Decidability of admissibility:On a problem by friedman and its solution by rybakov

Rybakov (1984) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility

Admissibility and unifiability in contact logics

Contact logics are logics for reasoning about the contact relations between regular subsets in a topological space. Admissible inference rules can be used to improve the performance of any algorithm that handles provability within the context of contact logics. The decision problem of unifiability can be seen as a special case of the decision problem of admissibility. In this paper, we examine the decidability of admissibility problems and unifiability problems in contact logics

About the type of modal logics for the unification problem

Dans cette thèse, nous étudierons le problème de l'unification dans les logiques modales ordinaires, les fusions de deux logiques modales et les logiques épistémiques multi-modales. Relativement à une logique propositionnelle L, étant donnée une formule A, nous devons trouver des substitutions s telle que s(A) est dans L. Lorsqu'elles existent, ces substitutions sont appelées unifieurs de A dans L. Nous étudions différentes méthodes pour construire des ensembles minimaux complets d'unifieurs d'une formule donnée A et, en fonction de la cardinalité des ces ensembles minimaux complets, nous discutons du type de l'unification de A. Enfin, nous déterminons les types de l'unification de plusieurs logiques propositionnelles.In this thesis, we shall investigate on the unification problem in ordinary modal logics, fusions of two modal logics and multi-modal epistemic logics. With respect to a propositional logic L, given a formula A, we have to find substitutions s such that s(A) is in L. When they exist, these substitutions are called unifiers of A in L. We study different methods for the construction of minimal complete sets of unifiers of a given formula A and according to the cardinality of these minimal complete sets, we shall discuss on the unification type of A. Then, we determine the unification types of several propositional logics

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

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

An extended interval temporal logic and a framing technique for temporal logic programming

PhD ThesisTemporal logic programming is a paradigm for specification and verification of concurrent programs in which a program can be written, and the properties of the program can be described and verified in a same notation. However, there are many aspects of programming in temporal logics that are not well-understood. One such an aspect is concurrent programming, another is framing and the third is synchronous communication for parallel processes. This thesis extends the original Interval Temporal Logic (ITL) to include infinite models, past operators, and a new projection operator for dealing with concurrent computation, synchronous communication, and framing in the context of temporal logic programming. The thesis generalizes the original ITL to include past operators such as previous and past chop, and extends the model to include infinite intervals. A considerable collection of logic laws regarding both propositional and first order logics is formalized and proved within model theory. After that, a subset of the extended ITL is formalized as a programming language, called extended Tempura. These extensions, as in their logic basis, include infinite models, the previous operator, projection and framing constructs. A normal form for programs within the extended Tempura is demonstrated. Next, a new projection operator is introduced. In the new construct, the sub-processes are autonomous; each process has the right to specify its own interval over which it is executed. The thesis presents a framing technique for temporal logic programming, which includes the definitions of new assignments, the assignment flag and the framing operator, the formalization of algebraic properties of the framing operator, the minimal model semantics of framed programs, as well as an executable framed interpreter. The synchronous communication operator await is based directly on the proposed framing technique. It enables us to deal with concurrent computation. Based on EITL and await operator, a framed concurrent temporal logic programming language, FTLL, is formally defined within EITL. Finally, the thesis describes a framed interpreter for the extended Tempura which has been developed in SICSTUS prolog. In the new interpreter, the implementation of new assignments, the frame operator, the await operator, and the new projection operator are all included

Hierarchical contextual reasoning

Computer supported development of proofs requires user interaction even for theorems that are simple by human standards. In this thesis we define a communication infrastructure as a mediator between the user and the automatic reasoning procedures. It is based on a new uniform meta proof theory for contextual reasoning and encompasses most aspects of communication from the presentation of the proof state, via the supply of relevant contextual information about possible proof continuations, to the support for a hierarchical proof development. The proof theory is uniform for a variety of logics. It exploits proof theoretic annotations in formulas for a contextual reasoning style that is as far as possible intuitive for the user while at the same time still adequate for automatic reasoning procedures. Furthermore, concepts are defined to accomodate both the use and the explicit representation of hierarchies that are inherent in problem solving in general.Das computergestuetzte Beweisen von Theoremen erfordert den Eingriff des menschlichen Benutzers selbst fuer nach menschlichen Maßstaeben einfache Theoreme. Diese Arbeit definiert eine Kommunikationsplattform, die eine synergetische Kooperationsform des Benutzers mit dem Beweisverfahren ermöglicht

Raisonnement sur les modèles : détection et isolation d'anomalies dans les systèmes de diagnostic

Dans le cadre du diagnostic à base de Modèle, un ensemble de règles d'inférence est typiquement exploité pour calculer des diagnostics, ceci en utilisant une théorie scientifique et mathématique sur le système à diagnostiquer, ainsi qu'un ensemble d'observations. Contrairement aux hypothèses classiques, les Modèles sont souvent anormaux vis-à-vis d'un ensemble de propriétés requises. Naturellement, cela affecte la qualité des diagnostics [à Airbus]. Une théorie sur la réalité, l'information et la cognition est créé pour redéfinir, dans une perspective basée sur la théorie des modèles, le cadre classique de diagnostic à base de Modèle. Ceci rend possible la formalisation des anomalies et de leur relation avec des propriétés des diagnostics. Avec ce travail et avec l'idée qu'un système de diagnostic implémenté peut être vu comme un objet à diagnostiquer, une théorie de méta-diagnostic est développée, permettant la détection et isolation d'anomalies dans les Modèles des systèmes de diagnostic. Cette théorie est mise en pratique à travers d'un outil, MEDITO; et est testée avec succès à travers un ensemble de problèmes industriels, à Airbus. Comme des différents systèmes de diagnostic Airbus, souffrant d'anomalies variées, peuvent calculer des diagnostics différents, un ensemble de méthodes et outils et développé pour: 1) déterminer la cohérence entre diagnostics et 2) valider et comparer la performance de ces systèmes de diagnostic. Ce travail dépend d'un pont original entre le cadre de diagnostic Airbus et son équivalent académique. Finalement, la théorie de méta-diagnostic est généralisée pour prendre en compte des méta-systèmes autres que des systèmes de diagnostic implémentés.In Model-Based Diagnosis, a set of inference rules is typically used to compute diagnoses using a scientific and mathematical theory about a system under study and some observations. Contrary to the classical hypothesis, it is often the case that these Models are abnormal with respect to a series of required properties, hence affecting the quality of the computed diagnoses with possibly huge economical consequences, in particular at Airbus. A thesis on reality and cognition is firstly used to redefine the classic framework of model-based diagnosis from a formal model-theoretic perspective. This, in turn, enables the formalisation of abnormalities and of their relation with the properties diagnoses. With such material and the idea that an implemented diagnostic system can be seen a real-world artefact to be diagnosed, a theory of meta-diagnosis is developed, enabling the detection and isolation of abnormalities in Models of diagnostic systems and explanation in general. Such theory is then encoded in a tool, called MEDITO, and successfuly tested against Airbus real-world industrial problems. Moreover, as different heterogeneous implemented Airbus diagnostic systems, suffering from distinct abnormalities, may compute different diagnoses, methods and tools are developed for: 1) checking the consistency between subsystem-level diagnoses and 2) validating and comparing the performance of these diagnostic systems. Such work relies on an original bridge between the Airbus framework of diagnosis and its academic counterpart. Finally, meta-diagnosis is generalised to handle meta-systems other than implemented diagnostic systems