Genuine Process Logic

The Genuine Process Logic described here (abbreviation: GPL) places the object-bound process itself at the center of formalism. It should be suitable for everyday use, i.e. it is not primarily intended for the formalization of computer programs, but instead, as a counter-conception to the classical state logics. The new and central operator of the GPL is an action symbol replacing the classical state symbols, e.g. of equivalence or identity. The complete renunciation of object-language state expressions also results in a completely new metalinguistic framework, both regarding the axioms and the expressive possibilities of this system. A mixture with state logical terms is readily possible

Information sharing among ideal agents

Multi-agent systems operating in complex domains crucially require agents to interact with each other. An important result of this interaction is that some of the private knowledge of the agents is being shared in the group of agents. This thesis investigates the theme of knowledge sharing from a theoretical point of view by means of the formal tools provided by modal logic. More specifically this thesis addresses the following three points. First, the case of hypercube systems, a special class of interpreted systems as defined by Halpern and colleagues, is analysed in full detail. It is here proven that the logic S5WDn constitutes a sound and complete axiomatisation for hypercube systems. This logic, an extension of the modal system S5n commonly used to represent knowledge of a multi-agent system, regulates how knowledge is being shared among agents modelled by hypercube systems. The logic S5WDn is proven to be decidable. Hypercube systems are proven to be synchronous agents with perfect recall that communicate only by broadcasting, in separate work jointly with Ron van der Meyden not fully reported in this thesis. Second, it is argued that a full spectrum of degrees of knowledge sharing can be present in any multi-agent system, with no sharing and full sharing at the extremes. This theme is investigated axiomatically and a range of logics representing a particular class of knowledge sharing between two agents is presented. All the logics but two in this spectrum are proven complete by standard canonicity proofs. We conjecture that these two remaining logics are not canonical and it is an open problem whether or not they are complete. Third, following a influential position paper by Halpern and Moses, the idea of refining and checking of knowledge structures in multi-agent systems is investigated. It is shown that, Kripke models, the standard semantic tools for this analysis are not adequate and an alternative notion, Kripke trees, is put forward. An algorithm for refining and checking Kripke trees is presented and its major properties investigated. The algorithm succeeds in solving the famous muddy-children puzzle, in which agents communicate and reason about each other's knowledge. The thesis concludes by discussing the extent to which combining logics, a promising new area in pure logic, can provide a significant boost in research for epistemic and other theories for multi-agent systems

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

Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL

The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic

Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL

The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic

Arguments to believe and beliefs to argue. Epistemic logics for argumentation and its dynamics

Arguing and believing are two skills that have typically played a crucial role in the analysis of human cognition. Both notions have received notable attention from a broad range of disciplines, including linguistics, philosophy, psychology, and computer science. The main goal of this dissertation consists in studying from a logical perspective (that is, focused on reasoning) some of the existing relations between beliefs and argumentation. From a methodological point of view, we propose to combine two well-known families of formalisms for knowledge representation that have been relatively disconnected (with some salient exceptions): epistemic logic (Fagin et al., 2004; Meyer and van der Hoek, 1995) together with its dynamic extensions (van Ditmarsch et al., 2007; van Benthem, 2011), on the one hand, and formal argumentation (Baroni et al., 2018; Gabbay et al., 2021), on the other hand. This choice is arguably natural. Epistemic logic provides well-known tools for qualitatively representing epistemic attitudes (belief, among them). Formal argumentation, on its side, is the broad research field where mathematical representations of argumentative phenomena are investigated. Moreover, the notion of awareness, as treated in the epistemic logic tradition since Fagin and Halpern (1987), can be used as a theoretical bridge among both areas. This dissertation is presented as a collection of papers [compendio de publicaciones], meaning that its main contributions are contained in the reprint of six works that have been previously published, placed in Chapter 4. In chapter 1, we pursue a general introduction to the research problem. Chapter 2 is devoted to the presentation of the technical tools employed through the thesis. Chapter 3 explains how the contributions approach the research problem. Chapter 5 provides a general discussion of results, by analysing closely related work. We conclude in Chapter 6 with some remarks and open paths for future research

Logic, Rational Agency, and Intelligent Interaction

... make a plea for recasting logic as a theory of interactive agency, and show how this perspective fits both old achievements and new broader ambitions for the field