and ψ = 2i−1 b if ϕ = 2j [a, ψ], j ≥ i, and ψ = 3i−1 b if ϕ = 2j 3i−1 b and j ≥ i otherwise
Note that for a tableau rule of CL, if the numerator is a set of L-clauses then each of the denominators of the rule is also a set of L-clauses. A tableau calculus CL has the analytic superformula property iff to every finite ∗ such that X ∗ contains all clauses set X of L-clauses we can assign a finite set XCL CL that may appear in any CL tableau for X. Our tableau calculi have the analytic ∗ calculated as follows. Let Sf (ϕ) be the set of all superformula property, with XCL subformulae of ϕ (including ϕ), and let e = Sf (X) ∪ ¬Sf (X) ∪ {⊥}. Sf (X) = ∪{Sf (ϕ) | ϕ ∈ X}, ¬X = {¬ϕ | ϕ ∈ X}, X ∗ = X. e For L ∈ {KDI4, KDI4s , KDI4s 5, KD4s 5s , KD45(m) }, choose XCL For L ∈ {KDI45, KD4Ig 5a }, let 2X = {2i ϕ | ϕ ∈ X and 1 ≤ i ≤ m}, let X|2 be the set of all clauses with modal depth ≤ 2, containing at most 2 primitive ∗ = propositions, which are primitive propositions occurring in X, and choose XCL e ∪ 2Sf (X) ∪ X|2 . X
8
Lemma 1. The tableau calculi defined in this section are sound. Proof. Let L be one of our seven logics. We show that CL contains only rules sound with respect to L. Consider the only nontrivial case, when L ∈ {KDI45, KD4Ig 5a }. For each static rule δ of CL, if δ is not (∨) then the denominator of δ is equivalent in L to the numerator of δ. Hence all static rules of CL are sound with respect to L. The transitional rules of CL are sound with respect to L because if ψ ∈ trans(L, i, ϕ) then ϕ → 2i ψ is L-valid.
5
Completeness of the Calculi
In this section, L is still one of our seven logics of belief and CL is the corresponding tableaux calculus. We prove completeness of our calculi via model graphs in a similar way as in [31, 11, 26, 12]. To show completeness of CL we give an algorithm that, given a finite CL-consistent set X of L-clauses, constructs an L-model graph (defined in Section 5.2) for X that satisfies every one of its formulae at the corresponding world. 5.1
Saturation
For a static rule δ, let δ 0 be the rule obtained from δ by adding the principal formulae to each of the denominators. Let SCL = {δ 0 | δ is a static rule of CL}. Note: for any rule of SCL, the numerator is included in each of its denominators. A set X is closed w.r.t. a tableau rule if, whenever the rule is applicable to X, one of the corresponding instances of the denominators is equal to X. For a finite CL-consistent set X of L-clauses, Y is a CL-saturation of X if Y is a maximal CLconsistent set of L-clauses obtainable from X by applications of the rules of SCL. In some cases, instead of CL-saturation we need a slightly different notion. For a finite CL-consistent set X of L-clauses, Y is a minimally CL-saturated set of X if Y is a minimal finite CL-consistent superset of X that is closed w.r.t. every rule of SCL. As stated by the following lemma, CL-saturations and minimally CL-saturated sets have the same nature as “downward saturated sets” defined in the works by Hintikka [15] and Gor´e [11]. Lemma 2. Let X be a finite CL-consistent set of L-clauses and Y a CL-saturation ∗ and Y is closed w.r.t. the or a minimally CL-saturated set of X. Then X ⊆ Y ⊆ XCL rules of SCL. Furthermore, there is an effective procedure that, given a finite CLconsistent set X of L-clauses, constructs some CL-saturation and some minimally CL-saturated set of X. The assertions about CL-saturations can be proved as in [27], and the assertions about minimally CL-saturated sets are immediate consequences. 9
5.2
Proving Completeness via Model Graphs
A model graph is a tuple hW, τ, R1 , . . . , Rm , Hi, where hW, τ, R1 , . . . , Rm i is a Kripke frame, and H is a function that maps each world to a set of clauses. We sometimes treat model graphs as models with h(w) as the restriction of H(w) to the set of primitive propositions. A model graph is called an L-model graph if its frame satisfies all L-frame restrictions. A model graph hW, τ, R1 , . . . , Rm , Hi is saturated if it satisfies the following conditions for every w ∈ W : ∨: if [ϕ1 , . . . , ϕk ] ∈ H(w), then some ϕi with 1 ≤ i ≤ k belongs to H(w); 2: if 2i ϕ ∈ H(w), then for every u such that Ri (w, u), ϕ ∈ H(u); 3: if 3i a ∈ H(w), then there exists u ∈ W s.t. Ri (w, u) and a ∈ H(u). A saturated model graph is consistent if no world contains ⊥ nor {p, ¬p} for any p. We use the term “model graph” merely to denote a data structure, while Rautenberg’s model graphs are required to be saturated and consistent. Lemma 3. If M = hW, τ, R1 , . . . , Rm , Hi is a consistent saturated model graph, then M is a model of H(τ ). Proof. By induction on the construction of ϕ that if ϕ ∈ H(w) then M, w ϕ. Given a finite CL-consistent set X of L-clauses, we construct a consistent saturated L-model graph M = hW, τ, R1 , . . . , Rm , Hi such that X ⊆ H(τ ) as a putative L-model for X. If for every w ∈ W , H(w) is a CL-saturation of some set, then the first condition of being a saturated model graph is satisfied. 5.3
Completeness
In the following algorithm, the worlds of the constructed model graph are marked either as unresolved or as resolved. Algorithm 1 Input: a finite CL-consistent set X of formulae. Output: an L-model graph M = hW, τ, R1 , . . . , Rm , Hi satisfying X. 1. Let W = {τ }, Ri0 = ∅ for 1 ≤ i ≤ m, H0 (τ ) = X, and H(τ ) be a CL-saturation of X. Mark τ as unresolved. 2. While there are unresolved worlds, take one, say w, and do: (a) For every formula 3i a in H(w): i. If there exists u ∈ W s.t. H0 (u) = {a} ∪ trans(L, i, H(w)), then add the pair (w, u) to Ri0 . ii. Else, add a new world u to W , let H0 (u) = {a} ∪ trans(L, i, H(w)) and H(u) be a CL-saturation of H0 (u), mark u as unresolved, and add the pair (w, u) to Ri0 . (b) Mark w as resolved. 3. Let Ri , for 1 ≤ i ≤ m, be the least extensions of Ri0 such that hW, τ, R1 , . . . , Rm i is an L-frame. ∗ for This algorithm always terminates because H0 is one-to-one and H0 (w) ⊆ XCL every w. Here is the main lemma concerning its properties:
10
Lemma 4. For L 6= KD4Ig 5a , let X be a finite CL-consistent set of L-clauses, and M = hW, τ, R1 , . . . , Rm , Hi be the model graph constructed by Algorithm 1 for X. Then M is a consistent L-model graph satisfying X. Proof. It is clear that M is a consistent L-model graph. Let Ri0 , for 1 ≤ i ≤ m, be the relations mentioned in the algorithm. Suppose that L 6= KDI45. Since X ⊆ H(τ ), by Lemma 3, it suffices to prove that M is a saturated model graph. For this, we only need to prove that, for every w ∈ W , if 2i ϕ ∈ H(w) then for every u such that Ri (w, u), ϕ ∈ H(u). Suppose that 2i ϕ ∈ H(w) and Ri (w, u) holds and consider the following cases: L = KDI4: Since Ri (w, u) holds, there exist w0 , . . . , wk such that w0 = w, wk = u, and for 1 ≤ j ≤ k there exists ij ≤ i such that Ri0 j (wj−1 , wj ) holds. Since 2i ϕ ∈ H(w0 ) and i1 ≤ i, we have {ϕ, 2i ϕ} ⊆ trans(KDI4, i1 , H(w0 )). Hence {ϕ, 2i ϕ} ⊆ H(w1 ). Repeating this argument, we can derive that {ϕ, 2i ϕ} ∈ H(wk ). Hence ϕ ∈ H(u). L = KDI4s : Since Ri (w, u) holds, there exist w0 , . . . , wk such that w0 = w, wk = u, and for 1 ≤ j ≤ k, Ri0 j (wj−1 , wj ) holds for some ij , with ik ≤ i. Since 2i ϕ ∈ H(w0 ), we have 2i ϕ ∈ trans(KDI4s , i1 , H(w0 )). Hence 2i ϕ ∈ H(w1 ). Repeating this argument, we can derive that 2i ϕ ∈ H(wk−1 ). Consequently, since ik ≤ i, we have ϕ ∈ H(wk ), i.e. ϕ ∈ H(u). L = KD4s 5s : Because of the form of KD4s 5s -clauses, τ is the only world that may contain formulae with modal operators. Since 2i ϕ ∈ H(w), we have w = τ . Since Ri (τ, u) is equivalent to Ri0 (τ, u), it follows that ϕ ∈ H(u). L = KDI4s 5: As for the above case, we have w = τ . Since Ri (τ, u) holds, Rj0 (τ, u) holds for some j ≤ i. Hence ϕ ∈ H(u). L = KD45(m) : Because of the form of KD45(m) -clauses and the assumption that 2i ϕ ∈ H(w), there is no v such that Ri0 (v, w) holds. Hence there is no v such that Ri (v, w) holds. Consequently, since Ri (w, u) holds, we derive that Ri0 (w, u) holds. This implies ϕ ∈ H(u). Remark 1. Now suppose that L = KDI45. Observe that: (a) For every u, v, ϕ, if Ri0 (u, v) holds and ϕ ∈ H(v) then every modal index occurring in ϕ is less than i. This follows from the form of KDI45-clauses and the definition of trans(KDI45, , ). (b) From (a), for every u, v, w, if Ri0 (u, v) and Rj0 (v, w) hold then i > j. (c) For every u, v, w1 , w2 , if Ri0 (u, v) holds, and w1 belongs to the subtree with root v created by {Rk0 | 1 ≤ k ≤ m}, but w2 does not, and Rj (w1 , w2 ) or Rj (w2 , w1 ) holds, then j ≥ i. Similarly, for every u, v, w, if Ri0 (u, v) and Rj (w, v) hold, then j ≥ i. These assertions follow because, at the beginning, when extending {Rk0 | 1 ≤ k ≤ m} to {Rk | 1 ≤ k ≤ m} using the KDI45-frame conditions, u is the only outside node which is connected to the mentioned subtree and this connection is via Ri0 . Let ϕ ∈ H(w). We prove by induction on the structure of ϕ that M, w ϕ. It suffices to consider the case when ϕ = ψ with modal context = 2i1 . . . 2ik and k ≥ 1. Since ϕ is a KDI45-clause, we have i1 > . . . > ik . Let w0 = w, w1 , . . . , wk be worlds such that Ris (ws−1 , ws ) holds for all 1 ≤ s ≤ k. We show that M, wk ψ. 11
