Cyclic Involutive Distributive Full Lambek Calculus is Decidable

Kozak, Michał

doi:10.1093/logcom/exq021

Cited by 3 publications

(12 citation statements)

References 13 publications

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“…Because of the strong completeness theorem that covers all basic variants of CyInFL with ‘intuitionistic’ sequents, we rename this sequent system to symmetric constructive full Lambek calculus (

boldSymConFL

). We verify the decidability of this system and its basic variants, as we did in the case of their distributive cousins . As a consequence we obtain that the corresponding theories of (distributive and nondistributive) symmetric constructive FL‐algebras are decidable.…”

mentioning

confidence: 71%

“…Now we shall show that the cut rule is admissible in

boldSymConFL

\begin{matrix} right (cut) & right \frac{Γ, A, Δ \Rightarrow α Ψ \Rightarrow A}{Γ, Ψ, Δ \Rightarrow α} \end{matrix}

Like in we shall carry out the proof of cut admissibility simultaneously proving that the following variants are admissible as well.

\begin{matrix} right {(cut)}_{1} & right \frac{Γ \Rightarrow \sim A Ψ \Rightarrow A}{Γ, Ψ \Rightarrow} & {(cut)}_{2} & \frac{Γ \Rightarrow A Ψ \Rightarrow \sim A}{Γ, Ψ \Rightarrow} \end{matrix}

Theorem Consider the system

boldSymConFL

enriched with the rules (cut) 1 , (cut) 2 and (cut).…”

Section: Symmetric Constructive Full Lambek Calculusmentioning

confidence: 98%

“…However only weak completeness, since according to the equivalence

{| C |}_{\equiv} \leq {(||, D)}_{\equiv} \Leftrightarrow C ≺ D and \sim D ≺ \sim C,

a sequent

E \Rightarrow F

frakturX

does not imply that

{(||, E)}_{\equiv} \leq {(||, F)}_{\equiv}

holds in

(〈〉 {double-struckL}_{\equiv F}, f)

. In ⇒ was interpreted as the partial order relation; here we interpret it as the quasi‐order relation. Nevertheless, strong completeness with respect to CyInFL holds when we limit

frakturX

to sequents with the empty antecedent (cf.…”

Section: Symmetric Constructive Full Lambek Calculusmentioning

confidence: 99%

“…Because of this rule, involutive FL‐algebras satisfying the identity

\sim x = - x

are called cyclic involutive FL‐algebras . This rule, however, turns out to be superfluous in the sequent system mentioned in as CyInFL with ‘intuitionistic’ sequents. It means that the completeness theorem can be shown without this rule—but only weak completeness!…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Strong negation in intuitionistic style sequent systems for residuated lattices

Kozak

2014

Mathematical Logic Qtrly

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We study the sequent system mentioned in the author's work [18] as CyInFL with 'intuitionistic' sequents. We explore the connection between this system and symmetric constructive logic of Zaslavsky [40] and develop an algebraic semantics for both of them. In contrast to the previous work, we prove the strong completeness theorem for CyInFL with 'intuitionistic' sequents and all of its basic variants, including variants with contraction. We also show how the defined classes of structures are related to cyclic involutive FL-algebras and Nelson FL ew -algebras. In particular, we prove the definitional equivalence of symmetric constructive FL ewc -algebras (algebraic models of symmetric constructive logic) and Nelson FL ew -algebras (algebras introduced by Spinks and Veroff [33], [34] as the termwise equivalent definition of Nelson algebras). Because of the strong completeness theorem that covers all basic variants of CyInFL with 'intuitionistic' sequents, we rename this sequent system to symmetric constructive full Lambek calculus (SymConFL). We verify the decidability of this system and its basic variants, as we did in the case of their distributive cousins [18]. As a consequence we obtain that the corresponding theories of (distributive and nondistributive) symmetric constructive FL-algebras are decidable.

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boldSymConFL

mentioning

confidence: 71%

“…Now we shall show that the cut rule is admissible in

boldSymConFL

\begin{matrix} right (cut) & right \frac{Γ, A, Δ \Rightarrow α Ψ \Rightarrow A}{Γ, Ψ, Δ \Rightarrow α} \end{matrix}

Like in we shall carry out the proof of cut admissibility simultaneously proving that the following variants are admissible as well.

\begin{matrix} right {(cut)}_{1} & right \frac{Γ \Rightarrow \sim A Ψ \Rightarrow A}{Γ, Ψ \Rightarrow} & {(cut)}_{2} & \frac{Γ \Rightarrow A Ψ \Rightarrow \sim A}{Γ, Ψ \Rightarrow} \end{matrix}

Theorem Consider the system

boldSymConFL

enriched with the rules (cut) 1 , (cut) 2 and (cut).…”

Section: Symmetric Constructive Full Lambek Calculusmentioning

confidence: 98%

“…However only weak completeness, since according to the equivalence

{| C |}_{\equiv} \leq {(||, D)}_{\equiv} \Leftrightarrow C ≺ D and \sim D ≺ \sim C,

a sequent

E \Rightarrow F

frakturX

does not imply that

{(||, E)}_{\equiv} \leq {(||, F)}_{\equiv}

holds in

(〈〉 {double-struckL}_{\equiv F}, f)

. In ⇒ was interpreted as the partial order relation; here we interpret it as the quasi‐order relation. Nevertheless, strong completeness with respect to CyInFL holds when we limit

frakturX

to sequents with the empty antecedent (cf.…”

Section: Symmetric Constructive Full Lambek Calculusmentioning

confidence: 99%

“…Because of this rule, involutive FL‐algebras satisfying the identity

\sim x = - x

Section: Introductionmentioning

confidence: 99%

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Strong negation in intuitionistic style sequent systems for residuated lattices

Kozak

2014

Mathematical Logic Qtrly

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“…It is also known that cyclic InFL-algebras [19,18], cyclic distributive InFL-algebras [10], commutative InFL-algebras and lattice-ordered groups have decidable equational theories. This, together with Theorem 22 and the above definition of V, V , yields the following result.…”

Section: Decidability Of Quasi Relation Algebrasmentioning

confidence: 99%

Relation algebras as expanded FL-algebras

Galatos

Jipsen

2012

Algebra Univers.

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Abstract. This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined, and is shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.

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Non-associative, Non-commutative Multi-modal Linear Logic

Blaisdell

Kanovich

Kuznetsov

et al. 2022

Lecture Notes in Computer Science

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Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system ($$\mathsf {acLL}_\varSigma $$ acLL Σ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of $$\mathsf {acLL}_\varSigma $$ acLL Σ .

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Cyclic Involutive Distributive Full Lambek Calculus is Decidable

Cited by 3 publications

References 13 publications

Strong negation in intuitionistic style sequent systems for residuated lattices

Strong negation in intuitionistic style sequent systems for residuated lattices

Relation algebras as expanded FL-algebras

Non-associative, Non-commutative Multi-modal Linear Logic

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