Cut Elimination for Shallow Modal Logics ⋆ Bj¨ orn Lellmann and Dirk Pattinson Department of Computing, Imperial College London, UK Abstract. Motivated by the fact that nearly all conditional logics are axioma- tised 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 ap- plications 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 ex- tended 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. 1 Introduction Cut elimination is without doubt a central theme in proof theory. Not only do cut-free sequent systems provide for reasonably simple syntactical proofs of results like inter- polation, they also pave the way for decision procedures via backwards proof search. While there are a variety of methods to construct a cut-free sequent system for specific logics (and at least as many different sequent calculi), the general approach is to come up with a sequent system tailored to the logic at hand, and then show cut elimination for this particular system. While this approach works very well for specific logics, a good deal of ingenuity is required to construct the actual system. Since this method consumes both a lot of time and effort, this raises the question whether there is a generic method to construct cut-free calculi, and in particular, whether we can delegate the task of con- structing these systems. Our motivation for investigating this question mainly stems from automated proof search and questions of complexity, where the shape and struc- ture of the rules of a cut-free system are not important, as long as we can recognise rule instances fast enough. Our ultimate aim in this somewhat radical endeavour is to synthesise algorithms that recognise instances of a cut-free sequent system, given an axiomatisation of the logic under consideration. This paper reports on our first results on this programme in the context of modal logic: we study the question to what extent we can convert a Hilbert-style axiomatisa- tion of a general, not necessarily normal modal logic into a cut-free sequent system such ⋆ Supported by EPSRC-Project EP/H016317/1