Continuous T-norms Research Articles

Representations of T-similarity relations

Convergence of sequences of triangular norms and their quasi-inverses, stability of the representation theorem for T-similarity relations, and the density property of the family of Z-similarity relations are investigated. It is shown, in particular, that under some hypothesis the representation theorem is stable under taking limits of continuous t-norms, and that any strictly reflexive and symmetric fuzzy relation can be approximated up to any precision level by a T-similarity relation with T being a strict t-norm.

A note on “Some results on the IF-normed spaces”

Recently, Lael and Nourouzi [Some results on the IF-normed spaces. Chaos, Solitons & Fractals 2006; doi:10.1016/j.chaos.2006.10.019] introduced and studied a new notation of IF-normed spaces by using the idea of intuitionistic fuzzy normed spaces due to Saadati and Park [On the intuitionistic fuzzy topological spaces. Chaos, Solitons & Fractals 2006;27:331–44], a special continuous t-norm i.e. min and a special continuous s-norm i.e. max. In this note, we consider the modified definition of IF-normed space i.e. L F -normed spaces and prove the open mapping and closed graph theorems for this space using arbitrary continuous t-norm.

Infinite fuzzy relation equations with continuous t-norms

This study develops a concept of infinite fuzzy relation equations with a continuous t-norm. It extends the work by Xiong and Wang [Q.Q. Xiong, X.P. Wang, Some properties of sup-min fuzzy relational equations on infinite domains, Fuzzy Sets and Systems 151 (2005) 393–402]. We describe attainable (respectively, unattainable) solutions, which are closely related to minimal solutions to the equations. It is shown that a solution set comprises both attainable and unattainable solutions. The study offers a characterization of these solutions. Under some assumptions, the solution set is presented and discussed. Two applications are also given.

On the representation of fuzzy rules

In fuzzy logic, connectives have a meaning that, can frequently be known through the use of these connectives in a given context. This implies that there is not a universal-class for each type of connective, and because of that several continuous t-norms, continuous t-conorms and strong negations, are employed to represent, respectively, the and, the or, and the not. The same happens with the case of the connective If/then for which there is a multiplicity of models called T-conditionals or implications. To reinforce that there is not a universal-class for this connective, four very simple classical laws translated into fuzzy logic are studied.

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.

A representation of t-norms in interval-valued L-fuzzy set theory

In this paper we consider the lattice L I which has the closed subintervals of a complete lattice as elements. We give a representation theorem of t-norms on this lattice for which the partial mappings are join-morphisms in terms of t-norms on the underlying lattice. In fuzzy logic, t-norms which satisfy the residuation principle play an important role. Using our representation theorem, we represent t-norms on L I which satisfy the residuation principle and two border conditions in terms of t-norms on the underlying lattice. In the case that the underlying lattice of L I is the unit interval, we obtain characterizations of continuous t-norms which are natural extensions of t-norms on the unit interval and which have join-morphisms as partial mappings or alternatively satisfy the residuation principle.

Unified full implication algorithms of fuzzy reasoning

This paper discusses the full implication inference of fuzzy reasoning. For all residuated implications induced by left continuous t-norms, unified α-triple I algorithms are constructed to generalize the known results. As the corollaries of the main results of this paper, some special algorithms can be easily derived based on four important residuated implications. These algorithms would be beneficial to applications of fuzzy reasoning. Based on properties of residuated implications, the proofs of the many conclusions are greatly simplified.

Convex combinations of continuous t-norms with the same diagonal function

The problem of whether a non-trivial convex combination of two continuous t-norms with the same diagonal function can be a t-norm is studied. It is shown that in both cases–of two nilpotent and of two strict t-norms–a non-trivial convex combination of t-norms with common diagonal function is associative only if the two t-norms involved coincide. For general continuous t-norms a similar result follows. An example of a convex class of non-continuous t-norms is also included.

