34 research outputs found
On a notion of abduction and relevance for first-order logic clause sets
I propose techniques to help with explaining entailment and non-entailment in first-order logic respectively relying on deductive and abductive reasoning. First, given an unsatisfiable clause set, one could ask which clauses are necessary for any possible deduction (\emph{syntactically relevant}), usable for some deduction (\emph{syntactically semi-relevant}), or unusable (\emph{syntactically irrelevant}). I propose a first-order formalization of this notion and demonstrate a lifting of this notion to the explanation of an entailment w.r.t some axiom set defined in some description logic fragments. Moreover, it is accompanied by a semantic characterization via \emph{conflict literals} (contradictory simple facts). From an unsatisfiable clause set, a pair of conflict literals are always deducible. A \emph{relevant} clause is necessary to derive any conflict literal, a \emph{semi-relevant} clause is necessary to derive some conflict literal, and an \emph{irrelevant} clause is not useful in deriving any conflict literals. It helps provide a picture of why an explanation holds beyond what one can get from the predominant notion of a minimal unsatisfiable set. The need to test if a clause is (syntactically) semi-relevant leads to a generalization of a well-known resolution strategy: resolution equipped with the set-of-support strategy is refutationally complete on a clause set and SOS if and only if there is a resolution refutation from using a clause in . This result non-trivially improves the original formulation. Second, abductive reasoning helps find extensions of a knowledge base to obtain an entailment of some missing consequence (called observation). Not only that it is useful to repair incomplete knowledge bases but also to explain a possibly unexpected observation. I particularly focus on TBox abduction in \EL description logic (still first-order logic fragment via some model-preserving translation scheme) which is rather lightweight but prevalent in practice. The solution space can be huge or even infinite. So, different kinds of minimality notions can help sort the chaff from the grain. I argue that existing ones are insufficient, and introduce \emph{connection minimality}. This criterion offers an interpretation of Occam's razor in which hypotheses are accepted only when they help acquire the entailment without arbitrarily using axioms unrelated to the problem at hand. In addition, I provide a first-order technique to compute the connection-minimal hypotheses in a sound and complete way. The key technique relies on prime implicates. While the negation of a single prime implicate can already serve as a first-order hypothesis, a connection-minimal hypothesis which follows \EL syntactic restrictions (a set of simple concept inclusions) would require a combination of them. Termination by bounding the term depth in the prime implicates is provable by only looking into the ones that are also subset-minimal. I also present an evaluation on ontologies from the medical domain by implementing a prototype with SPASS as a prime implicate generation engine.Ich schlage Techniken vor, die bei der Erklärung von Folgerung und Nichtfolgerung in der Logik erster Ordnung helfen, die sich jeweils auf deduktives und abduktives Denken stützen. Erstens könnte man bei einer gegebenen unerfüllbaren Klauselmenge fragen, welche Klauseln für eine mögliche Deduktion notwendig (\emph{syntaktisch relevant}), für eine Deduktion verwendbar (\emph{syntaktisch semi-relevant}) oder unbrauchbar (\emph{syntaktisch irrelevant}). Ich schlage eine Formalisierung erster Ordnung dieses Begriffs vor und demonstriere eine Anhebung dieses Begriffs auf die Erklärung einer Folgerung bezüglich einer Reihe von Axiomen, die in einigen Beschreibungslogikfragmenten definiert sind. Außerdem wird sie von einer semantischen Charakterisierung durch \emph{Konfliktliteral} (widersprüchliche einfache Fakten) begleitet. Aus einer unerfüllbaren Klauselmenge ist immer ein Konfliktliteralpaar ableitbar. Eine \emph{relevant}-Klausel ist notwendig, um ein Konfliktliteral abzuleiten, eine \emph{semi-relevant}-Klausel ist notwendig, um ein Konfliktliteral zu generieren, und eine \emph{irrelevant}-Klausel ist nicht nützlich, um Konfliktliterale zu generieren. Es hilft, ein Bild davon zu vermitteln, warum eine Erklärung über das hinausgeht, was man aus der vorherrschenden Vorstellung einer minimalen unerfüllbaren Menge erhalten kann. Die Notwendigkeit zu testen, ob eine Klausel (syntaktisch) semi-relevant ist, führt zu einer Verallgemeinerung einer bekannten Resolutionsstrategie: Die mit der Set-of-Support-Strategie ausgestattete Resolution ist auf einer Klauselmenge und SOS widerlegungsvollständig, genau dann wenn es eine Auflösungswiderlegung von unter Verwendung einer Klausel in gibt. Dieses Ergebnis verbessert die ursprüngliche Formulierung nicht trivial. Zweitens hilft abduktives Denken dabei, Erweiterungen einer Wissensbasis zu finden, um eine implikantion einer fehlenden Konsequenz (Beobachtung genannt) zu erhalten. Es ist nicht nur nützlich, unvollständige Wissensbasen zu reparieren, sondern auch, um eine möglicherweise unerwartete Beobachtung zu erklären. Ich konzentriere mich besonders auf die TBox-Abduktion in dem leichten, aber praktisch vorherrschenden Fragment der Beschreibungslogik \EL, das tatsächlich ein Logikfragment erster Ordnung ist (mittels eines modellerhaltenden Übersetzungsschemas). Der Lösungsraum kann riesig oder sogar unendlich sein. So können verschiedene Arten von Minimalitätsvorstellungen helfen, die Spreu vom Weizen zu trennen. Ich behaupte, dass die bestehenden unzureichend sind, und führe \emph{Verbindungsminimalität} ein. Dieses Kriterium bietet eine Interpretation von Ockhams Rasiermesser, bei der Hypothesen nur dann akzeptiert werden, wenn sie helfen, die Konsequenz zu erlangen, ohne willkürliche Axiome zu verwenden, die nichts mit dem vorliegenden Problem zu tun haben. Außerdem stelle ich eine Technik in Logik erster Ordnung zur Berechnung der verbindungsminimalen Hypothesen in zur Verfügung korrekte und vollständige Weise. Die Schlüsseltechnik beruht auf Primimplikanten. Während die Negation eines einzelnen Primimplikant bereits als Hypothese in Logik erster Ordnung dienen kann, würde eine Hypothese des Verbindungsminimums, die den syntaktischen Einschränkungen von \EL folgt (einer Menge einfacher Konzeptinklusionen), eine Kombination dieser beiden erfordern. Die Terminierung durch Begrenzung der Termtiefe in den Primimplikanten ist beweisbar, indem nur diejenigen betrachtet werden, die auch teilmengenminimal sind. Außerdem stelle ich eine Auswertung zu Ontologien aus der Medizin vor, Domäne durch die Implementierung eines Prototyps mit SPASS als Primimplikant-Generierungs-Engine
