Real Unit Interval Research Articles

Gödel–Dummett linear temporal logic

We investigate a version of linear temporal logic whose propositional fragment is Gödel–Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural semantics: first a real-valued semantics, where statements have a degree of truth in the real unit interval, and second a ‘bi-relational’ semantics. We then show that these two semantics indeed define one and the same logic: the statements that are valid for the real-valued semantics are the same as those that are valid for the bi-relational semantics. This Gödel temporal logic does not have any form of the finite model property for these two semantics: there are non-valid statements that can only be falsified on an infinite model. However, by using the technical notion of a quasimodel, we show that every falsifiable statement is falsifiable on a finite quasimodel, yielding an algorithm for deciding if a statement is valid or not. Later, we strengthen this decidability result by giving an algorithm that uses only a polynomial amount of memory, proving that Gödel temporal logic is PSPACE-complete. We also provide a deductive calculus for Gödel temporal logic, and show this calculus to be sound and complete for the above-mentioned semantics, so that all (and only) the valid statements can be proved with this calculus.

Exploring The Boundaries of Uncertainty: Interval Valued Pythagorean Neutrosophic Set and Their Properties

The interval-valued Pythagorean fuzzy set (IVPFS) presents a novel approach to tackling vagueness and uncertainty, while neutrosophic sets, a broader concept encompassing of fuzzy sets and intuitionistic fuzzy sets, are tailored to depict real-world data characterized by uncertainty, imprecision, inconsistency, and incompleteness. Additionally, the development of Interval Value Neutrosophic Sets (IVNS) enhances precision in handling problems involving a range of numbers within the real unit interval, rather than focusing solely on a single value. However, despite these advancements, there is a deficiency in research addressing practical implementation challenges, conducting comparative analyses with existing methods, and applying these concepts across various fields. This study aims to bridge this research gap by proposing a novel concept based on the Interval Valued Pythagorean Neutrosophic Set (IVPNS), which is a generalization of the IVPFS and INS. The development of IVPNS provides a more comprehensive framework for handling uncertainty, ambiguity, and incomplete information in various fields, leading to more robust decision-making processes, improved problem-solving capabilities, and better management of complex systems. Furthermore, this research introduces the algebraic operations for IVPNS, including addition, multiplication, scalar multiplication, and exponentiation and provides a comparative analysis with IVPFS and IVNS. The study incorporates illustrative numerical examples to demonstrate these operations in practice. Additionally, this study provides and rigorously proves the algebraic properties of IVPNS, specifically discussing their commutative and associative properties. This validation ensures compliance with established conditions for IVPNS, reinforcing their theoretical soundness and practical applicability.

Distorted copulas

In this paper, we deal with the problem of preserving the copula property by means of a transformation called distortion where automorphisms of the real unit interval are involved. A new methodological approach is followed by resorting to a special branch of group theory. The crucial role of absolute continuity of one-dimensional sections of a distorted copula is illustrated. Some necessary and sufficient conditions concerning the automorphisms which preserve the copula property for a large class of bivariate copulas are proved. Several examples are presented.

A Categorical Construction of the Real Unit Interval

The real unit interval is the fundamental building block for many branches of mathematics like probability theory, measure theory, convex sets and homotopy theory. However, a priori the unit interval could be considered an arbitrary choice and one can wonder if there is some more canonical way in which the unit interval can be constructed. In this paper we find such a construction by using the theory of effect algebras. We show that the real unit interval is the unique non-initial, non-final irreducible algebra of a particular monad on the category of bounded posets. The algebras of this monad carry an order, multiplication, addition and complement, and as such model much of the operations we need to do on probabilities. On a technical level, we show that both the categories of omega-complete effect algebras as well as that of omega-complete effect monoids are monadic over the category of bounded posets using Beck's monadicity theorem. The characterisation of the real unit interval then follows easily using a recent representation theorem for omega-complete effect monoids.