If w 6= τ , then there is v such that Rl0 (v, w) holds for some l. By Remark 1(a), l > i1 > . . . > ik . By Remark 1(c), it follows that w1 , . . . , wk must be nodes in the subtree with root w created by {Ri0 | 1 ≤ i ≤ m} and different from w (otherwise we would have is ≥ l for some 1 ≤ s ≤ k). Suppose that u0 = w, uh = wk , and Rj0 t (ut−1 , ut ) holds for 1 ≤ t ≤ h. By Remark 1(b), we have j1 > . . . > jh . By Remark 1(c), there exists 1 ≤ s ≤ k such that is ≥ j1 . Hence i1 ≥ j1 . By Remark 1(c), we also have ik ≥ jh . There are the following cases: ψ = [a1 , . . . , ap ] : Since 2i1 . . . 2ik ψ ∈ H(u0 ), i1 ≥ j1 and ik ≥ jh , by the definition of trans(KDI45, , ) and the tableau rule (I5a ), there exists i ≥ jh such that 2i ψ ∈ H(uh−1 ). Hence ψ ∈ H(uh ). By the inductive hypothesis, it follows that M, uh ψ, which means M, wk ψ. ψ = [a, 2i b] : If i < jh then apply the argument as in the above case. Suppose that i ≥ jh . Let 1 ≤ t ≤ h be the smallest index such that i ≥ jt . If t > 1 then i < jt−1 , and similarly to above case, we derive that either 2j ψ ∈ H(ut−1 ) for some j > i or [2i a, 2i b] ∈ H(ut−1 ). If t = 1 then, by the tableau rules (I5a ) and (I5c ), we derive that [2i a, 2i b] ∈ H(ut−1 ). Hence, by the tableau rules (I5c ) and (∨), 2i a ∈ H(ut−1 ) or 2i b ∈ H(ut−1 ). If 2i a ∈ H(ut−1 ) then we can derive that a ∈ H(uh ). If 2i b ∈ H(ut−1 ) then from the definition of trans(KDI45, , ) and Remark 1(c) we can derive that M, uh 2i b. Thus, in both of the cases, we have M, uh [a, 2i b], which means that M, wk ψ. ψ = [a, 3i b] : If i < jh then apply the argument as in the case when ψ is of the form [a1 . . . , ap ]. Suppose that i ≥ jh . Let 1 ≤ t ≤ h be the smallest index such that i ≥ jt . If t > 1 then we have i < jt−1 , and similarly as in the above cases, we derive that either 2j ψ ∈ H(ut−1 ) for some j > i or [2i a, 3i b] ∈ H(ut−1 ). If t = 1 then, by the tableau rules (I5a ) and (I5d ), we derive that [2i a, 3i b] ∈ H(ut−1 ). Hence, by the tableau rules (I5d ) and (∨), 2i a ∈ H(ut−1 ) or 3i b ∈ H(ut−1 ). If 2i a ∈ H(ut−1 ) then we can derive that a ∈ H(uh ). If 3i b ∈ H(ut−1 ) then there exists u such that Ri0 (ut−1 , u) holds and b ∈ H(u), and hence M, uh 3i b. Thus, in both of the cases, we have M, uh [a, 3i b], which means that M, wk ψ. ψ = 3i b : Similar to the above case, but use (I5b ) instead of (I5d ). Lemma 4 does not hold for L = KD4Ig 5a . To deal with KD4Ig 5a , we modify Algorithm 1 to obtain the following algorithm by replacing “CL-saturation” by “minimally CL-saturated set” and forbidding any sequence of two edges of the form Ri0 (u, v) and Ri0 (v, w) if g(i) is a singleton. Algorithm 2 Input: a finite CL-consistent set X of formulae, for L = KD4Ig 5a . Output: an L-model graph M = hW, τ, R1 , . . . , Rm , Hi satisfying X. 1. Let W = {τ }, Ri0 = ∅ for 1 ≤ i ≤ m, H0 (τ ) = X, and H(τ ) be a minimally CL-saturated set of X. Mark τ as unresolved. 2. While there are unresolved worlds, take one, say w, and do: (a) For every formula 3i a in H(w) such that either g(i) is not a singleton or there is no v s.t. Ri0 (v, w) holds: i. If there exists u ∈ W s.t. H0 (u) = {a} ∪ trans(L, i, H(w)), then add the pair (w, u) to Ri0 . 12
ii. Otherwise, add a new world u to W , let H0 (u) = {a} ∪ trans(L, i, H(w)) and H(u) be a minimally CL-saturated set of H0 (u), mark u as unresolved, and add the pair (w, u) to Ri0 . (b) Mark w as resolved. 3. Let Ri , for 1 ≤ i ≤ m, be the least extensions of Ri0 such that hW, τ, R1 , . . . , Rm i is an L-frame. As for Algorithm 1, this algorithm always terminates. We also have the following assertion, which is similar to Lemma 4: Lemma 5. For L = KD4Ig 5a , let X be a finite CL-consistent set of L-clauses, and M = hW, τ, R1 , . . . , Rm , Hi be the model graph constructed by Algorithm 2 for X. Then M is a consistent saturated L-model graph satisfying X. Proof. Clearly, M is a consistent L-model graph. It is sufficient to prove that M satisfies the two last conditions of being a saturated model graph. We first prove that M satisfies the second last condition of being a saturated model graph. Suppose that 3i b ∈ H(w). If g(i) is not a singleton or there is no v such that Ri0 (v, w) holds then, clearly, there exists u such that Ri0 (w, u) and Ri (w, u) hold and b ∈ H(u). Suppose that g(i) is a singleton and we have Ri0 (v, w) for some v. We show that 3i b ∈ H(v). Recall that H(w) is a minimally CLsaturated set of trans(L, i, H(v)) ∪ {a0 } for some a0 . Let 3i b ∈ H(w) have origin from ϕ ∈ trans(L, i, H(v)). There exists j such that g(j) ⊇ g(i) and either 2j ϕ ∈ H(v) or ϕ is of the form 2j ψ and 2j ψ ∈ H(v). Denote this property by (*). There are three cases: – If 3i b is added to H(w) not using the (∨) rule, then 3i b also belongs to H(v), because of (*) and that H(v) is a (minimally) CL-saturated set. – Consider the case when 3i b is added to H(w) using the (∨) rule for [a, 3i b] (a subformula of ϕ). First, we have that a ∈ / H(w) (otherwise it is not necessary to add 3i b to H(w)). Because of (*) and that H(v) is a (minimally) CL-saturated set, both 2i [a, 3i b] and [2i a, 3i b] belong to H(v). Since a ∈ / H(w), we have that 2i a ∈ / H(v). It follows that 3i b ∈ H(v). – Consider the case when 3i b is added to H(w) using the (∨) rule for [2i a, 3i b], which originates from 2i [a, 3i b], which in turn originates from ϕ. First, we have that 2i a ∈ / H(w) (otherwise it is not necessary to add 3i b to H(w)). Because of (*) and that H(v) is a (minimally) CL-saturated set, both 2i [a, 3i b] and [2i a, 3i b] belong to H(v). Since 2i a ∈ / H(w), we have that 2i a ∈ / H(v). It follows that 3i b ∈ H(v). We have shown that 3i b ∈ H(v). Thus, there exists u such that Ri0 (v, u) holds and b ∈ H(u). It follows that Ri (w, u) holds. We now prove that M satisfies the last condition of being a saturated model graph. Suppose that 2i ϕ ∈ H(w) and Ri (w, u) holds. We show that ϕ ∈ H(u). Consider the case when ϕ is a classical literal b, g(i) is a singleton, and Ri0 (v, w) holds for some v. We show that 2i b ∈ H(v). The proof is very similar to the given proof of 3i b ∈ H(v) when assuming 3i b ∈ H(w), but it has one additional subcase, which is considered in the follows. Let 2i b ∈ H(w) have origin from 13
ϕ0 ∈ trans(L, i, H(v)). There exists j such that g(j) ⊇ g(i) and either 2j ϕ0 ∈ H(v) or ϕ0 is of the form 2j ψ and 2j ψ ∈ H(v). Denote this property by (?). Consider the special subcase when 2i b is added to H(w) using the (∨) rule for [2i b, 3i c], which originates from 2i [b, 3i c], which in turn originates from ϕ0 . First, we have that 3i c ∈ / H(w) (otherwise it is not necessary to add 2i b to H(w)). Because of (?) and that H(v) is a (minimally) CL-saturated set, both 2i [b, 3i c] and [2i b, 3i c] belong to H(v). Since 3i c ∈ / H(w), we have that 2i 3i c ∈ / H(v) and 3i c ∈ / H(v) (because 3i c ∈ H(v) would imply that 2i 3i c ∈ H(v), due to the tableau rule (5ca )). It follows that 2i b ∈ H(v). In a similar way, for all the other subcases we can show that 2i b ∈ H(v). If Ri (w, u) holds then Ri0 (v, u) must hold, which implies that b ∈ H(u). Consider the case when ϕ is not a classical literal, g(i) is a singleton, and Ri0 (v, w) holds for some v. Analogously as for the above case, we can show that 2i ϕ ∈ H(v). If Ri (w, u) holds then Ri0 (v, u) must hold, which implies ϕ ∈ H(u). Consider the remaining case when g(i) is not a singleton or there is no v such that Ri0 (v, w) holds. The proof is similar to the case for KDI4. Since Ri (w, u) holds, there exist w0 , . . . , wk such that w0 = w, wk = u, and for 1 ≤ j ≤ k there exists ij such that g(ij ) ⊆ g(i) and Ri0 j (wj−1 , wj ) holds. In particular, if g(i) is a singleton then k = 1 and i1 = i. Since 2i ϕ ∈ H(w0 ) and g(i1 ) ⊆ g(i), we have {ϕ, 2i ϕ} ⊆ trans(L, i1 , H(w0 )). Hence {ϕ, 2i ϕ} ⊆ H(w1 ). Repeating this argument, we can derive that {ϕ, 2i ϕ} ∈ H(wk ). Hence ϕ ∈ H(u), which completes the proof. The following theorem follows from Lemmas 1, 4, and 5. Theorem 1. The tableau systems CL are sound and complete, for L ∈ {KDI4, KDI4s , KDI4s 5, KD4s 5s , KD45(m) , KDI45, KD4Ig 5a }.
