System and Method for Extremely Efficient Image and Pattern Recognition and Artificial Intelligence Platform

[0001] The current application is the Continuation of another pending application, named Zadeh-101-cipcipcip, Ser. No. 16/729,944, filed 30, Dec. 2019, titled “System and Method for Extremely Efficient Image and Pattern Recognition and Artificial Intelligence Platform”, now allowed (which is named “Our Continuation Application”, here). “Our Continuation Application” claims the benefit of and takes the priority of the earlier filing dates of the following U.S. provisional application No. 62/786,469, filed 30 Dec. 2018, called ZAdvanced-6-prov, titled “System and Method for Extremely Efficient image and Pattern Recognition and General-Artificial Intelligence Platform”. “Our Continuation Application” is also a CIP (Continuation-in-part) of another co-pending U.S. application Ser. No., 15/919170, filed 12 Mar. 2018, called Zadeh-101-cip-cip, now as patent Ser. No. 11/074,495, issued on 2021 Jul. 27, titled “System and Method for Extremely Efficient Image and Pattern Recognition and Artificial Intelligence Platform”, which is a CIP (Continuation-in-part) of another co-pending U.S. application Ser. No. 14/218,923, filed 18, Mar. 2014, called Zadeh-101-CIP, which is now issued as U.S. Pat. No. 9,916,538 on 13 Mar. 2018, which is a CIP (Continuation-in-part) of another co-pending U.S. application Ser. No., 13/781,303; filed Feb. 28, 2013, called ZAdvanced-1, now U.S. Pat. No. 8,873,813, issued on 28 Oct. 2014, which claims the benefit of and takes the priority of the earlier filing date of the following U.S. provisional application No. 61/701,789, filed Sep. 17, 2012, called ZAdvanced-1-prov. The application 14/218,923 also claims the benefit of and takes the priority of the earlier filing dates of the following U.S. provisional application numbers: 61/802,810, filed Mar. 18, 2013, called ZAdvanced-2-prov; and 61/832,816, filed Jun. 8, 2013, called ZAdvanced-3-prov; and 61/864,633, filed Aug. 11, 2013, called ZAdvanced-4-prov; and 61/871,860, filed Aug. 29, 2013, called ZAdvanced-5-prov. The application Ser. No. 14/218,923 is also a CIP (Continuation-in-part) of another co-pending U.S. application Ser. No. 14/201,974, filed 10Mar. 2014, called Zadeh-101-Cont-4, now as U.S. Pat. No. 8,949,170. issued on 3 Feb. 2015, which is a Continuation of another U.S. application Ser. No., 13/953,047, filed Jul. 29, 2013, called Zadeh-101-Cont-3, now U.S. Pat. No. 8,694,459, issued on 8 Apr. 2014, which is also a Continuation of another co-pending application, 13/621,135, filed Sep. 15, 2012, now issued as U.S. Pat. No. 8,515,890, on Aug. 20, 2013, which is also a Continuation of Ser. No. 13/621,164, filed Sep. 15, 2012, now issued as U.S. Pat. No. 8,463,735, which is a Continuation of another application, Ser. No. 13/423,758, filed Mar. 19, 2012, now issued as U.S. Pat. No. 8,311,973, which, in turn, claims the benefit of the U.S. provisional application No. 61/538,824, filed on Sep. 24, 2011. The current application incorporates by reference all of the applications and patents/provisionals mentioned above, including all their Appendices and attachments (Packages), and it claims benefits to and takes the priority of the earlier filing dates of all the provisional and utility applications or patents mentioned above. Please note that most of the Appendices and attachments (Packages) to the specifications for the above-mentioned applications and patents (such as U.S. Pat. No. 8,311,973) are available for public view, e.g., through Public Pair system at the USPTO web site (www.uspto.gov), with some of their listings given below in the next section:

[0002] (All incorporated by reference, herein, in the current application.)

[0003] In addition to the provisional cases above, the teachings of all 33 packages (the PDF files, named “Packages 1-33”) attached with some of the parent cases' filings (as Appendices) (such as U.S. Pat. No. 8,311,973 (i.e., Zadeh-101 docket)) are incorporated herein by reference to this current disclosure.

[0004] Furthermore, “Appendices 1-5” of Zadeh-10^-1-GIP (i.e., 14/218,923) are incorporated herein by reference to this current disclosure.

[0005] To reduce the size of the appendices/disclosure, these Packages (Packages 1-33) and Appendices (Appendices 1-5) are not repeated here again, but they may be referred to/incorporated. in, in the future from time to time in the current or the children/related applications, both in spec or claims, as our own previous teachings.

[0006] However, the new Appendices attached to this current application is now numbered after the appendices mentioned above, i.e., starting with Appendix 6, for this current application, to make it easier to refer to them in the future.

[0007] Please note that Appendices 1-5 (of Zadeh-101-CIP (i.e., Ser. No. 14/218,923)) are identified as:

• • [0008] Appendix 1: article about “Approximate Z-Number Evaluation based on Categorical Sets of Probability Distributions” (11 pages)
  • [0009] Appendix 2: hand-written technical notes, formulations, algorithms, and derivations (5 pages)
  • [0010] Appendix 3: presentation about “Approximate Z-Number Evaluation Based on Categorical Sets of Probability Distributions” (30 pages)
  • [0011] Appendix 4: presentation with FIGS. from B1 to B19 (19 pages)
  • [0012] Appendix 5: presentation about “SVM Classifier” (22 pages)

[0013] Please note that Appendices 6-10 (of Zadeh-101-CIP-C IP (i.e., the current application)) are identified as:

• • [0014] Appendix 6: article/journal/technical/research/paper about “The Information Principle”, by Prof. Lotfi Zadeh. Information Sciences, submitted 16 May 2014, published 2015 (10 pages)
  • [0015] Appendix 7: presentation/conference/talk/invited/keynote speaker/lecture about “Stratification, target set reachability, and incremental enlargement principle”, by Prof. Lotti Zadeh, UC Berkeley, World Conference on Soft Computing, May 22, 2016 (14 pages, each page including 9 slides, for a total of 126 slides) (first version prepared on Feb. 8, 2016)
  • [0016] Appendix 8: article about “Stratification, quantization, target set reachability, and incremental enlargement principle”, by Prof. Lotfi Zadeh, for Information Sciences, received 4 Jul. 2016 (17 pages) (first version prepared on Feb. 5, 2016)
  • [0017] Appendix 9: This shows the usage of visual search terms for our image search engine (1 page), which is the first in the industry. It shows an example for shoes (component or parts matching, from various shoes), using ZAC/our technology and platform. For example, it shows the search for: “side look like shoe number 1, heel look like shoe number 2, and toe look like shoe number 3”, based on what the user is looking/searching for. In general, we can have a combination of conditions, e.g.: (R₁ AND R₂ AND . . . AND R_n), or any logical search terms or combinations or operators, e.g., [R₁ OR (R₂ AND R₃)], which is very helpful for e-commerce or websites/e-stores.
  • [0018] Appendix 10: “Brief Introduction to AI and Machine Learning”, for conventional tools and methods, sometimes used or referred to in this invention, for completeness and as support of the main invention, or just for the purpose of comparison with the conventional tools and methods.

[0019] Please note that Appendices 11-13 (of ZAdvanced-6-prov) are identified as:

• • [0020] Appendix 11: “ZAC General-AI Platform for 31) Object Recognition & Search from any Direction (Revolutionary Image Recognition & Search Platform)”, for descriptions and details of General-AI Platform, which includes Explainable-AI (or XAI or X-AI or Explainable-Artificial Intelligence), as well. This also describes ZAC features and advantages over NN (or CNN or Deep CNN or Deep Convolutional Neural Net or ResNet). This also describes applications, markets, and use cases/examples/embodiments for ZAC tech/algotithms/platform.
  • [0021] Appendix 12: ZAC platform and operation, with features, architecture, modules, layers, and components. This also describes ZAC features and advantages over NN (or CNN or Deep CNN or Deep Convolutional Neural Net or ResNet).
  • [0022] Appendix 13: Some examples/embodiments/tech descriptions for ZAC tech/platform (General-AI Platform).

[0023] Please note that Appendix 14 (of Zadeh-101-cip-cip-cip) (i.e., the current ion) is identified as ZAC Explainable-AI, which is a component of ZAC General-AI Platform. This also describes applications, markets, and use cases/examples/embodiments for ZAC tech/algorithms/platform. This also describes ZAC features and advantages over NN (or CNN or Deep CNN or Deep Convolutional Neural Net or ResNet).

[0024] Please note that Packages 1-33 (of U.S. Pat. No. 8,311,973) are also one of the inventor's (Prof. Lotfi Zadeh's) own previous technical teachings, and thus, they may be referred to (from time-to-time) for further details or explanations, by the reader, if needed.

[0025] Please note that Packages 1-25 had already been submitted (and filed) with our provisional application for one of the parent cases.