On the law S(S(x, y), T(x, y)) = S(x, y) of fuzzy logic

Motivated by some functional models arising in fuzzy logic, when classical boolean relations between sets are generalized, we study the functional equation S(S(x, y), T(x, y)) = S(x, y), where S is a continuous t-conorm and T is a continuous t-norm. Some interesting methods for solving this type of equations are introduced.

On triangular norms and uninorms definable in [formula omitted

In this paper, we investigate the definability of classes of t-norms and uninorms in the logic Ł Π 1 2 . In particular we provide a complete characterization of definable continuous t-norms, weak nilpotent minimum t-norms, conjunctive uninorms continuous on [0, 1), and idempotent conjunctive uninorms, and give both positive and negative results concerning definability of left-continuous t-norms (and uninorms). We show that the class of definable uninorms is closed under construction methods as annihilation, rotation and rotation–annihilation. Moreover, we prove that every logic based on a definable uninorm is in PSPACE, and that any finitely axiomatizable logic based on a class of definable uninorms is decidable. Finally we show that the Uninorm Mingle Logic (UML) and the Basic Uninorm Logic (BUL) are finitely strongly standard complete w.r.t. the related class of definable left-continuous conjunctive uninorms.

Solutions of fuzzy relation equations based on continuous t-norms

This study is concerned with fuzzy relation equations with continuous t-norms in the form ATR = B, where A and B are the fuzzy subsets of X and Y, respectively; R ⊂ X × Y is a fuzzy relation, and T is a continuous t-norm. The problem is how to determine A from R and B. First, an “if and only if” condition of being solvable is presented. Novel algorithms are then presented for determining minimal solutions when X and Y are finite. The proposed algorithms generate all minimal solutions for the equations, making them efficient solving procedures.

Properties of Fuzzy Implications obtained via the Interval Constructor

This work considers an interval extension of fuzzy implication based on the best interval representation of continuous t-norms. Some related properties can be naturally extended and that extension preserves the behaviors of the implications in the interval endpoints.

De Finetti theorem and Borel states in [0, 1]-valued algebraic logic

In this paper de Finetti’s (no-Dutch-Book) criterion for coherent probability assignments is extended to large classes of logics and their algebras. Given a set A of “events” and a closed set W ⊆ [ 0 , 1 ] A of “possible worlds” we show that a map s : A → [ 0 , 1 ] satisfies de Finetti’s criterion if, and only if, it has the form s ( a ) = ∫ W V ( a ) d μ ( V ) for some probability measure μ on W . Our results are applicable to all logics whose connectives are continuous operations on [ 0 , 1 ] , notably (i) every [ 0 , 1 ] -valued logic with finitely many truth-values, (ii) every logic whose conjunction is a continuous t-norm, and whose negation is ¬ x = 1 - x , possibly also equipped with its t-conorm and with some continuous implication, (iii) any extension of Łukasiewicz logic with constants or with a product-like connective. We also extend de Finetti’s criterion to the noncommutative underlying logic of GMV-algebras.

Fuzzy logics based on [0,1)-continuous uninorms

Axiomatizations are presented for fuzzy logics characterized by uninorms continuous on the half-open real unit interval [0,1), generalizing the continuous t-norm based approach of Hajek. Basic uninorm logic BUL is defined and completeness is established with respect to algebras with lattice reduct [0,1] whose monoid operations are uninorms continuous on [0,1). Several extensions of BUL are also introduced. In particular, Cross ratio logic CRL, is shown to be complete with respect to one special uninorm. A Gentzen-style hypersequent calculus is provided for CRL and used to establish co-NP completeness results for these logics.

