Embedding theorems for LTL and its variants

Kamide, Norihiro

doi:10.1017/s0960129514000048

Cited by 4 publications

(1 citation statement)

References 38 publications

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“…We can also obtain a theorem for embedding

sans-serifTNQL

into

sans-serifINQL

. A similar translation has been used by Kamide to embed linear‐time temporal logic into infinitary logic . Definition We fix a countable non‐empty set Φ of propositional variables and define the sets

{normalΦ}_{i} : = false{p_{i} false| p \in normalΦ false}

(i \in ω)

of propositional variables where

p_{0} : = p \in Φ

.…”

Section: Embedding Theoremsmentioning

confidence: 99%

Extending paraconsistent quantum logic: a single‐antecedent/succedent system approach

Kamide

2018

Mathematical Logic Qtrly

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In this study, some conservative extensions of paraconsistent quantum logic, such as Nelsonian, modal, infinitary and temporal, are investigated by extending a single-antecedent/succedent sequent calculus PQL for paraconsistent quantum logic. A sequent calculus NQL, which is obtained from PQL by adding implication and co-implication, is introduced as a variant of Nelson's paraconsistent four-valued logic. Sequent calculi MPQL, IPQL and TPQL are introduced, respectively, as modal, infinitary and temporal extensions of PQL. The cut-elimination and duality theorems for these calculi are proved, and some extended calculi including NQL and MPQL, as well as their fragments, are shown to be decidable. A theorem for embedding NQL into its negation-free fragment and a theorem for embedding TPQL into IPQL are proved.The aim of this study is to obtain some novel conservative extensions of the paraconsistent quantum logic [4], which is a common denominator of minimal quantum logic (or orthologic) [3] and paraconsistent four-valued logic [2,5]. To obtain such extensions, we extend a simple single-antecedent/succedent sequent calculus PQL proposed in [9] for paraconsistent quantum logic. Paraconsistent quantum logic was introduced by Dalla Chiara and Giuntini [4] as a weak form of quantum logic [3], where the non-contradiction and the excluded-middle laws are violated. They introduced an axiomatization, a Kripke-style semantics, and an algebraic semantics for two forms of paraconsistent quantum logic. A cut-free sequent calculus for paraconsistent quantum logic was introduced originally by Faggian and Sambin [6], and they also referred to paraconsistent quantum logic as a basic orthologic. As mentioned, paraconsistent quantum logic itself has been studied, but a few novel conservative extensions of it have not yet been studied.The single-antecedent/succedent sequent calculus PQL and its first-order extension FPQL were introduced by Kamide in [9] for paraconsistent quantum logic and its first-order extension, respectively. The sequents ⇒ of PQL and FPQL have the restriction that both and are just a single formula rather than a set of formulas, i.e., ⇒ is of the form γ ⇒ δ, where γ and are formulas. This restriction of sequents is regarded as the intersection of the sequent restrictions on positive intuitionistic logic and positive dual-intuitionistic logic. The sequent calculus introduced by Faggian and Sambin was based on the usual sequent ⇒ , where and are regarded as a set of formulas. It was thus shown in [9] that the usual non-restricted sequents ⇒ of paraconsistent quantum logic can be restricted to the single antecedent/succedent sequents γ ⇒ δ. A comprehensive survey on sequent calculi for quantum logics, including paraconsistent quantum logic, is addressed in [9].By using the single antecedent/succedent sequents of PQL and FPQL, the proofs of the cut-elimination and decidability of PQL, FPQL and their extensions are made considerably simpler than other proposals. No double induction on grade and rank in the cut-elimina...

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“…We can also obtain a theorem for embedding