The consequence relation in the logic of commutative GBL-algebras is PSPACE-complete

Bova, Simone; Montagna, Franco

doi:10.1016/j.tcs.2008.10.024

Cited by 13 publications

(18 citation statements)

References 20 publications

(27 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, we reasonably expect that the problem of finding efficiently optimal closures (in a suitable sense, required to embed enforcing algorithms into branch and bound exhaustive search) can be formalized in purely algebraic and logical terms in the setting proposed in this paper. 3 We remark that analogous local consistency techniques have been investigated by Bistarelli and Gadducci over tropical residuated semirings [2], and we encourage a future comparison of the two settings in terms of unifying potential, structural insight, and computational viability. We also remark that the idea of formalizing soft constraints consistency techniques as many-valued logics refutations appears in the work of Ansótegui et al [1].…”

Section: Introductionmentioning

confidence: 70%

Soft Constraints Processing over Divisible Residuated Lattices

Bova

2009

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. We claim that divisible residuated lattices (DRLs) can act as a unifying evaluation framework for soft constraint satisfaction problems (soft CSPs). DRLs form the algebraic semantics of a large family of substructural and fuzzy logics [13,15], and are therefore natural candidates for this role. As a preliminary evidence in support to our claim, along the lines of Cooper et al. and Larrosa et al. [11,18], we describe a polynomial-time algorithm that enforces k-hyperarc consistency on soft CSPs evaluated over DRLs. Observed that, in general, DRLs are neither idempotent nor totally ordered, this algorithm accounts as a generalization of available enforcing algorithms over commutative idempotent semirings and fair valuation structures [4,11].

show abstract

Section: Introductionmentioning

confidence: 70%

Soft Constraints Processing over Divisible Residuated Lattices

Bova

2009

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…The decision problem for classical Lukasiewicz logic is known to be co-NPcomplete while the decision problem for minimal Lukasiewicz logic can be shown to reduce to the decision problem for the equational theory of commutative GBL-algebras, which is known to be PSPACE-complete [6]. In both cases, the known decision procedures are based on semantic methods and no effective proof search methods are known.…”

Section: Discussionmentioning

confidence: 99%

“…The logics LL m and LL i can be faithfully characterised algebraically using the classes of algebraic structures originally due to Büchi and Owen and known as hoops and bounded hoops, respectively (see [1,2,5,7]). Hoops are reducts of commutative GBL algebras, whose equational theory has been shown to be PSPACE-complete [6]. As the equational theory of GBL algebras is a conservative extension of that of hoops, it follows that the decision problems for LL m and LL i are also PSPACEcomplete 1 .…”

Section: (Dne) ¬¬A ⇒ Amentioning

confidence: 99%

A Curry-Howard Correspondence for the Minimal Fragment of Łukasiewicz Logic

Arthan¹,

Oliva²

2018

Preprint

View full text Add to dashboard Cite

In this paper we introduce a term calculus B which adds to the affine λ-calculus with pairing a new construct allowing for a restricted form of contraction. We obtain a Curry-Howard correspondence between B and the sub-structural logical system which we call "minimal Lukasiewicz logic", also known in the literature as the logic of hoops (a generalisation of MV-algebras). This logic lies strictly in between affine minimal logic and standard minimal logic. We prove that B is strongly normalising and has the Church-Rosser property. We also give examples of terms in B corresponding to some important derivations from our work and the literature. Finally, we discuss the relation between normalisation in B and cut-elimination for a Gentzen-style formulation of minimal Lukasiewicz logic.

show abstract

“…It follows from work on commutative GBL-algebras that ŁL i is decidable [5]. However the problem is PSPACE-complete.…”

Section: Homomorphism Properties Of Double Negation In łL Imentioning

confidence: 99%

Negative Translations for Affine and Lukasiewicz Logic

Arthan,

Oliva

2019

Preprint

View full text Add to dashboard Cite

We investigate four well-known negative translations of classical logic into intuitionistic logic within a substructural setting. We find that in affine logic the translation schemes due to Kolmogorov and Gödel both satisfy Troelstra's criteria for a negative translation. On the other hand, the schemes of Glivenko and Gentzen both fail for affine logic, but for different reasons: one can extend affine logic to make Glivenko work and Gentzen fail and vice versa. By contrast, in the setting of Łukasiewicz logic, we can prove a general result asserting that a wide class of formula translations including those of Kolmogorov, Gödel, Gentzen and Glivenko not only satisfy Troelstra's criteria with respect to a natural intuitionistic fragment of Łukasiewicz logic but are all equivalent.

show abstract

The consequence relation in the logic of commutative GBL-algebras is PSPACE-complete

Cited by 13 publications

References 20 publications

Soft Constraints Processing over Divisible Residuated Lattices

Soft Constraints Processing over Divisible Residuated Lattices

A Curry-Howard Correspondence for the Minimal Fragment of Łukasiewicz Logic

Negative Translations for Affine and Lukasiewicz Logic

Contact Info

Product

Resources

About