Logic KM: A Biography

Muravitsky, Alexei

doi:10.1007/978-94-017-8860-1_7

Cited by 31 publications

(6 citation statements)

References 41 publications

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“…As for KM lin , our proofs yield the finite model property for KM as an immediate consequence, although for KM this is already known [25].…”

Section: Terminating Proof Search For Kmmentioning

confidence: 63%

“…Linear intuitionistic frames are known to be captured by the intermediate logic Dummett's LC [8]; the validity of the LC axiom in the topos of trees was first observed by Litak [22]. Intuitionistic frames with converse-well-founded reflexive reduction are captured by the intuitionistic modal logic KM, first called I ∆ [25]. Hence the internal propositional modal logic of the topos of trees is semantically exactly their combination, which we call KM lin (Litak [23,Thm.…”

Section: Introductionmentioning

confidence: 99%

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Sequent Calculus in the Topos of Trees

Clouston

Goré

2015

Lecture Notes in Computer Science

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Abstract. Nakano's "later" modality, inspired by Gödel-Löb provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that the semantics of the propositional fragment of this logic can be given by linear converse-well-founded intuitionistic Kripke frames, so this logic is a marriage of the intuitionistic modal logic KM and the intermediate logic LC. We therefore call this logic KM lin . We give a sound and cut-free complete sequent calculus for KM lin via a strategy that decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence capture KM.

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“…As for KM lin , our proofs yield the finite model property for KM as an immediate consequence, although for KM this is already known [25].…”

Section: Terminating Proof Search For Kmmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Sequent Calculus in the Topos of Trees

Clouston

Goré

2015

Lecture Notes in Computer Science

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“…The class of unital logics was defined in (Muravitsky, 2014a), although the name for this class of logics was suggested by an anonymous referee of this paper. In fact, this notion is rooted in Definition 4 of (Muravitsky, 2014b), where "a calculus [that] admits the Lindenbaum-Tarski algebra" was defined, which is somewhat weaker than the notion of a unital logic.…”

Section: Historical Notesmentioning

confidence: 99%

Consequence Relations An Introduction to the Tarski-Lindenbaum Method

Citkin¹,

Muravitsky²

2021

Preprint

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“…On Heyting algebras the function S 1 is the successor function. For details about the successor function on Heyting algebras see [18,25,26]. On Heyting algebras the function G 1 is the Gabbay's function, which will be denoted G. The description of G as the minimum of certain set was proved in [11].…”

Section: Compatible Functionsmentioning

confidence: 99%