A Quasipolynomial Normalisation Procedure in Deep Inference Paola Bruscoli 1⋆ , Alessio Guglielmi 1⋆⋆ , Tom Gundersen 2⋆⋆⋆ , and Michel Parigot 2† 1 University of Bath (UK) 2 Laboratoire PPS, UMR 7126, CNRS & Université Paris 7 (France) Abstract. Jeˇ rábek showed in 2008 that cuts in propositional-logic deep-inference proofs can be eliminated in quasipolynomial time. The proof is an indirect one relying on a result of Atserias, Galesi and Pudlák about monotone sequent calculus and a correspondence between this system and cut-free deep-inference proofs. In this paper we give a direct proof of Jeˇ rábek’s result: we give a quasipolynomial- time cut-elimination procedure in propositional-logic deep inference. The main new ingredient is the use of a computational trace of deep-inference proofs called atomic flows, which are both very simple (they trace only structural rules and forget logical rules) and strong enough to faithfully represent the cut-elimination procedure. We also show how the technique can be extended to obtain a more general notion of normalisation called streamlining. 1 Introduction Deep inference is a deduction framework (see [Gug07,BT01,Brü04]), where deduction rules apply arbitrarily deep inside formulae, contrary to traditional proof systems such as natural deduction and the sequent calculus, where deduction rules deal only with the outermost structure of formulae. This greater freedom is both a source of immediate technical difficulty and the promise, in the long run, of new powerful proof-theoretic methods. A general methodology allows to design deep-inference deduction systems having more symmetries and finer structural properties than the sequent calculus. For instance, cut and identity become really dual of each other, whereas they are only morally so in the sequent calculus, and all structural rules can be reduced to their atomic form, whereas this is false for contraction in the sequent calculus. All usual logics have deep-inference deduction systems enjoying cut elimination (see [Gug] for a complete overview). The traditional methods of cut elimination of the ⋆ Supported by EPSRC grant EP/E042805/1 Complexity and Non-determinism in Deep Inference and by an ANR Senior Chaire d’Excellence titled Identity and Geometric Essence of Proofs. ⋆⋆ Supported by EPSRC grant EP/E042805/1 Complexity and Non-determinism in Deep Inference and by an ANR Senior Chaire d’Excellence titled Identity and Geometric Essence of Proofs. ⋆⋆⋆ Supported by an Overseas Research Studentship and a Research Studentship, both of the University of Bath, and by an ANR Senior Chaire d’Excellence titled Identity and Geometric Essence of Proofs. † Supported by Project INFER—Theory and Application of Deep Inference of the Agence Na- tionale de la Recherche.