Godel Algebras Research Articles

Fuzzy Bisimulations for Nondeterministic Fuzzy Transition Systems

Bisimulations are established forms of behavioral equivalences for discrete event systems. Recently, bisimulations have been introduced to fuzzy transition systems (FTSs) and developed quickly. Relational lifting by using weight functions is one of the most used methods in bisimulation research, which lifts a crisp relation on the state spaces to a crisp relation on the distributions of state spaces of FTSs. This paper proposes a method, called fuzzy relational lifting method, via fuzzy similarity measures induced by residua (implications) in complete residuated lattices. In contrast with the existing method, fuzzy relational lifting method relaxes the conditions of lifting relation by using weight functions and is a generalization of the method. Some properties of fuzzy lifting operation are presented. Based on the fuzzy lifting operation, a fuzzy bisimulation for nondeterministic fuzzy transition systems (NFTSs) is introduced, which is much more natural and robust than some behavioral measures, and can measure the degree of similarity between any two states which are deemed to be equivalent intuitively but simply distinguished by other behavioral measures. Then some properties of fuzzy bisimulations are provided. Moreover, a fixedpoint characterization of fuzzy bisimulations is given and the greatest fuzzy bisimulation is computed. Finally, a real-valued logic is introduced to characterize fuzzy bisimulations over Godel algebra.

An Algebraic Study of S5-Modal Gödel Logic

Fil: Castano, Diego Nicolas. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca. Instituto de Matematica Bahia Blanca. Universidad Nacional del Sur. Departamento de Matematica. Instituto de Matematica Bahia Blanca; Argentina

Axiomatization of Crisp Gödel Modal Logic

In this paper we consider the modal logic with both $$\Box $$ and $$\Diamond $$ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Godel algebra $$[0,1]_G$$ . We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given too. We also prove that in the studied logic it is not possible to get $$\Diamond $$ as an abbreviation of $$\Box $$ , nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature.

Pseudomonadic BL-algebras: an algebraic approach to possibilistic BL-logic

Fuzzy possibilistic logic is an important formalism for approximate reasoning. It extends the well-known basic propositional logic BL, introduced by Hajek, by offering the ability to reason about possibility and necessity of fuzzy propositions. We consider an algebraic approach to study this logic, introducing Pseudomonadic BL-algebras. These algebras turn to be a generalization of both Pseudomonadic algebras introduced by Bezhanishvili (Math Log Q 48:624–636, 2002) and serial, Euclidean and transitive Bimodal Godel algebras proposed by Caicedo and Rodriguez (J Log Comput 25:37–55, 2015). We present the connection between this class of algebras and possibilistic BL-frames, as a first step to solve an open problem proposed by Hajek (Metamathematics of fuzzy logic. Trends in logic, Kluwer, Dordrecht, 1998, Chap. 8, Sect. 3).

Representations for logics and algebras related to revised drastic product t-norm

We obtain a functional representation of RDP logic formulas and a combinatorial representation for finite RDP algebras. As the variety of RDP algebras includes the varieties of Godel, DP, and EMTL algebras, we derive analogous representations for such varieties and their corresponding logics.

Stabilizers in MTL-algebras

In the paper, we introduce some stabilizers and investigate related properties of them in MTL-algebras.Then, we also characterize some special classes of MTL-algebras, for example, IMTL-algebras, integral MTL-algebras, Godel algebras and MV-algebras, in terms of these stabilizers. Moreover, we discuss the relation between stabilizers and several special filters (ideals) in MTL-algebras. Finally, we discuss the relation between these stabilizers and prove that the right implicative stabilizer and right multiplicative stabilizer are order isomorphic. This results also give answers to some open problems, which were proposed by Motamed and Torkzadeh in [Soft Comput, {\bf 21} (2017) 686-693].

