Algebra Of Truth Values Research Articles

Fuzzy relational Galois connections between fuzzy transitive digraphs

We present a fuzzy version of the notion of relational Galois connection between fuzzy transitive directed graphs (fuzzy T-digraphs) on the specific setting in which the underlying algebra of truth values is a complete Heyting algebra. The components of such fuzzy Galois connection are fuzzy relations satisfying certain reasonable properties expressed in terms of the so-called full powering. Moreover, we provide a necessary and sufficient condition under which it is possible to construct a right adjoint for a given fuzzy relation between a fuzzy T-digraph and an unstructured set.

Subalgebras of the truth value algebra of type-2 fuzzy sets

In this paper, we first introduce some subalgebras in a reduct of the truth value algebra of type-2 fuzzy sets and prove one of them is a bisemilattice. Then we also obtain three necessary and sufficient conditions under which subalgebras of the truth value algebra of type-2 fuzzy sets form a bisemilattice. Next we investigate some subalgebras of this kind of bisemilattice and their characterizations. Finally, we introduce some subalgebras different from the bisemilattice and show a necessary and sufficient condition that a function is convex.

Note on “On the extension of nullnorms and uninorms to fuzzy truth values” [Fuzzy Sets Syst. 352 (2018) 92-118

Xie recently extended nullnorms and uninorms to fuzzy truth values and investigated whether the extended nullnorms and extended uninorms form respectively type-2 nullnorms and type-2 uninorms on the algebra of fuzzy truth values. In this note, we show by counterexamples that there exist some flaws in a previous paper by Xie [Fuzzy Sets Syst. 352 (2018) 92-118], such as Corollaries 2.1, 3.1, 4.1 and 4.2, Lemmas 3.3, 3.4, 3.5, 3.6, 3.7 and 3.8, Propositions 3.4, 3.5, 3.10, 3.11, 4.1 and 4.4, Theorems 3.1, 3.3, 4.1 and 4.3, and then we provide the correct versions. In addition, Proposition 3.7 in that paper is correct, but its proof contains a flaw. We point out the flaw and give a more general result than Proposition 3.7 in that paper.

A note on the lattice structure for subalgebras of the algebra of truth values of type-2 fuzzy sets

The algebra of truth values of type-2 fuzzy sets is not a lattice, but some of its subalgebras may be lattices. In this paper, we give a necessary and sufficient condition under which subalgebras of the truth value algebra of type-2 fuzzy sets form lattices. Further, we prove that if a certain subalgebra is a lattice, then it is isomorphic to an appropriate subalgebra whose all elements are convex functions with the same height. Based on these results, if a certain subalgebra is a lattice, the equivalent characterizations of the partial order in it are given.

Девіантні алгебри істиннісних значень та девіантні класи загальних недетермінованих предикатів

In this paper we study new classes of program-oriented logical formalisms – logics of general non-deter­ministic (GND) predicates. These logics reflect such properties of programs as nondeterminism, partiality, and non-fixed arity. GND-predicates can be modeled as 7-valued total deterministic (TD7) predicates. The main attention is paid to algebras of truth values (TV-algebras) of TD7-predicates. The set of truth values TV7 = {T, F, T#, F#, #, TF, TF#} defines TV-algebrа ATV7 = (TV7, {Ø*, Ú*}). There 20 subalgebras of ATV7 exist, and each of them induces a corresponding algebra of GND-predicates. At the same time there is a very large number of 7-valued logics so a lot of TV7 subsets are not closed under Ø* or Ú* and do not form subalgebras of ATV7. We call such subsets with the corresponding classes of GND-predicates deviant, they are not closed under logical connectives of GND-predicates. In order for deviant TV Í TV7 to form an algebra we need to modify Ø* or Ú*. Modifications can be made in a large number of ways. Modification of Ø* leads to specific non-classical logics and lies outside the scope of this paper. Modifications of Ú* sa­tisfying the TFC condition of predicate algebras logical connectives correctness are the most important, otherwise we obtain deviant TV-algebra which does not induce an algebra of GND-predicates. For all TV7 subsets we study the possibility of Ú* modification with TFC condition. Such modifica­tion induces corresponding classes of GND-predicates. We describe “natural” modifications of Ú* without TFC condition obtaining a number of deviant TV-algebras. There are no modifications with TFC condition for de­viant sets {#, TF, TF#}, {TF, #}, {TF#, #}, so for them we specify “relatively natural” deviant TV-algebras. Problems in programming 2019; 1: 14-26

Comment on: “RETRACTED: Rift-related active fault-system and a direction of maximum horizontal stress in the Cairo-Suez district, northeastern Egypt: A new approach from EMR-Technique and Cerescope data” by

