LBI Cut Elimination Proof with BI-MultiCut

Abstract

Cut elimination in sequent calculus is indispensable in bounding the number of distinct formulas to appear during a backward proof search. A usual approach to prove cut admissibility is permutation of derivation trees. Extra care must be taken, however, when contraction appears as an explicit inference rule. In G1i for example, a simple-minded permutation strategy comes short around contraction interacting directly with cut formulas, which entails irreducibility of the derivation height of Cut instances. One of the practices employed to overcome this issue is the use of MultiCut (the "mix" rule) which takes into account the effect of contraction within. A more recent substructural logic BI inherits the characteristics of the intuitionistic logic but also those of multiplicative linear logic (without exponentials). Following Pym's original work, the cut admissibility in LBI (the original BI sequent calculus) is supposed to hold with the same tweak. However, there is a critical issue in the approach: MultiCut does not take care of the effect of structural contraction that LBI permits. In this paper, we show a proper proof of the LBI cut admissibility based on another derivable rule BI-MultiCut.

Structural Interactions and Absorption of Structural Rules in BI Sequent Calculus

“…Just as in the case of intuitionistic logic, cut admissibility proof for a contraction-free BI sequent calculus is simpler than that for LBI [1]. Since we have already proved depthpreserving weakening admissibility, the following context sharing cut, Cut CS , is easily verified derivable in LBIZ + Cut:…”

“…There is a direct cut elimination procedure which proves admissibility of Cut in LBI (sketched in [11]; corrected in [1]).…”

Semantic cut elimination for the logic of bunched implications, formalized in Coq

“…In the case of BI, the deep nested structure of bunches and explicit structural rules contribute to the complexity and the level of details. For example, a proof of cut elimination for BI given in [38,Chapter 6] had a gap, that was later fixed in [3]. The issue seems to arise from the treatment of the contraction rule.…”

“…To counterbalance informal pen-and-paper proofs of cut elimination for BI, we provide a fully formalized proof in the Coq proof assistant. However, instead of trying to formalize an intricate Gentzen-style process, as in [3], we approach cut elimination using the ideas of algebraic proof theory: a research area aimed at making tight connections between structural proof theory and algebraic semantics of logics. In our proof we adapt the methods of algebraic semantic cut elimination for linear logic [33,34], in which cut elimination is obtained by constructing a special model for linear logic that is universal w.r.t.…”

“…Unfortunately, this generalization was originally done in a way that only works for some of the cases. See [3] for more details. 1 This is not the only instance of erroneous proofs of cut elimination slipping in.…”