Free and Projective Bimodal Symmetric Gödel Algebras

Godel logic (alias Dummett logic) is the extension of intuitionistic logic by the linearity axiom. Symmetric Godel logic is a logical system, the language of which is an enrichment of the language of Godel logic with their dual logical connectives. Symmetric Godel logic is the extension of symmetric intuitionistic logic (L. Esakia, C. Rauszer). The proof-intuitionistic calculus, the language of which is an enrichment of the language of intuitionistic logic by modal operator was investigated by Kuznetsov and Muravitsky. Bimodal symmetric Godel logic is a logical system, the language of which is an enrichment of the language of Godel logic with their dual logical connectives and two modal operators. Bimodal symmetric Godel logic is embedded into an extension of (bimodal) Godel---Lob logic (provability logic), the language of which contains disjunction, conjunction, negation and two (conjugate) modal operators. The variety of bimodal symmetric Godel algebras, that represent the algebraic counterparts of bimodal symmetric Godel logic, is investigated. Description of free algebras and characterization of projective algebras in the variety of bimodal symmetric Godel algebras is given. All finitely generated projective bimodal symmetric Godel algebras are infinite, while finitely generated projective symmetric Godel algebras are finite.

FILTERS OF RESIDUATED LATTICES BASED ON SOFT SET THEORY

Strong uni-soft filters and divisible uni-soft filters in residuated lattices are introduced, and several properties are investigated. Characterizations of a strong and divisible uni-soft filter are discussed. Conditions for a uni-soft filter to be divisible are established. Relations between a divisible uni-soft filter and a strong uni-soft filter are considered.

Prime ideals and Godel ideals of BL-algebras

In this paper we give further properties of ideals of a BL-algebra. The concepts of prime ideals, irreducible ideals and Godel ideals are introduced. We prove that the concept of prime ideals coincides with one of irreducible ideals, and establish the Prime Ideal Theorem in BL-algebras. As applications of Prime ideal Theorem we give several representation and decomposition properties of ideals in BL-algebras. In particular, we give some equivalent conditions of Godel ideals and prove that a BL-algebra A satisfying condition (C) is a Godel algebra i the ideal f0g is a Godel ideal i all ideals of A are Godel ideals i (a] = fx 2 A : a􀀀 x􀀀g for any a 2 A:

Исследование качества решения задачи классификации нейронными нечeткими продукционными сетями на основе модели вывода Мамдани - Заде

The article considers solving the problem of object recognition of intersected classes using fuzzy inference systems and neural networks. New multi-output network of Wang-Mendel is compared to a new architecture of neural fuzzy production network based on the model of Mamdani-Zadeh. Learning results of these models are given in the interpretation of logical operations provided by Godel, Goguen and Lukasiewicz algebras. New Wang-Mendel’s network can use minimum or sum-based formula as T-norm operation in accordance with an appropriate algebra rather than the standard multiplication only. Mamdani-Zadeh's network is designed as a cascade of T-norm, implication and S-norm operations defined by selected algebra. Moreover defuzzification layer is not presented in Mamdani-Zadeh’s network. Both networks have several outputs in accordance with the number of subject area classes what differs them from the basic realizations. Compliance degrees of an input vector to defined classes are formed at the network outputs. To compare the models the standard Fisher’s irises and Italian wines classification problems were used. This article presents the results calculated by training the networks by backpropagation algorithm. Classification error analysis shows that the use of these algebras as interpreting fuzzy logic operations proposed in this paper can reduce the classification error for both multi-output network of Wang-Mendel and a new network of Mamdani-Zadeh. The best learning results are shown by Godel algebra, but Lukasiewicz algebra demonstrates better generalizing properties while testing, what leads to a less number of classification errors.

Semi-Heyting Algebras Term-equivalent to Gödel Algebras

In this paper we investigate those subvarieties of the variety \(\mathcal {SH}\) of semi-Heyting algebras which are term-equivalent to the variety \(\mathcal L_{\mathcal H}\) of Godel algebras (linear Heyting algebras). We prove that the only other subvarieties with this property are the variety \(\mathcal L^{\rm Com}\) of commutative semi-Heyting algebras and the variety \(\mathcal L^{\vee}\) generated by the chains in which a < b implies a → b = b. We also study the variety \(\mathcal C\) generated within \(\mathcal{SH}\) by \(\mathcal L_{\mathcal H}\), \(\mathcal L_\vee\) and \(\mathcal L_{\rm Com}\). In particular we prove that \(\mathcal C\) is locally finite and we obtain a construction of the finitely generated free algebra in \(\mathcal C\).