Exploiting Uncertainty for Querying Inconsistent Description Logics Knowledge Bases
The necessity to manage inconsistency in Description Logics Knowledge
Bases~(KBs) has come to the fore with the increasing importance gained by the
Semantic Web, where information comes from different sources that constantly
change their content and may contain contradictory descriptions when considered
either alone or together. Classical reasoning algorithms do not handle
inconsistent KBs, forcing the debugging of the KB in order to remove the
inconsistency. In this paper, we exploit an existing probabilistic semantics
called DISPONTE to overcome this problem and allow queries also in case of
inconsistent KBs. We implemented our approach in the reasoners TRILL and BUNDLE
and empirically tested the validity of our proposal. Moreover, we formally
compare the presented approach to that of the repair semantics, one of the most
established semantics when considering DL reasoning tasks
Approximate Assertional Reasoning Over Expressive Ontologies
In this thesis, approximate reasoning methods for scalable assertional reasoning are provided whose computational properties can be established in a well-understood way, namely in terms of soundness and completeness, and whose quality can be analyzed in terms of statistical measurements, namely recall and precision. The basic idea of these approximate reasoning methods is to speed up reasoning by trading off the quality of reasoning results against increased speed
Semiring Provenance for Lightweight Description Logics
We investigate semiring provenance--a successful framework originally defined
in the relational database setting--for description logics. In this context,
the ontology axioms are annotated with elements of a commutative semiring and
these annotations are propagated to the ontology consequences in a way that
reflects how they are derived. We define a provenance semantics for a language
that encompasses several lightweight description logics and show its
relationships with semantics that have been defined for ontologies annotated
with a specific kind of annotation (such as fuzzy degrees). We show that under
some restrictions on the semiring, the semantics satisfies desirable properties
(such as extending the semiring provenance defined for databases). We then
focus on the well-known why-provenance, which allows to compute the semiring
provenance for every additively and multiplicatively idempotent commutative
semiring, and for which we study the complexity of problems related to the
provenance of an axiom or a conjunctive query answer. Finally, we consider two
more restricted cases which correspond to the so-called positive Boolean
provenance and lineage in the database setting. For these cases, we exhibit
relationships with well-known notions related to explanations in description
logics and complete our complexity analysis. As a side contribution, we provide
conditions on an ELHI_bot ontology that guarantee tractable reasoning.Comment: Paper currently under review. 102 page
Automated Reasoning
This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book
Developing Ontological Background Knowledge for Biomedicine
Biomedicine is an impressively fast developing, interdisciplinary field of
research. To control the growing volumes of biomedical data, ontologies are
increasingly used as common organization structures. Biomedical ontologies
describe domain knowledge in a formal, computationally accessible way. They
serve as controlled vocabularies and background knowledge in applications
dealing with the integration, analysis and retrieval of heterogeneous types
of data. The development of biomedical ontologies, however, is hampered by
specific challenges. They include the lack of quality standards, resulting
in very heterogeneous resources, and the decentralized development of
biomedical ontologies, causing the increasing fragmentation of domain
knowledge across them.
In the first part of this thesis, a life cycle model for biomedical
ontologies is developed, which is intended to cope with these challenges.
It comprises the stages "requirements analysis", "design and
implementation", "evaluation", "documentation and release" and
"maintenance". For each stage, associated subtasks and activities are
specified. To promote quality standards for biomedical ontology
development, an emphasis is set on the evaluation stage. As part of it,
comprehensive evaluation procedures are specified, which allow to assess
the quality of ontologies on various levels. To tackle the issue of
knowledge fragmentation, the life cycle model is extended to also cover
ontology alignments. Ontology alignments specify mappings between related
elements of different ontologies. By making potential overlaps and
similarities between ontologies explicit, they support the integration of
ontologies and help reduce the fragmentation of knowledge.
In the second part of this thesis, the life cycle model for biomedical
ontologies and alignments is validated by means of five case studies. As a
result, they confirm that the model is effective. Four of the case studies
demonstrate that it is able to support the development of useful new
ontologies and alignments. The latter facilitate novel natural language
processing and bioinformatics applications, and in one case constitute the
basis of a task of the "BioNLP shared task 2013", an international
challenge on biomedical information extraction. The fifth case study shows
that the presented evaluation procedures are an effective means to check
and improve the quality of ontology alignments. Hence, they support the
crucial task of quality assurance of alignments, which are themselves
increasingly used as reference standards in evaluations of automatic
ontology alignment systems. Both, the presented life cycle model and the
ontologies and alignments that have resulted from its validation improve
information and knowledge management in biomedicine and thus promote
biomedical research