Björn Lellmann

Discrete and Continuous Models for Partitioning Problems

Recently, variational relaxation techniques for approximating solutions of partitioning problems on continuous image domains have received considerable attention, since they introduce significantly less artifacts than established graph cut-based techniques. This work is concerned with the sources of such artifacts. We discuss the importance of differentiating between artifacts caused by discretization and those caused by relaxation and provide supporting numerical examples. Moreover, we consider in depth the consequences of a recent theoretical result concerning the optimality of solutions obtained using a particular relaxation method. Since the employed regularizer is quite tight, the considered relaxation generally involves a large computational cost. We propose a method to significantly reduce these costs in a fully automatic way for a large class of metrics including tree metrics, thus generalizing a method recently proposed by Strekalovskiy and Cremers (IEEE conference on computer vision and pattern recognition, pp. 1905–1911, 2011).

Correspondence between Modal Hilbert Axioms and Sequent Rules with an Application to S5

Which modal logics can be ‘naturally’ captured by a sequent system? Clearly, this question hinges on what one believes to be natural, i.e. which format of sequent rules one is willing to accept. This paper studies the relationship between the format of sequent rules and the corresponding syntactical shape of axioms in an equivalent Hilbert-system. We identify three different such formats, the most general of which captures most logics in the S5-cube. The format is based on restricting the context in rule premises and the correspondence is established by translating axioms into rules of our format and vice versa. As an application we show that there is no set of sequent rules of this format which is sound and cut-free complete for S5 and for which cut elimination can be shown by the standard permutation-of-rules argument.

Axioms vs Hypersequent Rules with Context Restrictions: Theory and Applications

We introduce transformations between hypersequent rules with context restrictions and Hilbert axioms extending classical (and intuitionistic) propositional logic and vice versa. The introduced rules are used to prove uniform cut elimination, decidability and complexity results as well as finite axiomatisations for many modal logics given by simple frame properties. Our work subsumes many logic-tailored results and allows for new results. As a case study we apply our methods to the logic of uniform deontic frames.

Constructing Cut Free Sequent Systems with Context Restrictions Based on Classical or Intuitionistic Logic

We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic \(\mathbb{VA}\).

Cut elimination for shallow modal logics

Motivated by the fact that nearly all conditional logics are axiomatised by so-called shallow axioms (axioms with modal nesting depth ≤ 1) we investigate sequent calculi and cut elimination for modal logics of this type. We first provide a generic translation of shallow axioms to (one-sided, unlabelled) sequent rules. The resulting system is complete if we admit pseudo-analytic cut, i.e. cuts on modalised propositional combinations of subformulas, leading to a generic (but sub-optimal) decision procedure. In a next step, we show that, for finite sets of axioms, only a small number of cuts is needed between any two applications of modal rules. More precisely, completeness still holds if we restrict to cuts that form a tree of logarithmic height between any two modal rules. In other words, we obtain a small (PSPACE-computable) representation of an extended rule set for which cut elimination holds. In particular, this entails PSPACE decidability of the underlying logic if contraction is also admissible. This leads to (tight) PSPACE bounds for various conditional logics.

Linear Nested Sequents, 2-Sequents and Hypersequents

We introduce the framework of linear nested sequent calculi by restricting nested sequents to linear structures. We show the close connection between this framework and that of 2-sequents, and provide linear nested sequent calculi for a number of modal logics as well as for intuitionistic logic. Furthermore, we explore connections to backwards proof search for sequent calculi and to the hypersequent framework, including a reinterpretation of various hypersequent calculi for modal logic S5 in the linear nested sequent framework.

Proof Search in Nested Sequent Calculi

We propose a notion of focusing for nested sequent calculi for modal logics which brings down the complexity of proof search to that of the corresponding sequent calculi. The resulting systems are amenable to specifications in linear logic. Examples include modal logic

Grafting hypersequents onto nested sequents

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