Xie recently extended nullnorms and uninorms to fuzzy truth values and investigated whether the extended nullnorms and extended uninorms form respectively type-2 nullnorms and type-2 uninorms on the algebra of fuzzy truth values. In this note, we show by counterexamples that there exist some flaws in a previous paper by Xie [Fuzzy Sets Syst. 352 (2018) 92-118], such as Corollaries 2.1, 3.1, 4.1 and 4.2, Lemmas 3.3, 3.4, 3.5, 3.6, 3.7 and 3.8, Propositions 3.4, 3.5, 3.10, 3.11, 4.1 and 4.4, Theorems 3.1, 3.3, 4.1 and 4.3, and then we provide the correct versions. In addition, Proposition 3.7 in that paper is correct, but its proof contains a flaw. We point out the flaw and give a more general result than Proposition 3.7 in that paper.

Семантичні властивості п’ятизначних логік

A new class of program-oriented logical formalisms – propositional five-valued logics and five-valued logics of quasiary predicates is proposed and studied in the article. The quasiarity of predicates means that their arity is not fixed. A special feature of such five-valued logics is the use of special truth values, indicating errors or uncertainties in various systems, in particular, software systems. The article gives an ontological justification of five-valued logics and examines their examples. For the propositional level, an algebra of truth values is constructed and its semantic properties are studied. For the predicate level, the five-valued logic of quasiary predicates is constructed, its semantic properties are studied, the relation of equivalence of formulas and the relation of logical consequence are introduced, the principle of duality is proved and various normal forms are considered. Further research is focused on constructing algorithms for checking satisfiability and refutability of formulas and constructing various types of calculi for the introduced logics. Problems in programming 2018; 1: 22-35

On Extended Representable Uninorms and Their Extended Fuzzy Implications (Coimplications)

In this work, by Zadeh’s extension principle, we extend representable uninorms and their fuzzy implications (coimplications) to type-2 fuzzy sets. Emphatically, we investigate in which algebras of fuzzy truth values the extended operations are type-2 uninorms and type-2 fuzzy implications (coimplications), respectively.

Note on the absorption laws in the algebra of truth values of type-2 fuzzy sets

This note gives a necessary and sufficient condition under which the absorption laws in the algebra of truth values of type-2 fuzzy sets are true.

Equations in Type-2 Fuzzy Sets

The main concern of this paper is with the equations satisfied by the algebra of truth values of type-2 fuzzy sets. That algebra has elements all mappings from the unit interval into itself with operations given by certain convolutions of operations on the unit interval. There are a number of positive results. Among them is a decision procedure, similar to the method of truth tables, to determine when an equation holds in this algebra. One particular equation that holds in this algebra implies that every subalgebra of it that is a lattice is a distributive lattice. It is also shown that this algebra is locally finite. Many questions are left unanswered. For example, we do not know whether or not this algebra has a finite equational basis, that is, whether or not there is a finite set of equations from which all equations satisfied by this algebra follow. This and various other topics about the equations satisfied by this algebra will be discussed.

Overview of Lattices of Convex Normal Functions

The algebra of truth values of type-2 fuzzy sets is the set of maps from the unit interval to itself with convolution ordering. In applications of type-2 fuzzy sets, the full algebra is seldom used, but rather certain subalgebras that satisfy useful algebraic properties. The algebra of truth values of type-2 fuzzy sets is not itself a lattice, but the subalgebras considered here are lattices and, in fact, are complete distributive lattices. The subalgebras of special interest are the lattice of convex normal maps, the lattice of convex strongly normal maps, and the lattice of upper semicontinuous convex normal maps. We review and summarize some interesting properties of these subalgebras. A special feature of our treatment is a representation of these algebras as sets of monotone functions with pointwise order, making the operations more intuitive.

Generalized aggregation of fuzzy truth values

This paper deals with generalized extended aggregation operations in accordance with generalized extension principle and provides a theoretical basis for generalized extended aggregation operations. Generalized extended aggregation operations are studied on the algebra of convex (resp. normal) fuzzy truth values. When an aggregation operation is continuous, the generalized extended aggregation operation is proven to be an aggregation operation on fuzzy truth values (FTV aggregation operation, for short) for arbitrary t-norm on the algebra of convex normal fuzzy truth values.