6
Complexity
In [14], Halpern and Moses proved that the satisfiability problem in KD45(m) is PSPACE-complete. In this section, using the algorithms of constructing L-model graphs given in the previous section, we analyse the time complexity of the satisfiability problem in the remaining multimodal logics KDI4s 5, KD4s 5s , KDI45, KDI4, KDI4s , KD4Ig 5a . Theorem 2. For KDI4s 5, KD4s 5s , and KDI45, the satisfiability problem is NPcomplete, and for KDI4 and KDI4s , it is PSPACE-complete. Proof. First, observe that computing a CL-saturation of a CL-consistent formula set can be nondeterministically done in polynomial time as follows: while the set is not closed w.r.t. some tableau rule δ from SCL, apply δ to the set, guess the right denominator and update the set to that denominator. If L ∈ {KDI4s 5, KD4s 5s , KDI45} and X is a CL-consistent formula set then an L-model graph constructed by Algorithm 1 for X has a polynomial size: the content of ∗ and has a size bounded by a polynomial each of the possible worlds is a subset of XCL in the size of X; the number of possible worlds of the model graph is bounded by a polynomial in the size of X because each path via {Ri0 | 1 ≤ i ≤ m} has length 0 or 1 for L ∈ {KDI4s 5, KD4s 5s } and a length not greater than m for L = KDI45. This together with the above observation imply that if X is a CL-consistent formula 14
set then an L-model of X can be nondeterministically constructed in polynomial time. Hence the problem of checking satisfiability in KDI4s 5, KD4s 5s , and KDI45 is NP-complete (the lower bound NP-hard is clear). Because KDI4 and KDI4s are “conservative extensions” of the monomodal logic KD4, the satisfiability problem in these logics is PSPACE-hard. We show that the problem is in the PSPACE class. Let X be a formula set with size n. To check whether X is L-satisfiable, where L ∈ {KDI4, KDI4s }, we do not try to construct a consistent L-model graph satisfying X as in Algorithm 1, but just check whether such a model graph exists. For our complexity analysis, we assume the following modifications for Algorithm 1: – An input for this algorithm is not required to be CL-consistent anymore. – “CL-saturation” is replaced by “candidate for CL-saturation”, which is nondeterministically constructed in polynomial time as follows: while the set is not closed w.r.t. some tableau rule δ from SCL, apply δ to the set, guess the right denominator and update the set to that denominator. – When loop-checking, instead of “cross-tree” edges, we allow only backward edges. (Lemma 4 still holds for this modification.) – If a world is inconsistent (in the sense that it contains both p and ¬p for some p) then the computation fails. – If all possible computations fail, then return that the input formula set is Lunsatisfiable, else return that the input formula set is L-satisfiable. During run time of the modified algorithm, we need to keep only the content of the current path in the model graph. To show that checking satisfiability in L ∈ {KDI4, KDI4s } can be done in PSPACE, it is sufficient to show that the current path (via {Ri0 | 1 ≤ i ≤ m}) in the model graph constructed for X always has a length bounded by a polynomial of n (the size of X). Consider the case L = KDI4s . Denote Y2 = {ϕ ∈ Y | ϕ is of the form 2i ψ}. Suppose that there is a path from u to w via {Ri0 | 1 ≤ i ≤ m} in an L-model graph constructed by the modified algorithm for X. Then we have H(w)2 ⊇ H(u)2 . Because of loop-checking, if H(w)2 = H(u)2 then the path from u to w has no more than n nodes. This implies that every path via {Ri0 | 1 ≤ i ≤ m} in an L-model graph constructed for X has a length not greater than n2 . Consider the case L = KDI4. Denote Y2i = {ϕ ∈ Y | ϕ is of the form 2i ψ}. Suppose that there is a path from u to w via {Ri0 | 1 ≤ i ≤ m} in an L-model graph constructed by the modified algorithm for X. Then we have H(w)2m ⊇ H(u)2m . Because of loop-checking, if H(w)2m = H(u)2m then the path from u to w has no 0 (v 0 , v) holds for some v 0 . This implies that every more than n nodes v such that Rm 0 path via {Ri | 1 ≤ i ≤ m} in an L-model graph constructed by the modified algorithm 0 (v 0 , v) holds for some v 0 . Analogously, for X has no more than n2 nodes v such that Rm 0 (v 0 , v) holds contains no more than every (sub)path without v 0 and v such that Rm 2 0 0 n nodes v such that Rm−1 (v , v) holds for some v 0 . Repeating this argument for (m − 2), . . . , 1, we derive that every path via {Ri0 | 1 ≤ i ≤ m} in an L-model graph constructed by the modified algorithm for X has no more than n2m nodes. Because KD4Ig 5a is an extension of the monomodal logic KD4, the satisfiability problem in KD4Ig 5a is PSPACE-hard. Using Algorithm 2 and the argument as in 15
[12], we can derive that the satisfiability problem in KD4Ig 5a is in EXPTIME. We do not know whether the problem is in PSPACE or EXPTIME-hard. We conjecture that it is EXPTIME-hard when m ≥ 7 (i.e. when there are at least 3 single agents)!
7
Conclusions
The multimodal logics considered in this paper are useful for reasoning about belief. The logics KDI4, KDI4s , KDI4s 5, and KDI45 can be used for reasoning about multi-degree belief [28], the logic KD4s 5s can be used for distributed systems of belief [28], and the logic KD4Ig 5a can be used for reasoning about belief and common belief of agents in multi-agent systems [28, 29]. Although the used axioms are widely known, most of the logics considered in this paper, namely, KDI4s , KDI4s 5, KDI45, KD4s 5s , and KD4Ig 5a have not been studied by other authors. Thus, there were no (labelled or non-labelled) tableau systems for these four logics. Only labelled tableau calculi for KDI4 and KD45(m) have previously been developed by other authors [3, 35], although Shvarts’ non-labelled calculus for (KD45) can be extended trivially to KD45(m) [11]. We have given (non-labelled) tableau calculi for all 7 of our multimodal logics of belief. Our calculi are sound, complete, cut-free and have the analytic superformula property. Thus, they give decision procedures. Our tableau calculi for KDI4, KDI4s , KDI4s 5, KD4s 5s , and KD45(m) are simple. The calculi for KDI4 and KDI4s are very similar to the standard tableau calculus for KD4, and clausal form is not essential for these two logics. On the other hand, the simplicity of the tableau calculi for KDI4s 5, KD4s 5s , and KD45(m) is due to the clausal forms for these logics. Traditional tableaux for modal logics do not use clausal form. What advantages might we gain by using clausal form? First, handling only clauses may speed up the search process. We do not have experimental results for this claim yet, but the hope follows from the fact that most efficient SAT solvers for classical propositional logic handle only clauses. Second, by using clauses in “normal form” such as in the case of KDI4s 5, KD4s 5s , KDI45, KD45(m) , the task of developing non-labelled tableau calculi is significantly simplified. For simple logics like KDI4 or KDI4s , clausal form does not matter, but for more complicated logics like KDI45 or KD4Ig 5a , clausal form helps enormously, even when we do not provide a “normal form” for clauses as in the case of KD4Ig 5a .1 Third, clausal tableau calculi sometimes bring interesting results. For example, using them, Nguyen [24] obtained the currently best space bound O(n. log(n)) for the modal logics K4, KD4, and S4. In [25], he also gave (non-labelled) cut-free clausal tableau calculi for the symmetric modal logics KB, KDB, and B. Finally, using our calculi, we have shown that the satisfiability problem for KDI4s 5, KD4s 5s , and KDI45 is NP-complete, and for KDI4 and KDI4s it is PSPACE-complete. The complexity lower bound of KD4Ig 5a remains open. 1
That is, adopting restrictions for KDI4s 5, KD4s 5s , KDI45, KD45(m) is like providing a normal form for clauses in these logics. For KD4Ig 5a , we could adopt the restriction that if g(i) is a singleton then 2i 2i is ruled out. We did not do this, but guaranteed it “on-the-fly” in our algorithms.
16
References 1. H. Aldewereld, W. van der Hoek, and J.-J.Ch. Meyer. Rational teams: Logical aspects of multiagent systems. Fundamenta Informaticae, 63(2–3):159–183, 2004. 2. M. Baldoni, L. Giordano, and A. Martelli. A framework for a modal logic programming. In Joint International Conference and Symposium on Logic Programming, pages 52–66. MIT Press, 1996. 3. M. Baldoni, L. Giordano, and A. Martelli. A tableau calculus for multimodal logics and some (un)decidability results. In H. de Swart, editor, Proceeding of TABLEAUX’98, LNCS 1397, pages 44–59. Springer-Verlag, 1998. 4. F. Debart, P. Enjalbert, and M. Lescot. Multimodal logic programming using equational and order-sorted logic. Theoretical Comp. Science, 105:141–166, 1992. 5. L. Fari˜ nas del Cerro and M. Penttonent. Grammar logics. Logique et Analyse, 121-122:123–134, 1988. 6. S. Demri. The complexity of regularity in grammar logics and related modal logics. Journal of Logic and Computation, 11(6):933–960, 2001. 7. S. Demri and H. de Nivelle. Deciding regular grammar logics with converse through first-order logic. Journal of Logic, Language and Information, 14(3):289–329, 2005. 8. R. Fagin, J.Y. Halpern, Y. Moses, and M.Y. Vardi. Reasoning About Knowledge. MIT Press, 1995. 9. M. Fitting. Proof Methods for Modal and Intuitionistic Logics. Volume 169 of Synthese Library. D. Reidel, Dordrecht, Holland, 1983. 10. M. Fitting, L. Thalmann, and A. Voronkov. Term-modal logics. Studia Logica, 69:133–169, 2001. 11. R. Gor´e. Tableau methods for modal and temporal logics. In D’Agostino, Gabbay, H¨ ahnle, and Posegga, editors, Handbook of Tableau Methods, pages 297–396. Kluwer Academic Publishers, 1999. 12. R. Gor´e and L.A. Nguyen. A tableau system with automaton-labelled formulae for regular grammar logics. In B. Beckert, editor, Proceedings of TABLEAUX 2005, LNAI 3702, pages 138–152. Springer-Verlag, 2005. 13. G. Governatori. Labelled tableaux for multi-modal logics. In P. Baumgartner, R. H¨ ahnle, and J. Posegga, editors, Proceedings of TABLEAUX’1995, LNAI 918, pages 79–94. Springer-Verlag, 1995. 14. J.Y. Halpern and Y. Moses. A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54:319–379, 1992. 15. K.J.J. Hintikka. Form and content in quantification theory. Acta Philosophica Fennica, 8:3–55, 1955. 16. I. Horrocks and U. Sattler. Decidability of SHIQ with complex role inclusion axioms. Artificial Intelligence, 160(1-2):79–104, 2004. 17. J. Hudelmaier. Improved decision procedures for the modal logics K, T, S4. In H.K. B¨ uning, editor, Proc. of CSL’95, LNCS 1092, pages 320–334. Springer, 1996. 18. C. Lutz, H. Sturm, F. Wolter, and M. Zakharyaschev. Tableaux for temporal description logic with constant domain. In R. Gor´e, A. Leitsch, and T. Nipkow, editors, Proceedings of IJCAR, LNCS 2083, pages 121–136. Springer Verlag, 2001. 19. F. Massacci. Single step tableaux for modal logics. Journal of Automated Reasoning, 24(3):319– 364, 2000. 20. J.-J.Ch. Meyer, F.S. de Boer, R.M. van Eijk, K.V. Hindriks, and W. van der Hoek. On programming KARO agents. Logic Journal of the IGPL, 9(2), 2001. 21. J.-J.Ch. Meyer and W. van der Hoek. A modal logic for nonmonotonic reasoning, chapter 3. Ellis Horwood, 1992. 22. J.-J.Ch. Meyer and W. van der Hoek. Epistemic Logic for Computer Science and Artificial Intelligence. Cambridge University Press, 1995. 23. G.E. Mints. Gentzen-type systems and resolution rules. In P.Martin-L¨ of, G. Mints (eds.): COLOG-88, LNCS 417, pages 198–231. Springer, 1988. 24. L.A. Nguyen. A new space bound for the modal logics K4, KD4 and S4. In M. Kutylowski and L. Pacholski, editors, Proceedings of MFCS’99, LNCS 1672, pages 321–331. Springer, 1999. 25. L.A. Nguyen. Clausal tableau systems and space bounds for the modal logics K, KD, T, KB, KDB, and B. Technical Report TR 00-01(261), Institute of Informatics, University of Warsaw, 2000.
