1 Introduction
Contexts involving epistemic and doxastic [1, 2, 3], agencybased [4, 5] and coalitional [6, 7], as well as deontic [8, 9, 10], reasoning capabilities populate the wide spectrum of settings where modal logics have found natural applications. In such scenarios, modal operators can be used to represent and reason about what agents, or groups of agents, respectively know, believe, have the capability, or have the permission, to bring about.
The semantics of modal operators is usually given in terms of relational models, based on frames consisting of a set of possible worlds equipped with suitable accessibility relations. However, all the modal systems interpreted by means of this kind of semantics, known as normal, validate principles that have been considered problematic or debatable for the aforementioned applications, leading to counterintuitive or unacceptable conclusions. Among the unpleasant features discussed in the literature, one encounters for instance the problem of logical omniscience [3], as well as a number of socalled paradoxes in the representation of agents’ abilities [5] and obligations [11, 12, 13].
To avoid the unwanted consequences of the relational semantics, several nonnormal modal logics have been proposed and studied, tracing back to the seminal works by C.I. Lewis [14], Lemmon [15], Kripke [16], Scott [17], Montague [18], Segerberg [19], and Chellas [20]. The semantics of such systems can be given in terms of neighbourhood models, generalisations of the relational ones that were first introduced by Scott [17] and Montague [18]. In this setting, a frame consists of a set of worlds, each of which is associated with a set of subsets of worlds. Since a subset of worlds can be thought as a proposition (that is true in those worlds), this means that every world in a neighbourhood model is assigned to a set of propositions, those considered necessary with respect to that world. This semantics both generalises the relational one, and avoids the drawbacks of the latter, since the modal principles validated on relational frames that are deemed as problematic for epistemic, coalitional or deontic applications do not hold in general on neighbourhood models.
Nonnormal modalities have been widely investigated as a way to extend propositional logic. A further line of research focuses on the behaviour of modal operators interpreted on neighbourhood frames in combination with firstorder logic. In this direction, a few works have provided completeness results for firstorder nonnormal modal logics [21, 22]. In addition, nonnormal modal extensions of description logics, seen as fragments of firstorder logic with a good tradeoff between expressive power and computational complexity, have been considered for knowledge representation applications [23, 24], also in multiagent coalitional settings [25, 26].
In this paper, we investigate satisfiability of nonnormal modal extensions of description logics. In particular, we study the logics characterised by the class of all neighbourhood frames (), supplemented neighbourhood frames (), neighbourhood frames closed under intersection (), and neighbourhood frames containing the unit (), and combine them with the prototypical description logic. We provide a framework of terminating, correct, and complete tableau algorithms to check satisfiability in such logics interpreted in neighbourhood models with varying domains (in this kind of semantics, the domains of the interpretations at each world can differ; cf. Section 2 for details). We then investigate the satisfiability problems in fragments of these languages obtained by restricting the application of modal operators to formulas only, and provide complexity upper bounds with constant domains (in this case the domains of the interpretations at every world are the same). We leave satisfiability checking procedures for nonrestricted languages interpreted on models with constant domain as open problems.
2 Preliminaries
In this section, we provide preliminary definitions for nonnormal modal description logics, first introducing their syntax, and then giving their semantics based on neighbourhood models.
Syntax
Let and be countably infinite and pairwise disjoint sets of concept names and role names respectively. An concept is an expression of the form
where , , and , with , are modal operators called boxes. A concept inclusion (CI) is an expression of the form , where are concepts. An formula takes the form
where . We will use the following standard definitions for concepts: , ; ; ; (operators are called diamonds). Concepts of the form , are called modalised concepts. Analogous conventions also hold for formulas, for which we set .
Semantics
A neighbourhood frame, or simply frame, is a pair , where is a nonempty set of worlds and, for each , is called a neighbourhood function. A frame is: supplemented if, for all , , , and implies ; closed under intersection if, for all , , , and implies ; and contains the unit if, for all , . An varying domain neighbourhood model, or simply model, based on a neighbourhood frame is a pair , where is a neighbourhood frame and is a function associating with every an interpretation , with nonempty domain , and where is a function such that: for all , ; for all , . An constant domain neighbourhood model is defined in the same way, except that, for all , we have that . Given a model and a world of (or simply in ), the interpretation of a concept in is defined as:
where, for all , the set is called the truth set of with respect to . We say that a concept is satisfied in if there is in such that , and that is satisfiable (over varying or constant neighbourhood models, respectively) if there is a (varying or constant domain, respectively) neighbourhood model in which it is satisfied. The satisfaction of an formula in of , written , is defined as follows:
where is the truth set of . As a consequence of the above definition, we obtain the following condition for diamond formulas: iff . Given a neighbourhood frame and a neighbourhood model , we say that is satisfied in if there is such that , and that is satisfiable (over varying or constant domain neighbourhood models, respectively) if it is satisfied in some (varying or constant domain, respectively) neighbourhood model.
Given a class of frames , by the formula satisfiability problem on (varying or constant domain, respectively) neighbourhood models based on a frame in we mean the problem of deciding whether an formula is satisfied in a (varying or constant domain, respectively) neighbourhood model based on a frame in . In the following, let . Given , the formula satisfiability problem on (varying or constant domain, respectively) neighbourhood models is the formula satisfiability problem on (varying or constant domain, respectively) neighbourhood models based on a frame in the class of:

