Generic Modal Cut Elimination Applied to Conditional Logics

We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature.

💡 Research Summary

The paper addresses a long‑standing challenge in proof theory: providing a uniform, modular criterion for cut elimination in sequent calculi that works not only for normal modal logics but also for a broad class of non‑normal and conditional logics. The authors observe that traditional cut‑elimination proofs are highly bespoke; each logical connective typically requires a dedicated set of structural rules (contraction, weakening, inversion) and a labor‑intensive meta‑proof that the cut rule can be eliminated. This approach becomes especially cumbersome for conditional logics, where the interaction between the conditional operator and structural rules has resisted a clean treatment, leaving important cases such as conditional logic with modus ponens (MP) and conditional excluded middle (CEM) open with respect to cut elimination and complexity.

To overcome these difficulties, the authors propose a “generic cut‑elimination criterion” that hinges on the notion of absorption. A sequent calculus is split into two layers: (i) a propositional layer handling the usual Boolean connectives (∧, ∨, →, ¬) with their standard structural rules, and (ii) a modal/conditional layer containing only the rules for modal operators (□, ◇) and the conditional operator (⇒). Crucially, the modal/conditional layer is required to absorb the structural rules—cut, contraction, weakening, and inversion—meaning that any application of these structural transformations after a modal/conditional rule can be simulated by a sequence of modal/conditional rules alone. In practice, this is expressed through two technical conditions: (a) closure of the modal part under the structural operations, and (b) regularity of the conditional part, which guarantees that premises of a conditional rule are preserved under weakening and contraction. When both conditions hold, the generic criterion guarantees that the whole calculus admits cut elimination without a separate, intricate proof for each connective.

The paper then instantiates the framework on a variety of logics, demonstrating its breadth and power:

1. Normal modal logics K45 and S4 – By placing the standard modal axioms (e.g., 4, 5, T) in the modal layer, the authors show that the absorption conditions are trivially satisfied, yielding a streamlined cut‑free calculus that is substantially simpler than existing ones.

2. Conditional logics CK, CK+MP, CK+CE, and CK+MP+CE – For each, the conditional rule is formulated in the modal/conditional layer. The authors verify absorption by constructing explicit derivations that simulate contraction and weakening inside the conditional rule. Notably, for the combined MP + CE system (previously listed as an open problem), the paper provides the first cut‑elimination proof and establishes that proof search remains in PSPACE, matching the known lower bound for many modal logics.

3. Hybrid systems – The framework also accommodates logics that mix modal and conditional operators, illustrating that the same absorption principle applies uniformly.

Beyond cut elimination, the authors analyze the proof‑search complexity of the resulting calculi. Because the modal/conditional layer contains far fewer rules and no explicit structural rules, proof trees are shallow and the search space is tightly bounded. This yields a direct PSPACE upper bound for all the studied logics, and in several cases the bound is optimal. The paper also discusses how the internalised calculi are amenable to implementation in automated theorem provers, where the reduced rule set leads to faster proof search and easier termination checking.

In summary, the contribution of the paper is threefold:

• It introduces a general, modular cut‑elimination criterion based on absorption, which abstracts away from the particulars of any given modal or conditional operator.
• It applies this criterion to a wide spectrum of logics, including previously unresolved conditional systems, thereby delivering the first cut‑free calculi and complexity results for those systems.
• It provides practically useful sequent calculi that are simpler, more uniform, and better suited for automation than the fragmented calculi that appeared in the literature.

The work thus bridges a gap between proof‑theoretic elegance and computational applicability, offering a robust template for future extensions to other non‑classical logics such as intuitionistic, substructural, or hybrid modal systems.