[0026] Packages 1-12 and 15-22 are marked accordingly at the bottom of each page or slide (as the identification). The other Packages (Packages 13-14 and 23-33) are identified here:

• • [0027] Package 13: 1 page, with 3 slides, starting with “FIG. 1, Membership function of A and probability density function of X”
  • [0028] Package 14: 1 page, with 5 slides, starting with “FIG. 1. f-transformation and f-geometry. Note that fuzzy figures, as shown, are not hand drawn. They should be visualized as hand drawn figures.”
  • [0029] Package 23: 2-page text, titled “The Concept of a Z-number—a New Direction in Computation, Lotfi A. Zadeh, Abstract” (dated Mar. 28, 2011)
  • [0030] Package 24: 2-page text, titled “Prof. Lotfi Zadeh. The Z-mouse—a visual means of entry and retrieval of fuzzy data”
  • [0031] Package 25: 12-page article, titled “Toward Extended Fuzzy Logic A First Step, Abstract”
  • [0032] Package 26: 2-page text, titled “Can mathematics deal with computational problems which are stated in a natural language?, Lotfi A. Zadeh, Sep. 30, 2011, Abstract” (Abstract dated Sep. 30, 2011)
  • [0033] Package 27: 15 pages, with 131 slides, titled “Can Mathematics Deal with Computational Problems Which are Stated in a Natural Language?, Lotfi A. Zadeh” (dated Feb. 2, 2012)
  • [0034] Package 28: 14 pages, with 123 slides, titled “Can Mathematics Deal with Computational Problems Which are Stated in a Natural Language?, Lotfi A. Zadeh” (dated Oct. 6, 201 1)
  • [0035] Package 29: 33 pages, with 289 slides, titled “Computing with Words—Principal Concepts and Ideas, Lotfi A. Zadeh” (dated Jan. 9, 2012)
  • [0036] Package 30: 23 pages, with 205 slides, titled “Computing with Words—Principal Concepts and Ideas, Lotfi A. Zadeh” (dated May 10, 2011)
  • [0037] Package 31: 3 pages, with 25 slides, titled “Computing with Words—Principal Concepts and Ideas, Lotfi A. Zadeh” (dated Nov. 29, 2011)
  • [0038] Package 32: 9 pages, with 73 slides, titled “Z-NUMBERS—A NEW DIRECTION IN THE ANALYSIS OF UNCERTAIN AND IMPRECISE SYSTEMS, Lotfi A. Zadeh” (dated Jan. 20, 2012)
  • [0039] Package 33: 15 pages, with 131 slides, titled “PRI CISIATION OF MEANING—A KEY TO SEMANTIC COMPUTING, Lotfi A. Zadeh” (dated Jul. 22, 2011)

[0040] Please note that all the Packages and Appendices (prepared by one or more of the inventors here) were also identified by their PDF file names, as they were submitted to the USPTO electronically.

[0041] Professor Lotfi A. Zadeh, one of the inventors of the current disclosure and some of the parent cases, is the “Father of Fuzzy Logic”. He first introduced the concept of Fuzzy Set and Fuzzy Theory in his famous paper, in 1965 (as a professor of University of California, at Berkeley). Since then, many people have worked on the Fuzzy Logic technology and science, Dr. Zadeh has also developed many other concepts related to Fuzzy Logic. He has invented Computation-with-Words (CWW or CW), e.g., for natural language processing (NLP) and analysis, as well as semantics of natural languages and computational theory of perceptions, for many diverse applications, which we address here, as well, as some of our new/innovative methods and systems are built based on those concepts/theories, as their novel/advanced extensions/additions/versions/extractions/branches/fields. One of his last revolutionary inventions is called Z-numbers, named after him (“Z” from Zadeh), which is one of the many subjects of the (many) current inventions. That is, some of the many embodiments of the current inventions are based on or related to Z-numbers. The concept of Z-numbers was first published in a recent paper, by Dr. Zadeh, called “A Note on Z-Nutnbers”, Information Sciences 181 (2011) 2923-2932.

[0042] However, in addition, there are many other embodiments in the current disclosure that deal with other important and innovative topics/subjects, e.g., related to General AI, versus Specific or Vertical or Narrow AI, machine learning, using/requiring only a small number of training samples (same as humans can do), learning one concept and use it in another context or environment (same as humans can do), addition of reasoning and cognitive layers to the learning module (same as humans can do), continuous learning and updating the learning machine continuously (same as humans can do), simultaneous learning and recognition (at the same time) (same as humans can do), and conflict and contradiction resolution (same as humans can do), with application, e.g., for image recognition, application for any pattern recognition, e.g., sound or voice, application for autonomous or driverless cars, application for security and biometrics, e.g., partial or covered or tilted or rotated face recognition, or emotion and feeling detections, application for playing games or strategic scenarios, application for fraud detection or verification/validation, e.g., for banking or cryptocurrency or tracking fund or certificates, application for medical imaging and medical diagnosis and medical procedures and drug developments and genetics, application for control systems and robotics, application for prediction, forecasting, and risk analysis, e.g., for weather forecasting, economy, oil price, interest rate, stock price, insurance premium, and social unrest indicators/parameters, and the like.

[0043] In the real world, uncertainty is a pervasive phenomenon. Much of the information on which decisions are based is uncertain. Humans have a remarkable capability to make rational decisions based on information which is uncertain, imprecise and/or incomplete. Formalization of this capability is one of the goals of these current inventions, in one embodiment.

[0044] Here are some of the publications on the related subjects, for some embodiments:

[0045] [1] R. Ash, Basic Probability Theory, Dover Publications, 2008.

[0046] [2] J-C. Buisson, Nutri-Educ, a nutrition software application for balancing meals, using fuzzy arithmetic and heuristic search algorithms, Artificial Intelligence in Medicine 42, (3), (2008) 213-227.

[0047] [3] E. Trillas, C. Moraga, S. Guadarrama, S. Cubillo and E. Castifieira, Computing with Antonyms. In: M, Nikravesh, J. Kacprzyk and L. A. Zadeh (Eds.). Forging New Frontiers: Fuzzy Pioneers I, Studies in Fuzziness and Soft Computing Vol 217, Springer-Verlag, Berlin Heidelberg 2007, pp. 133-153.

[0048] [4] R. R. Yager, On measures of specificity. In: O. Kaynak, L. A. Zadeh, B. Turksen, I. J. Rudas (Eds.), Computational Intelligence: Soft Computing and Fuzzy-Neuro Integration with Applications, Springer-Verlag, Berlin, 1998, pp. 94-113.

[0049] [5] L. A. Zadeh, Calculus of fuzzy restrictions. In: L. A. Zadeh, K. S. Fu, K. Tanaka, and M. Shimura (Eds.), Fuzzy sets and Their Applications to Cognitive and Decision Processes, Academic Press, N.Y., 1975, pp. 1-39.

[0050] [6] L. A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning,

[0051] Part I: Information Sciences 8 (1975) 199-249;

[0052] Part II: Information Sciences 8 (1975) 301-357;

[0053] Part III: Information Sciences 9 (1975) 43-80.

[0054] [7] L. A. Zadeh, Fuzzy logic and the calculi of fuzzy rules and fuzzy graphs, Multiple-Valued Logic 1, (1996) 1-38.

[0055] [8] L. A. Zadeh, From computing with numbers to computing with words—from manipulation of measurements to manipulation of perceptions, IEEE Transactions on Circuits and Systems 45, (1999) 105-119.

[0056] [9] L. A. Zadeh. The Z-mouse—a visual means of entry and retrieval of fuzzy data, posted on BISC Forum, Jul. 30, 2010. A more detailed description may be found in Computing with Words—principal concepts and ideas, Colloquium PowerPoint presentation, University of Southern California, Los Angeles, Calif., Oct. 22, 2010.

[0057] As one of the applications mentioned here in this disclosure, for comparisons, some of the search engines or question-answering engines in the market (in the recent years) are (or were): Google®, Yahoo®, Autonomy, IBM®, Fast Search, Powerset® (by Xerox® PARC and bought by Microsoft®), Microsoft® Bing, Wolfram®, AskJeeves, Collarity, Vivisimeo®,Endeca,®, Media River, Hakia®, Ask.com®, AltaVista, Excite, Go Network, HotBot®, Lycos®, Northern Light, and Like.com.

[0058] Other references on some of the related subjects are:

[0059] [1] A. R. Aronson, B. E. Jacobs, J. Minker, A note on fuzzy deduction, J. ACM27 (4) (1980). 599-603.

[0060] [2] A. Bardossy, L. Duckstein, Fuzzy Rule-based Modelling with Application to Geophysical, Biological and Engineering Systems, CRC Press, 1995.

[0061] [3] T. Berners-Lee, J. Hendler, O. Lassila. The semantic web, Scientific American 284 (5) (2001), 34-43.

[0062] [4] S. Brin, L. Page. The anatomy of a large-scale hypertextual web search engine, Computer Networks 30 (1-7) (1998), 107-117.

[0063] [5] W. J. H. J. Bronnenberg, M. C. Bunt, S. P. J. Lendshergen, R. H. J. Scha,W. J. Schoenmakers, E. P. C. van Utteren. The question answering system PHLIQA1, in: L. Bola (Ed.), Natural Language Question Answering Systems, Macmillan, 1980.

[0064] [6] L. S. Coles, Techniques for information retrieval using an inferential question-answering system with natural language input, SRI Report, 1972.

[0065] [7] A. Di Nola, S. Sessa., W. Pedrycz,W. Pei-Zhuang, Fuzzy relation equation under a class of triangular norms: a survey and new results, in: Fuzzy Sets for intelligent Systems, Morgan Kaufmann Publishers, San Mateo, Calif., 1993, pp. 166-189.

[0066] [8] A. Di Nola, S. Sessa, W. Pedrycz, F Sanchez, Fuzzy Relation Equations and their Applications to Knowledge Engineering, Kluwer Academic Publishers, Dordrecht, 1989.

[0067] [9] D. Dubois, H. Prade, Gradual inference rules in approximate reasoning. Inform. Sci. 61 (1-2) 1992), 103-122.

[0068] [10] D. Hey, R. R. Yager, Essentials of Fuzzy Modeling and Control, Wiley-Interscience, 1994.

[0069] [11] J. A. Goguen. The logic of inexact concepts, Synthese 19 (1969), 325-373.

[0070] [12] M. Jarnshidi, A. Titli, L. A. Zadeh, S. Boverie (Eds.), Applications of Fuzzy Logic—Towards High Machine Intelligence Quotient Systems, Environmental and Intelligent Manufacturing Systems Series, vol. 9, Prentice-Hall, Upper Saddle River, N.J., 1997.

[0071] [13] A. Kaufmann, M. M. Gupta. Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand, N.Y., 1985.

[0072] [14] D. B. Lenat, CYC: a large-scale investment in knowledge infrastructure, Comm.ACM38 (η) (1995), 32-38.

[0073] [15] E. H. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. Man-Machine Studies 7 (1975), 1-13.

[0074] [16] J. R. McSkimin, J. Minker. The use of a semantic network in a deductive question-answering system, in: IJCAI, 1977, pp. 50-58.

[0075] [17] R. E. Moore. Interval Analysis, SIAM Studies in Applied Mathematics, vol. 2, Philadelphia, Pa., 1979.

[0076] [18] M. Nagao, J. Tsujii, Mechanism of deduction in a question-answering system with natural language input, in: ICJAI, 1973, pp. 285-290.

[0077] [19] B. H. Partee (Ed.), Montague Grammar, Academic Press, N.Y., 1976.

[0078] [20] W. Pedrycz, F. Gomide. Introduction to Fuzzy Sets, MIT Press, Cambridge, Ma., 1998.

[0079] [21] F. Rossi, P. Codognet (Eds.), Soft Constraints, Special issue on Constraints, vol. 8, N. 1, Kluwer Academic Publishers, 2003.

[0080] [22] G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, N.J., 1976.

[0081] [23] M. K. Smith, C. Welty, D. McGuinness (Eds.), OWL Web Ontology Language Guide, W3C Working Draft 31,2003.

[0082] [24] L. A. Zadeh. Fuzzy sets. Inform, and Control 8 (1965), 338-353.

[0083] [25] L. A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl. 23 (1968), 421-427.

[0084] [26] L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. on Systems Man Cybernet. 3 (1973), 28-44.

[0085] [27] L. A. Zadeh, On the analysis of large scale systems, in: H. Gottinger (Ed.), Systems Approaches and Environment Problems, Vandenhoeck and Ruprecht, Gottingen, 1974, pp. 23-37.

[0086] [28] L. A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning, Part I. Inform. Sci. 8 (1975), 199-249;Part II. Inform. Sci. 8 (1975), 301-357;Part III, Inform. Sci. 9 (1975), 43-80.

[0087] [29] L. A. Zadeh, Fuzzy sets and information granularity, in: M. Gupta, R. Ragade, R. Yager (Eds.), Advances in Fuzzy Set Theory and Applications, North-Holland Publishing Co, Amsterdam, 1979, pp. 3-18.

[0088] [30] L. A. Zadeh, A theory of approximate reasoning, in: J. Hayes, D. Michie, L. I. Mikulich (Eds.), Machine Intelligence, vol. 9,Halstead Press, N.Y., 1979, pp. 149-194.

[0089] [31] L. A. Zadeh, Test-score semantics for natural languages and meaning representation via PRUF, in: B. Rieger (Ed.), Empirical Semantics, Brockmeyer, Bochum, W. Germany, 1982, pp. 281-349. Also Technical Memorandum 246, AI Center, SRI International, Menlo Park, Calif., 1981.

[0090] [32] L. A. Zadeh, A computational approach to fuzzy quantifiers in natural languages, Computers and Mathematics 9 (1983), 149-184.

[0091] [33] L. A. Zadeh, A fuzzy-set-theoretic approach to the compositionality of meaning: propositions, dispositions and canonical forms, J. Semantics 3 (1983), 253-272.

[0092] [34] L. A. Zadeh, Precisiation of meaning via translation into PRUF, in: L. Vaina, J. Hintikka (Eds.), Cognitive Constraints on Communication, Reidel, Dordrecht, 1984, pp. 373-402.

[0093] [35] L. A. Zadeh, Outline of a computational approach to meaning and knowledge representation based on a concept of a generalized assignment statement, in: M. Thoma, A. Wyner (Eds.), Proceedings of the International Seminar on Artificial Intelligence and Man-Machine Systems, Springer-Verlag, Heidelberg, 1986, pp. 198-211.

[0094] [36] L, A. Zadeh, Fuzzy logic and the calculi of fuzzy rules and fuzzy graphs, Multiple-Valued Logic 1 (1996), 1-38.

[0095] [37] L. A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems 90 (1997), 111-127.

[0096] [38] L. A. Zadeh, From computing with numbers to computing with words from manipulation of measurements to manipulation of perceptions, IEEE Trans. on Circuits and Systems 45 (1) (1999), 105-119.

[0097] [39] L. A. Zadeh, Toward a perception-based theory of probabilistic reasoning with imprecise probabilities, J. Statist. Plana. inference 105 (2002), 233-264.

[0098] [40] L. A, Zadeh, Precisiated natural language (PNL), AI Magazine 25 (3) (2004), 74-91.

[0099] [41] L. A. Zadeh, A note on web intelligence, world knowledge and fuzzy logic, Data and Knowledge Engineering 50 (2004), 291-304.

[0100] [42] L. A. Zadeh, Toward a generalized theory of uncertainty (GTIJ)—an outline, inform Sci. 172 (2005), 1-40.

[0101] [43] J. Ationa, R. Corchuelo, J. Pena, D. Ruiz, Coping with web knowledge, in: Advances in Web Intelligence, Springer-Verlag, Berlin, 2003, pp. 165-178.

[0102] [44] A. Bargiela, W. Pedrycz, Granular Computing—An Introduction, Kluwer Academic Publishers, Boston, 2003.

[0103] [45] Z. Bubnicki, Analysis and Decision Making in Uncertain Systems, Springer-Verlag, 2004.

[0104] [46] P. P. Chen, Entity-relationship Approach to Information Modeling and Analysis, North-Holland, 1983.

[0105] [47] M. Craven, D. DiPasquo, D. Freitag, A. McCallum, T. Mitchell, K. Nigam, S. Slattery, Learning to construct knowledge bases from the world wide web, Artificial Intelligence 118 (1-2) (2000), 69-113.

[0106] [48] M. J. Cresswell, Logic and Languages, Methuen, London, UK, 1973.

[0107] [49] D. Dubois, H. Prade, On the use of aggregation operations in information fusion processes, Fuzzy Sets and Systems 142 (1) (2004), 143-161.

[0108] [50] T. F. Gamat, Language, Logic and Linguistics, University of Chicago Press, 1996.

[0109] [51] M. Mares, Computation over Fuzzy Quantities, CRC, Boca Raton, Fla., 1994.

[0110] [52] V. Novak, I. Perfilieva, J. Mockor, Mathematical Principles of Fuzzy Logic, Kluwer Academic Publishers, Boston, 1999.

[0111] [53] V. Novak, I. Perfilieva (Eds.), Discovering the World with Fuzzy Logic, Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, 2000.

[0112] [54] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991.

[0113] [55] M. K. Smith, C. Welty, What is ontology? Ontology: towards a new synthesis, in: Proceedings of the Second International Conference on Formal Ontology in Information Systems, 2002.

[0114] However, none of the prior art, teaches the features mentioned in our invention disclosure.

[0115] There are a lot of research going on today, focusing on the search engine, analytics, Big Data processing, natural language processing, economy forecasting, dealing with reliability and certainty, medical diagnosis, pattern recognition, object recognition, biometrics, security analysis, risk analysis, fraud detection, satellite image analysis, machine generated data, machine learning, training samples, and the like.

[0116] For example, see the article by Technology Review, published by MIT, “Digging deeper in search web”, Jan. 29, 2009, by Kate Greene, or search engine by σ_x)OGLE®, MICROSOFT® (BINGT), or YAHOO®, or APPLE® SIRI, or WOLFRAM® ALPHA computational knowledge engine, or AMAZON engine, or FACEBOOK® engine, or ORACLE® database, or YANDEX® search engine in Russia, or PICASA® (σ_x)OGLE®) web albums, or YOUTUBE® (σ_x)OGLE®) engine, or ALIBABA (Chinese supplier connection), or SPUNK® (for Big Data), or MICROSTRATEGY® (for business intelligence), or QUID (or KAGGLE, ZESTFINANCE, APIXIO, DATAMEER, BLUEKAI, GNIP, RETAILNEXT, or RECOMMIND) (for Big Data), or paper by Viola-Jones, Viola et al., at Conference on Computer Vision and Pattern Recognition, 2001, titled “Rapid object detection using a boosted cascade of simple features”, from Mitsubishi and Compaq research labs, or paper by AIex Pentland et al., Feb 2000, at Computer, IEEE, titled “Face recognition for smart environments”, or σ_x)OGLE® official bldg; publication, May 16, 2012, titled “Introducing the knowledge graph: things, not strings”, or the article by Technology Review, published by MIT, “The future of search”, Jul. 16, 2007, by Kate Greene, or the article by Technology Review, published by MIT, “Microsoft searches for group advantage”, Jan. 30, 2009, by Robert Lemos, or the article by Technology Review, published by MIT, “WOLFRAM ALPHA and σ_x)OGLE face off”, May 5, 2009, by David Talbot, or the paper by Devarakonda et at, at International Journal of Software Engineering (JJSE), Vol. 2, Issue 1, 2011, titled “Next generation search engines for information retrieval”, or paper by Nair-Hinton, titled “Implicit mixtures of restricted Boltzmann machines”, NIPS, pp. 1145-1152, 2009, or paper by Nair, V. and Hinton, G. E., titled “3-D Object recognition with deep belief nets”, published in Advances in Neural Information Processing Systems 22, (Y. Bengio, D. Schuurmans, J. lafferty, C. K. I. Williams, and A. Culotta (Eds.)), pp 1339-1347. Other research groups include those headed by Andrew Ng, Yoshna Bengio, Fei Fei Li, Ashutosh Saxena, LeCun, Michael I. Jordan, Zoubin Ghahramani, and others in companies and universities around the world.

[0117] However, none of the prior art teaches the features mentioned in our invention disclosure, even in combination.

[0118] For one embodiment: Decisions are based on information. To be useful, information must be reliable. Basically, the concept of a Z-number relates to the issue of reliability of information. A Z-number, Z, has two components, Z=(A, B). The first component, A, is a restriction (constraint) on the values which a real-valued uncertain variable, X, is allowed to take. The second component, B, is a measure of reliability (certainty) of the first component. Typically, A and B are described in a natural language. Example: (about 45 minutes, very sure). An important issue relates to computation with Z-numbers. Examples are: What is the sum of (about 45 minutes, very sure) and (about 30 minutes, sure)? What is the square root of (approximately 100, likely)? Computation with Z-numbers falls within the province of Computing with Words (CW or CWW). In this disclosure, the concept of a Z-number is introduced and methods of computation with Z-numbers are shown. The concept of a Z-number has many applications, especially in the realms of economics, decision analysis, risk assessment, prediction, anticipation, rule-based characterization of imprecise functions and relations, and biomedicine. Different methods, applications, and systems are discussed. Other Fuzzy inventions and concepts are also discussed. Many non-Fuzzy-related inventions and concepts are also discussed.

[0119] For other embodiments: Specification also covers new algorithms, methods, and systems for artificial intelligence, soft computing, and deep/detailed learning/recognition, e.g., image recognition (e.g., for action, gesture, emotion, expression, biometrics, fingerprint, facial, OCR (text), background, relationship, position, pattern, and object), large number of images (“Big Data”) analytics, machine learning, training schemes, crowd-sourcing (using experts or humans), feature space, clustering, classification, similarity measures, optimization, search engine, ranking, question-answering system, soft (fuzzy or unsharp) boundaries/impreciseness/ambiguities/fuzziness in language, Natural Language Processing (NLP), Computing-with-Words (CWW), parsing, machine translation, sound and speech recognition, video search and analysis (e.g. tracking), image annotation, geometrical abstraction, image correction, semantic web, context analysis, data reliability (e.g.,using Z-number (e.g., “About 45 minutes; Very sure”)), rules engine, control system, autonomous vehicle (e.g., self-parking), self-diagnosis and self-repair robots, system diagnosis, medical diagnosis, biomedicine, data mining, event prediction, financial forecasting, economics, risk assessment, e-mail management, database management, indexing and join operation, memory management, and data compression.

