13 research outputs found
Parameterized aspects of team-based formalisms and logical inference
Parameterized complexity is an interesting subfield of complexity theory that has received a lot of attention in recent years. Such an analysis characterizes the complexity of (classically) intractable problems by pinpointing the computational hardness to some structural aspects of the input. In this thesis, we study the parameterized complexity of various problems from the area of team-based formalisms as well as logical inference.
In the context of team-based formalism, we consider propositional dependence logic (PDL). The problems of interest are model checking (MC) and satisfiability (SAT). Peter Lohmann studied the classical complexity of these problems as a part of his Ph.D. thesis proving that both MC and SAT are NP-complete for PDL. This thesis addresses the parameterized complexity of these problems with respect to a wealth of different parameterizations.
Interestingly, SAT for PDL boils down to the satisfiability of propositional logic as implied by the downwards closure of PDL-formulas. We propose an interesting satisfiability variant (mSAT) asking for a satisfiable team of size m. The problem mSAT restores the ‘team semantic’ nature of satisfiability for PDL-formulas. We propose another problem (MaxSubTeam) asking for a maximal satisfiable team if a given team does not satisfy the input formula.
From the area of logical inference, we consider (logic-based) abduction and argumentation. The problem of interest in abduction (ABD) is to determine whether there is an explanation for a manifestation in a knowledge base (KB). Following Pfandler et al., we also consider two of its variants by imposing additional restrictions over the size of an explanation (ABD and ABD=). In argumentation, our focus is on the argument existence (ARG), relevance (ARG-Rel) and verification (ARG-Check) problems. The complexity of these problems have been explored already in the classical setting, and each of them is known to be complete for the second level of the polynomial hierarchy (except for ARG-Check which is DP-complete) for propositional logic. Moreover, the work by Nord and Zanuttini (resp., Creignou et al.) explores the complexity of these problems with respect to various restrictions over allowed KBs for ABD (ARG). In this thesis, we explore a two-dimensional complexity analysis for these problems. The first dimension is the restrictions over KB in Schaefer’s framework (the same direction as Nord and Zanuttini and Creignou et al.). What differentiates the work in this thesis from an existing research on these problems is that we add another dimension, the parameterization.
The results obtained in this thesis are interesting for two reasons. First (from a theoretical point of view), ideas used in our reductions can help in developing further reductions and prove (in)tractability results for related problems. Second (from a practical point of view), the obtained tractability results might help an agent designing an instance of a problem come up with the one for which the problem is tractable
Parametrised enumeration
In this thesis, we develop a framework of parametrised enumeration complexity.
At first, we provide the reader with preliminary notions such as machine models and complexity classes besides proving them to be well-chosen.
Then, we study the interplay and the landscape of these classes and present connections to classical enumeration classes.
Afterwards, we translate the fundamental methods of kernelisation and self-reducibility into equivalent techniques in the setting of parametrised enumeration.
Subsequently, we illustrate the introduced classes by investigating the parametrised enumeration complexity of Max-Ones-SAT and strong backdoor sets as well as sharpen the first result by presenting a dichotomy theorem for Max-Ones-SAT.
After this, we extend the definitions of parametrised enumeration algorithms by allowing orders on the solution space.
In this context, we study the relations ``order by size'' and ``lexicographic order'' for graph modification problems and observe a trade-off between enumeration delay and space requirements of enumeration algorithms.
These results then yield an enumeration technique for generalised modification problems that is illustrated by applying this method to the problems closest string, weak and strong backdoor sets, and weighted satisfiability.
Eventually, we consider the enumeration of satisfying teams of formulas of poor man's propositional dependence logic.
There, we present an enumeration algorithm with FPT delay and exponential space which is one of the first enumeration complexity results of a problem in a team logic.
Finally, we show how this algorithm can be modified such that only polynomial space is required, however, by increasing the delay to incremental FPT time.In diesem Werk begründen wir die Theorie der parametrisierten Enumeration, präsentieren die grundlegenden Definitionen und prüfen ihre Sinnhaftigkeit.
Im nächsten Schritt, untersuchen wir das Zusammenspiel der eingeführten Komplexitätsklassen und zeigen Verbindungen zur klassischen Enumerationskomplexität auf.
Anschließend übertragen wir die zwei fundamentalen Techniken der Kernelisierung und Selbstreduzierbarkeit in Entsprechungen in dem Gebiet der parametrisierten Enumeration.
Schließlich untersuchen wir das Problem Max-Ones-SAT und das Problem der Aufzählung starker Backdoor-Mengen als typische Probleme in diesen Klassen.
Die vorherigen Resultate zu Max-Ones-SAT werden anschließend in einem Dichotomie-Satz vervollständigt.
