Display to Labeled Proofs and Back Again for Tense Logics

Abstract

We introduce translations between display calculus proofs and labeled calculus proofs in the context of tense logics. First, we show that every derivation in the display calculus for the minimal tense logic Kt extended with general path axioms can be effectively transformed into a derivation in the corresponding labeled calculus. Concerning the converse translation, we show that for Kt extended with path axioms, every derivation in the corresponding labeled calculus can be put into a special form that is translatable to a derivation in the associated display calculus. A key insight in this converse translation is a canonical representation of display sequents as labeled polytrees. Labeled polytrees, which represent equivalence classes of display sequents modulo display postulates, also shed light on related correspondence results for tense logics.

Nested Sequents for Intuitionistic Modal Logics via Structural Refinement

“…2 as logical rules. Our use of the term structural rules in reference to (d) and (S n,k ) is consistent with the use of the term in the literature on proof systems for modal and related logics [2,6,16,17] and is based on the fact that such rules manipulate the underlying data structure of sequents as opposed to introducing more complex logical formulae. Also, we point out that the (S n,k ) rules form a proper subclass of Simpson's (S χ ) geometric structural rules (see [31, p. 126]) used to generate labelled sequent systems for IK extended with any number of geometric axioms.…”

“…We show how to structurally refine the labelled systems introduced in the previous section, that is, we implement a methodology introduced and applied in [6,21,22,23] (referred to as structural refinement, or refinement more simply) for simplifying labelled systems and/or permitting the extraction of nested systems. The methodology consists of eliminating structural rules (viz.…”

“…The propagation rules we introduce are based on those of [6,17,22,23,33], and operate by viewing a labelled sequent as an automaton, allowing for the propagation of a formula (when applied bottom-up) from a label w to a label u given that a certain path of relational atoms exists between w and u (corresponding to a string generated by an A-grammar). We note that Simpson likewise introduced a variation of these rules, named (♦R) TH and ( L) TH (see [31, p. 126]), by closing the relational atoms of a sequent under the frame conditions related to each HSL φ(n, k) ∈ A.…”

“…Also, the propagation rules we use are largely based upon the work of [17,33], where such rules were used in the setting of display and nested calculi. These rules were then transported to the labelled setting to prove the decidability of agency logics [23], to establish translations between calculi within various proof-theoretic formalisms [6], and to provide a basis for the structural refinement methodology [21]. This paper accomplishes the following: First, we show that structural refinement can be used to extract nested sequent systems from Simpson's labelled sequent systems [31] with proofs in the latter formalism algorithmically translatable into proofs of the nested formalism.…”

Nested Sequents for Intuitionistic Grammar Logics via Structural Refinement

“…The propagation rules we use are largely based upon the work of [26,49], where such rules were used in the setting of display and nested calculi. These rules were then transported to the labeled setting to prove the decidability of agency (STIT) logics [34], to establish translations between calculi within various proof-theoretic formalisms [11], and to provide a basis for the structural refinement methodology [32]. In this paper, we apply this methodology in the setting of intuitionistic grammar logics, obtaining analytic nested systems for these logics, which are then put to use to establish conservativity and (un)decidability results.…”