We introduce a novel Mamdani-type fuzzy model, referred to as the all-permutations fuzzy rule-base, and show that it is mathematically equivalent to a standard feedforward neural network. We describe several applications of this equivalence between a neural network and our fuzzy rule base, including knowledge extraction from and knowledge insertion into neural networks.

The design of artificial systems inspired by biological behavior is recently attracting considerable interest. Many biological agents such as plants or animals were forced to develop sophisticated mechanisms in order to tackle various problems they encounter in their habitat. For example,

We apply fuzzy modeling to derive a mathematical model for a biological phenomenon: the regulation of population size in the Australian sheep-blowfly Lucilia cuprina. This behavior was described by several ethologists and fuzzy modeling allows us to transform their verbal descriptions into a well-defined mathematical model. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature. We believe that the fuzzy modeling approach demonstrated here may supply a suitable framework for biomimicry, that is, the design of artificial systems based on mimicking natural behavior.

Considerable research has been devoted to the integration of fuzzy logic (FL) tools with classic artificial intelligence (AI) paradigms. One reason for this is that FL provides powerful mechanisms for handling and processing symbolic information stated using natural language.

We develop a new approach for designing a recurrent neural network (RNN) that is suitable for solving a given problem. Initial information on the problem domain is stated in terms of symbolic If-Then rules. These rules have a special structure and inferring them yields a mapping that is equivalent to that of a net of sigmoid activated neurons with feedback connections. Thus, inferring the rules automatically yields a suitable RNN. We demonstrate the e#ciency of our approach by using it to design an RNN that recognizes a formal language.

We apply fuzzy modeling to derive a mathematical model for a biological phenomena: the orientation to light of the planarian Dendrocoleum lacteum. This behavior was described linguistically by several ethologists and fuzzy modeling allows us to transform their descriptions into a mathematical model. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature.

Summary. In many fields of science human observers have provided verbal descriptions and explanations of various systems. A formal mathematical model is indispensable when we wish to rigorously analyze these systems. In this chapter, we survey some recent results on transforming verbal descriptions into mathematical models using fuzzy modeling. This is a simple and direct approach that offers a unique advantage–the close relationship between the verbal description and the mathematical model can be used to verify the validity of the verbal explanation suggested by the observer. We review two applications of this approach from the field of ethology: the territorial behavior of the fish and the orientation to light of a flat worm. We believe that the fuzzy modeling approach demonstrated here may supply a suitable framework for biomimicry, that is, the design of artificial systems based on mimicking a natural behavior observed in nature.

A major drawback of artificial neural networks is their black-box character. Even when the trained network performs adequately, it is very di#cult to understand its operation. In this paper, we use the mathematical equivalence between artificial neural networks and a specific fuzzy rule base to extract the knowledge embedded in the network. We demonstrate this using a benchmark problem: the recognition of digits produced by a LED device. The method provides a symbolic and comprehensible description of the knowledge learned by the network during its training.

Gene regulation plays a central role in the development and functioning of living organisms. Developing a deeper qualitative and quantitative understanding of gene regulation is an important scientific challenge. The λ switch is commonly used as a paradigm of gene regulation. Verbal descriptions of the structure and functioning of the λ switch have appeared in biological textbooks. We apply fuzzy modeling to transform one such verbal description into a well-defined mathematical model. The resulting model is a piecewise-quadratic second-order differential equation. It demonstrates functional fidelity with known results while being simple enough to allow a rather detailed analysis. Properties such as the number, location, and domain of attraction of equilibrium points can be studied analytically. Furthermore, the model provides a rigorous explanation for the so-called stability puzzle of the λ switch.

Fuzzy systems address the imprecision of the input and output variables, which formally describe notions like “rather warm” or “pretty cold”, while provide a behaviour that depends on fuzzy data. This class of systems are classically represented by means of Fuzzy Inference Systems (FIS), a computing framework based on the concepts of fuzzy if-then rules and fuzzy reasoning. Even if FIS are largely used, these lack in compositionality. Moreover, the analysis of modeled behaviuors needs complex analytic tools. In this paper we propose a process algebraic approach to specification and analysis of fuzzy behaviours. Indeed, we introduce a Fuzzy variant of CCS (Calculus of Communicating Processes), that permits compositionally describing fuzzy behaviours. Moreover, we also show how standard process algebra formal tools, like modal logics and behavioural equivalences, can be used for supporting fuzzy reasoning.