On the characterizations of fuzzy implications satisfying [formula omitted

Iterative boolean-like laws in fuzzy logic have been studied by Alsina and Trillas [C. Alsina, E. Trillas, On iterative boolean-like laws of fuzzy sets, in: Proc. 4th Conf. Fuzzy Logic and Technology, Barcelona, Spain, 2005, pp. 389–394] for functional equations with boolean background in which only fuzzy conjunctions, fuzzy disjunctions and fuzzy negations are contained. In this paper we study an iterative boolean-like law with fuzzy implications, more precisely we derive characterizations of some classes of fuzzy implications satisfying I ( x , y ) = I ( x , I ( x , y ) ) , for all ( x , y ) ∈ [ 0 , 1 ] 2 . Our discussion mainly focuses on the three important classes of implications: S-implications, R-implications and QL-implications. We prove the sufficient and necessary conditions for an S-implication generated by any t-conorm and any fuzzy negation, an R-implication generated by a left-continuous t-norm, a QL-implication generated by a continuous t-conorm, a continuous t-norm and a strong fuzzy negation to satisfy I ( x , y ) = I ( x , I ( x , y ) ) , for all ( x , y ) ∈ [ 0 , 1 ] 2 .

Compatibility of t-norms with the concept of ϵ-partition

In this paper we study the behaviour of a kind of partitions formed by fuzzy sets, the ϵ-partitions, with respect to three important operations: refinement, union and product of partitions. In the crisp set theory, the previous operations lead to new partitions: every refinement of a partition is also a partition; the union of partitions of disjoint sets is a partition of the union set; the product of two partitions of two sets is a partition of the intersection of the partitioned sets. It has been proven that ϵ-partitions extend the three previous properties when the intersection of fuzzy sets is defined by the minimum t-norm and the union by the maximum t-conorm. In this paper we consider any t-norm defining the intersection of fuzzy sets and we characterize those t-norms for which refinements, unions and products of ϵ-partitions are ϵ-partitions. We pay special attention to these characterizations in the case of continuous t-norms.

Adding truth-constants to logics of continuous t-norms: Axiomatization and completeness results

In this paper we study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for Łukasiewicz logic and more recent results for Gödel and Product logics. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants { r ¯ | r ∈ C } , for a suitable countable C ⊆ [ 0 , 1 ] , and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of Łukasiewicz, Gödel and Product components, (ii) the set of truth-constants covers all the unit interval in the sense that each component of the t-norm contains at least one value of C different from the bounds of the component, and (iii) the truth-constants in Łukasiewicz components behave as rational numbers.

Conditional probability and fuzzy information

The main subject of this paper is the embedding of fuzzy set theory—and related concepts—in a coherent conditional probability scenario. This allows to deal with perception-based information—in the sense of Zadeh—and with a rigorous treatment of the concept of likelihood, dealing also with its role in statistical inference. A coherent conditional probability is looked on as a general non-additive “uncertainty” measure m ( · ) = P ( E | · ) of the conditioning events. This gives rise to a clear, precise and rigorous mathematical frame, which allows to define fuzzy subsets and to introduce in a very natural way the counterparts of the basic continuous T-norms and the corresponding dual T-conorms, bound to the former by coherence. Also the ensuing connections of this approach to possibility theory and to information measures are recalled.

Distributivity and conditional distributivity of a uninorm and a continuous t-conorm

The open problem recalled by Klement in the Linz2000 closing session, related to distributivity and conditional distributivity of a uninorm and a continuous t-conorm, is solved for the most usual known classes of uninorms. From the obtained results, it is deduced that distributivity and conditional distributivity are equivalent for these cases. It is remarkable that solutions appear involving not only strict t-conorms but also ordinal sums of the maximum with a strict t-conorm. Conversely, the distributivity of a t-conorm over a uninorm is also studied leading only to already known solutions. Moreover, the dual case of distributivity and conditional distributivity involving uninorms and continuous t-norms is also solved, proving again the equivalence of both kinds of distributivities.

Computational complexity of t-norm based propositional fuzzy logics with rational truth constants

If a continuous t-norm on [ 0 , 1 ] maps pairs of rationals into rationals then the corresponding fuzzy propositional calculus can be extended by rational truth constants and “bookkeeping” axioms for them. (Łukasiewicz t-norm is the classical example.) Computational complexity of such logics is studied. Consequences for fuzzy description logic are formulated.