We start from the observation that Zadeh’s compositional rule of inference (CRI) is a strategy to determine approximately a roughly given control function. From this point of view the problem to solve a system of fuzzy relation equations also becomes the problem to determine approximately such a control strategy. This gives a natural interpretation for approximate solutions of unsolvable system...

In this paper, we have developed an algebraic theory, suitable for the analysis of fuzzy systems. We have used the notions of semiring and semimodule, introduced the notion of semilinear space and gave numerous examples of them and defined also the notions of linear dependence and independence. Then, we have shown that the composition operation, which plays an essential role in the analysis of ...

The concept of approximate reasoning as it was con ceived by Zadeh provides a framework which al lows to model and process vague linguistic informa tion The idea is to model linguistic terms by fuzzy sets their logical relationship by fuzzy relations and their composition by the so called compositional rule of inference As an important eld of applica tions we refer to control processes for whic...

Defining the standard Boolean operations on fuzzy Booleans with the compositional rule of inference (CRI) or Zadeh's extension principle gives counter-intuitive results. We introduce and motivate a slight adaptation of the CRI, which only effects the results for non-normal fuzzy sets. It is shown that the adapted CRI gives the expected results for the standard Boolean operations on fuzzy Boolea...

The concept of an image of a fuzzy set under a fuzzy relation has proved to be a very powerful tool in fuzzy set theoretical applications. In this paper, we explain how it can be used to model linguistic expressions. For the representation of expressions, such as “at least middle-aged”, “brighter than average”, we will use fuzzy ordering relations, while resemblance relations will be suitable t...

Simpson’s paradox, also known as amalgamation or aggregation paradox, appears when dealing with proportions. Proportions are by construction parts of a whole, which can be interpreted as compositions assuming they only carry relative information. The Aitchison inner product space structure of the simplex, the sample space of compositions, explains the appearance of the paradox, given that amalg...

In this paper, a novel stock trading framework based on a neuro-fuzzy associative memory (FAM) architecture is proposed. The architecture incorporates the approximate analogical reasoning schema (AARS) to resolve the problem of discontinuous (staircase) response and inefficient memory utilization with uniform quantization in the associative memory structure. The resultant structure is conceptua...