all neighbourhood frames, for ;

supplemented neighbourhood frames, for ;

neighbourhood frames closed under intersection, for ; and

neighbourhood frames containing the unit, for .
3 Tableaux for Nonnormal Modal Description Logics
In this section, we provide terminating, sound and complete tableau algorithms to check satisfiability of formulas in varying domain neighbourhood models. The notation partly adheres to that of Gabbay et al. [27], while the model construction in the soundness proof is based on the strategy of Dalmonte et al. [28].
We require the following preliminary notions. For a concept or formula , we denote by the negation of put in negation normal form (NNF), defined as usual. Given an formula , we assume without loss of generality that is in NNF, it contains CIs only of the form , and every concept occurring in is also in NNF. We define the weight of a concept in NNF as follows: ; ; . The weight of a formula in NNF is defined as: ; ; . Observe that, for a concept or formula , we have that . We denote by and the set of subconcepts and subformulas of , respectively, and then we set and . The set is the set of role names occurring in . Let . Note that, by our assumption on the form of CIs in , we have .
Moreover, let be a countable set of variables, wellordered by the relation , and let be a countable set of labels. Given an formula , an labelled constraint for takes the form , or , or , where , , , , and . An labelled constraint system for is a set of labelled constraints for . (A labelled constraint for is an labelled constraint for , for some , and similarly for a labelled constraint system for ). A completion set is a nonempty union of labelled constraint system for , and we set .
Concerning variables, we adopt the following terminology. A variable occurs in if contains labelled constraints of the form or , where , or , and . In addition, is said to be fresh for if does not occur in and , for every that occurs in . (These notions can be used with respect to , whenever ). Without loss of generality, we assume that, whenever occurs in , the labelled constraint is in . Also, if , we call an successor of with respect to . Finally, given variables in an labelled constraint system , we say that is blocked by in if and .
A completion set contains a clash if , or , for some , and formula or concept . A completion set with no clash is clashfree. Given , a completion set is complete if no rule from Figure 1 is applicable to , where is either or , with , for , and is either or , with , with respect to the following application conditions associated to each rule:

; ;

; ;

is not blocked by any variable in , there is no such that , and is the minimal variable fresh for ;

;

occurs in an labelled constraint in and ;

is the minimal variable fresh for , and there is no such that ;

