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14
Are Artificial Neural Networks White Boxes?
, 2004
"... We introduce a novel Mamdanitype fuzzy model, referred to as the allpermutations fuzzy rulebase, 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, inclu ..."
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We introduce a novel Mamdanitype fuzzy model, referred to as the allpermutations fuzzy rulebase, 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.
Extracting symbolic knowledge from recurrent neural networksA fuzzy logic approach
 Fuzzy Sets and Systems, Volume 160, Issue
, 2009
"... 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. In this respect, fuzzy ..."
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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. In this respect, fuzzy rulebased systems are whiteboxes, as they process information in a form that is easy to understand, verify and, if necessary, refine. The synergy between artificial neural networks (ANNs), which are notorious for their blackbox character, and FL proved to be particularly successful. Such a synergy allows combining the powerful learningfromexamples capability of ANNs with the highlevel symbolic information processing of FL systems. In this paper, we present a new approach for extracting symbolic information from recurrent neural networks (RNNs). The approach is based on the mathematical equivalence between a specific fuzzy rulebase and functions composed of sums of sigmoids. We show that this equivalence can be used to provide a comprehensible explanation of the RNN functioning. We demonstrate the applicability of our approach by using it to extract the knowledge embedded within an RNN trained to recognize a formal language.
Nicholson's Blowflies Revisited: A Fuzzy Modeling Approach
 FUZZY SETS SYSTEMS
, 2007
"... We apply fuzzy modeling to derive a mathematical model for a biological phenomenon: the regulation of population size in the Australian sheepblowfly Lucilia cuprina. This behavior was described by several ethologists and fuzzy modeling allows us to transform their verbal descriptions into a wel ..."
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Cited by 6 (5 self)
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We apply fuzzy modeling to derive a mathematical model for a biological phenomenon: the regulation of population size in the Australian sheepblowfly Lucilia cuprina. This behavior was described by several ethologists and fuzzy modeling allows us to transform their verbal descriptions into a welldefined 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.
The fuzzy ant
 IEEE Computational Intelligence Magazine
, 2007
"... 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, animals mu ..."
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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, animals must develop efficient mechanisms for orienting themselves in space. Similar problems arise in the design of artificial systems. For example, planning and realizing oriented movements is a crucial problem in the design of autonomous robots. Thus, lessons from biological behavior may inspire suitable artificial designs. In some cases, ethologists provided verbal descriptions of the relevant animal behavior. Fuzzy modeling is the most suitable tool for transforming these verbal descriptions into mathematical models or computer algorithms that can be used in artificial systems. We demonstrate this by using fuzzy modeling to develop a mathematical model for the foraging behavior of ants. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature.
A new approach to knowledgebased design of recurrent neural networks
 IEEE Trans. Neural Networks
, 2008
"... Abstract — A major drawback of artificial neural networks (ANNs) is their blackbox character. This is especially true for recurrent neural networks (RNNs) because of their intricate feedback connections. In particular, given a problem and some initial information concerning its solution, it is not ..."
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Abstract — A major drawback of artificial neural networks (ANNs) is their blackbox character. This is especially true for recurrent neural networks (RNNs) because of their intricate feedback connections. In particular, given a problem and some initial information concerning its solution, it is not at all clear how to design an RNN that is suitable for solving this problem. In this paper, we consider a fuzzy rulebase with a special structure, referred to as the fuzzy allpermutations rulebase (FARB). Inferring the FARB yields an inputoutput mapping that is mathematically equivalent to that of an RNN. We use this equivalence to develop two new knowledgebased design methods for RNNs. The first method, referred to as the direct approach, is based on stating the desired functioning of the RNN in terms of several sets of symbolic rules, each one corresponding to a subnetwork. Each set is then transformed into a suitable FARB. The second method is based on first using the direct approach to design a library of simple modules, such as counters or comparators, and realize them using RNNs. Once designed, the correctness of each RNN can be verified. Then, the initial design problem is solved by using these basic modules as building blocks. This yields a modular and systematic approach for knowledgebased design of RNNs. We demonstrate the efficiency of these approaches by designing RNNs that recognize both regular and nonregular formal languages.
Mathematical modeling of natural phenomena: a fuzzy logic approach
 In
, 2007
"... 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 ..."
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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.
How Does the Dendrocoleum lacteum Orient to Light? A Fuzzy Modeling Approach
"... 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 m ..."
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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.
Knowledge Extraction from Neural Networks Using the AllPermutations Fuzzy Rule Base
, 2005
"... A major drawback of artificial neural networks is their blackbox 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 ..."
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Cited by 2 (1 self)
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A major drawback of artificial neural networks is their blackbox 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.
MATHEMATICAL MODELING OF THE λ SWITCH A FUZZY LOGIC APPROACH
"... 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 ..."
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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 welldefined mathematical model. The resulting model is a piecewisequadratic secondorder 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 socalled stability puzzle of the λ switch.
A Process Algebra Approach to Fuzzy Reasoning
 IFSAEUSFLAT
, 2009
"... 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 computin ..."
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Cited by 2 (0 self)
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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 ifthen 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.