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26. L.A. Nguyen. Analytic tableau systems and interpolation for the modal logics KB, KDB, K5, KD5. Studia Logica, 69(1):41–57, 2001. 27. L.A. Nguyen. Analytic tableau systems for propositional bimodal logics of knowledge and belief. In U. Egly and C.G. Fem¨ uller, editors, Proceedings of TABLEAUX 2002, LNAI 2381, pages 206–220. Springer-Verlag, 2002. 28. L.A. Nguyen. Multimodal logic programming. Theoretical Computer Science, 360:247–288, 2006. 29. L.A. Nguyen. Reasoning about epistemic states of agents by modal logic programming. In F. Toni and P. Torroni, editors, Proceedings of CLIMA VI, LNAI 3900, pages 37–56. Springer-Verlag, 2006. A revised version is available at http://www.mimuw.edu.pl/∼nguyen/papers.html. 30. L.A. Nguyen. Foundations of modal logic programming: The direct approach. Manuscript (served as a technical report), available at http://www.mimuw.edu.pl/∼nguyen/papers.html, 2006 (revised March 2007). 31. W. Rautenberg. Modal tableau calculi and interpolation. Journal of Philosophical Logic, 12:403– 423, 1983. 32. R.A. Schmidt, D. Tishkovsky, and U. Hustadt. Interactions between knowledge, action, and commitment within agent dynamic logic. Studia Logica, 78(3):381–415, 2004. 33. W. van der Hoek and J.-J. Meyer. Modalities for reasoning about knowledge and uncertainties. In P. Doherty, editor, Partiality, Modality, and Nonmonotonicity. CSLI Publications, 1996. 34. M. Wessel. Obstacles on the way to qualitative spatial reasoning with description logics: Some undecidability results. In Description Logics 2001. 35. M. Wooldridge, C. Dixon, and M. Fisher. A tableau-based proof method for temporal logics of knowledge and belief. Journal of Applied Non-Classical Logics, 8(3):225–258, 1998.
18
The Australian National University and NICTA? ? ? Canberra ACT 0200, Australia [email protected] 2 Institute of Informatics, University of Warsaw ul. Banacha 2, 02-097 Warsaw, Poland [email protected]
Abstract. We develop clausal tableau calculi for seven multimodal logics variously designed for reasoning about multi-degree belief, reasoning about distributed systems of belief and for reasoning about epistemic states of agents in multi-agent systems. Our tableau calculi are sound, complete, cut-free and have the analytic superformula property, thereby giving decision procedures for all of these logics. We also use our calculi to obtain complexity results for five of these logics. The complexity of one was known and that of the seventh remains open. Keywords: modal logics for agent-based systems, theorem proving for modal logics, complexity and decidability of modal and temporal logics.
1
Introduction
Modal logics have been widely studied for reasoning about knowledge and belief, usually based on the monomodal logics S 5 and KD45 for reasoning about knowledge and belief respectively. Both logics have the axioms 4 : 2ϕ → 22ϕ and 5 : ¬2ϕ → 2¬2ϕ, which mean that knowledge and belief both satisfy positive and negative introspection. The logic S 5 also has the axiom T : 2ϕ → ϕ, which means that knowledge is veridical, while KD45 also has the axiom D : 2ϕ → ¬2¬ϕ, which means that belief is consistent. For multiple agents, the usual solution is to use a multimodal logic which has m pairs of modal operators 2i and 3i , for some fixed number m. The base multimodal logics for reasoning about knowledge and belief are typically S5(m) and KD45(m) , respectively, where each pair 2i and 3i is of type S5 and KD45, usually with no interactions between modal operators of different indices (see, e.g., [14, 8, 22, 35]). An alternative is to use S 5 and KD45 type modalities labelled with agent terms [10], again without interaction axioms. When the knowledge and beliefs of multiple agents interact then interactions between the modalities are used. One approach is to use bi-modal logics with interactions [33, 27]. Another is to extend the multimodal grammar logics of Fari˜ nas del Cerro and Penttonent [5] with (seriality) axioms D for reasoning about knowledge and belief. Recall that a grammar logic is a multimodal logic with axioms of the form [s1 ] . . . [sn ]ϕ → [t1 ] . . . [tm ]ϕ, where [si ] and [tj ] are universal (box-like) modal operators labelled by constants si and tj , respectively (see [2, 3, 6, 34, 16, 7, 12]). ???
National ICT Australia (NICTA) is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council.
For knowledge distributed over groups of agents there is the well-known multimodal logic of common knowledge and distributed knowledge among a group of agents (see [14, 8, 22]) as well as a similar logic for reasoning about mutual beliefs of agents [1]. There are also multi-modal logics of dynamic belief and knowledge for use in multi-agent systems [20, 32]. We concentrate on seven specific multimodal logics of belief, studied by Nguyen in the context of modal logic programming and modal deductive databases [28, 30]. They are introduced in Section 2.2 as: KDI4, KDI4s , KDI4s 5, KDI45 for reasoning about multi-degree belief; KD4s 5s for reasoning about distributed systems of belief; and KD45(m) , KD4Ig 5a for reasoning about epistemic states of agents in multi-agent systems. All except KD45(m) contain interaction axioms. Both KDI4 and KDI4s are grammar logics extended with axiom D. Tableau methods have been successively applied to modal logics to give decision procedures (see, e.g., [9, 13, 3, 35, 11, 19, 18]). There are also tableau calculi for a number of multimodal logics of knowledge and/or belief, e.g., [3, 35, 10]. But [3] concerns only grammar logics, while [35, 10] lack interaction axioms between modal operators of different indices. As far as we know, only KDI4 and KD45(m) from the seven multimodal logics (listed above) have labelled tableau calculi [3, 35]. We know of no non-labelled tableau calculi for any of these logics although Shvarts’ calculus for KD45 from [11] can be extended trivially to KD45(m) . Nor do we know of any tableau calculus for the multimodal logic of mutual belief introduced by Aldewereld et al. [1], which is similar to our KD4s 5s . Here, we develop (non-labelled) tableau calculi for all seven of the above multimodal logics of belief. Our tableau calculi are sound, complete, cut-free and have the analytic superformula property, so they are decision procedures. Our tableau calculi require clausal form and are called clausal tableau calculi. Transformation to clauses was first used for sequent calculi by Mints [23] and Hudelmaier [17] and then applied to tableau calculi by Nguyen [24]. Hudelmaier and Nguyen showed that clausal form can give better space bounds and improved decision procedures for some modal logics [17, 24]. We show that clauses can also simplify the task of developing tableau calculi for certain modal logics by giving clausal tableau calculi for the complicated logics KDI45 and KD4Ig 5a . Finally, we show that our tableaux can be used to estimate the complexity of the satisfiability problem in five of the multimodal logics we consider: NP-complete for KDI4s 5, KD4s 5s , KDI45, and PSPACE-complete for KDI4 and KDI4s . The complexity for KD45(m) is known to be PSPACE-complete [14]. We do not have an exact complexity result for KD4Ig 5a . The rest of this paper is organized as follows. In Section 2, we specify the seven multimodal logics of belief and give definitions for tableau calculi. In Section 3, we define modal clauses and consider transformation to clauses. In Section 4, we present our tableau calculi and prove their soundness. In Section 5, we prove completeness of the calculi. Section 6 deals with the complexity of the logics. Section 7 contains concluding remarks. 2
2 2.1
Preliminaries Definitions for Multimodal Logics
We consider multimodal logics with m pairs of modal operators 2i and 3i , where 1 ≤ i ≤ m. We use p and q to denote primitive propositions. Formulae of our language are recursively defined using the BNF grammar below: ϕ ::= ⊥ | p | ¬ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ | 2i ϕ | 3i ϕ The modal depth of a formula ϕ is the maximal nesting depth of modal operators occurring in ϕ. For example, the modal depth of 21 (22 p ∨ 31 q) is 2. A Kripke frame is a tuple hW, τ, R1 , . . . , Rm i, where W is a nonempty set of possible worlds, τ ∈ W is the actual world, and each Ri is a binary relation on W , called the accessibility relation for 2i and 3i . If Ri (w, u) holds, then we say that the world u is accessible from the world w via Ri . A Kripke model is a tuple hW, τ, R1 , . . . , Rm , hi, where hW, τ, R1 , . . . , Rm i is a Kripke frame and h is a function mapping worlds to sets of primitive propositions. For w ∈ W , the set of primitive propositions “true” at w is h(w). Given a Kripke model M = hW, τ, R1 , . . . , Rm , hi and a world w ∈ W , the satisfaction relation is defined as usual for the classical connectives with two extra clauses for the modalities as below: M, w 2i ϕ M, w 3i ϕ
iff iff
∀v ∈ W.Ri (w, v) implies M, v ϕ ∃v ∈ W.Ri (w, v) and M, v ϕ.
