Sebastian Enqvist

Interrogative Belief Revision in Modal Logic

The well known AGM framework for belief revision has recently been extended to include a model of the research agenda of the agent, i.e. a set of questions to which the agent wishes to find answers (Olsson & Westlund in Erkenntnis, 65, 165–183, 2006). The resulting model has later come to be called interrogative belief revision. While belief revision has been studied extensively from the point of view of modal logic, so far interrogative belief revision has only been dealt with in the metalanguage approach in which AGM was originally presented. In this paper, I show how to model interrogative belief revision in a modal object language using a class of operators for questions. In particular, the solution I propose will be shown to capture the notion of K-truncation, a method for agenda update in the case of expansion constructed by Olsson & Westlund. Two case studies are conducted: first, an interrogative extension of Krister Segerberg’s system DDL, and then a similar extension of Giacomo Bonanno’s modal logic for belief revision. Sound and complete axioms will be provided for both of the resulting logics.

Instantial neighbourhood logic

This paper explores a new language of neighbourhood structures where existential information can be given about what kind of worlds occur in a neighbourhood of a current world. The resulting system of ‘instantial neighbourhood logic’ INL has a nontrivial mix of features from relational semantics and from neighbourhood semantics. We explore some basic model-theoretic behavior, including a matching notion of bisimulation, and give a complete axiom system for which we prove completeness by a new normal form technique. In addition, we relate INL to other modal logics by means of translations, and determine its precise SAT complexity. Finally, we discuss proof-theoretic fine-structure of INL in terms of semantic tableaux and some expressive fine-structure in terms of fragments, while discussing concrete illustrations of the instantial neighborhood language in topological spaces, in games with powers for players construed in a new way, as well as in dynamic logics of acquiring or deleting evidence. We conclude with some coalgebraic perspectives on what is achieved in this paper. Many of these final themes suggest follow-up work of independent interest.

A Structuralist Framework for the Logic of Theory Change

Ever since the 1980s, the dominant trend in the field within philosophical logic concerning the logic of theory change has been to work with models lying rather close to the well-known AGM approach. In particular, most of the (normative, formal) theories on the subject of theory change that can be found in the literature seem to be in relatively close agreement on how to represent theories; roughly, it is assumed that theories are representable as sets of statements. In this paper, I will try to draw the outlines of a model for theory change in which this assumption is revised: instead of the usual representation of theories, the foundation of the model will be based on the structuralistic notion of a theory net. Structuralism proceeds from the idea that scientific theories have what we may call a deep structure, and aims to provide a formal representation of theories that respects this structure and describes it in as much detail as possible. The result is a formal notion of “theory” which is considerably more fine-grained than the AGM-style representation in terms of logically closed sets of sentences, and my hope is that it will help shed some new light on the problems studied in the preexisting frameworks for theory change, as well as open up new and interesting research problems in the field.

Monadic Second-Order Logic and Bisimulation Invariance for Coalgebras

Generalizing standard monadic second-order logic for Kripke models, we introduce monadic second-order logic MSO(T) interpreted over co algebras for an arbitrary set functor T. Similar to well-known results for monadic second-order logic over trees, we provide a translation of this logic into a class of automata, relative to the class of T-co algebras that admit a tree-like supporting Kripke frame. We then consider invariance under behavioral equivalence of MSO(T)-formulas, more in particular, we investigate whether the co algebraic mu-calculus is the bisimulation-invariant fragment of MSO(T). Building on recent results by the third author we show that in order to provide such a co algebraic generalization of the Janin-Walukiewicz Theorem, it suffices to find what we call an adequate uniform construction for the functor T. As applications of this result we obtain a partly new proof of the Janin-Walukiewicz Theorem, and bisimulation invariance results for the bag functor (graded modal logic) and all exponential polynomial functors. Finally, we consider in some detail the monotone neighborhood functor M, which provides co algebraic semantics for monotone modal logic. It turns out that there is no adequate uniform construction for M, whence the automata-theoretic approach towards bisimulation invariance does not apply directly. This problem can be overcome if we consider global bisimulations between neighborhood models: one of our main results provides a characterization of the monotone modal mu-calculus extended with the global modalities, as the fragment of monadic second order logic for the monotone neighborhood functor that is invariant for global bisimulations.

A New Game Equivalence and its Modal Logic

We revisit the crucial issue of natural game equivalences, and semantics of game logics based on these. We present reasons for investigating finer concepts of game equivalence than equality of stan ...

Homomorphisms of coalgebras from predicate liftings

This paper assumes the concept of a predicate lifting from coalgebraic modal logic, and associates with every set Λ of predicate liftings for a set functor T a category \(\mathbb{C}^\Lambda_T\) of T-coalgebras and socalled Λ-homomorphisms. From this construction, some natural constructions on models such as products of models and submodels can be defined. A relationship with simulations of coalgebras arising from lax extensions is established, and the main technical result gives a condition under which the category \(\mathbb{C}^\Lambda_T\) is both complete and cocomplete.

A general Lindström theorem for some normal modal logics

There are several known Lindström-style characterization results for basic modal logic. This paper proves a generic Lindström theorem that covers any normal modal logic corresponding to a class of Kripke frames definable by a set of formulas called strict universal Horn formulas. The result is a generalization of a recent characterization of modal logic with the global modality. A negative result is also proved in an appendix showing that the result cannot be strengthened to cover every first-order elementary class of frames. This is shown by constructing an explicit counterexample.

Generalized Vietoris bisimulations

We introduce and study bisimulations for coalgebras on Stone spaces (Kupke et al., 2004, Theoretical Computer Science, 327, 109-134), motivated by previous work on ultrafilter extensions for coalge ...

A Propositional Dynamic Logic for Instantial Neighborhood Models

We propose a new perspective on logics of computation by combining instantial neighborhood logic INL with bisimulation safe operations adapted from PDL and dynamic game logic. INL is a recently proposed modal logic, based on a richer extension of neighborhood semantics which permits both universal and existential quantification over individual neighborhoods. We show that a number of game constructors from game logic can be adapted to this setting to ensure invariance for instantial neighborhood bisimulations, which give the appropriate bisimulation concept for INL. We also prove that our extended logic IPDL is a conservative extension of dual-free game logic, and its semantics generalizes the monotone neighborhood semantics of game logic. Finally, we provide a sound and complete system of axioms for IPDL, and establish its finite model property and decidability.

A new coalgebraic Lindström theorem

In a recent article, Alexander Kurz and Yde Venema establish a Lindstrom theorem for coalgebraic modal logic that is shown to imply a modal Lindstrom theorem by Maarten de Rijke. A later modal Lindstrom theorem has been established by Johan van Benthem, and this result still lacks a coalgebraic formulation. The main obstacle has so far been the lack of a suitable notion of ‘submodels’ in coalgebraic semantics, and the problem is left open by Kurz and Venema. In this article, we propose a solution to this problem and derive a general coalgebraic Lindstrom theorem along the lines of van Benthems result. We provide several applications of the result.