Quasipolynomial Normalisation in Deep Inference via Atomic Flows and Threshold Formulae

Je\v{r}'abek showed that cuts in classical propositional logic proofs in deep inference can be eliminated in quasipolynomial time. The proof is indirect and it relies on a result of Atserias, Galesi and Pudl'ak about monotone sequent calculus and a correspondence between that system and cut-free deep-inference proofs. In this paper we give a direct proof of Je\v{r}'abek’s result: we give a quasipolynomial-time cut-elimination procedure for classical propositional logic in 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 only trace structural rules and forget logical rules) and strong enough to faithfully represent the cut-elimination procedure.

💡 Research Summary

The paper presents a direct, quasipolynomial‑time cut‑elimination algorithm for classical propositional logic formulated in the deep‑inference system SKS (and its variant SKSg). Previously, Jeřábek (2009) showed that cuts in deep‑inference proofs could be eliminated in quasipolynomial time, but his proof was indirect: it relied on a result about monotone sequent calculus (Atserias‑Galesi‑Pudlák) and on a polynomial‑size simulation between monotone sequent proofs and deep‑inference proofs. The authors of the present work replace this indirect construction with a fully internal method that works entirely within deep inference.

Two main technical ingredients are introduced. First, the authors adapt the Atserias‑Galesi‑Pudlák technique to deep inference, simplifying the use of threshold functions. Their version of threshold formulae is smaller and structurally more transparent than the original ones, which reduces the overhead in the transformation. Second, they employ “atomic flows”, a graphical trace that records only the structural rules (contraction, weakening, cut, etc.) while ignoring logical rules. Atomic flows can be seen as a specialised form of Buss flow graphs; they capture the creation, duplication, and annihilation of atoms throughout a proof. Because they abstract away from logical connectives, atomic flows provide a clean geometric picture of the proof dynamics and allow the authors to control the cut‑elimination process precisely.

The overall normalization procedure consists of two phases. In the first phase (Section 4) any SKS derivation is transformed into a “simple form”. This transformation uses only deep‑inference manipulations and atomic‑flow analysis; no threshold formulae are introduced, and the size blow‑up is at most polynomial. The simple form arranges the structural rules in a disciplined way that makes subsequent processing straightforward.

In the second phase (Section 6) the authors perform the actual cut‑elimination on proofs already in simple form. Here the threshold formulae play a crucial role: they encode majority‑type conditions that allow the systematic removal of cuts by replacing them with a bounded number of contraction and weakening steps. Atomic flows guide the placement of these replacements, ensuring that each cut is eliminated without causing an exponential explosion of the proof size. After all cuts have been removed, a final cleanup (Section 7) eliminates the remaining non‑analytic rule, co‑weakening, by a standard deep‑inference reduction using the switch and medial rules. This yields a fully cut‑free, analytic SKS proof.

Complexity analysis shows that each phase can be carried out in time O(n·logⁿ) where n is the size of the original proof. Consequently, the whole cut‑elimination procedure runs in quasipolynomial time, matching Jeřábek’s bound but with a direct, internal construction. Moreover, the use of atomic flows makes the algorithm conceptually transparent and potentially adaptable to other logics or extensions of deep inference, because the flow‑based reasoning is largely syntax‑independent as long as linearity conditions are respected.

The paper also discusses related work, including the equivalence of SKS and its fragments with Frege systems, the role of the KS subsystem, and previous exponential‑time cut‑elimination methods. It highlights that the present approach not only improves the theoretical bound but also provides a geometric, flow‑based perspective that could inspire further research on proof complexity, proof compression, and the design of efficient proof‑search algorithms in deep inference.