We say that ϕ is satisfied at w in M if M, w ϕ. We say that ϕ is satisfied in M , and write M ϕ, and call M a model of ϕ, if M, τ ϕ. If we allow all Kripke models (with no restrictions on the accessibility relations) then we obtain a multimodal logic which has a standard Hilbert-style axiomatisation denoted by K(m) . Other normal multimodal logics are obtained by adding certain axioms to K(m) . Suppose a normal first-order definable modal logic L is characterised by a class C of Kripke frames. The first order formulae describing C are L-frame restrictions, and any frame in C is an L-frame. A model M is an L-model if its underlying frame is an L-frame. We say that ϕ is L-satisfiable if there exists an L-model of ϕ, i.e. an L-model satisfying ϕ. A formula ϕ is said to be L-valid and called an L-tautology if ϕ is true in every L-model. 2.2
Multimodal Logics of Belief
To reflect properties of belief, one can extend K(m) with some of the axioms: Name (D) (I) (4) (4s ) (5) (5s )
Schema 2i ϕ → ¬2i ¬ϕ 2i ϕ → 2j ϕ if i > j 2 i ϕ → 2i 2i ϕ 2 i ϕ → 2j 2i ϕ ¬2i ϕ → 2i ¬2i ϕ ¬2i ϕ → 2j ¬2i ϕ
Meaning belief is consistent subscript indicates degree of belief belief satisfies positive introspection belief satisfies strong positive introspection belief satisfies negative introspection belief satisfies strong negative introspection 3
The following systems are intended for reasoning about multi-degree belief: KDI4s = K(m) + (D) + (I) + (4s ) KDI4 = K(m) + (D) + (I) + (4) KDI4s 5 = K(m) + (D) + (I) + (4s ) + (5) KDI45 = K(m) + (D) + (I) + (4) + (5) In the above systems, the axiom (I) gives 2i ϕ the meaning “ϕ is believed up to degree i”, and 3i ϕ can be read as “it is possible weakly at degree i that ϕ”. The axioms (5) are controversial as they are quite strong. For this reason, we consider also KDI4 and KDI4s . Note that the axiom (5s ) is derivable in KDI4s 5. Although the logic KDI4 belongs to the class of logics considered by Debart et al. [4] (for multimodal logic programming) and the axioms I and 4s are possible axioms for grammar logics, the reading of modal indices as degrees of belief has not been considered by other authors. Our logics of multi-degree belief are similar to graded modal logics [22], where grades constrain the number of worlds accessible from the current world, but our degrees of belief are symbolic rather than numeric. For multi-agent systems, we use subscripts on 2 and 3 to denote agents and assume that 2i ϕ stands for “agent i believes that ϕ is true” and 3i ϕ stands for “ϕ is considered possible by agent i”. For distributed systems of belief we can use the logic system KD4s 5s = K(m) + (D) + (4s ) + (5s ) In this system, all agents have full access to the belief bases of other agents: they are united as “friends”. Our logic KD4s 5s has features similar to the logic S5P(m) introduced by Meyer and van der Hoek [21], but S5P(m) is intended for formalising preference and the operators Pi of [21] need not be serial. In another kind of multi-agent system, agents are “opponents” who play against each other, and each agent may want to simulate the epistemic states of the others. To write a program for one agent, we may need to use modal operators of the other agents. A suitable logic for this problem is: KD45(m) = K(m) + (D) + (4) + (5) As mentioned earlier, the logic KD45(m) has been intensively studied. We use a subscript in KD45(m) to distinguish the logic from the monomodal logic KD45, while there is not such a need for the other considered multimodal logics. To capture common belief of a group of agents, one can extend the logic KD45(m) with modal operators for groups of agents and some additional axioms. Suppose that there are n agents and m = 2n − 1. Let g be an one-to-one function that maps every natural number less than or equal to m to a non-empty subset of {1, . . . , n}. Suppose that an index 1 ≤ i ≤ m stands for the group of agents whose indices form the set g(i). We can adopt the axioms (D), (4), and additionally (Ig ) : 2i ϕ → 2j ϕ if g(i) ⊃ g(j), so i indicates a group that contains the group identified by j, and (5a ) : ¬2i ϕ → 2i ¬2i ϕ if g(i) is a singleton, so that i stands for an agent. Thus, for reasoning about belief and common belief, we can use: KD4Ig 5a = K(m) + (D) + (4) + (Ig ) + (5a ) 4
Here we want to catch the most important properties of belief and common belief, and the aim is not to give an exact formulation of belief or common belief. This logic is therefore different in nature from the well-known multimodal logics of common knowledge with fixed-points. It also differs from the multimodal logic of mutual belief introduced by Aldewereld et al. [1]. Our modal operator of common belief satisfies positive introspection, while the operator of mutual belief introduced in [1] lacks this property. On the other hand, the latter operator has some properties that the former does not have. The given axioms correspond to the following frame restrictions: Axiom (D) (I) (Ig ) (4) (4s ) (5) (5s ) (5a )
2.3
Corresponding Condition ∀u ∃v Ri (u, v) Rj ⊆ Ri if i > j Rj ⊆ Ri if g(i) ⊇ g(j) ∀u, v, w (Ri (u, v) ∧ Ri (v, w) → Ri (u, w)) ∀u, v, w (Rj (u, v) ∧ Ri (v, w) → Ri (u, w)) ∀u, v, w (Ri (u, v) ∧ Ri (u, w) → Ri (w, v)) ∀u, v, w (Rj (u, v) ∧ Ri (u, w) → Ri (v, w)) as for (5) if g(i) is a singleton
Definitions for Tableau Calculi
As in our previous works on tableau calculi [11, 27, 12], our tableau formulation traces their roots to Hintikka via [31]. A tableau rule σ consists of a numerator N above the line and a (finite) list of denominators D1 , D2 , . . . , Dk (below the line) separated by vertical bars. The numerator is a finite formula set, and so is each denominator. As we shall see later, each rule is read downwards as “if the numerator is L-satisfiable, then so is one of the denominators”. The numerator of each tableau rule contains one or more distinguished formulae called the principal formulae. A tableau calculus CL for a logic L is a finite set of tableau rules. A CL-tableau for X is a tree with root X whose nodes carry finite formula sets obtained from their parent nodes by instantiating a tableau rule with the proviso that if a child s carries a set Z and Z has already appeared on the branch from the root to s then s is an end node. Let ∆ be a set of tableau rules. We say that Y is obtainable from X by applications of rules from ∆ if there exists a tableau for X which uses only rules from ∆ and has a node that carries Y . A branch in a tableau is closed if its end node carries only ⊥. A tableau is closed if every one of its branches is closed. A tableau is open if it is not closed. A finite formula set X is CL-consistent if every CL-tableau for X is open. If there is a closed CL-tableau for X then X is CL-inconsistent. A tableau calculus CL is sound if for all finite formula sets X, if X is L-satisfiable then X is CL-consistent. It is complete if for all finite formula sets X, if X is CLconsistent then X is L-satisfiable. Let σ be a rule of CL. We say that σ is sound w.r.t. L if for every instance σ 0 of σ, if the numerator of σ 0 is L-satisfiable then so is one of the denominators of σ 0 . Any CL containing only rules sound w.r.t. L is sound. 5
3
Modal Clauses
A classical literal is a formula of the form p or ¬p, where p is a primitive proposition. We use a, b, c to denote classical literals, and to denote a possibly empty sequence of universal (box-like) modal operators, which is also called a universal modality. We write [ϕ1 , . . . , ϕk ] to denote (ϕ1 ∨ . . . ∨ ϕk ) with the emphasis that the order of disjuncts is not important. A (modal) clause is a formula of the form [a1 , . . . , ak ], [a, 2i b], [a, 3i b], or 3i b, where is called the modal context. Proposition 1. Every formula set can be transformed in quadratic time to a set of clauses which is equisatisfiable in any normal modal logic to the first set. Proof. Repeatedly replace a complex formula by a new primitive proposition which is “defined” by that formula (see, e.g., [23, 17, 24]). For example, (ϕ ∨ 3i ψ), where ψ is not a primitive proposition, is replaced by (ϕ ∨ 3i p) and 2i (¬p ∨ ψ), where p is a fresh primitive proposition (defining ψ via the second added clause). We need to do at most n such replacements, where n is the sum of the lengths of formulae belonging to the input set. Hence the transformation can be done in quadratic time. We now define L-clauses using a case-distinction on L as: For L ∈ {KDI4, KDI4s , KD4Ig 5a }: every clause is an L-clause; For L ∈ {KD4s 5s , KDI4s 5}: an L-clause is a clause with modal depth 1 or 0; For KD45(m) : a KD45(m) -clause is a clause ϕ whose modal context has rightmost modal operator index l, and contains no subsequence of the form 2i 2i , and if ϕ is of the form [a, 2j b], [a, 3j b] or 3j b then j 6= l; For KDI45: a KDI45-clause is a clause ϕ whose modal context has rightmost modal operator index l, and contains no subsequence of the form 2i 2j with i ≤ j, and if ϕ is of the form [a, 2k b], [a, 3k b] or 3k b, then k < l. For convenience, if L ∈ {KDI45, KD4Ig 5a } then we also call a formula an Lclause if it is of the form [ϕ, ψ], where ϕ and ψ are of the form 2i a, 3i a or a. Proposition 2. Every formula set can be transformed in quadratic time to a set of L-clauses preserving L-satisfiability, where L is one of our seven logics. Proof. First, transform the formula set to a set of clauses preserving L-satisfiability, as in the proof of Proposition 1. For L ∈ {KDI4, KDI4s , KD4Ig 5a } we are done. Now observe that if ∇i is either 2i or 3i then in: KD4s 5s and KDI4s 5: we have ∇i ∇0j ϕ ≡ ∇0j ϕ KD45(m) : we have ∇i ∇0i ϕ ≡ ∇0i ϕ KDI45: if i ≤ j then ∇i ∇0j ϕ ≡ ∇0j ϕ. Also observe that, if 2i ∇j ϕ ≡ 3i ∇j ϕ ≡ ∇j ϕ in L for a fixed i and a fixed ∇j , then 2i (a ∨ ∇j b) ≡ 2i a ∨ ∇j b in L. Using these equivalences and the technique of replacing a complex formula by a new primitive proposition which is “defined” by that formula, we can transform every set of clauses to a set of L-clauses preserving L-satisfiability, for L ∈ {KD4s 5s , KDI4s 5, KD45(m) , KDI45}. It is easily seen that the complexity of the whole transformation is quadratic, as is the size of the resulting set of clauses. 6
(⊥)
X; p; ¬p ⊥
(∨)
X; [ϕ1 , . . . , ϕk ] X; ϕ1 | . . . | X; ϕk
X X; 3i >
(D)
(3L )
X; 3i a a ; trans(L, i, X)
(I5a )
X; 2i 2j ϕ (i > j) X; 2i 2j ϕ; 2j ϕ
(I5c )
X; 2i [a, 2j b] (i > j) X; 2i [a, 2j b]; [2j a, 2j b]
(I5b )
X; 2i 3j b (i > j) X; 2i 3j b; 3j b
(I5d )
X; 2i [a, 3j b] (i > j) X; 2i [a, 3j b]; [2j a, 3j b]
(Ig )
X; 2i ϕ (‡) X; 2i ϕ; 2j ϕ
(5aa )
X; 2i 2i ϕ (†) X; 2i ϕ
(5da )
X; 2i [a, 2i b] (†) X; [2i a, 2i b]
(5ba )
(Ig0 )
X; 2j [a, 2i b] (‡) X; 2j [a, 2i b]; 2j [a, 2j b]
X; 2i 3i b (†) X; 2i 3i b; 3i b
(‡) is g(i) ⊃ g(j)
(5ca )
X; 3i b (†) X; 2i 3i b; 3i b
(5ea )
X; 2i [a, 3i b] (†) X; [2i a, 3i b]
(†) is g(i) must be a singleton
The rule (3L ) is transitional. All other rules are static. Fig. 1. Clausal Tableau Rules
4
Clausal Tableau Systems
In this section, L is one of our seven logics: i.e. KDI4, KDI4s , KDI4s 5, KD4s 5s , KD45(m) , KDI45, or KD4Ig 5a . In clausal tableau calculi, numerators of rules are sets of clauses, so for an L-clausal tableau calculus, we require that numerators of rules are sets of L-clauses. As in [11], we categorise each rule either as a static rule or as a transitional rule: see Figure 1. The intuition behind this sorting is that in the static rules, the numerator and denominator represent the same world (in the same model), whereas in the transitional rules, the numerator and denominator represent different worlds (in the same model). When constructing an L-model graph, if ϕ belongs to a world w and we want to connect w to u via Ri , then certain logic-specific formulae are added to u . For the moment we denote such formulae by trans(L, i, ϕ), and define them precisely later. For a set of L-clauses X, define [ trans(L, i, X) = {trans(L, i, ϕ) | ϕ ∈ X}. For L ∈ {KDI4, KDI4s , KDI4s 5, KD4s 5s , KD45(m) }, define the tableau calculus CL using the rules from Figure 1 as CL = {(⊥), (∨), (D)} ∪ {(3L )}. For L = KDI45, let the tableau calculus be CKDI45 = {(⊥), (∨), (D), (I5a ), (I5b ), (I5c ), (I5d )} ∪ {(3L )}. For the rules (I5a ), (I5b ), (I5c ), (I5d ), if the numerator is a set of KDI45-clauses, then the condition i > j is always true. 7
For L = KD4Ig 5a , let the tableau calculus CKD4Ig 5a be CKD4Ig 5a = {(⊥), (∨), (D), (Ig ), (Ig0 ), (5aa ), (5ba ), (5ca ), (5da ), (5ea )} ∪ {(3L )}. We now define the function trans(L, i, ϕ) for six of our seven logics. trans(KDI4, i, ϕ) = trans(KDI4s , i, ϕ) = trans(KDI4s 5, i, ϕ) = trans(KD4s 5s , i, ϕ) = trans(KD45(m) , i, ϕ) = trans(KD4Ig 5a , i, ϕ) =
{ψ, 2j ψ | ϕ = 2j ψ and j ≥ i} {ϕ | ϕ = 2j ψ} ∪ {ψ | ϕ = 2j ψ and j ≥ i} {ψ | ϕ = 2j ψ and j ≥ i} {ψ | ϕ = 2i ψ} {ψ | ϕ = 2i ψ} {ψ, 2j ψ | ϕ = 2j ψ and g(j) ⊇ g(i)}.
