From Display to Labelled Proofs for Tense Logics

Lecture Notes in Computer Science

2019

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This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated labelled calculus.

“…The cases of permuting (ref ) and (tra) above (⊃ * l ) and (lif t) are similar to Lem. 3,5,and Thm. 3.…”

Section: Deriving Nintqc From G3intqcmentioning

confidence: 99%

“…Deriving the calculus NInt from G3Int depends on a crucial observation made in [5] concerning labelled derivations: rules such as (ref ) and (tra) allow for theorems to be derived in proofs containing non-treelike labelled sequents. To demonstrate this fact, observe the following derivation in G3Int:…”

Section: ⊓ ⊔mentioning

confidence: 99%

On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

Lecture Notes in Computer Science

2019

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“…A labeled sequent can be viewed as a directed graph (defined using R) with formulae decorating each node [9]. Note that in a labeled sequent Λ = R, Γ commas between relational atoms are interpreted conjunctively, the comma between R and Γ is interpreted as an implication, and the commas between the labeled formulae in Γ are interpreted disjunctively.…”

Section: Labeled Calculi For Tense Logicsmentioning

confidence: 99%

“…In [9] we obtained translations from shallow nested calculi to labeled calculi for Scott-Lemmon axiomatic extensions ( ℎ → with ℎ, , , ∈ N) of Kt. This paper extends these results to a larger set of tense logics, and answers an open question posed in that paper regarding the existence of labeled to nested translations for extensions of Kt.…”

Section: Introductionmentioning

confidence: 99%

Display to Labeled Proofs and Back Again for Tense Logics

Ciabattoni

ACM Trans. Comput. Logic

Ramanayake

et al. 2021

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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.

“…⊓ ⊔ Additionally, from a computational viewpoint, it is interesting to know if completeness is preserved under a restricted class of sequents (cf. [6]). Indeed, Lem.…”

Section: Proof-search and Decidabilitymentioning

confidence: 99%

Automating Agential Reasoning: Proof-Calculi and Syntactic Decidability for STIT Logics

Lecture Notes in Computer Science

Berkel

2019

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This work provides proof-search algorithms and automated counter-model extraction for a class of STIT logics. With this, we answer an open problem concerning syntactic decision procedures and cut-free calculi for STIT logics. A new class of cut-free complete labelled sequent calculi G3Ldm m n , for multi-agent STIT with at most n-many choices, is introduced. We refine the calculi G3Ldm m n through the use of propagation rules and demonstrate the admissibility of their structural rules, resulting in the auxiliary calculi Ldm m n L. In the single-agent case, we show that the refined calculi Ldm m n L derive theorems within a restricted class of (forestlike) sequents, allowing us to provide proof-search algorithms that decide single-agent STIT logics. We prove that the proof-search algorithms are correct and terminate.