Elsevier

Fuzzy Sets and Systems

Volume 425, 30 November 2021, Pages 140-156
Fuzzy Sets and Systems

A logical characterization of multi-adjoint algebras

https://doi.org/10.1016/j.fss.2021.02.003Get rights and content

Abstract

This paper introduces a logical characterization of multi-adjoint algebras with a twofold contribution. On the one hand, the study of multi-adjoint algebras, from a logical perspective, will allow us to discover both the core and new features of these algebras. On the other hand, the axiomatization of multi-adjoint algebras will be useful to take advantage of the properties of the logical connectives considered in the corresponding deductive system. The mechanism considered to carry out the mentioned axiomatization follows the one given by Petr Hájek for residuated lattices. Specifically, the paper presents the bounded poset logic (BPL) as an axiomatization of a bounded poset, since this algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. In the following, the language of BPL is enriched with a family of pairs, composed of a conjunctor and an implication, and its axiomatic system is endowed with new axioms, giving rise to the multi-adjoint logic (ML). The soundness and completeness of BPL and ML are proven. Finally, a comparison between the axiomatization of the multi-adjoint logic and the one given for the BL logic is introduced.

Introduction

Fuzzy logic is an important mathematical area for handling vague, imperfect and incomplete information in data sets, which has widely been studied from its introduction in 1965 by Lotfi A. Zadeh [38]. One of the most interesting branches has been its use in logic programming. In this fuzzy area, one of the most general frameworks is multi-adjoint logic programming [33], [34]. This framework arose as a general mathematical setting for handling imprecise, incomplete or imperfect information, from which many well-known (fuzzy) logic programming frameworks are particular cases, such as residuated and monotonic logic programming, fuzzy logic programming, probabilistic logic programming, etc. [14], [25], [27], [15], [32]. Specifically, the semantics of this general logic programming framework is based on a complete lattice in which, for example, the operators can be neither commutative nor associative, and different implications can be considered in the same logic program. One of the main property of these operators is that they generalize the modus ponens in a fuzzy framework, as Hájek justified in [23]. This general structure is a particular case of a multi-adjoint algebra [9], [10], [11] and it allows a more flexible simulation of datasets to be modeled and so, a more suitable set of rules can be computed with a better accommodation to the behavior of the considered database.

On the other hand, the axiomatization of (fuzzy) logics allow to establish a precise picture of the deductive use of vague or imperfect concepts. Among other examples on these kind of logics, we can consider the basic logic BL introduced by Hájek [23], the logic for left-continuous t-norms presented by Esteva and Godo in [20], its extension considering right-continuous t-conorms [22], or the logics of subresiduated lattices from Epstein and Horn [19], whose axiomatizations are sound and complete.

In this paper, we present an axiomatization of multi-adjoint algebras based on two main goals. The first one is to study these algebras from another point of view for discovering the core and new features of these algebras. The second one is to accommodate these algebras in an axiomatic system for taking advantage of the different procedures and algorithms developed in these systems. The procedure considered for the axiomatization follows the one given in [23] for residuated lattices.

First of all, it is needed to present an axiomatization of a bounded poset. This algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. The obtained axiomatized logic is introduced in Section 3 and will be called bounded poset logic (BPL). We will also prove in this section that BPL is sound and complete. On this logic, the multi-adjoint logic (ML) is introduced in Section 4. A family of pairs, composed of a conjunctor and an implication, are considered in the language and a new set of axioms complements the ones given for BPL. The new logic captures the heart of right multi-adjoint algebras. An analogous procedure can be done to define logics for the rest of kinds of multi-adjoint algebras [11]. The new introduced logic is also sound and complete, and we will prove in Section 5 that it is weaker than the BL logic presented by Hájek in [23], which is based on the propositional calculus PC(⁎) given by a left-continuous triangular norm (t-norm).

Section snippets

Preliminaries

This section includes preliminary notions and results in order to make the paper self-contained. Let us start by recalling the definition of residuated lattice, which was introduced by Dilworth and Ward in [18].

Definition 1

A residuated lattice (L,,,,,0,1) is a tuple composed of four binary operators and two constants such that:

  • (1)

    (L,,,0,1) is a lattice where ≤ is the usual order defined from ∨, ∧ and where 0,1 are the bottom and top elements, respectively.

  • (2)

    (L,,1) is a commutative monoid, that is, ⁎ is a

Bounded poset logic

In this section, we will define a many-valued propositional logic framework on a bounded poset, which will be called bounded poset logic, following the philosophy given in [23].

From now on, we will consider a bounded poset (P,,0,1), where 0 and 1 denote the bottom and the top elements of P, together with the characteristic mapping of the ordering relation ⪯, that is:zdy={1 if yz0 otherwise 

Clearly, the consideration of this highlighted extra operator does not limit the bounded poset

Multi-adjoint logic

This section provides a more general many-valued propositional logic approach called multi-adjoint logic. Specifically, the multi-adjoint logic is a propositional logic associated with an arbitrary order-right multi-adjoint algebra (P,,&1,1,,&n,n) and the extra operator d defined in Equation (1) on the poset (P,). To carry out this study, we will employ a similar scheme to the one used in Section 3.

We will start introducing the notions associated with the syntax of the multi-adjoint logic

Comparison with BL logic

This section will show that ML has axioms less restrictive than the ones given to the basic logic BL [23]. As a consequence, it also contains for instance the logic for subresiduated lattices given in [19], the logic for left-continuous t-norms presented in [20], and the ones considering hedges [21], [24], [37].

Given a t-norm ⁎ and its residuum ⇒ on the unit interval, the propositional calculus given by ⁎, and denoted as PC(⁎), has Π as the set of propositional variables, the connectives ⋆, →

Conclusions and future work

We have presented an axiomatization for a bounded poset, which has been called bounded poset logic (BPL). We have proven that the obtained axiomatized logic is sound and complete. On this logic, we have introduced the multi-adjoint logic (ML). To reach this goal, a family of pairs, composed of a conjunctor and an implication, has been included in the language and a new set of axioms complementing the ones given for BPL has been considered. We have also proven the soundness and completeness of

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    Partially supported by the 2014-2020 ERDF Operational Programme in collaboration with the State Research Agency (AEI) in projects TIN2016-76653-P and PID2019-108991GB-I00, and with the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia in project FEDER-UCA18-108612, and by the European Cooperation in Science & Technology (COST) Action CA17124.

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