Rings and Gödel algebras

In this paper, we study those rings whose semiring of ideals can be given the structure of a Godel algebra. Such rings are called Godel rings. We investigate such structures both from an algebraic and a topological point of view. Our main result states that every Godel ring R is a subdirect product of prime Godel rings R i , and the Godel algebra Id(R) associated to R is subdirectly embeddable as an algebraic lattice into $${{\prod_{i}}Id(R_{i})}$$ , where each Id(R i ) is the algebraic lattice of ideals of R i that can be equipped with the structure of a Godel algebra. We see that the mapping associating to each Godel ring its Godel algebra of ideals is functorial from the category of Godel rings with epimorphisms into the full subcategory of frames whose objects are Godel algebras and whose morphisms are complete epimorphisms.

Standard Gödel Modal Logics

We prove strong completeness of the □-version and the ◊-version of a Godel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Godel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.

Basic Hoops: an Algebraic Study of Continuous t-norms

A continuous t-norm is a continuous map * from [0, 1]2 into [0, 1] such that $$\langle [0, 1], *, 1 \rangle$$ is a commutative totally ordered monoid. Since the natural ordering on [0, 1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and $$\langle [0, 1], *,\rightarrow, 1\rangle$$ becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras $$\langle [0, 1], *,\rightarrow, 1\rangle$$ , where * is a continuous t-norm. In this paper we investigate the structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hajek’s BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [0, 1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL-algebras, of Godel algebras and of product algebras.

On pseudo-BL algebras and BCC-algebras

We further study the filter theory of pseudo-BL algebras. We give some equivalent conditions of filter, normal filter and Boolean filter. We introduce the notion of pseudo MV-filter, pseudo-G filter and characterize Boolean algebras, pseudo-MV algebras and pseudo Godel algebras (i.e. Godel algebras) in pseudo-BL algebras. We establish the connections between BCC-algebras, pseudo-BCK algebras, pseudo-BL algebras and weak pseudo-BL algebras (pseudo-MTL algebras).

On the predicate logics of continuous t-norm BL-algebras

Given a class C of t-norm BL-algebras, one may wonder which is the complexity of the set Taut(C∀) of predicate formulas which are valid in any algebra in C. We first characterize the classes C for which Taut(C∀) is recursively axiomatizable, and we show that this is the case iff C only consists of the Godel algebra on [0,1]. We then prove that in all cases except from a finite number Taut(C∀) is not even arithmetical. Finally we consider predicate monadic logics Taut M (C∀) of classes C of t-norm BL-algebras, and we prove that (possibly with finitely many exceptions) they are undecidable.

BL-algebras and effect algebras

Although the notions of a BL-algebra and of an effect algebra arose in rather different contextes, both types of algebras have certain structural properties in common. To clarify their mutual relation, we introduce weak effect algebras, which generalize effect algebras in that the order is no longer necessarily determined by the partial addition. A subclass of the weak effect algebras is shown to be identifiable with the BL-algebras. Moreover, weak D-posets are defined, being based on a partial difference rather than a partial addition. They are equivalent to weak effect algebras. Finally, it is seen to which subclasses of the weak effect algebras certain subclasses of the BL-algebras, namely the MV-, product, and Godel algebras, correspond.

Pseudo-t-norms and pseudo-BL algebras

BL algebras were introduced by Hajek as algebraic structures for his Basic Logic, starting from continuous t-norms on [0,1]. MV algebras, product algebras and Godel algebras are particular cases of BL algebras. On the other hand, the pseudo-MV algebras extend the MV-algebras in the same way in which the arbitrary l-groups extend the abelian l-groups. We have generalized the BL algebras and pseudo-MV algebras, introducing the pseudo-BL algebras. In this paper we introduce weak-BL algebras and weak-pseudo-BL algebras. We also introduce non-commutative t-norms (we call them pseudo-t-norms) and use them in constructing pseudo-BL algebras and weak-pseudo-BL algebras.