Two types of ordinal sums of fuzzy implications on bounded lattices

In this paper, two kinds of ordinal sums of fuzzy implications on bounded lattices are provided. The first one is complementing a specific fuzzy implication to a given family of fuzzy implications defined on pairwise disjoint closed subintervals of a bounded lattice. The second way is defining specific values outside the given family of countably many subintervals, where the endpoints of the subintervals constitute a chain. For the first way, necessary and sufficient conditions for a fuzzy implication to be a complement of a given family of fuzzy implications such that the ordinal sum is a fuzzy implication are provided, which generalize some results of fuzzy implications on the real unit interval. Based on this, we deal with those elements that are incomparable with the endpoints of the given subintervals and provide a fuzzy implication as a complement such that for arbitrary family of fuzzy implications on the given subintervals, the ordinal sum is always a fuzzy implication. For the second ordinal sum, we provide two methods for defining values outside the given subintervals that are shown to be always fuzzy implications.

On local isometries between algebras of C(Y)-valued differentiable maps

Let K be either the real unit interval [0, 1] or the complex unit circle {mathbb {T}} and let C(Y) be the space of all complex-valued continuous functions on a compact Hausdorff space Y. We prove that the isometry group of the algebra C^1(K,C(Y)) of all C(Y)-valued continuously differentiable maps on K, equipped with the Sigma -norm, is topologically reflexive and 2-topologically reflexive whenever the isometry group of C(Y) is topologically reflexive.

Approximate positional analysis of fuzzy social networks

In order to facilitate the processing of vast amounts of data that emerges in the study of fuzzy social networks, scholars have developed various procedures for reducing these networks. Some procedures use regular and structural fuzzy relations to reduce such networks. In this article, we generalize the notions of regular and structural fuzzy relations to obtain even better reductions of fuzzy networks. More precisely, for a fuzzy social network given by the set of social entities and the family of fuzzy relations between them, we define μ-approximate regular fuzzy relations, where μ is the degree taken from the underlying set of truth values, which is a complete Heyting algebra. Using these specific fuzzy relations, we show that it is possible to reduce a fuzzy social network in some cases when previously developed algorithms fail to reduce it. We investigate the properties of μ-approximate regular fuzzy relations. We show that the blockmodel of a fuzzy social network, which is its reduced fuzzy social network built via μ-approximate regular fuzzy preorder, retains specific structure-preserving properties. We give a method for calculating the greatest μ-approximate regular fuzzy relation on a given fuzzy social network. For fuzzy social networks defined over the real-unit interval [0,1], we give a procedure that determines all subintervals of [0,1] that share the greatest μ-approximate regular fuzzy relation. Analogous results are provided for μ-approximate structural fuzzy relations.

Idempotent uninorms on a complete chain

We characterize idempotent uninorms on a complete chain in terms of decreasing unary functions with a symmetry-related property. As a particular case, we retrieve and simplify a known characterization theorem for idempotent uninorms on the real unit interval. We also introduce the notion of left-continuity (resp. right-continuity) of an idempotent uninorm on a complete chain and characterize left-continuous (resp. right-continuous) idempotent uninorms in terms of decreasing unary functions with a second (resp. a third) symmetry-related property, unifying a characterization theorem for left-continuous (resp. right-continuous) idempotent uninorms on the real unit interval and a characterization theorem for idempotent uninorms on a finite chain known in the literature.

Bipolar equations on complete distributive symmetric residuated lattices: The case of a join-irreducible right-hand side

Bipolar max-⁎ equations, with ⁎ a triangular norm, have recently become a popular research topic embedded in the broad field of fuzzy relational equations. In this paper, we lift the work from the restrictive setting of the real unit interval — obfuscating the underlying lattice-theoretical essence — to the general setting of complete distributive symmetric residuated lattices, allowing to build upon the vast body of knowledge on unipolar sup-⁎ equations on complete distributive residuated lattices. We determine the full solution set, with particular emphasis on the extremal solutions, of a bipolar sup-⁎ equation in case the right-hand side is a join-irreducible element. The results are illustrated by means of ample examples.

