Distributive Full Lambek Calculus Has the Finite Model Property

Kozak, Michał

doi:10.1007/s11225-009-9172-7

Cited by 17 publications

(16 citation statements)

References 7 publications

(17 reference statements)

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“…The finite model property for DRL was established in [12] and for BI it was proved in [10]. We extend these results by proving the finite model property (FMP) for many simple extensions of DRL and of GBI, actually for many simple extensions nGBI, namely axiomatized by certain equations/sequents that do not involve divisions and implication, but otherwise can have any combination of the other connectives.…”

Section: Finite Model Propertymentioning

confidence: 62%

“…Although not stated explicitly in [12], it can be inferred from the proof that it is possible to use only the model-checker, since an upper-bound for a countermodel (if it exists) can be estimated. The proof of the FMP of DRL given there is based on a proof search for the given sequent, but because of the rule ( c) the naive exhaustive proof search is not finite; the FMP is established in [12] without showing or claiming that a finite proof search is possible. Our first result in this section is to show that a finite proof search is possible, and from there we easily deduce the FMP, for all the extensions mentioned above, including extensions of GBI.…”

Section: Finite Model Propertymentioning

confidence: 99%

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Distributive residuated frames and generalized bunched implication algebras

Galatos

Jipsen

2017

Algebra Univers.

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We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames.

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Section: Finite Model Propertymentioning

confidence: 62%

Section: Finite Model Propertymentioning

confidence: 99%

Distributive residuated frames and generalized bunched implication algebras

Galatos

Jipsen

2017

Algebra Univers.

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“…A theorem of Urquhart [69] implies that the equational theory of DMM is undecidable, whereas results in [10,36,47] show that the respective varieties of distributive and of square-increasing IRLs are generated by their finite members, whence their equational theories are decidable. (In the squareincreasing case, the complexity of any decision procedure is known to be immense [71].…”

Section: Lemma 42 a De Morgan Monoid Is Integral Iff It Is A Booleamentioning

confidence: 99%

Varieties of De Morgan monoids: Minimality and irreducible algebras

Moraschini

Raftery

Wannenburg

2019

Journal of Pure and Applied Algebra

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It is proved that every finitely subdirectly irreducible De Morgan monoid A (with neutral element e) is either (i) a Sugihara chain in which e covers ¬e or (ii) the union of an interval subalgebra [¬a, a] and two chains of idempotents, (¬a] and [a), where a = (¬e) 2 . In the latter case, the variety generated by [¬a, a] has no nontrivial idempotent member, and A/[¬a) is a Sugihara chain in which ¬e = e. It is also proved that there are just four minimal varieties of De Morgan monoids. This theorem is then used to simplify the proof of a description (due to K.Świrydowicz) of the lower part of the subvariety lattice of relevant algebras. The results throw light on the models and the axiomatic extensions of fundamental relevance logics.

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“…In , Buszkowski proved the FMP for BCI , which is the implicational fragment of MAILL . In , Kozak proved the FMP for Distributive Full Lambek Calculus ( DFL ) whose algebraic semantics is the class of distributive residuated lattices ( DRL ). For a review on the FMP for substructural logics, we refer to .…”

Section: Introductionmentioning

confidence: 99%

The finite model property for semilinear substructural logics

Wang

2013

Mathematical Logic Qtrly

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Key wordsModel theory, the finite model property, semilinear substructural logics, substructural fuzzy logics. MSC (2010) 03B47, 03B52In this paper, we show that the finite model property fails for certain non-integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo-uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members.The Hilbert system HpsUL of the logic of bounded representable residuated lattices is based on a countable propositional language with formulas built inductively as usual from a set of propositional variables, binary *

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Distributive Full Lambek Calculus Has the Finite Model Property

Abstract: We prove the Finite Model Property (FMP) for Distributive Full Lambek Calculus (DFL) whose algebraic semantics is the class of distributive residuated lattices (DRL). The problem was left open in [8,5]. We use the method of nuclei and quasiembedding in the style of [10, 1].

Cited by 17 publications

References 7 publications

Distributive residuated frames and generalized bunched implication algebras

Distributive residuated frames and generalized bunched implication algebras

Varieties of De Morgan monoids: Minimality and irreducible algebras

The finite model property for semilinear substructural logics

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