Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics

We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for L. These include: hypersequent calculi for A and L and terminating versions of these calculi; labelled single sequent calculi for A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur

Bounding Resource Consumption with Gödel-Dummett Logics

International audienceGödel-Dummett logic LC and its finite approximations LCn are the intermediate logics complete w.r.t. linearly ordered Kripke models. In this paper, we use LCn logics as a tool to bound resource consumption in some process calculi. We introduce a non-deterministic process calculus where the consumption of a particular resource denoted * is explicit and provide an operational semantics which measures the consumption of this resource.We present a linear transformation of a process P into a formula f of LC. We show that the consumption of the resource by P can be bounded by the positive integer n if and only if the formula f admits a counter-model in LCn. Combining this result with our previous results on proof and counter-model construction for LCn, we conclude that bounding resource consumption is (linearly) equivalent to searching counter-models in LCn

Bounded-analytic sequent calculi and embeddings for hypersequent logics

A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory

Graph-based decision for Gödel-Dummett logics

International audienceWe present a graph-based decision procedure for Gödel-Dummett logics and an algorithm to compute counter-models. A formula is transformed into a conditional bi-colored graph in which we detect some specific cycles and alternating chains using matrix computations. From an instance graph containing no such cycle (resp. no (n+1)-alternating chain) we extract a counter-model in LC (resp. LCn)

Simplification rules for intuitionistic propositional tableaux

The implementation of a logic requires, besides the definition of a calculus and a decision procedure, the development of techniques to reduce the search space. In this article we introduce some simplification rules for Intuitionistic propositional logic that try to replace a formula with an equi-satisfiable \u201csimpler\u201d one with the aim to reduce the search space. Our results are proved via semantical techniques based on Kripke models. We also provide an empirical evaluation of their impact on implementations

Model and Proof Theory of Constructive ALC, Constructive Description Logics

Description logics (DLs) represent a widely studied logical formalism with a significant impact in the field of knowledge representation and the Semantic Web. However, they are equipped with a classical descriptive semantics that is characterised by a platonic notion of truth, being insufficiently expressive to deal with evolving and incomplete information, as from data streams or ongoing processes. Such partially determined and incomplete knowledge can be expressed by relying on a constructive semantics. This thesis investigates the model and proof theory of a constructive variant of the basic description logic ALC, called cALC. The semantic dimension of constructive DLs is investigated by replacing the classical binary truth interpretation of ALC with a constructive notion of truth. This semantic characterisation is crucial to represent applications with partial information adequately, and to achieve both consistency under abstraction as well as robustness under refinement, and on the other hand is compatible with the Curry-Howard isomorphism in order to form the cornerstone for a DL-based type theory. The proof theory of cALC is investigated by giving a sound and complete Hilbert-style axiomatisation, a Gentzen-style sequent calculus and a labelled tableau calculus showing finite model property and decidability. Moreover, cALC can be strengthened towards normal intuitionistic modal logics and classical ALC in terms of sound and complete extensions and hereby forms a starting point for the systematic investigation of a constructive correspondence theory.Beschreibungslogiken (BLen) stellen einen vieluntersuchten logischen Formalismus dar, der den Bereich der Wissensrepräsentation und das Semantic Web signifikant geprägt hat. Allerdings basieren BLen meist auf einer klassischen deskriptiven Semantik, die gekennzeichnet ist durch einen idealisierten Wahrheitsbegriff nach Platons Ideenlehre, weshalb diese unzureichend ausdrucksstark sind, um in Entwicklung befindliches und unvollständiges Wissen zu repräsentieren, wie es beispielsweise durch Datenströme oder fortlaufende Prozesse generiert wird. Derartiges partiell festgelegtes und unvollständiges Wissen lässt sich auf der Basis einer konstruktiven Semantik ausdrücken. Diese Arbeit untersucht die Model- und Beweistheorie einer konstruktiven Variante der Basis-BL ALC, die im Folgenden als cALC bezeichnet wird. Die Semantik dieser konstruktiven Beschreibungslogik resultiert daraus, die traditionelle zweiwertige Interpretation logischer Aussagen des Systems ALC durch einen konstruktiven Wahrheitsbegriff zu ersetzen. Eine derartige Interpretation ist die Voraussetzung dafür, um einerseits Anwendungen mit partiellem Wissen angemessen zu repräsentieren, und sowohl die Konsistenz logischer Aussagen unter Abstraktion als auch ihre Robustheit unter Verfeinerung zu gewährleisten, und andererseits um den Grundstein für eine Beschreibungslogik-basierte Typentheorie gemäß dem Curry-Howard Isomorphismus zu legen. Die Ergebnisse der Untersuchung der Beweistheorie von cALC umfassen eine vollständige und korrekte Hilbert Axiomatisierung, einen Gentzen Sequenzenkalkül, und ein semantisches Tableaukalkül, sowie Beweise zur endlichen Modelleigenschaft und Entscheidbarkeit. Darüber hinaus kann cALC zu normaler intuitionistischer Modallogik und klassischem ALC durch vollständige und korrekte Erweiterungen ausgebaut werden, und bildet damit einen Startpunkt für die systematische Untersuchung einer konstruktiven Korrespondenztheorie

Goal-directed proof theory

This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape