{"references": ["O. Ledion Bitincka, George E. Antoniou, \"PDA-based Boolean Function\nSimplification: A Useful Educational Tool,\" Informatika, 2004, Vol. 15,\nno. 3, pp. 329-336.", "Ledion Bitincka, George E. Antoniou, \"Pocket-PC Boolean Function\nSimplification,\" Journal of Electrical Engineering, vol. 56, no. 7-8,\n2005, pp. 209-212.", "TOMASZEWSKI, S. P.\u00d4\u00c7\u00f6ILGAZ, I. U.\u00d4\u00c7\u00f6ANTONIOU, G. E. :WWWBased\nBoolean Function Simplification, International Journal of\nMathematics and Computer Science 13 No. 4 (2003), 577-583.", "WAKERLY, J. F. : Digital Design, Prentice-Hall, New York, 2000.", "NELSON, V. P.\u00d4\u00c7\u00f6NAGLE, H. T.\u00d4\u00c7\u00f6CARROLL, B. D.\u00d4\u00c7\u00f6IRWIN, D. :\nDigital Logic Circuit Analysis and Design, Prentice-Hall, New Jersey,\n1995.", "KATZ, R. : Contemporary Logic Design, Benjamin/Cummings Publ,\nRedwood City, CA, 1994.", "BROWN, S.\u00d4\u00c7\u00f6VRANESIC, Z. : Fundamentals of Digital Logic with\nVHDL, McGraw-Hill, New York, 2003.", "HAYES, J. P. : Digital Logic Design, New York, 1993.", "KARNAUGH, M.: The Map Method for Synthesis of Com-binatorial\nLogic Circuits, Trans. AIEE, Communications and Electronics 72\n(1953), 593-598.\n[10] QUINE, W. V. : The Problem of Simplifying Truth Tables, Am. Math.\nMonthly 59 No. 8 (1952), 521-531.\n[11] McCLUSKEY, E. J. : Minimization of Boolean functions, Bell System\nTech. Journal 35 No. 5 (1956), 1417-1444.\n[12] GAJSKI, D. D. : Principles of Digital Design, Prentice-Hall, 1997.\n[13] CHIRLIAN, P. M. : Digital Circuits with Microprocessor Applications,\nMatrix Publishers, Oregon, 1982.\n[14] HILL, F. J,\u00d4\u00c7\u00f6PETERSON, G. R. : Computer Aided Logical Design with\nEmphasis on VLSI, Wiley, New York, 1993.\n[15] Alan B. Marcovitz, \"Introduction to Logic and Computer Design\n\"Hardcover, 2007.\n[16] Alan B. Marcovitz, \"Introduction to Logic Design,\" (2nd Economy\nEdition), Paperback, 2005.\n[17] Amy E. Arntson, \"Digital Design Basics,\" Paperback, 2005.\n[18] Frank Vahid, \" Digital Design,\" Hardcover, 2006.\n[19] Janaye M. Houghton and Robert S. Houghton, \"Circuit Sense for\nElementary Teachers and Students: Understanding and Building Simple\nLogic Circuits,\" Paperback, 1994.\n[20] John F. Wakerly, \"Digital Design: Principles and Practices Package (4th\nEdition), Hardcover, 2005.\n[21] Mano, Charles Kime, \"Logic and Computer Design Fundamentals\"\n(Third Edition), Hardcover, 2003.\n[22] Morris M. Mano, Michael D. Ciletti, \"Digital Design (4th Edition),\nHardcover, 2006.\n[23] Morris M. Mano, \"Digital Design,\" Hardcover, 1984.\n[24] Morris M. Mano, \"Computer System Architecture (3rd Edition)\",\nHardcover, 1992.\n[25] Nripendra N. Biswas, \"Computer aided minimization procedure for\nboolean functions,\" Proceedings of the 21st conference on Design\nautomation,\" Albuquerque, New Mexico, United States, Pages: 699 -\n702, 1984.\n[26] Randal E. Bryant, \"Graph-Based Algorithms for Boolean Function\nManipulation\", IEEE Transactions on Computers, C-35-8, pp. 677-691,\nAugust, 1986.\n[27] Robert Dueck, \"Digital Design with CPLD Applications and\nVHDL(Digital Design: Principles and Practices Package (2nd Edition),\nHardcover, 2004.\n[28] Thomas L. Floyd ,\"Electronics Fundamentals: Circuits, Devices and\nApplications\" (7 th Edition), Hardcover, 2006.\n[29] Thomas L. Floyd, Digital Fundamentals, Hardcover, 1994.\n[30] Victor P. Nelson, H. Troy Nagle, Bill D. Carroll, and David Irwin,\n\"Digital Logic Circuit Analysis and Design\", Paperback, 1995\n[31] William Kleitz, \"Digital and Microprocessor Fundamentals: Theory and\nApplication (4th Edition), Hardcover, 2002.\n[32] Yves Crama and Peter L. Hammer, \"Boolean Functions Theory,\nAlgorithms and Applications\", January 22, 2006.\n[33] http://www.asic-world.com/digital/kmaps.html, \"Simplification of\nBoolean Functions\", 2006.\n[34] Hazem M. El-Bakry, and Ahmed Atwan, \"Simplification and\nImplementation of Boolean Functions,\" International Journal of\nUniversal Computer Sciences, issue 1, vol. 1, 2010, pp. 19-33.\n[35] Hazem M. El-bakry, Ahmed Atwan, and Nikos Mastorakis \"A\u2518\u00c9 New\nTechnique for Realization of Boolean Functions, \" Proc. of Recent\nAdvances in Artificial Intelligence, Koweledge Engineering and\nDatabases, Cambridge, UK, February 20-22, 2010, pp. 260-270.\n[36] Hazem M. El-bakry, and Nikos Mastorakis \"A Fast Computerized\nMethod For Automatic Simplification of Boolean Functions,\" Proc. of\n9th WSEAS International Conference on SYSTEMS THEORY AND\nSCIENTIFIC COMPUTATION (ISTASC '09), Moscow, Russia, August\n26-28, 2009, pp. 99-107.\n[37] Hazem M. El-bakry, \"Fast Karnough Map for Simplification of\nComplex Boolean Functions,\" Proc. of 10th WSEAS International\nConference on Applied Computer Science (ACS'10), Japan, October 4-\n6, 2010, pp. 478-483.\n[38] Hazem M. El-Bakry, \"An Efficient Algorithm for Pattern Detection\nusing Combined Classifiers and Data Fusion,\" Information Fusion\nJournal, vol. 11, 2010, pp. 133-148..\n[39] Hazem M. El-Bakry, \"A Novel High Speed Neural Model for Fast\nPattern Recognition,\" Soft Computing Journal, vol. 14, no. 6, 2010, pp.\n647-666.\n[40] Hazem M. El-Bakry, \"Fast Virus Detection by using High Speed Time\nDelay Neural Networks,\" Journal of Computer Virology, vol.6, no.2,\n2010, pp.115-122.\n[41] Hazem M. El-Bakry, \"New Fast Principal Component Analysis For\nReal-Time Face Detection,\" MG&V Journal, vol. 18, no. 4, 2009, pp.\n405-426.\n[42] Hazem M. El-Bakry, \"A New Neural Design for Faster Pattern\nDetection Using Cross Correlation and Matrix Decomposition,\" Neural\nWorld journal, 2009, vol. 19, no. 2, pp. 131-164.\n[43] Hazem M. El-Bakry and M. Hamada, \"A New Implementation for High\nSpeed Neural Networks in Frequency Space,\" Lecture Notes in\nComputer Science, Springer, KES 2008, Part I, LNAI 5177, pp. 33-40.\n[44] Hazem M. El-Bakry and Mohamed Hamada, \" New Fast Decision Tree\nClassifier for Identifying Protein Coding Regions,\" Lecture Notes in\nComputer Science, Springer, ISICA 2008, LNCS 5370, 2008, pp. 489-\n500.\n[45] Hazem M. El-Bakry, and Nikos Mastorakis \"New Fast Normalized\nNeural Networks for Pattern Detection,\" Image and Vision Computing\nJournal, vol. 25, issue 11, 2007, pp. 1767-1784.\n[46] Hazem M. El-Bakry and Nikos Mastorakis, \"Fast Code Detection Using\nHigh Speed Time Delay Neural Networks,\" Lecture Notes in Computer\nScience, Springer, vol. 4493, Part III, May 2007, pp. 764-773.\n[47] Hazem M. El-Bakry, \"New Fast Time Delay Neural Networks Using\nCross Correlation Performed in the Frequency Domain,\"\nNeurocomputing Journal, vol. 69, October 2006, pp. 2360-2363.\n[48] Hazem M. El-Bakry, \"Face detection using fast neural networks and\nimage decomposition,\" Neurocomputing Journal, vol. 48, 2002, pp.\n1039-1046.\n[49] Hazem M. El-Bakry, \"Human Iris Detection Using Fast Cooperative\nModular Neural Nets and Image Decomposition,\" Machine Graphics &\nVision Journal (MG&V), vol. 11, no. 4, 2002, pp. 498-512.\n[50] Hazem M. El-Bakry, \"Automatic Human Face Recognition Using\nModular Neural Networks,\" Machine Graphics & Vision Journal\n(MG&V), vol. 10, no. 1, 2001, pp. 47-73.\n[51] Hazem M. El-Bakry, \"New Faster Normalized Neural Networks for Sub-\nMatrix Detection using Cross Correlation in the Frequency Domain and\nMatrix Decomposition,\" Applied Soft Computing journal, vol. 8, issue 2,\nMarch 2008, pp. 1131-1149.