Partial Orders on Fuzzy Truth Value Algebras

The elements of the truth value algebra of type-2 fuzzy sets are the mappings of the unit interval into itself, with operations given by various convolutions of the pointwise operations. This algebra can be specialized and generalized in various interesting ways. First, we consider the more general case of all mappings of a bounded chain with an involution into a complete chain, and delimit some of the properties of the resulting algebra. These include two binary operations each of which give a partial order on the elements of that algebra. These partial orders and their intersection are the principal objects of interest. We specialize this situation in two cases: (1) all mappings of the unit interval into itself, the original version of the truth value algebra of type-2 fuzzy sets introduced by Zadeh, and (2) all mappings of a finite chain into another finite chain. Again, each of these two cases yields two partial orders on the elements of the resulting algebras, and in each case, our principal interest is in these partial orders and their intersection.

Notes on aggregation of fuzzy truth values

Takáč recently introduced the notion of aggregation operations on fuzzy truth values (called FTV aggregation operations) and investigated extended aggregation operations on the algebra of convex normal fuzzy truth values with their left and right parts. In this paper, it is pointed out that the left (resp. right) part of an arbitrary convex normal fuzzy truth value is not necessarily its increasing (resp. decreasing) part. Moreover, an arbitrary convex normal fuzzy truth value is not equivalent to either intersection or union of its left and right parts. The conclusions of extended aggregation operations are revalidated in our paper, which were proven with uncorrect assertions about convex normal fuzzy truth values.

Generalized extended fuzzy implications

This paper is devoted to investigate generalized extended fuzzy implications in accordance with the generalized extension principle. The algebraic properties of generalized extended fuzzy implications are discussed on different algebras of fuzzy truth values. On this basis, fuzzy-valued fuzzy implications are constructed from arbitrary fuzzy implications and t-norms. Moreover, a fuzzy-valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values, when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t-norm.

On type-2 fuzzy relations and interval-valued type-2 fuzzy sets

This paper introduces new operations on the algebra of fuzzy truth values, extended supremum and extended infimum, which are generalizations of the extended operations of maximum and minimum between fuzzy truth values for type-2 fuzzy sets, respectively. Using these new operations, the properties of type-2 fuzzy relations are discussed, especially the compositions of type-2 fuzzy relations. On this basis, this paper introduces interval-valued type-2 fuzzy sets and interval-valued type-2 fuzzy relations, and discusses their properties.

Type-2 operations on finite chains

The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit interval into itself, with operations certain convolutions of these mappings with respect to pointwise max and min. This algebra generalizes the truth-value algebras of both type-1 and of interval-valued fuzzy sets, and has been studied rather extensively both from a theoretical and applied point of view. This paper addresses the situation when the unit interval is replaced by two finite chains. Most of the basic theory goes through, but there are several special circumstances of interest. These algebras are of interest on two counts, both as special cases of bases for fuzzy theories, and as mathematical entities per se.

EQ-algebras from the point of view of generalized algebras with fuzzy equalities

EQ-algebras introduced by Novák are algebras of truth values for a higher-order fuzzy logic (fuzzy type theory). In this paper, the compatibility of multiplication w.r.t. the fuzzy equality in an arbitrary EQ-algebra is examined. Particularly, an example indicates that the compatibility axiom does not always hold, and then a class of EQ-algebras satisfying the compatibility axiom is characterized by introducing a residuated integral-meet-semilattice-ordered-monoid-valued fuzzy algebra called a generalized algebra with a fuzzy equality.

On Boolean, Heyting and Brouwerian Algebras

Several models for logic have been developed in previous century. Each model has its own algebra of truth values. Frequently used algebras for truth values are Boolean Algebra for classical logic and Heyting Algebra and Brouwerian algebra for intuitionist logic logic and Brazilian logic respectively. Each of these algebras is a distributive lattice. In this paper we consider lattices which admit certain binary operations that force distributivity. For Boolean algebra this binary operation is induced by an endofunction which turns out to be negation for the Boolean algebra. These binary operations and corresponding negations for Heyting algebras and Browerian algebras, are discussed in detail. At the end we give a necessary and sufficient condition for a lattice to be a Boolean algebra.

On Boolean, Heyting and Brouwerian Algebras

Several models for logic have been developed in previous century. Each model has its own algebra of truth values. Frequently used algebras for truth values are Boolean Algebra for classical logic and Heyting Algebra and Brouwerian algebra for intuitionist logic logic and Brazilian logic respectively. Each of these algebras is a distributive lattice. In this paper we consider lattices which admit certain binary operations that force distributivity. For Boolean algebra this binary operation is induced by an endofunction which turns out to be negation for the Boolean algebra. These binary operations and corresponding negations for Heyting algebras and Browerian algebras, are discussed in detail. At the end we give a necessary and sufficient condition for a lattice to be a Boolean algebra.