Canonical models are very useful for determining simple representation formalism for qualitative relations. Allen's interval relations, e.g., can thereby be represented using the start and the end point of the intervals. Such a simple representation was not possible for regions of higher dimension as used by the Region Connection Calculus. In this paper we present a canonical model which allows regions and relations between them to be represented as points of the topological space and information about their neighbourhoods. With this formalism we are able to prove that whenever a set of RCC-8 formulas is consistent there exists a realization in any dimension, even when the regions are constrained to be (sets of) polytopes. For three- and higher dimensional space this is also true for internally connected regions. Using the canonical model we give algorithms for generating consistent scenarios. 1 Introduction The Region Connection Calculus (RCC) is a topological approach t...

In 1979, Ron Brachman discussed a classification of the various primitives used by KR systems at that time. He argued that they could be grouped in four levels, ranging from the implementational to the linguistic level (Fig. 1). Each level corresponds to an explicit set of primitives offered to the knowledge engineer. At the implementational level, primitives are merely pointers and memory cells, which allow us to construct data structures with no a priori semantics. At the logical level, primitives are propositions, predicates, logical functions and operators, which are given a formal semantics in terms of relations among objects in the real world. No particular assumption is made however as to the nature of such relations: classical predicate logic is a general, uniform, neutral formalism, and the user is free to adapt it to its own representation purposes. At th

The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satisfiability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satisfiability checking algorithms for these logics. We deal with the possible world semantics under the constant domain assumption and show that the expanding and varying domain assumptions are reducible to it. Models with both finite and arbitrary constant domains are investigated. We begin by considering description logics with only one modal operator and then prove a general transfer theorem which makes it possible to lift the obtained results to many systems of polymodal description logic.

Model Theory. That is, we have a non-empty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard Back-and-Forth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. (A complete argument is in [16].) An outline is reproduced here, for convenience. First, k-variable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism I 2 I which is defined on the A-values for all variables x 1 ; : : : ; x k , that M;A j= OE iff N; I ffi A j= OE: The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be restricted to size at most k \Gamma 1, whence a matching choice for the witness can be made on the opposite side. This proves "only if". Next, "if" has a proof analogous to...

. We consider the modal logics KT and S4, the tense logic K t , and the fragment IPC (^;!) of intuitionistic logic. For these logics backward proof search in the standard sequent or tableau calculi does not terminate in general. In terms of the respective Kripke semantics, there are several kinds of non-termination: loops inside a world (KT), innite resp. looping branches (S4, IPC (^;!) ), and innite branching degree (K t ). We give uniform sequent-based calculi that contain specically tailored loop-checks such that the eciency of proof search is not deteriorated. Moreover all these loop-checks are easy to implement and can be combined with optimizations. Note that our calculus for S4 is not a pure contraction-free sequent calculus, but this (theoretical) defect does not hinder its application in practice. 1 Introduction For many non-classical propositional logics, backward proof search in the usual sequent calculi does not terminate in general. For all the logics we consider in th...

. This chapter summarizes our ongoing research on topological spatial reasoning using the Region Connection Calculus. We are addressing different questions and problems that arise when using this calculus. This includes representational issues, e.g., how can regions be represented and what is the required dimension of the applied space. Further, it includes computational issues, e.g., how hard is it to reason with the calculus and are there efficient algorithms. Finally, we also address cognitive issues, i.e., is the calculus cognitively adequate. 1 Introduction When describing a spatial configuration or when reasoning about such a configuration, often it is not possible or desirable to obtain precise, quantitative data. In these cases, qualitative reasoning about spatial configurations may be used. Different aspects of space can be treated in a qualitative way. Among others there are approaches considering orientation, distance, shape, topology, and combinations of these. A summary o...

Multi-valued logics support the explicit modeling of uncertainty and disagreement by allowing additional truth values in the logic. Such logics can be used for verification of dynamic properties of systems where complete, agreed upon models of the system are not available. In this paper, we present an implementation of a symbolic model checker for multi-valued temporal logics. The model checker works for any multi-valued logic whose truth values form a quasiboolean lattice. Our models are generalized Kripke structures, where both atomic propositions and transitions between states may take any of the truth values of a given multi-valued logic. Properties to be model checked are expressed in CTL, generalized with a multi-valued semantics. The design of the model checker is based on the use of MDDs, a multi-valued extension of Binary Decision Diagrams. We describe MDDs and their use in the model checker. We also give its theoretical time complexity and some preliminary empirical performance data.

Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, we introduce a dynamic logic for hybrid programs, which is a program notation for hybrid systems. As a verification technique that is suitable for automation, we introduce a free variable proof calculus with a novel combination of real-valued free variables and Skolemisation for lifting quantifier elimination for real arithmetic to dynamic logic. The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts. Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations. In a case study with cooperating traffic agents of the European Train Control System, we further show that our calculus is well-suited for verifying realistic hybrid systems with parametric system dynamics.

Formalizing the ontological commitment of a logical language means offering a way to specify the intended meaning of its vocabulary by constraining the set of its models, giving explicit information about the intended nature of the modelling primitives and their a priori relationships. We present here a formal definition of ontological commitment which aims to capture the very basic ontological assumptions about the intended domain, related to issues such as identity and internal structure. To tackle such issues, a modal framework endowed with mereo-topological primitives has been adopted. The paper is mostly based on a revisitation of philosophical (and linguistic) literature in the perspective of knowledge representation.

. After a short analysis of the requirements that a knowledge representation language must satisfy, we introduce Description Logics, Modal Logics, and Nonmonotonic Logics as formalisms for representing terminological knowledge, time-dependent or subjective knowledge, and incomplete knowledge respectively. At the end of each section, we briefly comment on the connection to Logic Programming. 1 Introduction This section is concerned with the question under which conditions one may rightfully claim to have represented knowledge about an application domain, and not just stored data occurring in this domain. 1 In the early days of Artificial Intelligence and Knowledge Representation, there was a heated discussion on whether logic can at all be used as a formalism for Knowledge Representation (see e.g. [135, 91, 92]). One aspect of the requirements on knowledge representation formalisms that can be derived from the considerations in this section is very well satisfied by logical for...