James C Bezdek
email: jcbezdek@gmail.com

Didier Dubois
email: dubois@irit.fr

Henri Prade
email: prade@irit.fr

The posterity of Zadeh's 50-year-old paper: A retrospective in 101 Easy Pieces -and a Few More

Keywords: fuzzy pattern recognition, fuzzy control fuzzy systems, fuzzy models, list of 101 fuzzy citations

of Fuzzy Systems (FUZZ-IEEE) to celebrate the 50 th Anniversary of Lotfi Zadeh's seminal 1965 paper on fuzzy sets. In addition to Lotfi's original paper, this note itemizes 100 citations of books and papers deemed "important (significant, seminal, etc.)" by 20 of the 21 living IEEE CIS Fuzzy Systems pioneers. Each of the 20 contributors supplied 5 citations, and Lotfi's paper makes the overall list a tidy 101, as in "Fuzzy Sets 101". This note is not a survey in any real sense of the word, but the contributors did offer short remarks to indicate the reason for inclusion (e.g., historical, topical, seminal, etc.) of each citation. Citation statistics are easy to find and notoriously erroneous, so we refrain from reporting themalmost. The exception is that according to Google scholar on April 9, 2015, Lotfi's 1965 paper has been cited 55,479 times.

You may ask "the 101 best books and papers according to whom?" And in fact one of our pioneers did ask this very question, and refused to participate because he felt such lists were completely arbitrary and therefore entirely useless. Well, perhaps they areis there any value to such a list at all? If you believe that history is important -that the way forward is in some sense better understood if presaged by an understanding of the road already travelled, our list may be helpful.

In the age of internet search, we know that all these references are at your fingertips -as long as you ask the right question or know what to look for. Our hope is that the citations given here encourage you to move in a direction you may not have been interested in before seeing them.

II. HOW THE LIST WAS BUILT

Table I lists the 23 pioneers, arranged in the chronological order in which they received the award. Here is our collection algorithm. Each of the 21 living pioneers was invited to submit up to five citations subject to these constraints: (i) no more than three self-citations; (ii) no more than one citation involving another pioneer; and (iii) at least one citation for a non-pioneer. The response was hardly uniform! Indeed, Abe Kandel, one of the 2012 pioneers, refused to participate at all. Here is his statement of declination, reproduced verbatim from his email to us dated September 7. 2014: Abe wrote: "I am very sorry but I will decline this invitation due to the following reasons: 1) The concept of an Important publication is not really well defined. Important to whom, the author? His friends ? His students ? World peace ? Applications to improve society ? Etc. 2) why as fuzzy logicians we select a binary number of 100 ? What about the paper in location 101 ? And why not 1000 or just 3 "most important"? 3) who made US [eds: "US" is not the USA here]-the fuzzy pioneers the "God of Fuzziness " to make these kinds of decisions ? Why not to include also other very good and promising researchers in the field. Just because we were on this bus does not imply anything as evaluators in this entirely fuzzy process. I think that we should all consider this Idea and not just spend 5 minutes as recently suggested." And who's to say Abe is wrong? Some pioneers supplied five citations, some supplied less than five citations, and of course the two deceased pioneers supplied none. We exercised our editorial prerogative to fill in the empty slots. Some of the explanatory comments supplied to us were too long or seemed confusing, so in a few cases, we edited them for brevity and/or clarity. Finally, there is little value in knowing which pioneer suggested which citations, so that information is not reported here.

Section III contains the 101 citations and remarks, ordered alphabetically and within author, chronologically, by the last name of the first author. The references are given in a modified form of the standard IEEE format which we think is self-explanatory, brief, and enables alphabetization. Abbreviations for commonly occuring journals in the citations are listed in Table II. There is not much we can say about this paper that has not already been said. Without it, there is no 101 list, and many of us would be herding cows, painting houses, riding motorcycles, drinking beer (ok, some of us would be doing that anyway) or playing guitars in seedy juke joints. So, on to the subsequent 100 papers and books supplied by the 20 pioneers.

[1] Atanassov, K. T., "Intuitionistic fuzzy sets," FSS, 20, 87-96, 1986. In this paper Atanassov introduced his ideas about intuitionistic fuzzy sets to the fuzzy set community, and the basic definitions.!

[2] Baldwin, J. F., "A new approach to approximate reasoning using fuzzy logic," FSS, 2, 1979, 309-325. This is one of the first papers, that focused on the extension of fuzzy logic to approximate reasoning on the basis of logical considerations.

In contrast to fuzzy control, Baldwin used human argumentation rather than the control of artificial systems (machines etc.). It is still computationally simple and efficient and eventually led to the development of the fuzzy computer language Fril.

[3] Bellman, R. E. and L. A. Perhaps the first publication to define and use the term "Computational Intelligence," subsequently adopted by the Neural Networks Council (NNC). The NNC attached the term to its triumvirate of flagship conferences (WCCI), and eventually changed their name to the IEEE Computational Intelligence Society. For more information on the history of this term and its relationship to the Canadian journal Computational Intelligence published by Wiley, visit ieee-cis.sightworks.net/documents/History/Bezdek-eolss-CIhistory.pdf.!

[6] Bezdek, J. C. and Harris, J. D., "Fuzzy partitions and relations: an axiomatic basis for clustering, " FSS, 1, 1978, 111-127. This paper derives a hierarchy of fuzzy similarity relation spaces (FSRs) whose minimal member is the set of crisp equivalence relations, and whose maximal member is the set of max" # transitive FSRs. A transformation of fuzzy partitions based on sum-min matrix multiplication is shown to induce a pseudo metric on the data.

[7] Bezdek, J. C. and R. J. Hathaway, "Clustering with relational cmeans partitions from pairwise distance data, Math. Modelling, 9(6), 1987, 435-439. This paper introduced the idea of relational duals for the hard and fuzzy c-means algorithms. It is the basis for the branch of soft clustering that includes possibilistic and non-Euclidean versions of relational c-means.

[8] Bonissone, P. "Soft computing: the convergence of emerging reasoning technologies", Soft Computing, 1(1), 1997, 6-18.

One of the first studies of Hybrid Soft Computing, jointly using fuzzy logic (FL), neural networks (NN) and genetic algorithms (GA). The paper presents several cases studies of hybridization of two or more soft computing techniques, such as the use of FL to control GAs and NNs parameters, the application of GAs to evolve NNs topologies or weights, or to tune FL controllers, and the implementation of FL controllers as NNs tuned by back-propagation type algorithm. This paper has inspired many other subsequent works in hybrid soft computing.

[9] Bonissone, P. and K. Decker, "Selecting uncertainty calculi and granularity: An experiment in trading-off precision and complexity", in Uncertainty in Artificial Intelligence, L. Kanal, and J. Lemmer (Eds.), 217-247, North-Holland, 1986. This paper is the first study of term sets granularity and triangular norms distinguishability. In the paper it is noted that, when using term sets typical for knowledge elicitation, many t-norms collapse into a small number of similarity classes. As a result, five t-norms are enough to cover most situations.

[10] Bosc, P. and O. Pivert, "SQLf: a relational database language for fuzzy querying," TFS, 3(1), 1995, pp. 1-17. This paper describes how to extend well-known languages and algorithms for handling queries to relational databases, when queries involve preferences described in terms of fuzzy sets.!

[11] Bouchon-Meunier, B., Rifqi, M. and S. Bothorel, "Towards general measures of comparison of objects," FSS 84 (2), 1996, pp. 143-153. This paper is an extensive study on indices of similarity between fuzzy sets, that bridges the gap between the fuzzy set literature and the mathematical psychology literature on similarity. !

[12] Buckles, B. P. and Petry, F. E., "A fuzzy representation of data for relational databases," FSS, 7(3), 1982, 213-226. One of the earliest and most influential papers on the use of fuzzy sets and models in the context of relational databases.

[13] De Luca, A. and S. Termini, "A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory", Inf. and Control, 20(4), [301][302][303][304][305][306][307][308][309][310][311][312]1972.

One of the earliest papers to consider the concept of entropy, defined in the context of fuzzy information.

