Complexity of Deep Inference via Atomic Flows Anupam Das  Abstract. We consider the fragment of deep inference free of compression mech- anisms and compare its proof complexity to other systems, utilising ‘atomic flows’ to examine size of proofs. Results include a simulation of Resolution and dag-like cut-free Gentzen, as well as a separation from bounded-depth Frege. 1 Introduction Deep inference differs from other proof formalisms by allowing derivations themselves to be composed by logical connectives. There has recently been a lot of activity in the proof complexity of deep inference [2], most notably that a cut-free system, KS + , quasipolynomially simulates Frege systems [12] [3]. It is conjectured that this can be improved to a polynomial simulation, so finding lower bounds for KS + is probably as hard as finding one for Frege, which has escaped proof complexity theorists for years. However this quasipolynomial simulation relies crucially on the presence of dag- like behaviour, manifested in deep inference by a particular rule, cocontraction: A ------ A ∧ A . Without it we have a minimal complete system closed under deep inference, KS. This system is free of compression mechanisms, in that a proof of a conjunction can be ‘partitioned’ into proofs of each conjunct, unlike proofs that are dag-like or contain cut. It is conjectured that KS is unable to polynomially simulate KS + [2], raising the question of exactly where it fits in the hierarchy of proof systems. In this paper we focus on upper bounds and simulations to demonstrate the relative strength of KS. Our arguments employ atomic flows [10], diagrams that track struc- tural changes in a proof but forget logical information, to show that cocontraction, and certain other steps, can be sometimes eliminated from a proof in polynomial time. A comprehensive introduction to atomic flows can be found in [11]. Our main result is a polynomial simulation of dag-like cut-free Gentzen systems (dag-Gen - ) in KS, improving on the simulation of the tree-like system in [2]. This also places KS in the gap between dag-Gen - and a variation augmented with elimina- tion rules (Gen  ), shown in [7] to simulate KS + , thereby quasipolynomially simulating Frege by the aforementioned result. This is discussed further in conclusion 7.2. Fig. 1 summarises our results, and full proofs of results can be found in [8] 2 Deep Inference We work in propositional logic over the basis { ¯ ·, ∧, ∨} with formulae in negation nor- mal form. Syntactic equivalence of formulae is denoted ≡. Definition 1 (Rules and Systems). An inference rule is a binary relation on formulae decidable in polynomial time, and a system is a set of rules. We define the rules we use  a.das@bath.ac.uk. Department of Computer Science, University of Bath.