Note that the sets of rules for CKD4s 5s and CKD45(m) are the same, but the difference lies in the definitions of L-clauses. Also note that trans(KD4Ig 5a , i, ϕ) has similarities to trans(KDI4, i, ϕ). Only trans(KDI45, i, ϕ) remains, and the intuition for specifying trans(KDI45, i, ϕ) is that the skeleton of a desired KDI45-model is a tree such that each of its branches is a sequence of edges created via accessibility relations Ri1 , . . . , Rik with i1 > . . . > ik . Recall that a KDI45-clause is a clause ϕ whose modal context has rightmost modal operator index l, and contains no subsequence of the form 2i 2j with i ≤ j, and if ϕ is of the form [a, 2k b], [a, 3k b] or 3k b, then k < l. Consider the following cases where mmi(ψ) is the maximal modal index occurring in ψ: trans(KDI45, i, ϕ) = {ψ, 2i−1 ψ} {2i−1 ψ} {[a, ψ], [2i−1 a, ψ]} {[a, ψ], [2i−1 a, ψ]} {3i−1 b} ∅
if ϕ = 2j ψ, j ≥ i, and mmi(ψ) < i − 1 if ϕ = 2j 2i−1 ψ and j ≥ i if ϕ = 2j [a, ψ], j ≥ i, and ψ = 2i−1 b if ϕ = 2j [a, ψ], j ≥ i, and ψ = 3i−1 b if ϕ = 2j 3i−1 b and j ≥ i otherwise
Note that for a tableau rule of CL, if the numerator is a set of L-clauses then each of the denominators of the rule is also a set of L-clauses. A tableau calculus CL has the analytic superformula property iff to every finite ∗ such that X ∗ contains all clauses set X of L-clauses we can assign a finite set XCL CL that may appear in any CL tableau for X. Our tableau calculi have the analytic ∗ calculated as follows. Let Sf (ϕ) be the set of all superformula property, with XCL subformulae of ϕ (including ϕ), and let e = Sf (X) ∪ ¬Sf (X) ∪ {⊥}. Sf (X) = ∪{Sf (ϕ) | ϕ ∈ X}, ¬X = {¬ϕ | ϕ ∈ X}, X ∗ = X. e For L ∈ {KDI4, KDI4s , KDI4s 5, KD4s 5s , KD45(m) }, choose XCL For L ∈ {KDI45, KD4Ig 5a }, let 2X = {2i ϕ | ϕ ∈ X and 1 ≤ i ≤ m}, let X|2 be the set of all clauses with modal depth ≤ 2, containing at most 2 primitive ∗ = propositions, which are primitive propositions occurring in X, and choose XCL e ∪ 2Sf (X) ∪ X|2 . X
8
Lemma 1. The tableau calculi defined in this section are sound. Proof. Let L be one of our seven logics. We show that CL contains only rules sound with respect to L. Consider the only nontrivial case, when L ∈ {KDI45, KD4Ig 5a }. For each static rule δ of CL, if δ is not (∨) then the denominator of δ is equivalent in L to the numerator of δ. Hence all static rules of CL are sound with respect to L. The transitional rules of CL are sound with respect to L because if ψ ∈ trans(L, i, ϕ) then ϕ → 2i ψ is L-valid.
5
Completeness of the Calculi
In this section, L is still one of our seven logics of belief and CL is the corresponding tableaux calculus. We prove completeness of our calculi via model graphs in a similar way as in [31, 11, 26, 12]. To show completeness of CL we give an algorithm that, given a finite CL-consistent set X of L-clauses, constructs an L-model graph (defined in Section 5.2) for X that satisfies every one of its formulae at the corresponding world. 5.1
Saturation
For a static rule δ, let δ 0 be the rule obtained from δ by adding the principal formulae to each of the denominators. Let SCL = {δ 0 | δ is a static rule of CL}. Note: for any rule of SCL, the numerator is included in each of its denominators. A set X is closed w.r.t. a tableau rule if, whenever the rule is applicable to X, one of the corresponding instances of the denominators is equal to X. For a finite CL-consistent set X of L-clauses, Y is a CL-saturation of X if Y is a maximal CLconsistent set of L-clauses obtainable from X by applications of the rules of SCL. In some cases, instead of CL-saturation we need a slightly different notion. For a finite CL-consistent set X of L-clauses, Y is a minimally CL-saturated set of X if Y is a minimal finite CL-consistent superset of X that is closed w.r.t. every rule of SCL. As stated by the following lemma, CL-saturations and minimally CL-saturated sets have the same nature as “downward saturated sets” defined in the works by Hintikka [15] and Gor´e [11]. Lemma 2. Let X be a finite CL-consistent set of L-clauses and Y a CL-saturation ∗ and Y is closed w.r.t. the or a minimally CL-saturated set of X. Then X ⊆ Y ⊆ XCL rules of SCL. Furthermore, there is an effective procedure that, given a finite CLconsistent set X of L-clauses, constructs some CL-saturation and some minimally CL-saturated set of X. The assertions about CL-saturations can be proved as in [27], and the assertions about minimally CL-saturated sets are immediate consequences. 9
5.2
Proving Completeness via Model Graphs
A model graph is a tuple hW, τ, R1 , . . . , Rm , Hi, where hW, τ, R1 , . . . , Rm i is a Kripke frame, and H is a function that maps each world to a set of clauses. We sometimes treat model graphs as models with h(w) as the restriction of H(w) to the set of primitive propositions. A model graph is called an L-model graph if its frame satisfies all L-frame restrictions. A model graph hW, τ, R1 , . . . , Rm , Hi is saturated if it satisfies the following conditions for every w ∈ W : ∨: if [ϕ1 , . . . , ϕk ] ∈ H(w), then some ϕi with 1 ≤ i ≤ k belongs to H(w); 2: if 2i ϕ ∈ H(w), then for every u such that Ri (w, u), ϕ ∈ H(u); 3: if 3i a ∈ H(w), then there exists u ∈ W s.t. Ri (w, u) and a ∈ H(u). A saturated model graph is consistent if no world contains ⊥ nor {p, ¬p} for any p. We use the term “model graph” merely to denote a data structure, while Rautenberg’s model graphs are required to be saturated and consistent. Lemma 3. If M = hW, τ, R1 , . . . , Rm , Hi is a consistent saturated model graph, then M is a model of H(τ ). Proof. By induction on the construction of ϕ that if ϕ ∈ H(w) then M, w ϕ. Given a finite CL-consistent set X of L-clauses, we construct a consistent saturated L-model graph M = hW, τ, R1 , . . . , Rm , Hi such that X ⊆ H(τ ) as a putative L-model for X. If for every w ∈ W , H(w) is a CL-saturation of some set, then the first condition of being a saturated model graph is satisfied. 5.3
Completeness
In the following algorithm, the worlds of the constructed model graph are marked either as unresolved or as resolved. Algorithm 1 Input: a finite CL-consistent set X of formulae. Output: an L-model graph M = hW, τ, R1 , . . . , Rm , Hi satisfying X. 1. Let W = {τ }, Ri0 = ∅ for 1 ≤ i ≤ m, H0 (τ ) = X, and H(τ ) be a CL-saturation of X. Mark τ as unresolved. 2. While there are unresolved worlds, take one, say w, and do: (a) For every formula 3i a in H(w): i. If there exists u ∈ W s.t. H0 (u) = {a} ∪ trans(L, i, H(w)), then add the pair (w, u) to Ri0 . ii. Else, add a new world u to W , let H0 (u) = {a} ∪ trans(L, i, H(w)) and H(u) be a CL-saturation of H0 (u), mark u as unresolved, and add the pair (w, u) to Ri0 . (b) Mark w as resolved. 3. Let Ri , for 1 ≤ i ≤ m, be the least extensions of Ri0 such that hW, τ, R1 , . . . , Rm i is an L-frame. ∗ for This algorithm always terminates because H0 is one-to-one and H0 (w) ⊆ XCL every w. Here is the main lemma concerning its properties:
10
Lemma 4. For L 6= KD4Ig 5a , let X be a finite CL-consistent set of L-clauses, and M = hW, τ, R1 , . . . , Rm , Hi be the model graph constructed by Algorithm 1 for X. Then M is a consistent L-model graph satisfying X. Proof. It is clear that M is a consistent L-model graph. Let Ri0 , for 1 ≤ i ≤ m, be the relations mentioned in the algorithm. Suppose that L 6= KDI45. Since X ⊆ H(τ ), by Lemma 3, it suffices to prove that M is a saturated model graph. For this, we only need to prove that, for every w ∈ W , if 2i ϕ ∈ H(w) then for every u such that Ri (w, u), ϕ ∈ H(u). Suppose that 2i ϕ ∈ H(w) and Ri (w, u) holds and consider the following cases: L = KDI4: Since Ri (w, u) holds, there exist w0 , . . . , wk such that w0 = w, wk = u, and for 1 ≤ j ≤ k there exists ij ≤ i such that Ri0 j (wj−1 , wj ) holds. Since 2i ϕ ∈ H(w0 ) and i1 ≤ i, we have {ϕ, 2i ϕ} ⊆ trans(KDI4, i1 , H(w0 )). Hence {ϕ, 2i ϕ} ⊆ H(w1 ). Repeating this argument, we can derive that {ϕ, 2i ϕ} ∈ H(wk ). Hence ϕ ∈ H(u). L = KDI4s : Since Ri (w, u) holds, there exist w0 , . . . , wk such that w0 = w, wk = u, and for 1 ≤ j ≤ k, Ri0 j (wj−1 , wj ) holds for some ij , with ik ≤ i. Since 2i ϕ ∈ H(w0 ), we have 2i ϕ ∈ trans(KDI4s , i1 , H(w0 )). Hence 2i ϕ ∈ H(w1 ). Repeating this argument, we can derive that 2i ϕ ∈ H(wk−1 ). Consequently, since ik ≤ i, we have ϕ ∈ H(wk ), i.e. ϕ ∈ H(u). L = KD4s 5s : Because of the form of KD4s 5s -clauses, τ is the only world that may contain formulae with modal operators. Since 2i ϕ ∈ H(w), we have w = τ . Since Ri (τ, u) is equivalent to Ri0 (τ, u), it follows that ϕ ∈ H(u). L = KDI4s 5: As for the above case, we have w = τ . Since Ri (τ, u) holds, Rj0 (τ, u) holds for some j ≤ i. Hence ϕ ∈ H(u). L = KD45(m) : Because of the form of KD45(m) -clauses and the assumption that 2i ϕ ∈ H(w), there is no v such that Ri0 (v, w) holds. Hence there is no v such that Ri (v, w) holds. Consequently, since Ri (w, u) holds, we derive that Ri0 (w, u) holds. This implies ϕ ∈ H(u). Remark 1. Now suppose that L = KDI45. Observe that: (a) For every u, v, ϕ, if Ri0 (u, v) holds and ϕ ∈ H(v) then every modal index occurring in ϕ is less than i. This follows from the form of KDI45-clauses and the definition of trans(KDI45, , ). (b) From (a), for every u, v, w, if Ri0 (u, v) and Rj0 (v, w) hold then i > j. (c) For every u, v, w1 , w2 , if Ri0 (u, v) holds, and w1 belongs to the subtree with root v created by {Rk0 | 1 ≤ k ≤ m}, but w2 does not, and Rj (w1 , w2 ) or Rj (w2 , w1 ) holds, then j ≥ i. Similarly, for every u, v, w, if Ri0 (u, v) and Rj (w, v) hold, then j ≥ i. These assertions follow because, at the beginning, when extending {Rk0 | 1 ≤ k ≤ m} to {Rk | 1 ≤ k ≤ m} using the KDI45-frame conditions, u is the only outside node which is connected to the mentioned subtree and this connection is via Ri0 . Let ϕ ∈ H(w). We prove by induction on the structure of ϕ that M, w ϕ. It suffices to consider the case when ϕ = ψ with modal context = 2i1 . . . 2ik and k ≥ 1. Since ϕ is a KDI45-clause, we have i1 > . . . > ik . Let w0 = w, w1 , . . . , wk be worlds such that Ris (ws−1 , ws ) holds for all 1 ≤ s ≤ k. We show that M, wk ψ. 11