[14] Dubois, D. and H. Prade, "Operations on fuzzy numbers," Int. J. Systems Science, 9(6), 1978, pp. 613-626. An influential paper in the arithmetic of fuzzy intervals, studying the four operations, as well as the maximum and the minimum. While the basic definitions were proposed by Zadeh and had been studied by some scholars in Japan and the United States, this paper proposed a parametric representation (LR-fuzzy numbers) of fuzzy intervals and showed how to compute practical results with it. It also proved a general shape-invariance result for the addition of fuzzy numbers.

[15] Dubois, D. and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, 1980. This is the first extensive monograph describing the state of the art of the field after 15 years of fuzzy set research. It covers all aspects of the theory and its applications and contains a very extensive list of references on fuzzy sets at the time. Moreover it provides for the first time extensive accounts on topics such as the arithmetic of fuzzy intervals and fuzzy analysis, possiblity theory and its relation to the theory of evidence, fuzzy linear programming and fuzzy logic control.

[ , 3, 1994, pp. 439-513. This paper defines an extension of classical logic to the case where propositions have various levels of certainty. It is based on the old principle that the validity of a reasoning chain is the validity of its weakest link. The model-theoretic semantics is in terms of fuzzy sets of models. This logic is inconsistency-tolerant. This work demonstrates a close connection between fuzzy sets and the literature on nonmonotonic reasoning and belief revision in artificial intelligence.

[19] Dunn, J. C., "A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, Jo. Cyber., 3(3), 1974, 32-57.

The first paper to derive a fuzzy version of the classical batch hard cmeans (aka k-means) clustering model and alternating optimization algorithm, which was subsequently generalized as described in [4].

[20] Filev, D. "Fuzzy modeling of complex systems", IJAR, 5(3), 1991, 281-290. This paper introduces state space and polytopic Takagi-Sugeno type models as alternative to the conventional dynamic state space models of nonlinear systems.!

[21] Fodor, J. C. and M. R. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Springer Theory and Decision Library, 14, 1994. This monograph achieved a breakthrough in the study of fuzzy relations meant to model the idea of preference, a topic pioneered by Sergei Orlowski in the 1970's. It relied on the state of the art in fuzzy aggregation operations, especially t-norms and co-norms, and applied it to the decomposition of a preference relation into its strict part, its equivalence part and its incomparability part. It shows the difficulty of carrying preference modeling techniques over to the valued case in the max-min setting, indicating the need for algebraic structures like MValgebras.

[22] Goguen, J. A., "L-fuzzy sets," JMAA, 18, 1967, 145-174. This paper laid bare the mathematical nature of fuzzy sets as mappings from a set to a complete lattice. It can be considered as the seminal work that motivated much of the mathematical literature on fuzzy sets.

[23] Goguen, J. A., "The logic of inexact concepts," This paper describes a very useful algorthm which is a staple in the Matlab TM Fuzzy Toolbox. The author presents a Takagi-Sugeno (TS) fuzzy system in network form and combines it with a backpropagation like learning algorithm to provide automated tuning of membership functions and polynomial coefficients. This algorithm enabled a large number of useful applications during the 1990s.

[31] Kacprzyk, J. Group decision making with a fuzzy majority, FSS, 18, 1986, 105-118.

Introduction of a fuzzy majority -equated with fuzzy linguistic quantifiers and dealt with in terms of a calculus of linguistically quantified propositions. One of the primary references for fuzzy models of group decisions, social choice, and voting schemes.

[32] Kacprzyk, J. This monograph develops an approach to fuzzy random variables originally proposed by Huibert Kwakernaak in the late 1970's. A fuzzy random variable is viewed as an ill-known random variable in contrast with the Madan Puri -Dan Ralescu approach. !

[45] Lee, S. C. and E. T. Lee, "Fuzzy neural networks," Math. Biosciences, 23, 1975, 151-177. This was the first paper to define the idea of a fuzzy neuron as a generalization of the McCulloch-Pitts neuron. Although cast in the more formal language of automata theory, it is the first paper about a fuzzy neural network.!

[46] Lin, C. T. and George Lee, C. S. , "Neural-network-based fuzzy logic control and decision system," TC, 40( 12), 1991, 1320-1336. This paper introduces an innovative five-layer neural architecture for realizing a fuzzy rule based system for control and other decision making applications. It uses a hybrid learning scheme involving an unsupervised phase for defining the membership functions and a supervised phase for refining neuro-fuzzy system is proposed.

[47] Mamdani, E. H. and Assilian, S. "An experiment in linguistic synthesis with a fuzzy logic controller," IJMMS, 7, 1975, 1-13. The starting point of fuzzy control whose continuation was a turning point for the acceptance of fuzzy logic in engineering.!

[48] Marinos, P. N., "Fuzzy logic and its applications to switching systems," TC, 18(4), 1969, 343-348. This is the first paper that presents a technique for analysis and synthesis of fuzzy logic functions with implementation in terms of logic gates. This paper led to the implementation of real fuzzy information processing hardware systems such as high-speed fuzzy logic controllers.!

[49] Mendel, J. M., Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, 2001. This textbook offers comprehensive coverage of both type-1 and type-2 fuzzy sets and rule-based systems for singleton and non-singleton fuzzifications.

[50] Mendel, J. M. and R. John, "Type-2 fuzzy sets made simple," TFS, 10, 2002, 117-127. This paper provides a representation theorem that shows a new way to represent a type-2 fuzzy set in terms of simpler embedded type-2 fuzzy sets.!

[51] Mizumoto, M., "Fuzzy controls by product-sum-gravity method dealing with fuzzy rules of emphatic and suppressive types," Int. Jo. of Uncertainty, Fuzziness and Knowledge-Based Systems, 2(3), 1994, 305-319. This paper shows that emphatic or suppressive effects on fuzzy inference results are observed under the product-sum-gravity method by using fuzzy control rules whose consequent part is characterized by a membership function whose grades are greater than 1, or a negativevalued membership function. The use of negative-valued membership functions is beneficial to the construction of fuzzy control rules.

[52] Mizumoto, M. and Tanaka, K., "Some properties of fuzzy sets of type 2," Inf. and Control, 31(4), 1976, 312-340. This paper investigates the algebraic structure of Type 2 fuzzy sets under set operations defined by means of the extension principle on fuzzy numbers on the unit intervals serving as fuzzy membership grades. !

[53] Murofushi. T and Sugeno, M. "An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure", FSS, 29, 1989, 201-227. This paper first showed with concrete examples that (1) a non-additive measure (capacity in the sense of Choquet or fuzzy measure in the sense of Sugeno) represents interactions among subsets and (2) the Choquet integral is a reasonable integral with respect to such a nonadditive measure.

[54] Negoita, C. V. and Ralescu, D. A., Application of Fuzzy Sets to Systems Analysis, Wiley, 1975. This is the first book written on the basics of fuzzy sets, fuzzy theories (categories, topologies, etc.) and fuzzy logic, and also their possible applications to systems, automata, clustering, etc.

[55] Nguyen H. T. "A note on the extension principle for fuzzy sets," JMAA, 64, 1978, 369-380. This pioneering paper describes the connection between the extension principle and the calculation of functions with set-valued arguments using alpha-cuts. It shows that fuzzy number calculations commute with cuts.!

[56] Pal, N. R. and J. C. Bezdek, "Measuring fuzzy uncertainty." TFS, 2(2), 1994, 107-118. This paper introduces two new classes, additive and multiplicative classes, of measures of fuzziness, which satisfy the five axioms of such measures. This paper also introduces the concept of weighted fuzziness to incorporate subjectivity in measures of fuzziness.

[57] Pedrycz, W., "Algorithms of fuzzy clustering with partial supervision," Pattern Recognition Letters, 3, 1985, 13 -20. This paper introduced the concept of partial supervision for fuzzy clustering and proposed algorithms that used it to do clustering in presence of partially labeled data.

[58] Pedrycz, W., Fuzzy Control and Fuzzy Systems, John Wiley. 1991 This research monograph is one of the first comprehensive and innovative publications that focuses on fuzzy control and fuzzy systems within a framework of fuzzy relational equations.

[59] Puri, M. L. and D. A. Ralescu, "Fuzzy random variables," JMAA, 114

(2), 1986, pp. 409-422.

This seminal paper proposed a mathematical extension of the theory of random sets to fuzzy random sets, 10 years after pioneering but largely ignored works by Robert Féron. In this approach, a fuzzy random variable is viewed as a mapping from a probability space to a space of membership functions, equipped with a suitable metric structure. Since then many scholars have followed this line to handle random linguistic variables.

