Foundations for Uniform Interpolation and Forgetting in Expressive Description Logics

We study uniform interpolation and forgetting in the description logic ALC. Our main results are model-theoretic characterizations of uniform inter- polants and their existence in terms of bisimula- tions, tight complexity bounds for deciding the existence of uniform interpolants, an approach to computing interpolants when they exist, and tight bounds on their size. We use a mix of model- theoretic and automata-theoretic methods that, as a by-product, also provides characterizations of and decision procedures for conservative extensions

Rewritability in Monadic Disjunctive Datalog, MMSNP, and Expressive Description Logics

We study rewritability of monadic disjunctive Datalog programs, (the complements of) MMSNP sentences, and ontology-mediated queries (OMQs) based on expressive description logics of the ALC family and on conjunctive queries. We show that rewritability into FO and into monadic Datalog (MDLog) are decidable, and that rewritability into Datalog is decidable when the original query satisfies a certain condition related to equality. We establish 2NExpTime-completeness for all studied problems except rewritability into MDLog for which there remains a gap between 2NExpTime and 3ExpTime. We also analyze the shape of rewritings, which in the MMSNP case correspond to obstructions, and give a new construction of canonical Datalog programs that is more elementary than existing ones and also applies to formulas with free variables

Complexity and Succinctness of Public Announcement Logic

There is a recent trend of extending epistemic logic (EL) with dynamic operators that allow to express the evolution of knowledge and belief induced by knowledge-changing actions. The most basic such extension is public announcement logic (PAL), which is obtained from EL by adding an operator for truthful publix announcements. In this paper, we consider the computational complexity of PAL and show that it coincides with that of EL. This holds in the single- and multi-agent case, and also in the presence of common knowledge operators. We also prove that there are properties that can be expressed exponentially more succint in PAL than in EL. This shows that, despite the known fact that PAL and EL have the same expressive power, ther eis a benefit in adding the public announcement operator to EL: it exponentially increases the succinctness of formulas without having negative effects on computational complexity

PDL with Intersection and Converse is Decidable

In its many guises and variations, propositional dynamic logic (PDL) plays an important role in various areas of computer science such as databases, artificial intelligence, and computer linguistics. One relevant and powerful variation is ICPDL, the extension of PDL with intersection and converse. Although ICPDL has several interesting applications, its computational properties have never been investigated. In this paper, we prove that ICPDL is decidable by developing a translation to the monadic second order logic of infinite trees. Our result has applications in information logic, description logic, and epistemic logic. In particular, we solve a long-standing open problem in information logic. Another virtue of our approach is that it provides a decidability proof that is more transparent than existing ones for PDL with intersection (but without converse)

Interval-based Temporal Reasoning with General TBoxes

From the Motivation: „Description Logics (DLs) are a family of formalisms well-suited for the representation of and reasoning about knowledge. Whereas most Description Logics represent only static aspects of the application domain, recent research resulted in the exploration of various Description Logics that allow to, additionally, represent temporal information, see [4] for an overview. The approaches to integrate time differ in at least two important aspects: First, the basic temporal entity may be a time point or a time interval. Second, the temporal structure may be part of the semantics (yielding a multi-dimensional semantics) or it may be integrated as a so-called concrete domain. Examples for multi-dimensional point-based logics can be find in, e.g., [21;29], while multi-dimensional interval-based logics are used in, e.g., [23;2]. The concrete domain approach needs some more explanation. Concrete domains have been proposed by Baader and Hanschke as an extension of Description Logics that allows reasoning about 'concrete qualities' of the entities of the application domain such as sizes, length, or weights of real-worlds objects [5]. Description Logics with concrete domains do usually not use a fixed concrete domain; instead the concrete domain can be thought of as a parameter to the logic. As was first described in [16], if a 'temporal' concrete domain is employed, then concrete domains may be point-based, interval-based, or both. ...

Adding Numbers to the SHIQ Description Logic - First Results

Recently, the Description Logic (DL) SHIQ has found a large number of applications. This success is due to the fact that SHIQ combines a rich expressivity with efficient reasoning, as is demonstrated by its implementation in DL systems such as FaCT and RACER. One weakness of SHIQ, however, limits its usability in several application areas: numerical knowledge such as knowledge about the age, weight, or temperature of real-world entities cannot be adequately represented. In this paper, we propose an extension of SHIQ that aims at closing this gap. The new Description Logic Q-SHIQ, which augments SHIQ by additional, 'concrete domain' style concept constructors, allows to refer to rational numbers in concept descriptions, and also to define concepts based on the comparison of numbers via predicates such as < or =. We argue that this kind of expressivity is needed in many application areas such as reasoning about the semantic web. We prove reasoning with Q-SHIQ to be EXPTIME-complete (thus not harder than reasoning with SHIQ) by devising an automata-based decision procedure

Conjunctive Query Answering for the Description Logic SHIQ

Conjunctive queries play an important role as an expressive query language for Description Logics (DLs). Although modern DLs usually provide for transitive roles, conjunctive query answering over DL knowledge bases is only poorly understood if transitive roles are admitted in the query. In this paper, we consider unions of conjunctive queries over knowledge bases formulated in the prominent DL SHIQ and allow transitive roles in both the query and the knowledge base. We show decidability of query answering in this setting and establish two tight complexity bounds: regarding combined complexity, we prove that there is a deterministic algorithm for query answering that needs time single exponential in the size of the KB and double exponential in the size of the query, which is optimal. Regarding data complexity, we prove containment in co-NP