Non-Well-Founded Proofs for the Grzegorczyk Modal Logic

Abstract

We present a sequent calculus for the Grzegorczyk modal logic $\mathsf {Grz}$ allowing cyclic and other non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs. As an application, we establish the Lyndon interpolation property for the logic $\mathsf {Grz}$ proof-theoretically.

Abstract cyclic proofs

“…For further examples of modelling of the trace conditions of cyclic proof systems in trace categories, in particular, those of cyclic arithmetic (Simpson 2017), HFL N (Kori et al 2021) and Grzegorczyk modal logic (Savateev and Shamkanov 2021), we refer the reader to Wehr (2021).…”

“…An application of the result above is establishing the equivalence between cyclic proof systems with and without a Cut-rule. Many Cut-elimination procedures for cyclic proof systems in the literature are corecursive algorithms which lazily transform cyclic proofs with Cut-applications into ∞-proofs without Cuts (e.g., Baelde et al 2016;Fortier and Santocanale 2013;Savateev and Shamkanov 2021). If the Cut-free fragment of the derivation system is finite, the result above can then be applied to conclude that there must also exist a Cut-free cyclic proof.…”

“…If the Cut-free fragment of the derivation system is finite, the result above can then be applied to conclude that there must also exist a Cut-free cyclic proof. An example of the result being applied in this way is given by Savateev and Shamkanov (2021). A disadvantage of this method of Cut-elimination is that the Cut-free cyclic proof need not be related to the original Cutfree ∞-proof.…”

Metric fixed point theory and partial impredicativity

“…Another fixed point theorem that was originally proven via infinitary methods (stated in terms of an explicit invocation of Zorn's lemma) is the Priess-Crampe & Ribenboim theorem, which deals with spherically complete ultrametric spaces. This result is partially motivated by logic programming [6] and has recently found applications in cut-elimination for ill-founded proofs [7]. Definition 1.4.…”

Cut Elimination for the Weak Modal Grzegorczyk Logic via Non-well-Founded Proofs

“…Recently a new proof-theoretic presentation for the logic GL in the form of a sequent calculus allowing non-well-founded proofs was given in [10,4]. Later, the same ideas were applied to the modal Grzegorczyk logic Grz in [8,7], where it allowed to prove several proof-theoretic properties of this logic syntactically.…”