[60] Rodriguez, R. O., Esteva, F., Garcia, P. and Godo, L., "On implicative closure operators in approximate reasoning," IJAR, 33, 2003, 159-184. This paper clarifies the notions of graded implication and, through the imposition of reasonable constraints, characterization of the nature of implication operators.

[61] Ruspini, E. H. ,"A new approach to clustering," Inform. Control, 15(1), 1969, 22-32. This is the first paper to define the notion of a fuzzy c-partition of data. As such, it is the root paper for the entire field of fuzzy clustering, which is now a very large part of the pattern recognition landscape.!

[62] Ruspini, E. H., "On the semantics of fuzzy logic," IJAR, 5, 1991, 45-88. This paper presents a formal characterization of the major concepts and constructs of fuzzy logic in terms of notions of distance, closeness, and similarity between pairs of possible worlds. The similarity logic developed in the paper allows a form of logical "'extrapolation'" between possible worlds. It is shown to have connections with possibility theory, in the setting of metric spaces.

[63] Ruspini, E. H., P. Bonissone, and W. Pedrycz, Handbook of Fuzzy Computing, Institute of Physics, 1998.

A handbook on fuzzy sets, systems, and applications that was state of the art in 1998. Co-edited by three fuzzy pioneers, it offers a coherent presentation and notation across multiple entries, which were written by a large number of other fuzzy pioneers.

[64] Saffiotti, A., Konolige, K., and Ruspini, E. H., "A multivalued logic approach to integrating planning and control," Artificial Intelligence, 76, 1981, 481-522. This paper presents the first significant application of fuzzy logic methods to the planning and control of autonomous robots. This approach led to the SAPPHIRA architecture, which, until recently, was employed in many commercial autonomous mobile robots. The multilevel hierarchical, supervisor-controller, architecture introduced in this paper has been widely applied to other control systems.

[65] Sanchez, E. "Resolution of composite fuzzy relation equations," Inf. and Control, 30, 1976, 38-48. This is the first, highly original and influential publication in the area of fuzzy relational equations. It is the root paper for a large ongoing research effort in relational theory.

[66] Seki, H. and Mizumoto, M., "On the equivalence conditions of fuzzy inference methods -part 1: Basic concept and definition," TFS, 19(6), 2011, 1097-1106. This paper addresses equivalence conditions of a number of fuzzy inference methods such as the product-sum-gravity method, simplified fuzzy inference method, fuzzy singleton-type inference method, SIRMs inference method, and SIC inference method.!

[67] Sugeno, M. Theory of Fuzzy Integrals and Its Applications, Ph.D. Thesis, Tokyo Institute of Technology, 1974.

Starting point of the fertile subject of fuzzy measures and the so-called Sugeno's integral. This paper introduced a family of measures (the lambda ones) that are either additive, or subadditive, or superadditive.!

[68] Sugeno, M., and Yasukawa, T., "A fuzzy-logic-based approach to qualitative modeling. " TFS, 1(1), 1993, 7-31. This paper proposes a two-step process, fuzzy modelling and its linguistic approximation, to generate qualitative models of systems based on input-output numerical data. Although the primary emphasis of this is on qualitative modelling, it also introduces another very important concept, the use of clustering to find fuzzy rules from numerical data, which drastically reduces the complexity of fuzzy rule generation.

[69] Tahani, H. and J. Keller, "Information fusion in computer vision using the fuzzy integral", TSMC, 20(3), 1990, 733-741. This was the first journal paper (preceded by 2 conference papers) that framed the pattern recognition problem in terms of fuzzy integral fusion of information.

[70] Takagi, T. and M. Sugeno, "Fuzzy identification of systems and its applications to modeling and control", TSMC, 15, 1985, 116-132. This paper established a link between conventional and fuzzy systems models and paved the way for the use of machine learning and control theory in fuzzy systems.

[71] Tanaka, H., Uejima, S. and Asai, K., "Linear regression analysis with fuzzy model," TSMC, 12(6), 1982, 903-907. This paper was the first to propose a study of fuzzy linear regression (FLR), by adding fuzziness to regression analysis. It considered parameter estimation of FLR models under two factors: (i) the degree of fit; and (ii) the vagueness of the model. This paper inspired many subsequent works in fuzzy linear regression models.!

[72] Tanaka, K and Wang, H., Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, John Wiley & Sons, 2004. This book offers a systematic approach to the analysis and synthesis of stable fuzzy control systems based on Takagi-Sugeno type models.

[73] Trillas, E. and Riera, T. "Entropies in finite fuzzy sets", Inf. Sciences, 15(2), 1978, 159-168. This paper first studied fuzzy entropies which are different from a Shannon-type (employed by De Luca and Termini) and considered relations between entropies and fuzzy integrals.

[ FSS, 17, 1985, 313-328. Valverde's paper on the structure of fuzzy similarities brought clarity, through a principled approach, to the structure of fuzzy preorders and fuzzy similarity equations. Furthermore, this paper clarified the relationship between the notions of fuzzy preference and fuzzy similarity.

[76] Wang, L-X., and Mendel, J. M., "Generating fuzzy rules by learning from examples," TSMC, 22(6), 1992, 1414-1427. This paper proposes a useful scheme for generating a fuzzy rule based system from numerical data for function-approximation type systems. It also proves that such a fuzzy rule based system has the universal approximation capability, which can approximate any nonlinear continuous function on a compact set to an arbitrary accuracy.

[77] Wang, L.-X. , "Fuzzy systems are universal approximators," Proc. FUZZ-IEEE, 1992. This paper provided the first rigorous proof that a Mamdani fuzzy logic system is a universal approximator. cf. E. P Klement, "Are fuzzy systems universal approximators? IJGS, 28(2/3), 1999, 259-282.

[78] Wang, X., De Baets, B. and E. Kerre. "A comparative study of similarity measures," FSS, 73 (2), 1995, 259-268.

A systematic study of the notion of similarity between fuzzy sets and the properties of such similarity indices. This is used as a basis for defining a notion of approximate equality between fuzzy sets.!

[79] Yager, R. R., "On a general class of fuzzy connectives," FSS, 4, 1980, 235-242.

One of the earliest papers to provide a generalization of the union and intersection operators used in fuzzy sets.

[80] Yager, R. R., "A procedure for ordering fuzzy subsets of the unit interval,", Inf. Sci., 24, 1981, 143-161. This early work deals with the issue of comparing fuzzy sets of the unit interval, with an approach compatible with fuzzy arithmetics.!

[81] Yager, R.R., "A new approach to the summarization of data", Inf. Sci., 28, 1982, 69-86.

A breakthrough paper that introduces the concept of a linguistic data summary, which is equated to a linguistically quantified proposition with a fuzzy linguistic quantifier. As opposed to linguistic summarization of data (previously known for many years), this scheme accounts for imprecision in data and enables us to grasp the very essence of data in a human consistent way.

[82] Yager, R.R., "On Ordered Weighted Averaging aggregation operators," TSMC, 18, 1988, 183-190. Perhaps the central paper that fueled many subsequent studies of aggregation functions in fuzzy logic. OWA operators were, and are, widely used in various applications.!

[83] Yager, R. R., "Quantifier guided aggregation using OWA operators, IJIS, 11, 1996, 49-73. In this paper Yager provides an approach for going from a linguistic specification of an aggregation imperative to its manifestation in terms of an OWA operator. It gives us an example of the concept of computing with words applied to aggregation.!

[84] Yager, R. R. and Filev, D. P., "Essentials of Fuzzy Modeling and Control", John Wiley, 1994.

A textbook containing a systematic approach to fuzzy models and control, methods for developing and learning fuzzy models from data, and their applications.

[85] Yamakawa, T., "High-speed fuzzy controller hardware system : The mega-FIPS machine," Inf. Sci., 45, 1988, 113-128. This article describes a high-speed fuzzy controller hardware system which facilitates approximate reasoning at 1,000,000 FIPS (fuzzy inferences per second). This was the first step in an approach to a fuzzy computer.!

[86] Yamakawa, T., "A fuzzy inference engine in nonlinear analog mode and its application to a fuzzy logic control," TNN, 4(3), 1993, 496-522. This is a tutorial on the utility of fuzzy systems that provides a broad scope overview of analog mode hardware.!

[87] Yamakawa, T., "Silicon implementation of a fuzzy neuron," TFS, 4(4), 1996, 488-501. This paper describes a fuzzy neuron chip which modifies an ordinary neuron model by fuzzy logic and facilitates high speed recognition (less than 0.5 microseconds) of handwritten characters.!

[88] Zadeh, L. A., "Similarity relations and fuzzy orderings," Inf. Sci., 1971, 177-200. The first paper that showed how to decompose a fuzzy similarity relation to discover cluster substructure in a partition tree on relational (usually dissimilarity) data. Also introduced the idea of transitive closures for fuzzy similarity relations.!

[89] Zadeh, L. A., "Fuzzy logic and approximate reasoning," Synthese 30, 1975, 407-428. This paper introduces two basic formalisms: fuzzy logic and approximate reasoning. Basically, fuzzy logic is a system of reasoning and computation in which the objects of reasoning and computation are classes with unsharp (fuzzy) boundaries. Fuzzy logic is much more than a logical system. !

[90] Zadeh, L. A. "Outline of a new approach to the analysis of complex systems and decision processes", TSMC, 3(1), 1973, 28-44. This paper introduces to the concepts of fuzzy systems, algorithms, models, and optimization from the perspective of conventional systems theory. It is the genesis of the fuzzy logic control literature.

[91] Zadeh, L. A. "The concept of a linguistic variable and its application to approximate aeasoning," Parts 1-3, Inf. Sci., p1: 8, 1975, 199-249;1975, p2: 8, 301-357;1976;p3: 9, 43- Boston, 1984, 372-402. This is the best paper that clearly and completely describes one of Lotfi Zadeh's greatest ideas -the one of precisiating the meaning of utterances in natural language by translating them into the meaning representation language PRUF. The language is based on a fuzzy-set interpretation of the theory of graded possibilities, whose expressive power is comparable to that of natural languages.

[95] Zadeh, L. A., "Fuzzy logic = computing with words," TFS, 2, 1996, 103-111. Prof. Zadeh led the fuzzy community with innovative ideas that possessed deep insights. There were two phases: 1) propose fuzzy sets and their mathematical foundations and 2) propose of the idea "computing with words," which had significant value in expanding fuzzy logic from a scientific tool to the liberal arts. This was the first paper in that direction.

[96] Zadeh, L. A., "Generalized theory of uncertainty (GTU) -Principal concepts and ideas," Comp. Stat. and Data Analysis, 51, 2006, 15-46. A basic premise in this paper is that there are many different kinds of uncertainty. The three principal kinds are possibilistic uncertainty, probabilistic uncertainty and bimodal uncertainty. GTU addresses the three principal kinds and others. GTU is a challenge to the Bayesian doctrine which posits that any kind of uncertainty can and should be dealt with through the use of probability theory. GTU has a unique capability--the capability to compute with probabilities, possibilities, events and relations which are described in natural language.

[97] Zadeh, L. A., "Towards a restriction-centered theory of truth and meaning (RCT), Information Sciences, 248, 2013, 1-14. This paper is a radical departure from traditional approaches to representation of meaning and definition of truth. The meaning of a proposition is expressed as a restriction. A proposition is associated with two truth values: internal truth value and external truth value.

[98] Zimmermann, H.-J., "Fuzzy programming and linear programming with several objective functions". FSS, 1, 45-55, 1978. This paper paved the way for many developments and applications in Operations Research. For example, classical linear programming requires crisp constraints that are often unrealistic. This paper showed how to soften the constraints, obtaining a more realistic model.!!

[99] Zimmermann, H.-J.. and Zysno, P., "Latent Connectives in Human Decision Making," FSS, 4, 1980, 37-51.

A paper published before t-norms and t-conorms were broadly used in fuzzy logic. The authors showed that fuzzy connectives cannot belong to universal classes, but should be contextually chosen. It also suggested the use of aggregation functions.

[100] Zimmermann, H.-J., "Fuzzy Sets, Decision Making, and Expert Systems", Kluwer, 1987. This book was one of the first texts that discusses how modeling with mathematics and empirical findings can be used to turn expert systems based on classical dichotomous logics into fuzzy expert systems.

IV. DISCUSSION AND SUPPLEMENTAL READING

The 101 list was compiled using a very constrained method of sampling (i.e., only IEEE CIS pioneers were consulted). Consequently we feel justified in expanding the list a bit by adding some remarks and citations that might otherwise go unrecognized.

(i) Many important papers have been written in fields that are not directly germane to engineering applications. As you might expect, since our contributors are IEEE pioneers, this 101 list is heavily weighted towards the theory and applications in pattern recognition and control. However, there are very important papers in areas that might be called "pure mathematics, logic, philosophy, etc." such as topology, category theory, etc., that fall outside the natural interests of most members of a professional engineering society. Here are a few early citations, in chronological order, which were overlooked by our IEEE pioneers:

TABLE I .

 I THE IEEE CIS FUZZY SYSTEMS PIONEERS: D~DECEASED

TABLE II

 II

		.	ABBREVIATIONS USED IN THE 101 LIST
	FSS	Fuzzy Sets and Systems
	IJAR	International Jo. of Approximate Reasoning
	IJGS	International Jo. of General Systems
	IJIS	International Jo. of Intelligent Systems
	IJMMS International Jo. of Man-Machine Studies
	JMAA	Jo. Math Analysis and Applications
	TC	IEEE Transactions on Computers
	TCS	IEEE Transactions on Circuits and Systems
	TEC	IEEE Transactions on Evolutionary Computation
	TFS	IEEE Transactions on Fuzzy Systems
	TNN	IEEE Transactions on Neural Networks
	TPAMI IEEE Transactions on Pattern Analysis and
		Machine Intelligence
	TSMC	IEEE Transactions on Systems, Man and
		Cybernetics
	III. THE 101 CITATIONS IN ALPHABETICAL ORDER

Our list of 101 begins with the root paper: Zadeh, L. A., "Fuzzy sets," Information and Control, 8(3), 1965, 338-353.

  Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998. This book is the culmination of seminal contributions by Peter Hajek to fuzzy logic in the narrow sense. The book is the first comprehensive axiomatic presentation of important fuzzy logics, each based on a distinct t-norm and its residuum, with the rigorous proofs that these fuzzy logics are both sound and complete. Contrary to other contributors to fuzzy logic in the narrow sense, Hajek has always considered advances in fuzzy logic in the broad sense as an important source of inspiration for research in fuzzy logic in the narrow sense. !

	[28] Herrera, F., Herrera-Viedma, E. and J. L. Verdegay. "A model of
	consensus in group decision making under linguistic assessments,
	FSS, 78 (1), 1996, 73-87.
	This paper proposes the use of linguistic preferences to represent
	individuals' opinions, and a definition of fuzzy majority of
	consensus, represented by means of a linguistic quantifier. Several
	linguistic consensus degrees and linguistic distances are defined to
	indicate how far a group of individuals is from the maximum
	consensus, and how far each individual is from current consensus
	labels over the preferences.
	[29] Inuiguchi, M. and Ramık, J. "Possibilistic linear programming: a
	brief review of fuzzy mathematical programming and a comparison
	with stochastic programming in portfolio selection problems, FSS,
	111 (1), 2000, 3-28.
	This survey reviews the application of possibility theory to fuzzy
	optimization, augmenting flexible constraints in fuzzy linear
	programming with uncertainty about coefficients, represented by
	fuzzy numbers. Then degrees of possibility and necessity of satisfying
	constraints can be used in the spirit of chance-constrained
	programming. !
	[27] Hall, L. O.; Ozyurt, I. B. and J. C. Bezdek, "Clustering with a
	genetically optimized approach," TEC, 3(2), 1999, 103-112.
	This paper introduces a new way to optimize a fuzzy objective
	function for clustering. The evolutionary approach is shown to
	provide partitions with a better optimized objective function value
	than the classical alternating optimization scheme.!

Synthese, 19, 1968/69, 325-373.

This paper develops a remarkably sophisticated foundation for this new logic. The results provide the framework for both the development of Zadeh's agenda of fuzzy logic in the broad sense as well for the parallel development on the agenda of fuzzy logic in the narrow sense. !

[24] Grabisch, M. and Labreuche, Ch. "Bi-capacities, Part I: definition, Möbius transformation and interaction", FSS, 151, 211-236, 2005. Bi-capacities arise as a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours, encompassing models such as Kahneman and Tversky' Cumulative Prospect Theory. The paper extends all familiar notions used for fuzzy measures in this more general framework, and introduces the interaction index for bi-capacities, generalizing the Shapley value in a cooperative game theoretic perspective.

[25] Gustafson, D. E. and Kessel, W. C. "Fuzzy clustering with a fuzzy covariance matrix," Proc. IEEE

CDC, 1979, 761-766

. This is the first fuzzy clustering model with an objective function that attempts to match cluster shapes by adapting the individual norm associated with each cluster. As alternating optimization proceeds, the norm associated with each cluster adapts to fit the local structure of the cluster via the fuzzy covariance matrix.

[26] Hajek, P.

[30] Jang, J. S. R. "ANFIS: Adaptive-network-based-fuzzy-inferencesystem," TSMC,

23, 1993, 665-685.

  Linguistic database summaries and their protoforms: towards natural language based knowledge discovery tools. Information Sciences, 173 (4), 2005, 281-304. This paper puts together an approach to linguistic summaries of databases after Yager and Zadeh's notion of protoform, in connection with the handling of queries in fuzzy databases.! This is a foundational paper that established many of the basic concepts in the field of type-2 fuzzy logic systems. Introduction to the Theory of Fuzzy Subsets, Academic Press, 1975. This book is the English translation of the first monograph ever written (in French) on fuzzy set theory. It contains elementary definitions of fuzzy sets and related topics, covering the first papers by Zadeh, with special emphasis on max-min-transitive fuzzy similarity relations. This book is tutorial and contains many examples and exercises.

	[36] Kasabov N. and Qun Song, "DENFIS: Dynamic Evolving Neural-
	Fuzzy Inference System and its application for time-series
	prediction," TFS, 10(2), 2002, 1-37.
	This paper introduces a new type of fuzzy inference system, DENFIS,
	for adaptive on-line and off-line learning, and shows how to apply it
	to dynamic time series prediction.!
	[37] Kasabov, N., Foundations of Neural Networks, Fuzzy Systems and
	Knowledge Engineering, MIT Press, 1996.
	This book provides an understandable approach to knowledge-based
	systems for problem solving by combining different methods of AI,
	fuzzy systems, and neural networks.
	[38] Kaufmann A. [42] Kosko, B. Neural Networks and Fuzzy Systems: A Dynamical Systems
	Approach to Machine Intelligence. Prentice Hall, 1992.
	This book is of historical significance due to its important role in the
	genesis of neurofuzzy systems.!
		[34] Kandel, A. Fuzzy Techniques in Pattern Recognition, John Wiley &
	[43] Krishnapuram, R. and J. M. Keller, "A possibilistic approach to	Sons, New York, 1982.
	clustering", TFS, 1(2), 1993, 98-110.	One of the first comprehensive and pioneering treatises of the subject
	This paper generalized (hard and fuzzy) c-means clustering by	of pattern recognition in the framework of fuzzy sets. The
	eliminating the constraint that the sum of cluster memberships for any	fundamentals of fuzzy sets are discussed in the framework of
	object must equal 1. It also introduced the idea of a possibilistic	constructive ways to use this technology to formulate and solve
	partition as one consisting of typicalities. !	certain pattern recognition problems. !

, Multistage Fuzzy Control: A Model-Based Approach to Control and Decision-Making, Wiley & Sons, 1997.

The first comprehensive coverage of multistage optimal fuzzy control, viz., fuzzy dynamic programming, for deterministic, stochastic and fuzzy systems under control. Real world applications include socioeconomic regional development, and power systems planning.!

[33] Kacprzyk, J., Zadrozny S.,

[35] Karnik, N., J. M. Mendel and Q. Liang, "Type-2 fuzzy logic systems," TFS, 7, 1999, 643-658.

[39] Keller, J., and Hunt, D., "Incorporating fuzzy membership functions into the perceptron algorithm," TPAMI, 7(6), 1985, 693-699. This paper develops a fuzzy perceptron model and algorithm that (unlike the classical crisp perceptron) terminates on non-linearly separable data sets. The article includes a proof of convergence for iterative optimization of the fuzzy perceptron objective function. ! [40] Klement, E. P., Mesiar, R. and E. Pap "Triangular Norms", Springer, 2000 This book gathers many mathematical results concerning fuzzy set connectives in an organized ways, with a stress on solving functional equations.! [41] Klir, G. J. and B. Yuan, "Fuzzy Sets and Fuzzy Logic: Theory and Applications", Prentice-Hall, 1995. A very complete and comprehensive textbook that covers the basic elements of fuzzy models from a mathematical point of view. ! [44] Kruse, R. and K. D. Meyer, Statistics with Vague Data, Springer, 1987.

  This paper introduces the concept of granularity and relates it to information. Fuzzy granularity is a concept which is unique to fuzzy logic. The linguistic variable is a granular variable. A concept which is introduced in this paper is that of a fuzzy-set-value random variable, with fuzzy probabilities. This concept provides a basis for a generalization of the Dempster-Shafer Theory of Evidence.! [94] Zadeh, L. A., "Precisiation of meaning via translation into PRUF," In Cognitive Constraints and Communication, Vaina, L. and Hintica, J. (eds.), D. Reidel,

	80.
	This three part publication develops the definition, theory and
	applications of linguistic variables for use in approximate reasoning.
	It is a superb treatment of an integral component of all subsequent
	work in fuzzy logic, linguistic data processing, and computing with
	words.
	[92] Zadeh, L. A., "A theory of approximate reasoning," in Machine
	Intelligence, 9, Hayes, J., Michie, D., and Mikulich, L. I., Eds., ed
	New York: Halstead Press, 1979, 149-194.
	In this paper Zadeh very elegantly puts together many of his ideas on
	approximate reasoning in a wholistic framework. It provides the basis
	of much of Zadeh subsequent work on computing with words.!
	[93] Zadeh, L. A., "Fuzzy sets and information granularity," in Advances
	in Fuzzy Set Theory and Applications, eds. M. Gupta, R. Ragade and
	R. R. Yager, North Holland, 1979, 3-18.

G. Shafer. "A Mathematical Theory of

 Evidence", 1976, PrincetonUniversity Press.

(iii) There are also some works that do not appear in the 101 list because they were not necessarily foundational (at least, for the 20 contributing IEEE pioneers). We want to mention three of them here, with historical footnotes of a sort, that explain in part why we wanted to include them. W. G. Wee and K. S. Fu. "A formulation of fuzzy automata and its application as a model of learning systems," IEEE Trans. Syst. Science and Cyberns, 5(3), 1969, 215-223. K. S. Fu was one of the really important "big guys" in the early history of fuzzy sets. The importance of his interest in the field at a time when it was quite embryonic and survival was a real issue cannot be overstated. He strongly encouraged the publication of the book [15]. He was also the first president of NAFIPS, the North American Fuzzy Information Processing Society, which in turn was the first professional society whose primary focus was fuzzy sets and models. (iv) The above lists, mainly oriented towards papers by "IEEE pioneers", do not give much credit to the newer generation of fuzzy set researchers, active in the last twenty years. We could give yet another partially arbitrary list recognizing these newer papers and books, but refrain from doing it. Yet they fully belong to the posterity of Zadeh's 50year-old paper. Many of them are named on the editorial boards of the numerous fuzzy sets and soft computing journals.

We conclude with this observation. Instead of the method of collection used here, we could have polled the past presidents of IFSA (the International Fuzzy Systems Association), all the editors of FSS, or only those researchers working in business, or for a government. Each poll would produce a somewhat different list. The intersection of our 101 list with any of these lists would in all likelihood not be empty. But, for example, a list of the 100 most cited references in fuzzy sets, would certainly not coincide with our 101 list either. But ... which citation engine? -each one would undoubtedly produce a slightly different set of rankings. Carrying this argument to its logical conclusion, there can obviously be an infinite number of lists, no two of which coincide. We can only hope that this list is of some value to readers and attendees at the 2015 FUZZ-IEEE.

That's all, Folks!


