Balasubramaniam Jayaram

(S, N)- and R-implications: A state-of-the-art survey

In this work we give a state-of-the-art review of two of the most established classes of fuzzy implications, viz., (S,N)- and R-implications. Firstly, we discuss their properties, characterizations and representations. Many new results concerning fuzzy negations and (S,N)-implications, notably their characterizations with respect to the identity principle and ordering property, are presented, which give rise to some representation results. Finally, using the presented facts, an almost complete characterization of the intersections that exist among some subfamilies of (S,N)- and R-implications are obtained.

On the Law of Importation

The law of importation, given by the equivalence (x Lambda y) rarr z equiv (xrarr (y rarr z)), is a tautology in classical logic. In A-implications defined by Turksen et aL, the above equivalence is taken as an axiom. In this paper, we investigate the general form of the law of importation J(T(x, y), z) = J(x, J(y, z)), where T is a t-norm and J is a fuzzy implication, for the three main classes of fuzzy implications, i.e., R-, S- and QL-implications and also for the recently proposed Yagers classes of fuzzy implications, i.e., f- and g-implications. We give necessary and sufficient conditions under which the law of importation holds for R-, S-, f- and g-implications. In the case of QL-implications, we investigate some specific families of QL-implications. Also, we investigate the general form of the law of importation in the more general setting of uninorms and t-operators for the above classes of fuzzy implications. Following this, we propose a novel modified scheme of compositional rule of inference (CRI) inferencing called the hierarchical CRI, which has some advantages over the classical CRI. Following this, we give some sufficient conditions on the operators employed under which the inference obtained from the classical CRI and the hierarchical CRI become identical, highlighting the significant role played by the law of importation.

(x \wedge y) \longrightarrow z \equiv (x \longrightarrow (y \longrightarrow z))

The characterization of S-implications generated from strong negations presented firstly by Trillas and Valverde in 1985 is well-known in the literature. In this paper we show that some assumptions are needless and present two characterizations of S-implications with mutually independent requirements. We also present characterizations of (S,N)-implications obtained from continuous fuzzy negations or strict negations. Besides these main results some new facts concerning fuzzy implications, fuzzy negations and laws of contraposition are proved.

in Fuzzy Logic

Fuzzy relational inference (FRI) systems form an important part of approximate reasoning schemes using fuzzy sets. The compositional rule of inference (CRI), which was introduced by Zadeh, has attracted the most attention so far. In this paper, we show that the FRI scheme that is based on the Bandler-Kohout (BK) subproduct, along with a suitable realization of the fuzzy rules, possesses all the important properties that are cited in favor of using CRI, viz., equivalent and reasonable conditions for their solvability, their interpolative properties, and the preservation of the indistinguishability that may be inherent in the input fuzzy sets. Moreover, we show that under certain conditions, the equivalence of first-infer-then-aggregate (FITA) and first-aggregate-then-infer (FATI) inference strategies can be shown for the BK subproduct, much like in the case of CRI. Finally, by addressing the computational complexity that may exist in the BK subproduct, we suggest a hierarchical inferencing scheme. Thus, this paper shows that the BK-subproduct-based FRI is as effective and efficient as the CRI itself.

On the characterizations of (S,N)-implications

Recently, we have presented characterizations of (S,N)-implications generated from t-conorms and continuous (strict, strong) negations. Uninorms were introduced by Yager and Rybalov in 1996 as a generalization of t-norms and t-conorms, thus they are another fertile source based on which one can define fuzzy implications. (U,N)-implications are a generalization of (S,N)-implications, where a t-conorm S is replaced by a (disjunctive) uninorm U. In this work we present characterizations of (U,N)-implications obtained from disjunctive uninorms and continuous negations as well as (U,N)-operations defined from uninorms and continuous negations.

