From paraconsistent three-valued logics to multiple-source epistemic logic

National audienceSeveral interpretations can be given to the third truth value in three-valued logics. Here, we consider the case when it refers to the epistemic notion of contradictory, or both true and false at the same time. We study several paraconsistent three-valued logics that carry this concern and show that they can be translated into a fragment of a simple epistemic logic where modalities can only appear in front of literals. This logic is unusual in the sense that necessity modalities distribute over disjunctions instead of conjunctions. An equivalent translation into a fragment of KD modal logic can be obtained by exchanging the role of possibility and necessity modalities, highlighting the perfect symmetry between three-valued logics of contradiction and three-valued logics of incomplete information

Structured Decompositions: Structural and Algorithmic Compositionality

We introduce structured decompositions: category-theoretic generalizations of many combinatorial invariants -- including tree-width, layered tree-width, co-tree-width and graph decomposition width -- which have played a central role in the study of structural and algorithmic compositionality in both graph theory and parameterized complexity. Structured decompositions allow us to generalize combinatorial invariants to new settings (for example decompositions of matroids) in which they describe algorithmically useful structural compositionality. As an application of our theory we prove an algorithmic meta theorem for the Sub_P-composition problem which, when instantiated in the category of graphs, yields compositional algorithms for NP-hard problems such as: Maximum Bipartite Subgraph, Maximum Planar Subgraph and Longest Path

A Semantics for Hybrid Iteration

The recently introduced notions of guarded traced (monoidal) category and guarded (pre-)iterative monad aim at unifying different instances of partial iteration whilst keeping in touch with the established theory of total iteration and preserving its merits. In this paper we use these notions and the corresponding stock of results to examine different types of iteration for hybrid computations. As a starting point we use an available notion of hybrid monad restricted to the category of sets, and modify it in order to obtain a suitable notion of guarded iteration with guardedness interpreted as progressiveness in time - we motivate this modification by our intention to capture Zeno behaviour in an arguably general and feasible way. We illustrate our results with a simple programming language for hybrid computations and interpret it over the developed semantic foundations

A Monadic Second-Order Version of Tarski’s Geometry of Solids

In this paper, we are concerned with the development of a general set theory using the single axiom version of Leśniewski’s mereology. The specification of mereology, and further of Tarski’s geometry of solids will rely on the Calculus of Inductive Constructions (CIC). In the first part, we provide a specification of Leśniewski’s mereology as a model for an atomless Boolean algebra using Clay’s ideas. In the second part, we interpret Leśniewski’s mereology in monadic second-order logic using names and develop a full version of mereology referred to as CIC-based Monadic Mereology (λ-MM) allowing an expressive theory while involving only two axioms. In the third part, we propose a modeling of Tarski’s solid geometry relying on λ-MM. It is intended to serve as a basis for spatial reasoning. All parts have been proved using a translation in type theory

Barry Smith an sich

Festschrift in Honor of Barry Smith on the occasion of his 65th Birthday. Published as issue 4:4 of the journal Cosmos + Taxis: Studies in Emergent Order and Organization. Includes contributions by Wolfgang Grassl, Nicola Guarino, John T. Kearns, Rudolf Lüthe, Luc Schneider, Peter Simons, Wojciech Żełaniec, and Jan Woleński

Formal Foundations for Information-Preserving Model Synchronization Processes Based on Triple Graph Grammars