Dichotomy between Deterministic and Probabilistic Models in Countably Additive Effectus Theory

Effectus theory is a relatively new approach to categorical logic that can be seen as an abstract form of generalized probabilistic theories (GPTs). While the scalars of a GPT are always the real unit interval [0,1], in an effectus they can form any effect monoid. Hence, there are quite exotic effectuses resulting from more pathological effect monoids. In this paper we introduce sigma-effectuses, where certain countable sums of morphisms are defined. We study in particular sigma-effectuses where unnormalized states can be normalized. We show that a non-trivial sigma-effectus with normalization has as scalars either the two-element effect monoid 0,1 or the real unit interval [0,1]. When states and/or predicates separate the morphisms we find that in the 0,1 case the category must embed into the category of sets and partial functions (and hence the category of Boolean algebras), showing that it implements a deterministic model, while in the [0,1] case we find it embeds into the category of Banach order-unit spaces and of Banach pre-base-norm spaces (satisfying additional properties), recovering the structure present in GPTs. Hence, from abstract categorical and operational considerations we find a dichotomy between deterministic and convex probabilistic models of physical theories.

(2102-6487) Ordinal sum constructions for aggregation functions on the real unit interval

We discuss ordinal sums as one of powerful tools in the aggregation theory serving, depending on the context, both as a construction method and as a representation, respectively. Up to recalling of several classical results dealing with ordinal sums, in particular dealing, e.g., with continuous t-norms, copulas, or recent results, e.g., concerning uninorms with continuous underlying functions, we present also several new results, such as the uniqueness of the link between t-norms or t-conorms, and related Archimedean components, problems dealing with the cardinality of the considered index sets in ordinal sums, or infinite ordinal sums of aggregation functions covering by one type of ordinal sums both t-norms and t-conorms ordinal sums.

One-variable fragments of intermediate logics over linear frames

A correspondence is established between one-variable fragments of (first-order) intermediate logics defined over a fixed countable linear frame and Gödel modal logics defined over many-valued equivalence relations with values in a closed subset of the real unit interval. It is also shown that each of these logics can be interpreted in the one-variable fragment of the corresponding constant domain intermediate logic, which is equivalent to a Gödel modal logic defined over (crisp) equivalence relations. Although the latter modal logics in general lack the finite model property with respect to their frame semantics, an alternative semantics is defined that has this property and used to establish co-NP-completeness results for the one-variable fragments of the corresponding intermediate logics both with and without constant domains.

The three types of normal sequential effect algebras

A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C∗-algebras. A SEA is called normal when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition convex, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when 0=1.We show that any normal SEA E splits as a direct sum E=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb, a convex normal SEA Ec, and a newly identified type of normal SEA Eac we dub purely almost-convex.Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.

A characterization of the classes Umin and Umax of uninorms on a bounded lattice

As generalizations of the classes Umin and Umax of uninorms on the real unit interval, we introduce the classes Umin and Umax of uninorms on a bounded lattice and lay bare the structure of their members. It is shown that uninorms in Umin (resp. Umax) are characterized by a triangular conorm (resp. triangular norm) and a triangular subnorm (resp. triangular superconorm). The characterization theorems unify all the existing construction methods for these two classes of uninorms known in the literature. We also introduce four new related classes of uninorms on a bounded lattice and present dedicated construction methods using triangular subnorms and triangular superconorms. Finally, connections among the introduced classes of uninorms are discussed.

Constructions for t-conorms and t-norms on interval-valued and interval-valued intuitionistic fuzzy sets by paving

Paving is a method for constructing new operations from a given one. Kalina and Kral in 2015 showed that on the real unit interval this method can be used to construct associative, commutative and monotone operations from particular given operations (from basic ‘paving stones’). In the present paper we modify the construction method for interval-valued fuzzy sets. From given (possibly representable) t-norms and t-conorms we construct new, non-representable operations. In the last section, we modify the presented construction method for interval-valued intuitionistic fuzzy sets.

