Inducing Syntactic Cut-Elimination for Indexed Nested Sequents

Abstract

Abstract. The key to the proof-theoretical study of a logic is a cutfree proof calculus. Unfortunately there are many logics of interest lacking suitable proof calculi. The proof formalism of nested sequents was recently generalised to indexed nested sequents in order to yield cutfree proof calculi for extensions of the modal logic K by Geach (LemmonScott) axioms. The proofs of completeness and cut-elimination therein were semantical and intricate. Here we identify a subclass of the labelled sequent formalism and show that it corresponds to the indexed nested sequent formalism. This correspondence is then exploited to induce syntactic proofs for indexed nested sequents using the elegant existing proofs in the labelled sequent formalism. A larger goal of this work is to demonstrate how specialising existing proof-theoretical transformations (adapting these as required to remain within the subclass) is an alternative means of argument which can alleviate the need for independent proofs from 'scratch' in each formalism. Moreover, such coercion can be used to induce new cutfree calculi. We demonstrate by presenting the first indexed nested sequent calculi for intermediate logics. Finally, we introduce a notion of duplicate subgraph using certain graph automorphisms-generalising the notion of contraction for labelled sequents whose underlying structure is more general than trees-yielding a normal form for derivations. We envisage that such notions might be employed in order to prove decidability of a logic using its restricted labelled calculus.

Proof Theory for Indexed Nested Sequents

“…As Fitting's original system does not use a cut rule, this result is actually entailed by his (semantical) completeness theorem. Using the translation mentioned above, one could also use the cut-elimination result for labelled tree sequents with equality, yielding an indirect proof [23]. However, only an internal cut-elimination proof makes a proof formalism a first-class citizen for structural proof theory.…”

“…Fitting recently introduced indexed nested sequents [7], an extension of nested sequents which goes beyond the tree structure to give a cut-free system for the classical modal logic K extended with an arbitrary set of Scott-Lemmon axioms. In some sense indexed nested sequents are more similar to labelled systems than pure nested sequents-in fact, the translation between nested sequents and labelled tree sequents mentioned above is naturally extended in [23] to a translation between indexed nested sequents and labelled tree sequents with equality, where some nodes of the underlying tree can be identified.…”

Hypersequents and Systems of Rules: Embeddings and Applications

“…Such embeddings can also provide useful reformulations of known calculi and allow the transfer of certain proof-theoretic results, thus alleviating the need for independent proofs in each system and avoiding duplicating work. Various embeddings between formalisms have appeared in the literature, see, e.g., [24,23,13,11,20,21,26] (and the bibliography thereof).…”