Zwischen verschiedenen Artefakten, die Informationen teilen, wieder Konsistenz herzustellen, nachdem eines von ihnen geändert wurde, ist ein wichtiges Problem, das in verschiedenen Bereichen der Informatik auftaucht. Mit dieser Dissertation legen wir eine Lösung für das grundlegende Modellsynchronisationsproblem vor. Bei diesem Problem ist ein Paar solcher Artefakte (Modelle) gegeben, von denen eines geändert wurde; Aufgabe ist die Wiederherstellung der Konsistenz. Tripelgraphgrammatiken (TGGs) sind ein etablierter und geeigneter Formalismus, um dieses und verwandte Probleme anzugehen. Da sie auf der algebraischen Theorie der Graphtransformation und dem (Double-)Pushout Zugang zu Ersetzungssystemen basieren, sind sie besonders geeignet, um Lösungen zu entwickeln, deren Eigenschaften formal bewiesen werden können. Doch obwohl TGG-basierte Ansätze etabliert sind, leiden viele von ihnen unter dem Problem des Informationsverlustes. Wenn ein Modell geändert wurde, können während eines Synchronisationsprozesses Informationen verloren gehen, die nur im zweiten Modell vorliegen. Das liegt daran, dass solche Synchronisationsprozesse darauf zurückfallen Konsistenz dadurch wiederherzustellen, dass sie das geänderte Modell (bzw. große Teile von ihm) neu übersetzen. Wir schlagen einen TGG-basierten Ansatz vor, der fortgeschrittene Features von TGGs unterstützt (Attribute und negative Constraints), durchgängig formalisiert ist, implementiert und inkrementell in dem Sinne ist, dass er den Informationsverlust im Vergleich mit vorherigen Ansätzen drastisch reduziert. Bisher gibt es keinen TGG-basierten Ansatz mit vergleichbaren Eigenschaften. Zentraler Beitrag dieser Dissertation ist es, diesen Ansatz formal auszuarbeiten und seine wesentlichen Eigenschaften, nämlich Korrektheit, Vollständigkeit und Termination, zu beweisen. Die entscheidende neue Idee unseres Ansatzes ist es, Reparaturregeln anzuwenden. Dies sind spezielle Regeln, die es erlauben, Änderungen an einem Modell direkt zu propagieren anstatt auf Neuübersetzung zurückzugreifen. Um diese Reparaturregeln erstellen und anwenden zu können, entwickeln wir grundlegende Beiträge zur Theorie der algebraischen Graphtransformation. Zunächst entwickeln wir eine neue Art der sequentiellen Komposition von Regeln. Im Gegensatz zur gewöhnlichen Komposition, die zu Regeln führt, die Elemente löschen und dann wieder neu erzeugen, können wir Regeln herleiten, die solche Elemente stattdessen bewahren. Technisch gesehen findet der Synchronisationsprozess, den wir entwickeln, außerdem in der Kategorie der partiellen Tripelgraphen statt und nicht in der der normalen Tripelgraphen. Daher müssen wir sicherstellen, dass die für Double-Pushout-Ersetzungssysteme ausgearbeitete Theorie immer noch gültig ist. Dazu entwickeln wir eine (kategorientheoretische) Konstruktion neuer Kategorien aus gegebenen und zeigen, dass (i) diese Konstruktion die Axiome erhält, die nötig sind, um die Theorie für Double-Pushout-Ersetzungssysteme zu entwickeln, und (ii) partielle Tripelgraphen als eine solche Kategorie konstruiert werden können. Zusammen ermöglichen diese beiden grundsätzlichen Beiträge es uns, unsere Lösung für das grundlegende Modellsynchronisationsproblem vollständig formal auszuarbeiten und ihre zentralen Eigenschaften zu beweisen.Restoring consistency between different information-sharing artifacts after one of them has been changed is an important problem that arises in several areas of computer science. In this thesis, we provide a solution to the basic model synchronization problem. There, a pair of such artifacts (models), one of which has been changed, is given and consistency shall be restored. Triple graph grammars (TGGs) are an established and suitable formalism to address this and related problems. Being based on the algebraic theory of graph transformation and (double-)pushout rewriting, they are especially suited to develop solutions whose properties can be formally proven. Despite being established, many TGG-based solutions do not satisfactorily deal with the problem of information loss. When one model is changed, in the process of restoring consistency such solutions may lose information that is only present in the second model because the synchronization process resorts to restoring consistency by re-translating (large parts of) the updated model. We introduce a TGG-based approach that supports advanced features of TGGs (attributes and negative constraints), is comprehensively formalized, implemented, and is incremental in the sense that it drastically reduces the amount of information loss compared to former approaches. Up to now, a TGG-based approach with these characteristics is not available. The central contribution of this thesis is to formally develop that approach and to prove its essential properties, namely correctness, completeness, and termination. The crucial new idea in our approach is the use of repair rules, which are special rules that allow one to directly propagate changes from one model to the other instead of resorting to re-translation. To be able to construct and apply these repair rules, we contribute more fundamentally to the theory of algebraic graph transformation. First, we develop a new kind of sequential rule composition. Whereas the conventional composition of rules leads to rules that delete and re-create elements, we can compute rules that preserve such elements instead. Furthermore, technically the setting in which the synchronization process we develop takes place is the category of partial triple graphs and not the one of ordinary triple graphs. Hence, we have to ensure that the elaborate theory of double-pushout rewriting still applies. Therefore, we develop a (category-theoretic) construction of new categories from given ones and show that (i) this construction preserves the axioms that are necessary to develop the theory of double-pushout rewriting and (ii) partial triple graphs can be constructed as such a category. Together, those two more fundamental contributions enable us to develop our solution to the basic model synchronization problem in a fully formal manner and to prove its central properties

Quine and Quantified Modal Logic – Against the Received View

The textbook-like history of analytic philosophy is a history of myths, re-ceived views and dogmas. Though mainly the last few years have witnessed a huge amount of historical work that aimed to reconsider our narratives of the history of ana-lytic philosophy there is still a lot to do. The present study is meant to present such a micro story which is still quite untouched by historians. According to the received view Kripke has defeated all the arguments of Quine against quantified modal logic and thus it became a respectful tool for philosophers. If we accept the historical interpreta-tion of the network between Quine, Kripke and modal logic, which is to be presented here, we have to conclude that Quine’s real philosophical animadversions against the modalities are still on the table: though Kripke has provided some important (formal-logical) answers, Quine’s animadversions are still viable and worthy of further consideration

Quine and Quantified Modal Logic

The textbook - like history of analytic philosophy is a history of myths, r e- ceived views and dogmas. Though mainly the last few years have witnessed a huge amo unt of historical work that aimed to reconsider our narratives of the history of an a- lytic philosophy there is still a lot to do. The present study is meant to present such a micro story which is still quite untouched by historians. According to the receiv ed view Kripke has defeated all the arguments of Quine against quantified modal logic and thus it became a respectful tool for philosophers. If we accept the historical interpret ation of the network between Quine, Kripke and modal logic, which is to be pre sented here, we have to conclude that Quine’s real philosophical animadversions against the modalities are still on the table: though Kripke has provided some important (formal - logical) answers, Quine’s animadversions are still viable and worthy of further consider ation