We see a systematic set of cut-free axiomatisations for all the basic normal modal logics formed by some combination the axioms d, t, b, 4, 5. They employ a form of deep inference but otherwise stay very close to Gentzen's sequent calculus, in particular they enjoy a subformula property in the literal sense. No semantic notions are used inside the proof systems, in particular there is no use of labels. All their rules are invertible and the rules cut, weakening and contraction are admissible. All systems admit a straightforward terminating proof search procedure as well as a syntactic cut elimination procedure.

Abstract. The calculus of structures is a framework for specifying logical systems, which is similar to the one-sided sequent calculus but more general. We present a system of inference rules for propositional classical logic in this new framework and prove cut elimination for it. The system enjoys a decomposition theorem for derivations that is not available in the sequent calculus. The main novelty of our system is that all the rules are local : contraction, in particular, is reduced to atomic form. This should be interesting for distributed proof-search and also for complexity theory, since the computational cost of applying each rule is bounded.

Abstract. System SKS is a set of rules for classical propositional logic presented in the calculus of structures. Like sequent systems and unlike natural deduction systems, it has an explicit cut rule, which is admissible. In contrast to sequent systems, the cut rule can easily be restricted to atoms. This allows for a very simple cut elimination procedure based on plugging in parts of a proof, like normalisation in natural deduction and unlike cut elimination in the sequent calculus. It should thus be a good common starting point for investigations into both proof search as computation and proof normalisation as computation.

Currently known sequent systems for temporal logics such as linear time temporal logic and computation tree logic either rely on a cut rule, an invariant rule, or an infinitary rule. The first and second violate the subformula property and the third has infinitely many premises. We present finitary cut-free invariant-free weakeningfree and contraction-free sequent systems for both logics mentioned. In the case of linear time all rules are invertible. The systems are based on annotating fixpoint formulas with a history, an approach which has also been used in game-theoretic characterisations of these logics.

In this paper we will see deductive systems for classical propositional and predicate logic in the calculus of structures. Like sequent systems, they have a cut rule which is admissible. In addition, they enjoy a top-down symmetry and some normal forms for derivations that are not available in the sequent calculus. Identity axiom, cut, weakening and also contraction can be reduced to atomic form. This leads to rules that are local : they do not require the inspection of expressions of unbounded size. Conclusions 47