is fresh for , and there is no such that , or , for some , where and are as in Figure 1.
for tree= calign=center, grow’=east, parent anchor=east, child anchor=west, [() [ ] [ ] [ ] [ ] ]
The rules essentially state how to extend a completion set on the basis of the information contained in it. Branching rules entail a nondeterministic choice in the expansion of the completion set. For each , we now define an algorithm based on rules for checking the formula satisfiability. We then prove that the algorithm terminates for every formula , and that it is sound and complete with respect to satisfiability.
Definition 1 ( tableau algorithm for ).
Given an formula , the tableau algorithm for runs as follows: set the initial completion set , and expand it by means of the rules until a clash or an complete completion set is obtained.
In the rest of this section, we prove termination, soundness and completeness of the tableau algorithms given above. We start by showing that the tableau algorithm terminates.
Theorem 1 (Termination).
Having started on the initial completion set , the tableau algorithm for terminates after at most steps, where is a polynomial function.
Proof.
We first require the following claims.
Claim 1.1.
Let be a completion set obtained by applying the tableau algorithm for . For each , the number of labelled constraints for in does not exceed , where is a polynomial function.
Proof of Claim.
We remark that, for each , the tableaux algorithm behaves exactly like a standard (nonmodal) tableaux algorithm (cf. e.g. [27, Theorem 15.4], noting also that in our case we do not have to deal with individual names). ∎
Claim 1.2.
Let be a completion set obtained by applying the tableau algorithm for . For , . For , .
Proof of Claim.
Labels are generated in by means of the application of the rule . For , this rule is applied to two labelled contraints (for possibly also to a single constraint ), while for it is applied to labelled contraints . By the application condition of , each such combination of constraints generates at most one label . Therefore, the number of labels that can be generated in is bounded by the number of possible such combinations, which is at most , for , and at most , for . ∎
The theorem is then a consequence of the following observations. Given a completion set constructed by the tableau algorithm, we have by Claim 1.2 that the number of applications of rule is bounded by , which is at most , for , and at most , for . Moreover, since every application of the rules and introduces a new formula to an labelled constraint, the total number of such rule applications is bounded by . Finally, by Claim 1.1, the number of applications of rules per label is bounded by , where is a polynomial function, since these rules add a new constraint to an labelled constraint system. Thus, the overall number of such rule applications is bounded by . ∎
We now proceed to prove that the tableau algorithm is sound.
Theorem 2 (Soundness).
If, having started on the initial completion set , the tableau algorithm constructs an complete and clashfree completion set for , then is satisfiable.
Proof.
Given an complete and clashfree completion set for , define, for , , , and occurring in ,
= ,  = , 
= ,  = . 
Moreover, define and let range over formulas or concepts, where: , if , and , if ; and similarly for . We set , with and , for , defined as follows:

;

for every , we set such that:
– for : ;
– for :;
– for :;
– for :; 
;

;

for some blocking in }.
First, we observe the following.

For , we have that is such that is supplemented. Indeed, for all , , suppose that and . By definition, this implies that: for some , . Hence, .

For , we have that is such that is closed under intersection. Indeed, for all , , suppose that and . Now suppose that, for some and, for some . Then for some and some the following holds, which in turn implies that :

For , we have that , with , is such that contains the unit. Indeed, by construction, for all , .
We then require the following claims.
Claim 2.1.
For every , , and : if , then .
Proof of Claim.
We show the claim by induction on the weight of (in NNF). The base case of comes immediately from the definitions. For the base case of , suppose that . Since is clashfree, we have that , and thus by definition of , meaning . The inductive cases of and come from the fact that is closed under and , respectively, and straightforward applications of the inductive hypothesis. We show the remaining cases (cf. also [27, Claim 15.2]).
. Let , meaning that . We distinguish two cases.

is not blocked by any variable in . Since is closed under , there exists occurring in such that and . Thus, by definition, and . By inductive hypothesis, we obtain that .

is blocked by a variable in , implying that there exists a minimal (since is a wellordering) occurring in such that and . In turn, this implies that is not blocked by any other variable in (for otherwise would block , with , against the fact that is minimal). By reasoning as in the case above, since is not blocked and is closed under , we have a variable occurring in such that and . Since blocks , by definition we have that , and by inductive hypothesis we get from that . Thus, .
. Let , meaning that , and suppose that . By definition, either or , for some blocking in . In the former case, since is closed under , we get that . In the latter case, since blocks in , we obtain ; again, since is closed under , this implies that . Hence, in both cases, we have . By inductive hypothesis, this means that . Since was arbitrary, we conclude that .
. Let , meaning that . Consider .

[leftmargin=*, align=left]

We have by inductive hypothesis that . By inductive hypothesis (since ), we also have that . Hence, . In conclusion, we have such that . Thus, by definition, , as required.

We have by inductive hypothesis that . Thus, we have such that . By definition, this means , as required.

This cases are analogous to the case for .
. Let . Consider .

[leftmargin=*, align=left]

We distinguish two cases. There exists no . This means that . Thus, , meaning that . There exists . We then reason similarly to the case for .

We distinguish two cases. There exist no . As for , we obtain . There exist . Since is complete, there exists such that: and ; or and , for some . By inductive hypothesis, the previous step implies that there exists such that: and ; or and , for some . Equivalently, it is not the case that, for every : implies