Im nächsten Abschnitt erweitern wir die neuen Definitionen auf Ordnungen (auf dem Lösungsraum) und erforschen insbesondere die zwei Relationen \glqq Größenordnung\grqq\ und \glqq lexikographische Reihenfolge\grqq\ im Kontext von Graphen-Modifikationsproblemen.
Hierbei scheint es, als müsste man zwischen Delay und Speicheranforderungen von Aufzählungsalgorithmen abwägen, wobei dies jedoch nicht abschließend gelöst werden kann.
Aus den vorherigen Überlegungen wird schließlich ein generisches Enumerationsverfahren für allgemeine Modifikationsprobleme entwickelt und anhand der Probleme Closest String, schwacher und starker Backdoor-Mengen sowie gewichteter Erfüllbarkeit veranschaulicht.
Im letzten Abschnitt betrachten wir die parametrisierte Enumerationskomplexität von Erfüllbarkeitsproblemen im Bereich der Poor Man's Propositional Dependence Logic und stellen einen Aufzählungsalgorithmus mit FPT Delay vor, der mit exponentiellem Platz arbeitet.
Dies ist einer der ersten Aufzählungsalgorithmen im Bereich der Teamlogiken.
Abschließend zeigen wir, wie dieser Algorithmus so modifiziert werden kann, dass nur polynomieller Speicherplatz benötigt wird, bezahlen jedoch diese Einsparung mit einem Anstieg des Delays auf inkrementelle FPT Zeit (IncFPT)
Efficient paraconsistent reasoning with rules and ontologies for the semantic web
Ontologies formalized by means of Description Logics (DLs) and rules in the form of Logic Programs (LPs) are two prominent formalisms in the field of Knowledge Representation and Reasoning. While DLs adhere to the OpenWorld Assumption and are suited for taxonomic reasoning, LPs implement reasoning under the Closed World Assumption, so that default knowledge can be expressed. However, for many applications it is useful to have a means that allows reasoning over an open domain and expressing rules with exceptions at the same time. Hybrid MKNF knowledge bases make such a means available by formalizing DLs and LPs in a common logic, the Logic of Minimal Knowledge and Negation as Failure (MKNF).
Since rules and ontologies are used in open environments such as the Semantic Web,
inconsistencies cannot always be avoided. This poses a problem due to the Principle of Explosion, which holds in classical logics. Paraconsistent Logics offer a solution to this issue by assigning meaningful models even to contradictory sets of formulas. Consequently, paraconsistent semantics for DLs and LPs have been investigated intensively. Our goal is to apply the paraconsistent approach to the combination of DLs and LPs in hybrid MKNF knowledge bases.
In this thesis, a new six-valued semantics for hybrid MKNF knowledge bases is introduced, extending the three-valued approach by Knorr et al., which is based on the wellfounded semantics for logic programs. Additionally, a procedural way of computing paraconsistent well-founded models for hybrid MKNF knowledge bases by means of an alternating fixpoint construction is presented and it is proven that the algorithm is sound and complete w.r.t. the model-theoretic characterization of the semantics. Moreover, it is shown that the new semantics is faithful w.r.t. well-studied paraconsistent semantics for DLs and LPs, respectively, and maintains the efficiency of the approach it extends
Change impact analysis for evolving ontology-based content management
Ontologies have become ubiquitous tools to embed semantics into content and applications on the semantic web. They are used to define concepts in a domain and allow us to reach at a common understanding on subjects of interest. Ontologies cover wide range of topics enabling both humans and machines to understand meanings and to reason in different contexts. They cover topics such as semantic web, artificial intelligence, information retrieval, machine translation, software development, content management, etc. We use ontologies for semantic annotation of content to facilitate understandability of the content by humans and machines. However, building ontology and annotations is often a manual process which is error prone and time consuming. Ontologies and ontology-driven content management systems (OCMS) evolve due to a change in conceptualization, the representation or the specification of the domain knowledge. These changes are often immense and frequent. Implementing the changes and adapting the OCMS accordingly require a huge effort. This is due to complex impacts of the changes on the ontologies, the content and dependent applications. Thus, evolving the OCMS with minimum and predictable impacts is among the top priorities of evolution in OCMS. We approach the problem of evolution by proposing a framework which clearly represents the interactions of the components of an OCMS. We proposed a layered OCMS framework which contains an ontology layer, content layer and annotation layer. Further, we propose a novel approach for analysing impacts of change operations. Impacts of atomic change operations are assigned individually by analysing the target entity and all the other entities that are structurally or semantically dependent on it. Impacts of composite change operations are analysed following three stage process. We use impact cancellation, impact balancing and impact transformation to analyse the impacts when two or more atomic changes are executed as part of a composite or domain specific change operation. We build a model which estimates the impacts of a complete change operation enabling the ontology engineer to specify the weight associated with each optimization criteria. Finally, the model identifies the implementation strategy with minimum cost of evolution. We evaluate our system by building a prototype as a proof of concept and find out encouraging results