Constructions of overlap functions on bounded lattices

In this paper, we present two methods for constructing new overlap functions on bounded lattices from given ones. At first, we introduce the notion of overlap functions on bounded lattices, which is a generalization of overlap functions on the real unit interval. Then we provide the ∧-extension of an overlap function on a subinterval and give the necessary and sufficient conditions for the ∧-extension to be an overlap function. Finally, we propose a definition of ordinal sum of finitely many overlap functions on subintervals of a bounded lattice, where the endpoints of the subintervals constitute a chain. Necessary and sufficient conditions for the ordinal sum yielding again an overlap function are provided.

General definitions for the union and intersection of ordered fuzzy multisets

Since its original formulation, the theory of fuzzy sets has spawned a number of extensions where the role of membership values in the real unit interval $[0, 1]$ is handed over to more complex mathematical entities. Amongst the many existing extensions, two similar ones, the fuzzy multisets and the hesitant fuzzy sets, rely on collections of several distinct values to represent fuzzy membership, the key difference being that the fuzzy multisets allow for repeated membership values whereas the hesitant fuzzy sets do not. But in neither case are these collections of values ordered, as they are simply represented through multisets or sets. In this paper, we study ordered fuzzy multisets, where the membership value can be an ordered $n$-tuple of values, thus accounting for both order and repetition. We present some basic definitions and results and explore the relation between these ordered fuzzy multisets and the fuzzy multisets and hesitant fuzzy sets.

Necessary and sufficient conditions for the existence of solution of generalized fuzzy relation equations A ⇔X = B

In 2013 Li and Jin studied a particular type of fuzzy relational equations on finite sets, where the introduced min-bi-implication composition is based on Łukasiewicz equivalence. In this paper such fuzzy relation equations are studied on a more general level, namely complete residuated lattice valued fuzzy relation equations of type ⋀y∈Y(A(x,y)↔X(y)=B(x) are analyzed, and the existence of solutions S is studied. First a necessary condition for the existence of solution is established, then conditions for lower and upper limits of solutions are given, and finally sufficient conditions for the existence of the smallest and largest solutions, respectively, are characterized. If such general or global solutions do not exist, there might still be partial or point wise solutions; this is a novel way to study fuzzy relation equations. Such point wise solutions are studied on Łukasiewicz, Product and Gödel t-norm based residuated lattices on the real unit interval.

Ordinal sums of triangular norms on a bounded lattice

The ordinal sum construction provides a very effective way to generate a new triangular norm on the real unit interval from existing ones. One of the most prominent theorems concerning the ordinal sum of triangular norms on the real unit interval states that a triangular norm is continuous if and only if it is uniquely representable as an ordinal sum of continuous Archimedean triangular norms. However, the ordinal sum of triangular norms on subintervals of a bounded lattice is not always a triangular norm (even if only one summand is involved), if one just extends the ordinal sum construction to a bounded lattice in a naïve way. In the present paper, appropriately dealing with those elements that are incomparable with the endpoints of the given subintervals, we propose an alternative definition of ordinal sum of countably many (finite or countably infinite) triangular norms on subintervals of a complete lattice, where the endpoints of the subintervals constitute a chain. The completeness requirement for the lattice is not needed when considering finitely many triangular norms. The newly proposed ordinal sum is shown to be always a triangular norm. Several illustrative examples are given.

(1904-5132) General Definitions for the Union and Intersection of Ordered Fuzzy Multisets

Since its original formulation, the theory of fuzzy sets has spawned a number of extensions where the role of membershipvalues in the real unit interval [0; 1] is handed over to more complex mathematical entities. Amongst the many existingextensions, two similar ones, the fuzzy multisets and the hesitant fuzzy sets, rely on collections of several distinct valuesto represent fuzzy membership, the key difference being that the fuzzy multisets allow for repeated membership valueswhereas the hesitant fuzzy sets do not. But in neither case are these collections of values ordered, as they are simplyrepresented through multisets or sets. In this paper, we study ordered fuzzy multisets, where the membership valuecan be an ordered n-tuple of values, thus accounting for both order and repetition. We present some basic de finitionsand results and explore the relation between these ordered fuzzy multisets and the fuzzy multisets and hesitant fuzzysets.