Nesrine Ouannes
email: nesrineouannes@gmail.com

Noureddine Djedi
email: n.djedi@univ-biskra.dz

Hervé Luga
email: herve.luga@univ-tlse1.fr

Yves Duthen
email: yves.duthen@univ-tlse1.fr

Modeling bacterial chemotaxis inside a cell

Keywords: Virtual bacteria, Chemotaxis network simulation, Evolved behaviors

This paper describes a bacterial system that reproduces a population of bacteria that behave by simulating the internal reactions of each bacterial cell. The chemotaxis network of a cell is modulated by a hybrid approach that uses an algebraic model for the receptor clusters activity and an ordinary differential equation for the adaptation dynamics. The experiments are defined in order to simulate bacterial growth in an environment where nutrients are regularly added to it. The results show analysis of the motion obtained by some bacteria and their effects on the population behaviors generated by evolution. This evolution allows bacteria to have the ability to adapt themselves to better growth in the available food existed in its environment and to survive.

I. INTRODUCTION

Simulation models in artificial life have focused metabolic, cellular systems and artificial chemistries. Artificial life research has also made progress in the study of adaptive behavior through computational models of artificial organisms.

Remarkably simple chemical reactions can perform movements toward some attractants, and are therefore capable of modulating the behavior of artificial organisms. The main goal of this work is to test the present knowledge about chemical reactions of a bacterial cell, in order to explain the evolution of organisms with a very simple development process (which is bacterial chemotaxis). We will also demonstrate whether a simple bacterial chemotaxis process of a cell can explain the evolution of more complicated behaviors such as bacterial population dynamics. One of the central questions of modern systems biology is the influence of microscopic parameters of a single cell on the behavior of a cell population. In terms of bacterial chemotaxis, this issue can be formulated as the influence of signaling network parameters on the spatiotemporal dynamics of bacteria that migrate towards chemical attractants and away from repellents. This chemotaxis is one of the simplest behaviors known, and it most likely is one of the first behaviors to have existed in the history of life on earth. Bacteria such as Escherichia coli, is a good candidate organism for chemotaxis modeling, thanks to the rich experimental information collected over years of extensive research. As many other bacteria, E. coli can migrate towards high concentrations of attractants and away from repellents [START_REF] Berg | Chemotaxis in escherichia coli analysed by three-dimensional tracking[END_REF].

Chemotaxis pathway of E. Color changes in attractant or repellent concentrations are sensed by a protein complex consisting of transmembrane receptors, which transduce information about the chemical environment in the cells. The chemoreceptors form complexes inside the cells with the kinase CheA (A) and CheW (W). These receptors are known as methyl-accepting chemotaxis proteins (or MCP, which contain the PAS sensor protein-domain, evolutionarily highly conserved), and are present on the bacterium membrane surface. These proteins act as chemoreceptors and bind with chemicals in the environment. The Autophosphorylation activity of CheA is inhibited by attractant binding and enhanced by repellent binding to receptors. The phosphoryl group is transferred from CheA to the response regulator CheY. The phosphorylated form of CheY interacts with the flagellar motors to induce tumbles. The rate of CheY dephosphorylation is greatly enhanced by CheZ (Z). The binding of attractants to the receptors decreases the rate of CheY phosphorylation and tumbling is reduced. Adaptation is provided by changes in the level of methylation of the chemoreceptors: methylation increases the rate of CheY phosphorylation. A pair of enzymes, CheR (R) and CheB (B), add and remove methyl (m) groups. To adapt to an attractant, methylation of the receptors must rise to overcome the suppression of receptor activity caused by the attractant binding. CheA enhances the demethylating activity of CheB [START_REF] Stewart | Interaction of cheb with chemotaxis signal transduction components in escherichia coli: modulation of the methylesterase activity and effects on cell swimming behavior[END_REF][START_REF] Lupas | Phosphorylation of an nterminal regulatory domain activates the cheb methylesterase in bacterial chemotaxis[END_REF]. This process is mediated by a protein network, as presented in figure 1.

In the bacterial chemotaxis process, when no attractant or repellant is present, or when the concentration of attractant or repellant is uniform, a bacterium such as E. coli tends to swim in a random walk, with periods of smooth swimming (or runs) interrupted by brief tumbles that changes the swimming direction. In response to attractant gradient, this random walk becomes biased and the bacteria tumble less frequently when encountering increasing concentrations of an attractant (i.e., they swim longer runs), and tumble more frequently when the attractant concentration is decreasing [START_REF] Berg | Chemotaxis in escherichia coli analysed by three-dimensional tracking[END_REF][START_REF] Adler | Chemotaxis in bacteria[END_REF].

The motivation for studying such small organisms lies in the belief that elucidating the mechanisms controlling their behavior will help in understanding more complex biological pathways and organisms. We provide here a comprehensive overview of the range of mathematical approaches used for modeling, within a single bacterium, chemotactic processes caused by changes to external gradients in its environment.

Phosphorylation cascade in a chemotaxis network was first simulated by Bray et al [START_REF] Bray | Computer simulation of the phosphorylation cascade controlling bacterial chemotaxis[END_REF], using a system of ODEs, and [START_REF] Levin | Origins of individual swimming behavior in bacteria[END_REF], a later version of their model, added adaptation. Spiro et al [START_REF] Spiro | A model of excitation and adaptation in bacterial chemotaxis[END_REF] also incorporated attractant binding, methylation, and phosphorylation and CheYp-motor interaction into their model. A major advance in chemotaxis modeling was achieved in [START_REF] Barkal | Robustness in simple biochemical networks[END_REF]. Later, in [START_REF] Mello | Perfect and near-perfect adaptation in a model of bacterial chemotaxis[END_REF], a theoretical analysis of a full ODE system with included phosphorylation cascade are performed.

As deterministic models, Morton-Firth and Bray suggested a fully stochastic model of chemotaxis pathway [START_REF] Morton-Firth | Predicting temporal fluctuations in an intracellular signalling pathway[END_REF]. Some hybrid models of bacterial chemotaxis simulation were proposed. In [START_REF] Setayeshgar | Application of coarse integration to bacterial chemotaxis[END_REF] the authors describe the chemotactic excitation and adaptation with a simplified model of two ODEs. RapidCell [START_REF] Vladimirov | Dependence of bacterial chemotaxis on gradient shape and adaptation rate[END_REF] is a hybrid model of chemotactic Escherichia coli that combines the Monod-Wyman-Changeux signal processing by mixed chemoreceptor clusters, the adaptation dynamics described by ODEs, and a detailed model of cell tumbling. In [START_REF] Bray | The chemotactic behavior of computer-based surrogate bacteria[END_REF], Bray et al used a molecularly detailed reaction kinetic model of the chemotaxis pathway in Escherichia coli that simulates the responses of bacteria to twodimensional gradients of attractants. Other approaches used their individual model of bacterial chemotaxis to study population behaviors.

Here, we present a simulation of the bacteria chemotaxis network. The chemotactic Escherichia coli bacterium model describes the signal processing by mixed chemoreceptor clusters (MWC 'Monod-Wyman-Changeux signal processing' model), which is a rapid-equilibrium (algebraic) model, adaptation through methylation simulated by ordinary differential equations (ODEs), and the running and tumbling of a cell with a flagella motor [START_REF] Vladimirov | Dependence of bacterial chemotaxis on gradient shape and adaptation rate[END_REF].

The metabolism of this bacterium is a set of chemical reactions that occur in the cell. These chemical reactions are designed digitally to perform different functions as 'split', 'mutation' and 'death'.

II. BACTERIAL MODELLING

In this part, we aim to describe the proposed model of the bacteria Escherichia coli that combines a model of detailed chemotaxis simulation, a metabolic model, and a genetic process to simulate bacteria evolution in an artificial environment. The chemotactic response is driven by attractants, substances that the bacteria tend to move toward. Metabolism is affected by realizing some functions.

A. Bacterial Chemotaxis

The role of chemotaxis is not only to mobilize bacteria but also to allow them to detect chemical gradients with great sensitivity. The chemotaxis process consists of three stages: chemoreception, signaling, and adaptation [START_REF] Berg | Motile behavior of bacteria[END_REF].

