A PSpace-algorithm for ALCQI-satisfiability

The description logic ALCQI extends the 'standard' description logic ALC by qualifying number restrictions and converse roles. We show that concept satisfiability for this DL is still decidable in polynomial space. The presented algorithm combines techniques from [Tob99] to deal with qualifying number restrictions and from [HST99] to deal with converse roles

PSPACE Reasoning for Graded Modal Logics

We present a PSPACE algorithm that decides satisfiability of the graded modal logic Gr(K_R)---a natural extension of propositional modal logic K_R by counting expressions---which plays an important role in the area of knowledge representation. The algorithm employs a tableaux approach and is the first known algorithm which meets the lower bound for the complexity of the problem. Thus, we exactly fix the complexity of the problem and refute an ExpTime-hardness conjecture. We extend the results to the logic Gr(K_(R \cap I)), which augments Gr(K_R) with inverse relations and intersection of accessibility relations. This establishes a kind of ``theoretical benchmark'' that all algorithmic approaches can be measured against

Concept Descriptions with Set Constraints and Cardinality Constraints

We introduce a new description logic that extends the well-known logic ALCQ by allowing the statement of constraints on role successors that are more general than the qualified number restrictions of ALCQ. To formulate these constraints, we use the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA), in which one can express Boolean combinations of set constraints and numerical constraints on the cardinalities of sets. Though our new logic is considerably more expressive than ALCQ, we are able to show that the complexity of reasoning in it is the same as in ALCQ, both without and with TBoxes.The first version of this report was put online on April 6, 2017. The current version, containing more information on related work, was put online on July 13, 2017. This is an extended version of a paper published in the proceedings of FroCoS 2017

On the Complexity and Expressiveness of Description Logics with Counting

Simple counting quantifiers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a fixed natural number have been employed in Description Logics (DLs) for more than two decades under the respective names of number restrictions and cardinality restrictions on concepts. Recently, we have considerably extended the expressivity of such quantifiers by allowing to impose set and cardinality constraints formulated in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) on sets of role successors and concepts, respectively. We were able to prove that this extension does not increase the complexity of reasoning. In the present paper, we investigate the expressive power of the DLs obtained in this way, using appropriate bisimulation characterizations and 0–1 laws as tools to differentiate between the expressiveness of different logics. In particular, we show that, in contrast to most classical DLs, these logics are no longer expressible in first-order predicate logic (FOL), and we characterize their first-order fragments. In most of our previous work on DLs with QFBAPA-based set and cardinality constraints we have employed finiteness restrictions on interpretations to ensure that the obtained sets are finite, as required by the standard semantics for QFBAPA. Here we dispense with these restrictions to ease the comparison with classical DLs, where one usually considers arbitrary models rather than finite ones, easier. It turns out that doing so does not change the complexity of reasoning

A Description Logic with Transitive and Converse Roles, Role Hierarchies and Qualifying Number Restrictions

As widely argued [HG97; Sat96], transitive roles play an important role in the adequate representation of aggregated objects: they allow these objects to be described by referring to their parts without specifying a level of decomposition. In [HG97], the Description Logic (DL) ALCHR+ is presented, which extends ALC with transitive roles and a role hierarchy. It is argued in [Sat98] that ALCHR+ is well-suited to the representation of aggregated objects in applications that require various part-whole relations to be distinguished, some of which are transitive. However, ALCHR+ allows neither the description of parts by means of the whole to which they belong, or vice versa. To overcome this limitation, we present the DL SHI which allows the use of, for example, has part as well as is part of. To achieve this, ALCHR+ was extended with inverse roles. It could be argued that, instead of defining yet another DL, one could make use of the results presented in [DL96] and use ALC extended with role expressions which include transitive closure and inverse operators. The reason for not proceeding like this is the fact that transitive roles can be implemented more efficiently than the transitive closure of roles (see [HG97]), although they lead to the same complexity class (ExpTime-hard) when added, together with role hierarchies, to ALC. Furthermore, it is still an open question whether the transitive closure of roles together with inverse roles necessitates the use of the cut rule [DM98], and this rule leads to an algorithm with very bad behaviour. We will present an algorithm for SHI without such a rule. Furthermore, we enrich the language with functional restrictions and, finally, with qualifying number restrictions. We give sound and complete decision proceduresfor the resulting logics that are derived from the initial algorithm for SHI. The structure of this report is as follows: In Section 2, we introduce the DL SI and present a tableaux algorithm for satisfiability (and subsumption) of SI-concepts—in another report [HST98] we prove that this algorithm can be refined to run in polynomial space. In Section 3 we add role hierarchies to SI and show how the algorithm can be modified to handle this extension appropriately. Please note that this logic, namely SHI, allows for the internalisation of general concept inclusion axioms, one of the most general form of terminological axioms. In Section 4 we augment SHI with functional restrictions and, using the so-called pairwise-blocking technique, the algorithm can be adapted to this extension as well. Finally, in Section 5, we show that standard techniques for handling qualifying number restrictions [HB91;BBH96] together with the techniques described in previous sections can be used to decide satisfiability and subsumption for SHIQ, namely ALC extended with transitive and inverse roles, role hierarchies, and qualifying number restrictions. Although Section 5 heavily depends on the previous sections, we have made it self-contained, i.e. it contains all necessary definitions and proofs from scratch, for a better readability. Building on the previous sections, Section 6 presents an algorithm that decides the satisfiability of SHIQ-ABoxes

Studies on modal logics of time and space

This dissertation presents original results in Temporal Logic and Spatial Logic. Part I concerns Branching-Time Logic. Since Prior 1967, two main semantics for Branching-Time Logic have been devised: Peircean and Ockhamist semantics. Zanardo 1998 proposed a general semantics, called Indistinguishability semantics, of which Peircean and Ockhamist semantics are limit cases. We provide a finite axiomatization of the Indistinguishability logic of upward endless bundled trees using a non-standard inference rule, and prove that this logic is strongly complete. In Part II, we study the temporal logic given by the tense operators F for future and P for past together with the derivative operator , interpreted on the real numbers. We prove that this logic is neither strongly nor Kripke complete, it is PSPACE-complete, and it is finitely axiomatizable. In Part III, we study the spatial logic given by the derivative operator and the graded modalities {n | n in N}. We prove that this language, call it L, is as expressive as the first-order language Lt of Flum and Ziegler 1980 when interpreted on T3 topological spaces. Then, we give a general definition of modal operator: essentially, a modal operator will be defined by a formula of Lt with at most one free variable. If a modal operator is defined by a formula predicating only over points, then it is called point-sort operator. We prove that L, even if enriched with all point-sort operators, however enriched with finitely many modal operators predicating also on open sets, cannot express Lt on T2 spaces. Finally, we axiomatize the logic of any class between all T1 and all T3 spaces and prove that it is PSPACE-complete.Open Acces

A PSpace Algorithm for Graded Modal Logic

We present a PSpace algorithm that decides satisfiability of the graded modal logic Gr(K_R)---a natural extension of propositional modal logic KR by counting expressions---which plays an important role in the area of knowledge representation. The algorithm employs a tableaux approach and is the first known algorithm which meets the lower bound for the complexity of the problem. Thus, we exactly fix the complexity of the problem and refute a ExpTime-hardness conjecture. This establishes a kind of "theoretical benchmark" that all algorithmic approaches can be measured with