On the Suitability of the Bandler–Kohout Subproduct as an Inference Mechanism

Recently, many works have appeared in this very journal dealing with the distributivity of fuzzy implications over t-norms and t-conorms. These equations have a very important role to play in efficient inferencing in approximate reasoning, especially fuzzy control systems. Of all the four equations considered, the equation I(x, S1 (y,z)) = S2(I(x,y),I(x,z)), when S1,S2 are both t-conorms and I is an R-implication obtained from a strict t-norm, was not solved. In this paper, we characterize functions I that satisfy the previous functional equation when S1,S2 are either both strict or nilpotent t-conorms. Using the obtained characterizations, we show that the previous equation does not hold when S1,S2 are either both strict or nilpotent t-conorms, and I is a continuous fuzzy implication. Moreover, the previous equation does not hold when I is an R -implication obtained from a strict t-norm, and S1,S2 are both strict t-conorms, while it holds for an R-implication I obtained from a strict t-norm T if and only if the t-conorms S1 = S2 are Phi-conjugate to the Lukasiewicz t-conorm for some increasing bijection phi of the unit interval, which is also a multiplicative generator of T.

(U,N)-implications and their characterizations

In this paper, we attempt a systematic study of QL-implications. Towards this end, firstly, we investigate the conditions under which a QL-operation becomes a fuzzy implication without imposing any conditions on the underlying operations. Following this, we discuss the conditions under which this family satisfies some desirable algebraic properties. Based on the obtained results and existing characterization results, the intersections between QL-implications and the two most established families of fuzzy implications, viz., (S,N)- and R-implications are determined. It is shown that QL-implications contain the set of all R-implications obtained from left-continuous t-norms that are also (S,N)-implications. Finally, the overlaps between QL-implications and the recently proposed f- and g-implications are also studied.

On the Distributivity of Fuzzy Implications Over Nilpotent or Strict Triangular Conorms

The two most important models of inferencing in approximate reasoning with fuzzy sets are Zadehs Compositional Rule of Inference (CRI) and Similarity Based Reasoning (SBR). It is known that inferencing in the above models is resource consuming (both memory and time), since these schemes often consist of discretisation of the input and output spaces followed by computations in each point. Also an increase in the number of rules only exacerbates the problem. As the number of input variables and/or input/output fuzzy sets increases, there is a combinatorial explosion of rules in multiple fuzzy rule based systems. In this paper, given a fuzzy if--then rule base that is used in an SBR inference mechanism, we propose to reduce the number of rules by combining the antecedents of the rules that have the same consequent. We also present some sufficient conditions on the operators employed in SBR inference schemes such that the inferences obtained using the original rule base and the reduced rule base obtained as above are identical. Subsequently, these conditions are investigated and many solutions are presented for some specific SBR inference schemes.

QL-implications: Some properties and intersections

Fuzzy implications are a generalization of the classical two-valued implication to the multi-valued setting. They play a very important role both in the theory and applications, as can be seen from their use in, among others, multivalued mathematical logic, approximate reasoning, fuzzy control, image processing, and data analysis. The goal of this chapter is to present the evolution of fuzzy implications from their beginnings to the current days. From the theoretical point of view, we present the basic facts, as well as the main topics and lines of research around fuzzy implications. We also devote a specific section to state and recall a list of main application fields where fuzzy implications are employed, as well as another one to the main open problems on the topic.

Rule reduction for efficient inferencing in similarity based reasoning

Special implications were introduced by Hajek and Kohout [Fuzzy implications and generalized quantifiers, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 4 (1996) 225-233] in their investigations on some statistics on marginals. They have either suggested or only partially answered three important questions, especially related to special implications and residuals of t-norms. In this work we investigate these posers in-depth and give complete answers. Toward this end, firstly we show that many of the properties considered as part of the definition of special implications are redundant. Then, a geometric interpretation of the specialty property is given, using which many results and bounds for such implications are obtained. We have obtained a characterization of general binary operations whose residuals become special. Finally, some constructive procedures to obtain special fuzzy implications are proposed and methods of obtaining special implications from existing ones are given, showing that there are infinitely many fuzzy implications that are special but cannot be obtained as residuals of t-norms.