Provenance for the Description Logic ELHr

We address the problem of handling provenance information in ELHr ontologies. We consider a setting recently introduced for ontology-based data access, based on semirings and extending classical data provenance, in which ontology axioms are annotated with provenance tokens. A consequence inherits the provenance of the axioms involved in deriving it, yielding a provenance polynomial as an annotation. We analyse the semantics for the ELHr case and show that the presence of conjunctions poses various difficulties for handling provenance, some of which are mitigated by assuming multiplicative idempotency of the semiring. Under this assumption, we study three problems: ontology completion with provenance, computing the set of relevant axioms for a consequence, and query answering.Comment: This is the long version of IJCAI 2020 paper 2243 (24 pages

Provenance for the Description Logic ELHr

Reportée de juillet 2020 à janvier 2021 en raison de la COVIDInternational audienceWe address the problem of handling provenance information in ELHr ontologies. We consider a setting recently introduced for ontology-based data access, based on semirings and extending classical data provenance, in which ontology axioms are annotated with provenance tokens. A consequence inherits the provenance of the axioms involved in deriving it, yielding a provenance polynomial as annotation. We analyse the semantics for the ELHr case and show that the presence of conjunctions poses various difficulties for handling provenance, some of which are mitigated by assuming multiplicative idempotency of the semiring. Under this assumption, we study three problems: ontology completion with provenance, computing the set of relevant axioms for a consequence, and query answering

Provenance for the Description Logic ELHr (Extended Abstract)

International audienceThis extended abstract presents our work on provenance forthe description logic ELHr published at IJCAI 2

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

Semiring Provenance for B\"uchi Games: Strategy Analysis with Absorptive Polynomials

This paper presents a case study for the application of semiring semantics for fixed-point formulae to the analysis of strategies in B\"uchi games. Semiring semantics generalizes the classical Boolean semantics by permitting multiple truth values from certain semirings. Evaluating the fixed-point formula that defines the winning region in a given game in an appropriate semiring of polynomials provides not only the Boolean information on who wins, but also tells us how they win and which strategies they might use. This is well-understood for reachability games, where the winning region is definable as a least fixed point. The case of B\"uchi games is of special interest, not only due to their practical importance, but also because it is the simplest case where the fixed-point definition involves a genuine alternation of a greatest and a least fixed point. We show that, in a precise sense, semiring semantics provide information about all absorption-dominant strategies -- strategies that win with minimal effort, and we discuss how these relate to positional and the more general persistent strategies. This information enables further applications such as game synthesis or determining minimal modifications to the game needed to change its outcome. Lastly, we discuss limitations of our approach and present questions that cannot be immediately answered by semiring semantics.Comment: Full version of a paper submitted to GandALF 202

Generalized Completeness for {SOS} Resolution and its Application to a New Notion of Relevance

International audienceWe prove the SOS strategy for first-order resolution to be refutationally complete on a clause set $N$ and set-of-support $S$ if and only if there exists a clause in $S$ that occurs in a resolution refutation from $N\cup S$. This strictly generalizes and sharpens the original completeness result requiring $N$ to be satisfiable. The generalized SOS completeness result supports automated reasoning on a new notion of relevance aiming at capturing the support of a clause in the refutation of a clause set. A clause $C$ is relevant for refuting a clause set $N$ if $C$ occurs in every refutation of $N$. The clause $C$ is semi-relevant if it occurs in some refutation, i.e., if there exists an SOS refutation with set-of-support $S = \{C\}$ from $N\setminus \{C\}$. A clause that does not occur in any refutation from $N$ is irrelevant, i.e., it is not semi-relevant. Our new notion of relevance separates clauses in a proof that are ultimately needed from clauses that may be replaced by different clauses. In this way it provides insights towards proof explanation in refutations beyond existing notions such as that of an unsatisfiable core

Semantic Relevance

International audienceAbstract A clause C is syntactically relevant in some clause set N , if it occurs in every refutation of N . A clause C is syntactically semi-relevant, if it occurs in some refutation of N . While syntactic relevance coincides with satisfiability (if C is syntactically relevant then $$N\setminus \{C\}$$ N \ { C } is satisfiable), the semantic counterpart for syntactic semi-relevance was not known so far. Using the new notion of a conflict literal we show that for independent clause sets N a clause C is syntactically semi-relevant in the clause set N if and only if it adds to the number of conflict literals in N . A clause set is independent, if no clause out of the clause set is the consequence of different clauses from the clause set. Furthermore, we relate the notion of relevance to that of a minimally unsatisfiable subset (MUS) of some independent clause set N . In propositional logic, a clause C is relevant if it occurs in all MUSes of some clause set N and semi-relevant if it occurs in some MUS. For first-order logic the characterization needs to be refined with respect to ground instances of N and C

A survey of large-scale reasoning on the Web of data

As more and more data is being generated by sensor networks, social media and organizations, the Webinterlinking this wealth of information becomes more complex. This is particularly true for the so-calledWeb of Data, in which data is semantically enriched and interlinked using ontologies. In this large anduncoordinated environment, reasoning can be used to check the consistency of the data and of asso-ciated ontologies, or to infer logical consequences which, in turn, can be used to obtain new insightsfrom the data. However, reasoning approaches need to be scalable in order to enable reasoning over theentire Web of Data. To address this problem, several high-performance reasoning systems, whichmainly implement distributed or parallel algorithms, have been proposed in the last few years. Thesesystems differ significantly; for instance in terms of reasoning expressivity, computational propertiessuch as completeness, or reasoning objectives. In order to provide afirst complete overview of thefield,this paper reports a systematic review of such scalable reasoning approaches over various ontologicallanguages, reporting details about the methods and over the conducted experiments. We highlight theshortcomings of these approaches and discuss some of the open problems related to performing scalablereasoning

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 $N$ and SOS $M$ if and only if there is a resolution refutation from $N\cup M$ using a clause in $M$. 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 $N$ und SOS $M$ widerlegungsvollständig, genau dann wenn es eine Auflösungswiderlegung von $N\cup M$ unter Verwendung einer Klausel in $M$ 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

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