A logical characterization of multi-adjoint algebras☆
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 is a tuple composed of four binary operators and two constants such that: is a lattice where ≤ is the usual order defined from ∨, ∧ and where are the bottom and top elements, respectively. 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 , where 0 and 1 denote the bottom and the top elements of P, together with the characteristic mapping of the ordering relation ⪯, that is:
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 and the extra operator defined in Equation (1) on the poset . 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.
References (38)
- et al.
On residuation in multilattices: filters, congruences, and homomorphisms
Fuzzy Sets Syst.
(2014) - et al.
Multi-adjoint algebras versus non-commutative residuated structures
Int. J. Approx. Reason.
(2015) - et al.
Multi-adjoint fuzzy rough sets: definition, properties and attribute selection
Int. J. Approx. Reason.
(2014) - et al.
Monoidal t-norm based logic: towards a logic for left-continuous t-norms
Fuzzy Sets Syst.
(2001) - et al.
A logical approach to fuzzy truth hedges
Inf. Sci.
(2013) On very true
Fuzzy Logic
Fuzzy Sets Syst.
(2001)- et al.
On fuzzy unfolding: a multi-adjoint approach
Fuzzy Sets Syst.
(2005) Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices
Fuzzy Sets Syst.
(2012)Multi-adjoint property-oriented and object-oriented concept lattices
Inf. Sci.
(2012)- et al.
On the Dedekind-MacNeille completion and formal concept analysis based on multilattices
Fuzzy Sets Syst.
(2016)
Formal concept analysis via multi-adjoint concept lattices
Fuzzy Sets Syst.
Similarity-based unification: a multi-adjoint approach
Fuzzy Sets Syst.
Propositional calculus under adjointness
Fuzzy Sets Syst.
Truth-depressing hedges and bl-logic
Fuzzy Sets Syst.
Fuzzy sets
Inf. Control
Fuzzy Relational Systems: Foundations and Principles
Algebraizable Logics
Residuated relational systems
Asian-Eur. J. Math.
How to introduce the connective implication in orthomodular posets
Asian-Eur. J. Math.
Cited by (0)
- ☆
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.