If w 6= τ , then there is v such that Rl0 (v, w) holds for some l. By Remark 1(a), l > i1 > . . . > ik . By Remark 1(c), it follows that w1 , . . . , wk must be nodes in the subtree with root w created by {Ri0 | 1 ≤ i ≤ m} and different from w (otherwise we would have is ≥ l for some 1 ≤ s ≤ k). Suppose that u0 = w, uh = wk , and Rj0 t (ut−1 , ut ) holds for 1 ≤ t ≤ h. By Remark 1(b), we have j1 > . . . > jh . By Remark 1(c), there exists 1 ≤ s ≤ k such that is ≥ j1 . Hence i1 ≥ j1 . By Remark 1(c), we also have ik ≥ jh . There are the following cases: ψ = [a1 , . . . , ap ] : Since 2i1 . . . 2ik ψ ∈ H(u0 ), i1 ≥ j1 and ik ≥ jh , by the definition of trans(KDI45, , ) and the tableau rule (I5a ), there exists i ≥ jh such that 2i ψ ∈ H(uh−1 ). Hence ψ ∈ H(uh ). By the inductive hypothesis, it follows that M, uh ψ, which means M, wk ψ. ψ = [a, 2i b] : If i < jh then apply the argument as in the above case. Suppose that i ≥ jh . Let 1 ≤ t ≤ h be the smallest index such that i ≥ jt . If t > 1 then i < jt−1 , and similarly to above case, we derive that either 2j ψ ∈ H(ut−1 ) for some j > i or [2i a, 2i b] ∈ H(ut−1 ). If t = 1 then, by the tableau rules (I5a ) and (I5c ), we derive that [2i a, 2i b] ∈ H(ut−1 ). Hence, by the tableau rules (I5c ) and (∨), 2i a ∈ H(ut−1 ) or 2i b ∈ H(ut−1 ). If 2i a ∈ H(ut−1 ) then we can derive that a ∈ H(uh ). If 2i b ∈ H(ut−1 ) then from the definition of trans(KDI45, , ) and Remark 1(c) we can derive that M, uh 2i b. Thus, in both of the cases, we have M, uh [a, 2i b], which means that M, wk ψ. ψ = [a, 3i b] : If i < jh then apply the argument as in the case when ψ is of the form [a1 . . . , ap ]. Suppose that i ≥ jh . Let 1 ≤ t ≤ h be the smallest index such that i ≥ jt . If t > 1 then we have i < jt−1 , and similarly as in the above cases, we derive that either 2j ψ ∈ H(ut−1 ) for some j > i or [2i a, 3i b] ∈ H(ut−1 ). If t = 1 then, by the tableau rules (I5a ) and (I5d ), we derive that [2i a, 3i b] ∈ H(ut−1 ). Hence, by the tableau rules (I5d ) and (∨), 2i a ∈ H(ut−1 ) or 3i b ∈ H(ut−1 ). If 2i a ∈ H(ut−1 ) then we can derive that a ∈ H(uh ). If 3i b ∈ H(ut−1 ) then there exists u such that Ri0 (ut−1 , u) holds and b ∈ H(u), and hence M, uh 3i b. Thus, in both of the cases, we have M, uh [a, 3i b], which means that M, wk ψ. ψ = 3i b : Similar to the above case, but use (I5b ) instead of (I5d ). Lemma 4 does not hold for L = KD4Ig 5a . To deal with KD4Ig 5a , we modify Algorithm 1 to obtain the following algorithm by replacing “CL-saturation” by “minimally CL-saturated set” and forbidding any sequence of two edges of the form Ri0 (u, v) and Ri0 (v, w) if g(i) is a singleton. Algorithm 2 Input: a finite CL-consistent set X of formulae, for L = KD4Ig 5a . Output: an L-model graph M = hW, τ, R1 , . . . , Rm , Hi satisfying X. 1. Let W = {τ }, Ri0 = ∅ for 1 ≤ i ≤ m, H0 (τ ) = X, and H(τ ) be a minimally CL-saturated set of X. Mark τ as unresolved. 2. While there are unresolved worlds, take one, say w, and do: (a) For every formula 3i a in H(w) such that either g(i) is not a singleton or there is no v s.t. Ri0 (v, w) holds: i. If there exists u ∈ W s.t. H0 (u) = {a} ∪ trans(L, i, H(w)), then add the pair (w, u) to Ri0 . 12
ii. Otherwise, add a new world u to W , let H0 (u) = {a} ∪ trans(L, i, H(w)) and H(u) be a minimally CL-saturated set of H0 (u), mark u as unresolved, and add the pair (w, u) to Ri0 . (b) Mark w as resolved. 3. Let Ri , for 1 ≤ i ≤ m, be the least extensions of Ri0 such that hW, τ, R1 , . . . , Rm i is an L-frame. As for Algorithm 1, this algorithm always terminates. We also have the following assertion, which is similar to Lemma 4: Lemma 5. For L = KD4Ig 5a , let X be a finite CL-consistent set of L-clauses, and M = hW, τ, R1 , . . . , Rm , Hi be the model graph constructed by Algorithm 2 for X. Then M is a consistent saturated L-model graph satisfying X. Proof. Clearly, M is a consistent L-model graph. It is sufficient to prove that M satisfies the two last conditions of being a saturated model graph. We first prove that M satisfies the second last condition of being a saturated model graph. Suppose that 3i b ∈ H(w). If g(i) is not a singleton or there is no v such that Ri0 (v, w) holds then, clearly, there exists u such that Ri0 (w, u) and Ri (w, u) hold and b ∈ H(u). Suppose that g(i) is a singleton and we have Ri0 (v, w) for some v. We show that 3i b ∈ H(v). Recall that H(w) is a minimally CLsaturated set of trans(L, i, H(v)) ∪ {a0 } for some a0 . Let 3i b ∈ H(w) have origin from ϕ ∈ trans(L, i, H(v)). There exists j such that g(j) ⊇ g(i) and either 2j ϕ ∈ H(v) or ϕ is of the form 2j ψ and 2j ψ ∈ H(v). Denote this property by (*). There are three cases: – If 3i b is added to H(w) not using the (∨) rule, then 3i b also belongs to H(v), because of (*) and that H(v) is a (minimally) CL-saturated set. – Consider the case when 3i b is added to H(w) using the (∨) rule for [a, 3i b] (a subformula of ϕ). First, we have that a ∈ / H(w) (otherwise it is not necessary to add 3i b to H(w)). Because of (*) and that H(v) is a (minimally) CL-saturated set, both 2i [a, 3i b] and [2i a, 3i b] belong to H(v). Since a ∈ / H(w), we have that 2i a ∈ / H(v). It follows that 3i b ∈ H(v). – Consider the case when 3i b is added to H(w) using the (∨) rule for [2i a, 3i b], which originates from 2i [a, 3i b], which in turn originates from ϕ. First, we have that 2i a ∈ / H(w) (otherwise it is not necessary to add 3i b to H(w)). Because of (*) and that H(v) is a (minimally) CL-saturated set, both 2i [a, 3i b] and [2i a, 3i b] belong to H(v). Since 2i a ∈ / H(w), we have that 2i a ∈ / H(v). It follows that 3i b ∈ H(v). We have shown that 3i b ∈ H(v). Thus, there exists u such that Ri0 (v, u) holds and b ∈ H(u). It follows that Ri (w, u) holds. We now prove that M satisfies the last condition of being a saturated model graph. Suppose that 2i ϕ ∈ H(w) and Ri (w, u) holds. We show that ϕ ∈ H(u). Consider the case when ϕ is a classical literal b, g(i) is a singleton, and Ri0 (v, w) holds for some v. We show that 2i b ∈ H(v). The proof is very similar to the given proof of 3i b ∈ H(v) when assuming 3i b ∈ H(w), but it has one additional subcase, which is considered in the follows. Let 2i b ∈ H(w) have origin from 13
ϕ0 ∈ trans(L, i, H(v)). There exists j such that g(j) ⊇ g(i) and either 2j ϕ0 ∈ H(v) or ϕ0 is of the form 2j ψ and 2j ψ ∈ H(v). Denote this property by (?). Consider the special subcase when 2i b is added to H(w) using the (∨) rule for [2i b, 3i c], which originates from 2i [b, 3i c], which in turn originates from ϕ0 . First, we have that 3i c ∈ / H(w) (otherwise it is not necessary to add 2i b to H(w)). Because of (?) and that H(v) is a (minimally) CL-saturated set, both 2i [b, 3i c] and [2i b, 3i c] belong to H(v). Since 3i c ∈ / H(w), we have that 2i 3i c ∈ / H(v) and 3i c ∈ / H(v) (because 3i c ∈ H(v) would imply that 2i 3i c ∈ H(v), due to the tableau rule (5ca )). It follows that 2i b ∈ H(v). In a similar way, for all the other subcases we can show that 2i b ∈ H(v). If Ri (w, u) holds then Ri0 (v, u) must hold, which implies that b ∈ H(u). Consider the case when ϕ is not a classical literal, g(i) is a singleton, and Ri0 (v, w) holds for some v. Analogously as for the above case, we can show that 2i ϕ ∈ H(v). If Ri (w, u) holds then Ri0 (v, u) must hold, which implies ϕ ∈ H(u). Consider the remaining case when g(i) is not a singleton or there is no v such that Ri0 (v, w) holds. The proof is similar to the case for KDI4. Since Ri (w, u) holds, there exist w0 , . . . , wk such that w0 = w, wk = u, and for 1 ≤ j ≤ k there exists ij such that g(ij ) ⊆ g(i) and Ri0 j (wj−1 , wj ) holds. In particular, if g(i) is a singleton then k = 1 and i1 = i. Since 2i ϕ ∈ H(w0 ) and g(i1 ) ⊆ g(i), we have {ϕ, 2i ϕ} ⊆ trans(L, i1 , H(w0 )). Hence {ϕ, 2i ϕ} ⊆ H(w1 ). Repeating this argument, we can derive that {ϕ, 2i ϕ} ∈ H(wk ). Hence ϕ ∈ H(u), which completes the proof. The following theorem follows from Lemmas 1, 4, and 5. Theorem 1. The tableau systems CL are sound and complete, for L ∈ {KDI4, KDI4s , KDI4s 5, KD4s 5s , KD45(m) , KDI45, KD4Ig 5a }.
