On Bisimulations for Description Logics

We study bisimulations for useful description logics. The simplest among the considered logics is $\mathcal{ALC}_{reg}$ (a variant of PDL). The others extend that logic with inverse roles, nominals, quantified number restrictions, the universal role, and/or the concept constructor for expressing the local reflexivity of a role. They also allow role axioms. We give results about invariance of concepts, TBoxes and ABoxes, preservation of RBoxes and knowledge bases, and the Hennessy-Milner property w.r.t. bisimulations in the considered description logics. Using the invariance results we compare the expressiveness of the considered description logics w.r.t. concepts, TBoxes and ABoxes. Our results about separating the expressiveness of description logics are naturally extended to the case when instead of $\mathcal{ALC}_{reg}$ we have any sublogic of $\mathcal{ALC}_{reg}$ that extends $\mathcal{ALC}$. We also provide results on the largest auto-bisimulations and quotient interpretations w.r.t. such equivalence relations. Such results are useful for minimizing interpretations and concept learning in description logics. To deal with minimizing interpretations for the case when the considered logic allows quantified number restrictions and/or the constructor for the local reflexivity of a role, we introduce a new notion called QS-interpretation, which is needed for obtaining expected results. By adapting Hopcroft's automaton minimization algorithm and the Paige-Tarjan algorithm, we give efficient algorithms for computing the partition corresponding to the largest auto-bisimulation of a finite interpretation.Comment: 42 page

Integrating Reasoning Services for Description Logics with Cardinality Constraints with Numerical Optimization Techniques

Recent research in the field of Description Logic (DL) investigated the complexity of the satisfiability problem for description logics that are obtained by enriching the well-known DL ALCQ with more complex set and cardinality constraints over role successors. The algorithms that have been proposed so far, despite providing worst-case optimal decision procedures for the concept satisfiability problem (both without and with a terminology) lack the efficiency needed to obtain usable implementations. In particular, the algorithm for the case without terminology is non-deterministic and the one for the case with a terminology is also best-case exponential. The goal of this thesis is to use well-established techniques from the field of numerical optimization, such as column generation, in order to obtain more practical algorithms. As a starting point, efficient approaches for dealing with counting quantifiers over unary predicates based on SAT-based column generation should be considered.:1. Introduction 2. Preliminaries 2.1. First-order logic 2.2. Linear Programming 2.3. The description logic ALCQ 2.4. Extending ALCQ with expressive role successor constraints 2.4.1. The logic QFBAPA 2.4.2 The description logic ALCSCC 3. The description logic ALCCQU 3.1. A normal form for ALCCQU 3.2. ALCQt as an equivalent formulation of ALCCQU 3.2.1. ALCQt is a sublogic of ALCCQU 3.2.2. ALCCQU is a sublogic of ALCQt 3.3. Model-theoretic characterization of ALCQt 3.3.1. ALCQt-bisimulation and invariance for ALCQt 3.3.2. Characterization of ALCQt concept descriptions 3.4. Expressive power 3.4.1. Relative expressivity of ALCQ and ALCCQU 3.4.2. Relative expressivity of ALCCQU and ALCSCC 3.5. ALCCQU as the first-order fragment of ALCSCC 4. Concept satisfiability in ALCCQU 4.1. The first-order fragment CQU 4.2. Column generation with SAT oracle 4.2.1. Column generation and CQU 4.2.2. From linear inequalities to propositional formulae 4.2.3. Column generation and ALCCQU 4.3. Branch-and-Price for ALCCQU concept satisfiability 4.4. Correctness of ALCCQU-BB 4.4.1. Complexity of ALCCQU-BB 5. Conclusion - Bibliograph

A doctrinal approach to modal/temporal Heyting logic and non-determinism in processes

The study of algebraic modelling of labelled non-deterministic concurrent processes leads us to consider a category LB , obtained from a complete meet-semilattice B and from B-valued equivalence relations. We prove that, if B has enough properties, then LB presents a two-fold internal logical structure, induced by two doctrines definable on it: one related to its families of subobjects and one to its families of regular subobjects. The first doctrine is Heyting and makes LB a Heyting category, the second one is Boolean. We will see that the difference between these two logical structures, namely the different behaviour of the negation operator, can be interpreted in terms of a distinction between non-deterministic and deterministic behaviours of agents able to perform computations in the context of the same process. Moreover, the sorted first-order logic naturally associated with LB can be extended to a modal/temporal logic, again using the doctrinal setting. Relations are also drawn to other computational model