Methyl accepting chemotaxis proteins (MCPs) are located along the cell surface. These proteins act as chemoreceptors and bind with chemicals in the environment. They are simulated in our model with a gene describing the capacity of a bacterium to better detect the nutrients sources. If a nutrient attractant is detected outside of the cell, through MCP, the level of production of protein CheA decreases because the receptors state shifts to the off state. It has been shown that the activity of the receptor cluster depends on the local ligand concentration and the methylation level according to the MWC (Monod-Wyman-Changeux signal processing) model [START_REF] Endres | Precise adaptation in bacterial chemotaxis through "assistance neighborhoods[END_REF][START_REF] Keymer | Chemosensing in escherichia coli: two regimes of two-state receptors[END_REF]. CheA binds with phosphate in the cell (denoted CheA-P). And the phosphate group is transferred from the active CheA to the response regulator CheY. The concentration of CheY-P modulates the motor and its behavior makes the cell run or tumble.

1) MWC model:

We applied the MWC model for a mixed receptor cluster [START_REF] Endres | Precise adaptation in bacterial chemotaxis through "assistance neighborhoods[END_REF][START_REF] Keymer | Chemosensing in escherichia coli: two regimes of two-state receptors[END_REF][START_REF] Mello | An allosteric model for heterogeneous receptor complexes: understanding bacterial chemotaxis responses to multiple stimuli[END_REF]; were each receptor homodimer is described by a two-state model. The inactive state of a receptor has a higher affinity to the attractant than the active state. The entire complex exists with all of its receptor homodimers either active or inactive.

The probability A that receptor cluster is active is dependent on ligand concentration and the methylation state of the receptors.

A = 1 / (1+ e F )...................(1)
Where F=F on -F off , and where F on/off is the free energy of the cluster to be on/off as a whole. Hence, the average activity per receptor in the cluster is A. The total free-energy difference in the mean-field approximation is F=n r f r (m), which is just the sum of the individual free-energy differences between the receptor 'on' and 'off' states:

            + + + = - = off r on r r off r on r r K S K S m f f m f ] [ 1 ] [ 1 log ) ( ) ( ε ) 2 ( .... .. ..... .........
Where [S] is the ligand concentration, and K r on/off is, the dissociation constant for the ligand in the 'on' and 'off' states, respectively. The methylation state of the receptor enters via the 'offset energy' ε r (m).

2) Adaptation model: Adaptation is modeled according to the mean-field approximation of the assistance-neighborhood (AN) model [START_REF] Endres | Precise adaptation in bacterial chemotaxis through "assistance neighborhoods[END_REF][START_REF] Hansen | Chemotaxis in escherichia coli: a molecular model for robust precise adaptation[END_REF]. Adaptation in chemotaxis is mediated by two enzymes, methyltransferase CheR and methylesterase CheB. It is assuming that the demethylating enzyme CheB works only on active receptors and that the methylating enzyme CheR works only on inactive receptors within the AN.

Each bound CheR adds methyl groups at a rate a(1-A), and each bound CheB removes methyl groups at a rate bA. It is assumed that both enzymes work at saturation:

dm / dt = a(1-A) CheR [ ] -bA CheB [ ] .......(3)
The average methylation level evolves in time as:

m(t + ∆t) = m(t) + kV∆t...........

.(4)

The parameter k indicates the adaptation rate relative to the wild type adaptation rate V that is the rate of receptor methylation (see equation 3) [START_REF] Vladimirov | Dependence of bacterial chemotaxis on gradient shape and adaptation rate[END_REF].

3) Kinase activity: Both ligand binding and receptor methylation affect the activity of CheA. For example, the increase of an attractant inhibits CheA activity, but subsequently methylates a specific receptor. CheA kinase activity [START_REF] Vladimirov | Dependence of bacterial chemotaxis on gradient shape and adaptation rate[END_REF] is calculated as:

where A is the probability that receptor cluster is active, CheY tot is the total CheY concentration, K A =5 and K Y =100 are the rate constants according to [START_REF] Vladimirov | Dependence of bacterial chemotaxis on gradient shape and adaptation rate[END_REF].

4) CheY phosphorylation:

The concentration of CheYp is obtained as a function of active CheA from the steady-state equation [START_REF] Kollmann | Design principles of a bacterial signalling network[END_REF].

Where, CheY tot is the total CheY concentration, CheZ is the total CheZ concentration, CheA is the active [CheA], and K y =100µM-1s-1, K Z = 30/[CheZ]s-1, Y = 0.1 are the rate constants according to [START_REF] Kollmann | Design principles of a bacterial signalling network[END_REF][START_REF] Stewart | Rapid phosphotransfer to chey from a chea protein lacking the cheybinding domain[END_REF][START_REF] Sourjik | Binding of the Escherichia coli response regulator chey to its target measured in vivo by fluorescence resonance energy transfer[END_REF].

Receptor modification increases CheA activity and decreases sensitivity to attractants.

5) The CCW motor bias:

The CCW motor bias depends on CheYp concentration in the following form [START_REF] Cluzel | An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells[END_REF][START_REF] Shimizu | A spatially extended stochastic model of the bacterial chemotaxis signaling pathway[END_REF].

where, mb 0 is the steady-state motor bias.

B. Bacterial Metabolism

The metabolism is responsible for essential cycles of growth, development and reproduction. Genes and movement of a bacterium affect the majority of these cycles. In this model, every bacterium is represented by a genome from which it extracts its basic properties describing how it moves, gains energy, and expels toxins. These properties are updated in the genome at each time step, and mutation is applied after each "split" operation. The absorption of nutrient may invoke the production of a matter, (but it is not applied here). Each bacterium must survive while maintaining its energy level, which is calculated from its metabolism. Energy is accumulated by the absorption of nutrients from the environment and is decreased by the cost of movement; this cost depends on tumble frequency. This metabolic model encourages bacteria to stabilize their energy consumption in order to reach splitting threshold. Upon reaching its splitting threshold, the bacteria split into two daughter cells. The amount of energy in the bacterium is exposed to various changes during its lifespan; the following table lists the various modifications applied to the metabolism of a bacterium. First, the energy of each bacterium at birth is E 0 . The energy level of each bacterium is then increased by E, which is the rate of energy gained from the absorbed nutrient, and decreased by -T, -F, -S, or -P; which are, the toxin consumption rate, and the costs of movement, and splitting, respectively.

C. Genetic representation

In the bacterial chemotaxis, there is a processing system of moderate complexity within the cell, triggered by its inputs and producing an output response. In E.Coli bacterium, this response corresponds to a change in the flagella rotation. The bacterial chemotaxis shows properties of receptor function, adaptation, memory and motor bias.

To control these properties in order to simulate bacterial population behaviors, we use a genome that encodes the activities of each level in the chemotaxis network. By updating some of its parameters, we can control the response of a bacterium and evolve its behaviors in a virtual environment.

D. Mutation and split

When biological life-forms (such as bacteria) clone themselves asexually, the clones are not exact copies of their parents [START_REF] Adami | Evolution of biological complexity[END_REF]. The biological 'copy operation' does not work perfectly. Biology would not be improved by a perfect copy operation. Imperfect copying in biology causes the mutations and novelty that allow evolution to happen. In our work, an individual of the evolved population of bacteria will be divided; once a bacterium manages to accumulate enough energy to reach the division state, it divides immediately into two identical daughter cells, except that, the new bacterium copy will be mutated, in order to enable bacteria to evolve. This ensures that the parent's genetic material is preserved, and that simultaneously, new genetic material is introduced in the population. A small probability pm of mutation is proposed to be applied to the genome, by adding noise to a selected gene. It must be emphasized at this point that, after the division process, the amount of energy of the parent's cell will be distributed equally between the two copies. This will guarantee that the parent cell continues to exist, and it can create many different offsprings during its lifetime and does not 'die' after division. In general, the survival probability of an individual is directly related to its relative effectiveness in the population.

III. EXPERIMENTAL SETUP

The experiments in this paper address the simulation of a bacterial system that reproduces a population of bacteria that behave from simulating the internal reaction of each bacterium cell. The model describes bacterial properties including MCP capacities, metabolism, cell division, and death. The environment of simulation is a two-dimensional space subdivided into discrete grid squares in which the bacteria exist as individual entities. This environment additionally to bacteria, it contains resources that are an important element of the simulation, as they provide the energy that is required to sustain life.