\n[52] Hazem M. El-Bakry, \"Automatic Human Face Recognition Using\nModular Neural Networks,\" Machine Graphics & Vision Journal\n(MG&V), vol. 10, no. 1, 2001, pp. 47-73.\n[53] Hazem M. El-Bakry, \"Human Iris Detection Using Fast Cooperative\nModular Neural Nets and Image Decomposition,\" Machine Graphics &\nVision Journal (MG&V), vol. 11, no. 4, 2002, pp. 498-512.\n[54] Hazem M. El-Bakry \"Fast Iris Detection for Personal Verification Using\nModular Neural Networks,\" Lecture Notes in Computer Science,\nSpringer, vol. 2206, October 2001, pp. 269-283.\n[55] Hazem M. El-bakry, \"Complexity Reduction Using Modular Neural\nNetworks,\" Proc. of IEEE IJCNN-03, Portland, Oregon, pp. 2202-2207,\nJuly, 20-24, 2003.\n[56] R. Klette, and Zamperon, \"Handbook of image processing operators, \"\nJohn Wiley & Sonsltd, 1996.\n[57] J. Murre, Learning and Categorization in Modular Neural Networks,\nHarvester Wheatcheaf. 1992.\n[58] R. Jacobs, M. Jordan, A. Barto, Task Decomposition Through\nCompetition in a Modular Connectionist Architecture: The what and\nwhere vision tasks, Neural Computation 3, pp. 79-87, 1991.\n[59] J. W. Cooley, and J. W. Tukey, \"An algorithm for the machine\ncalculation of complex Fourier series,\" Math. Comput. 19, 297-301,\n1965.\n[60] G. Auda, and M. Kamel, CMNN: Cooperative Modular Neural\nNetworks for Pattern Recognition, Pattern Recognition Letters, Vol. 18,\npp. 1391-1398, 1997.\n[61] E. Alpaydin, , Multiple Networks for Function Learning, Int. Conf. on\nNeural Networks, Vol.1 CA, USA, pp. 9-14, 1993.\n[62] A. Waibel, Modular Construction of Time Delay Neural Networks for\nSpeach Recognition, Neural Computing 1, pp.39-46.\n[63] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Learning\nrepresentation by error backpropagation, Parallel distributed Processing:\nExplorations in the Microstructues of Cognition, Vol. 1, Cambridge,\nMA:MIT Press, pp. 318-362, 1986.\n[64] K. Joe, Y. Mori, S. Miyake, Construction of a large scale neural\nnetwork: Simulation of handwritten Japanese Character Recognition, on\nNCUBE Concurrency 2 (2), pp. 79-107."]}

In this paper a new fast simplification method is presented. Such method realizes Karnough map with large number of variables. In order to accelerate the operation of the proposed method, a new approach for fast detection of group of ones is presented. Such approach implemented in the frequency domain. The search operation relies on performing cross correlation in the frequency domain rather than time one. It is proved mathematically and practically that the number of computation steps required for the presented method is less than that needed by conventional cross correlation. Simulation results using MATLAB confirm the theoretical computations. Furthermore, a powerful solution for realization of complex functions is given. The simplified functions are implemented by using a new desigen for neural networks. Neural networks are used because they are fault tolerance and as a result they can recognize signals even with noise or distortion. This is very useful for logic functions used in data and computer communications. Moreover, the implemented functions are realized with minimum amount of components. This is done by using modular neural nets (MNNs) that divide the input space into several homogenous regions. Such approach is applied to implement XOR function, 16 logic functions on one bit level, and 2-bit digital multiplier. Compared to previous non- modular designs, a clear reduction in the order of computations and hardware requirements is achieved.