6
Complexity
In [14], Halpern and Moses proved that the satisfiability problem in KD45(m) is PSPACE-complete. In this section, using the algorithms of constructing L-model graphs given in the previous section, we analyse the time complexity of the satisfiability problem in the remaining multimodal logics KDI4s 5, KD4s 5s , KDI45, KDI4, KDI4s , KD4Ig 5a . Theorem 2. For KDI4s 5, KD4s 5s , and KDI45, the satisfiability problem is NPcomplete, and for KDI4 and KDI4s , it is PSPACE-complete. Proof. First, observe that computing a CL-saturation of a CL-consistent formula set can be nondeterministically done in polynomial time as follows: while the set is not closed w.r.t. some tableau rule δ from SCL, apply δ to the set, guess the right denominator and update the set to that denominator. If L ∈ {KDI4s 5, KD4s 5s , KDI45} and X is a CL-consistent formula set then an L-model graph constructed by Algorithm 1 for X has a polynomial size: the content of ∗ and has a size bounded by a polynomial each of the possible worlds is a subset of XCL in the size of X; the number of possible worlds of the model graph is bounded by a polynomial in the size of X because each path via {Ri0 | 1 ≤ i ≤ m} has length 0 or 1 for L ∈ {KDI4s 5, KD4s 5s } and a length not greater than m for L = KDI45. This together with the above observation imply that if X is a CL-consistent formula 14
set then an L-model of X can be nondeterministically constructed in polynomial time. Hence the problem of checking satisfiability in KDI4s 5, KD4s 5s , and KDI45 is NP-complete (the lower bound NP-hard is clear). Because KDI4 and KDI4s are “conservative extensions” of the monomodal logic KD4, the satisfiability problem in these logics is PSPACE-hard. We show that the problem is in the PSPACE class. Let X be a formula set with size n. To check whether X is L-satisfiable, where L ∈ {KDI4, KDI4s }, we do not try to construct a consistent L-model graph satisfying X as in Algorithm 1, but just check whether such a model graph exists. For our complexity analysis, we assume the following modifications for Algorithm 1: – An input for this algorithm is not required to be CL-consistent anymore. – “CL-saturation” is replaced by “candidate for CL-saturation”, which is nondeterministically constructed in polynomial time as follows: while the set is not closed w.r.t. some tableau rule δ from SCL, apply δ to the set, guess the right denominator and update the set to that denominator. – When loop-checking, instead of “cross-tree” edges, we allow only backward edges. (Lemma 4 still holds for this modification.) – If a world is inconsistent (in the sense that it contains both p and ¬p for some p) then the computation fails. – If all possible computations fail, then return that the input formula set is Lunsatisfiable, else return that the input formula set is L-satisfiable. During run time of the modified algorithm, we need to keep only the content of the current path in the model graph. To show that checking satisfiability in L ∈ {KDI4, KDI4s } can be done in PSPACE, it is sufficient to show that the current path (via {Ri0 | 1 ≤ i ≤ m}) in the model graph constructed for X always has a length bounded by a polynomial of n (the size of X). Consider the case L = KDI4s . Denote Y2 = {ϕ ∈ Y | ϕ is of the form 2i ψ}. Suppose that there is a path from u to w via {Ri0 | 1 ≤ i ≤ m} in an L-model graph constructed by the modified algorithm for X. Then we have H(w)2 ⊇ H(u)2 . Because of loop-checking, if H(w)2 = H(u)2 then the path from u to w has no more than n nodes. This implies that every path via {Ri0 | 1 ≤ i ≤ m} in an L-model graph constructed for X has a length not greater than n2 . Consider the case L = KDI4. Denote Y2i = {ϕ ∈ Y | ϕ is of the form 2i ψ}. Suppose that there is a path from u to w via {Ri0 | 1 ≤ i ≤ m} in an L-model graph constructed by the modified algorithm for X. Then we have H(w)2m ⊇ H(u)2m . Because of loop-checking, if H(w)2m = H(u)2m then the path from u to w has no 0 (v 0 , v) holds for some v 0 . This implies that every more than n nodes v such that Rm 0 path via {Ri | 1 ≤ i ≤ m} in an L-model graph constructed by the modified algorithm 0 (v 0 , v) holds for some v 0 . Analogously, for X has no more than n2 nodes v such that Rm 0 (v 0 , v) holds contains no more than every (sub)path without v 0 and v such that Rm 2 0 0 n nodes v such that Rm−1 (v , v) holds for some v 0 . Repeating this argument for (m − 2), . . . , 1, we derive that every path via {Ri0 | 1 ≤ i ≤ m} in an L-model graph constructed by the modified algorithm for X has no more than n2m nodes. Because KD4Ig 5a is an extension of the monomodal logic KD4, the satisfiability problem in KD4Ig 5a is PSPACE-hard. Using Algorithm 2 and the argument as in 15
[12], we can derive that the satisfiability problem in KD4Ig 5a is in EXPTIME. We do not know whether the problem is in PSPACE or EXPTIME-hard. We conjecture that it is EXPTIME-hard when m ≥ 7 (i.e. when there are at least 3 single agents)!
7
Conclusions
The multimodal logics considered in this paper are useful for reasoning about belief. The logics KDI4, KDI4s , KDI4s 5, and KDI45 can be used for reasoning about multi-degree belief [28], the logic KD4s 5s can be used for distributed systems of belief [28], and the logic KD4Ig 5a can be used for reasoning about belief and common belief of agents in multi-agent systems [28, 29]. Although the used axioms are widely known, most of the logics considered in this paper, namely, KDI4s , KDI4s 5, KDI45, KD4s 5s , and KD4Ig 5a have not been studied by other authors. Thus, there were no (labelled or non-labelled) tableau systems for these four logics. Only labelled tableau calculi for KDI4 and KD45(m) have previously been developed by other authors [3, 35], although Shvarts’ non-labelled calculus for (KD45) can be extended trivially to KD45(m) [11]. We have given (non-labelled) tableau calculi for all 7 of our multimodal logics of belief. Our calculi are sound, complete, cut-free and have the analytic superformula property. Thus, they give decision procedures. Our tableau calculi for KDI4, KDI4s , KDI4s 5, KD4s 5s , and KD45(m) are simple. The calculi for KDI4 and KDI4s are very similar to the standard tableau calculus for KD4, and clausal form is not essential for these two logics. On the other hand, the simplicity of the tableau calculi for KDI4s 5, KD4s 5s , and KD45(m) is due to the clausal forms for these logics. Traditional tableaux for modal logics do not use clausal form. What advantages might we gain by using clausal form? First, handling only clauses may speed up the search process. We do not have experimental results for this claim yet, but the hope follows from the fact that most efficient SAT solvers for classical propositional logic handle only clauses. Second, by using clauses in “normal form” such as in the case of KDI4s 5, KD4s 5s , KDI45, KD45(m) , the task of developing non-labelled tableau calculi is significantly simplified. For simple logics like KDI4 or KDI4s , clausal form does not matter, but for more complicated logics like KDI45 or KD4Ig 5a , clausal form helps enormously, even when we do not provide a “normal form” for clauses as in the case of KD4Ig 5a .1 Third, clausal tableau calculi sometimes bring interesting results. For example, using them, Nguyen [24] obtained the currently best space bound O(n. log(n)) for the modal logics K4, KD4, and S4. In [25], he also gave (non-labelled) cut-free clausal tableau calculi for the symmetric modal logics KB, KDB, and B. Finally, using our calculi, we have shown that the satisfiability problem for KDI4s 5, KD4s 5s , and KDI45 is NP-complete, and for KDI4 and KDI4s it is PSPACE-complete. The complexity lower bound of KD4Ig 5a remains open. 1
That is, adopting restrictions for KDI4s 5, KD4s 5s , KDI45, KD45(m) is like providing a normal form for clauses in these logics. For KD4Ig 5a , we could adopt the restriction that if g(i) is a singleton then 2i 2i is ruled out. We did not do this, but guaranteed it “on-the-fly” in our algorithms.
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