CheY = CheY tot K Y A / (K Y CheA + K Z CheZ + g y ).
..( 6)

CheA = CheA tot AK A / (AK A + K Y CheY tot ).....( 5
)
The artificial bacteria are represented by their genomes as described above. The genetic materials of each bacterium consist of a set of reactions that transfers the inputs (i.e. detected molecules) to its outputs (i.e. tumble frequency). This reactions modeled by the ODEs (see section II) determines the behavior of the bacterium when subjected to different types of stimuli. The objective of the genetic algorithm is to modify the set of the concentration inside a cell in such a way that a desired behavior is obtained. The choice of the interval of MCP capacity is important in the overall behavior of the system. If it is chosen too small, the bacteria may not be able to detect sufficient energy to function, and may quickly become inactive. On the other hand, if it is chosen too large, the resources may be depleted very rapidly before the system has time to explore the environment and evolve new and interesting behavioral patterns (or capacities). The simulated environment is defined by a 200 by 150 grid, consisting of 30,000 sites. The initial population size is 10 bacteria, the initial nutrient count is 10 and their concentrations are 1000. The simulation ends when the population biomass reaches zero. The most important parameters are listed in Table 1. The parameters presented at the top of this table are fixed over time while the others are variables.

IV. EXPERIMENTAL RESULTS

In this section we present a collection of simulation results and some tests that demonstrate how the bacterial system that simulates a bacterium cell, can be used in the context of evolution to obtain the desired behavior, that is one that maximizes the lifespan and reproductive success of a bacterium. We have conducted our experiments with the same initial conditions that were carefully chosen (after many of trials). The runs are obtained with a set of randomized genotypes.

A typical simulation run of the bacterial system is illustrated in the figure 3, which shows six snapshots of the environment in different stages of the simulation. The followed graph presented in figure 4 shows paths realized by the initial 10 bacteria, they are very sensitive to initial conditions, especially in the case where the resources are present in the environment knowing that the genomes are randomly initialized. The simulation has to been repeated many times with different conditions, until the population contained bacteria that are sufficiently fit, as they evolve faster and utilize the resources more efficiently. The figure 4 shows the path of (x,y) coordinates of the 10 initial bacteria bacteria borrowed from the simulation in the first 300 generations, when a bacterium follow nutrients existed in the environment.

Figure 5 presents the 'Growth rate' runs, where all experiments (i.e. runs) show a fast population increase in the twenty first simulation cycles (or generations). This increase lead to a population reproduction (or split), and then the population stays relatively constant for about 200 generations. From this level to the generation 300, the population decreases rapidly. This is a consequence of two facts. First a high number of bacteria die due to the depletion of food resources. Secondly, the speed of decrease is due to the bad MCP and toxin avoidance capacities. From generation 300, and every 300 cycles, the growth rate is often increased according to the capacities defined in the genome of each bacterium, which are also advanced. The number of species varies great during an experiment, which means that bacteria frequently split and die over time. It is important simply not relate this to food in the environment and to their own biomass, it is rather related to MCP and toxin avoidance capacities that are evolved after each split (i.e. over time).

In the simulation runs, where the sources of nutrient are regularly added each 300 generations, all the bacteria change oscillation degrees and move toward new detected sources with MCP and toxin avoidance capacities corresponding to each bacterium. This costs them energy, as explained above in the metabolism model.

In Figure 6 we observe that the metabolizable resources are consumed, while the population's collective energy decreases in the beginning of the run (as new cells are created), at the division process's maximum speed, division process the biomass is exponentially decreased. Knowing that all sources are depleted. Within thirty generations, while many bacteria die because they did not have enough energy for movement, but fortunately not all the population, as, all simulation will stop in this case (but this has been tested before choosing the environment parameters).

In iteration 300, when new nutrients resources are added to the environment, the bacteria consume nutrients, split, and when no more nutrient are present in the environment, their biomass decreases again but avidly than before. This means that the bacteria obtained after thousand of time steps are more stabilized and more effective in their use of energy. This effectiveness is due to the evolved capacities of detection (MCP capacities) of nutrients.

The figure 7 shows the path of (x,y) coordinates of some bacteria borrowed from the simulation, when a bacterium applies long runs and short tumbles in the presence of nutrient sources (as response to the nutrient). A tumble presents a reorientation of the bacterium, which is seen in the figure as the angles formed between two runs. The 'random walk' is a movement applied by bacteria that is presented with short runs and many tumbles, as observed in the graph when no nutrient sources are present from the twentieth generations. Some bacteria don't move because, they are not fit enough to exploit the available resources (they are not present in the graph).

V. DISCUSSION

The results show that a single simulation model of singlecelled creatures and biological mechanisms and simple chemical reactions allow us to model more complicated behaviors of a population of these bacteria. Therefore, the results have managed to answer the question posed at the beginning of this paper. Results show that the growth rate and biomass of the whole population in all runs tend to evolve over time; some are increased and others are stabilized.

We summarize that the growth rate continues to increase for several hundred epochs, until the resources are eventually present in the environment, and the population's collective lifespan is ameliorated because the evolved bacteria consume less energy with their optimal capacities and gather more sources. The evolution leads to a population composed of individuals well adapted to their environment. When environmental difficulties appear (food consumed, toxins present), results show a decrease in population number. The behavior of the system is thus to favor emergence of best capacities to detect food and avoid toxins, therefore to avoid death and to better reproduce and to survive longer. Concerning the chemoaxis network, when bacteria are moving, consuming and splitting? The different genes of a bacterium vary in intervals as presented in table 1. In the figure 8 we present the two states of a bacterium; an inactive state when bacteria are consuming sources (from generation 0 to 50) and an active state when bacteria are applying a 'random walk'.

VI. CONCLUSUION AND FUTURE WORK

Our model has been designed to simulate growth and behavior of bacterial system; it controls a group of bacteria cells at each time step.

To analyze the obtained behaviors, we present data that characterizes bacteria positions in space, biomass, and, state in the cellular reproduction cycle. These results demonstrate that bacteria are still able to evolve through mutation and adapt rapidly according to their changing environment. It was shown that a simple chemotaxis network of simulated bacteria could result in a set of highly fit bacteria with strong MCP capacities that can be used to evolve more interesting behaviors in bacteria colonies. In future work, we aim to improve the effect of the chemotaxis network to obtain more powerful bacteria that can emerge as new species which behaves differently from others, via the concept of colonies, and also to test this model on different environmental conditions and various changes. Finally, we plan to include this detailed system of simulating bacterial chemotaxis in an ecosystem with creatures of different natures such as foragers [START_REF] Ouannes | Following Food Sources by Artificial Creature in a Virtual Ecosystem[END_REF].

VII. REFERENCES

Figure 1 .

 1 Figure 1. Chemotaxis pathway of E. coli.

Figure 2 .

 2 Figure 2. The bacterium's genome

Figure 3 .

 3 Figure 3. Snapshots of the bacterial system at different stages of the experiment.

Figure 4 .

 4 Figure 4. Path realized by the initial 10 bacteria in 2D space for the first 300 cycles.

Figure 5 .

 5 Figure 5. The 'Growth rate' runs, which we have replicated 30 times with quantitatively the same results, representing the optimal values of the whole of the bacteria for 5000 generations.

Figure 6 .

 6 Figure 6. The 'Biomass' of the evolved population of bacteria for 30 runs at 5000 generations.

Figure 7 .

 7 Figure 7. Path realized by some bacteria in 2D space for the first 600 cycles.

Figure 8 .

 8 Figure 8. Graph of variations in concentrations of proteins and enzymes used inside bacteria during the chemotaxis process.

TABLE I .

 I

			THE SIMULATION PARAMETERS
	Parameters Values References Parameters Values	Ref
	Kon a	12	[10]	[CheB]	0.28	[26]
	Koff a	1.7	[10]	[CheYtot]	9.7	[26]
	Kon s	106	[15]	[CheR]	0.16	[26]
	Koff s	100	[15]	a	0.0625	[12]
	na	6	[15]	b	0.0714	[12]
	ns	12	[15]	E0	25	TW*
	A	1/3	[15]	ST**	50	TW*
	mb0	0.65	[21]	Energy	10	TW*
	Pm	0.1	TW*	A	[0-0164]	[12]
	CheA	[0-1]	[12]	meth	[1.9-2.8]	[12]
	Motor Bias	[0-1]	[12]	CheYP	[0-2.6]	[12]

*This Work **Splitting Threshold