[0120] Other topics/inventions covered are, e.g.:

• • [0121] Method and System for Identification or Verification for an Object, a Person, or their Attributes
  • [0122] System and Method for Image Recognition and Matching for Targeted Advertisement
  • [0123] System and Method for Analyzing Ambiguities in Language for Natural Language Processing
  • [0124] Application of Z-Webs and Z-factors to Analytics, Search Engine, Learning, Recognition, Natural Language, and Other Utilities
  • [0125] Method and System for Approximate Z-Number Evaluation based on Categorical Sets of Probability Distributions
  • [0126] Image and Video Recognition and Application to Social Network and Image and Video Repositories
  • [0127] System and Method for Image Recognition for Event-Centric Social Networks
  • [0128] System and Method for Image Recognition for Image Ad Network
  • [0129] System and Method for Increasing Efficiency of Support Vector Machine Classifiers

[0130] Other topics/inventions covered are, e.g.:

• • [0131] Information Principle
  • [0132] Stratification
  • [0133] Incremental Enlargement Principle
  • [0134] Deep/detailed Machine Learning and training schemes
  • [0135] Image recognition (e.g., for action, gesture, emotion, expression, biometrics, fingerprint, facial (e.g. using eigenface), monument and landmark, OCR, background, partial object, relationship, position, pattern, texture, and object)
  • [0136] Basis functions
  • [0137] Image and video auto-annotation
  • [0138] Focus window
  • [0139] Modified/Enhanced Boltzmann Machines
  • [0140] Feature space translation
  • [0141] Geometrical abstraction
  • [0142] Image correction
  • [0143] Semantic web
  • [0144] Context analysis
  • [0145] Data reliability
  • [0146] Correlation layer
  • [0147] Clustering
  • [0148] Classification
  • [0149] Support Vector Machines
  • [0150] Similarity measures
  • [0151] Optimization
  • [0152] Z-number
  • [0153] Z-factor
  • [0154] Z-web
  • [0155] Rules engine
  • [0156] Control system
  • [0157] Robotics
  • [0158] Search engine
  • [0159] Ranking
  • [0160] Question-answering system
  • [0161] Soft boundaries & Fuzziness in language
  • [0162] Natural Language Processing (NLP)
  • [0163] System diagnosis
  • [0164] Medical diagnosis
  • [0165] Big Data analytics
  • [0166] Event prediction
  • [0167] Financial forecasting
  • [0168] Computing with Words (CWW)
  • [0169] Parsing
  • [0170] Soft boundaries & Fuzziness in clustering & classification
  • [0171] Soft boundaries & Fuzziness in recognition
  • [0172] Machine translation
  • [0173] Risk assessment
  • [0174] e-mail management
  • [0175] Database management
  • [0176] Indexing and join operation
  • [0177] Memory management
  • [0178] Sound and speech recognition
  • [0179] Video search & analysis (e.g. tracking)
  • [0180] Data compression
  • [0181] Crowd sourcing (e.g., with experts or SMEs)
  • [0182] Event-centric social networking (based on image)
  • [0183] Energy
  • [0184] Transportation
  • [0185] Distribution of materials
  • [0186] Optimization
  • [0187] Scheduling

[0188] We have also introduced the first Image Ad Network, powered by our next generation image search engine.

[0189] We have introduced our novel “ZAC™ Image Recognition Platform”, which applies learning based on General-AI algorithms. This way, we need much smaller number of training samples to train (the same as humans do), e.g., for evaluating or analyzing a 3-D object/image, e.g., a complex object, such as a shoe, from any direction or angle. To our knowledge, nobody has solved this problem, yet. This is the “Holy Grail” of image recognition. Having/requiring much smaller number of training samples to train is also the “Holy Grail” of AI and machine learning. So, here, we have achieved 2 major scientific and technical milestones/breakthroughs that others have failed to obtain. (These results had been originally reported in our parent cases, as well.)

[0190] In addition, to our knowledge, this is the first successful example of application of General-AI algorithms, systems, and methods in any field, application, industry, university, research, paper, experiment, demo, or usage.

[0191] With other methods in the industry/universities, e.g., Deep Learning or Convolutional Neural Networks or Deep Reinforcement Learning (maximizing a cumulative reward function) or variations of Neural Networks (e.g., Capsule Networks, recently introduced by Prof. Hinton, Sara Sabour, and Nicholas Frosst, from Google and U. of Toronto), these cannot be done at all, even with much larger number of training samples and much larger CPU/GPU computing time/power and much longer training time periods.

[0192] So, we have a significant advantage over the other methods in the industry/universities, as these tasks cannot be done by other methods at all.

[0193] Even for the conventional/much easier/very specific tasks, where the other AI methods are applicable/useful, we still have a huge advantage over them, by some orders of magnitude, in terms of cost, efficiency, size, training time, computing/resource requirements, battery lifetime, flexibility, and detection/recognition/prediction accuracy.

[0194] These shortcomings/failures/limitations of the other methods/systems/algorithms/results in the AI/machine learning industry/universities have been expressed/confirmed by various AI/machine learning people/researchers. For example, Prof. Hinton, a Google Fellow and a pioneer in AI from U. of Toronto, in an interview (GIGAOM, Jan. 16, 2017), stated that, “One problem we still haven't solved is getting neural nets to generalize well front small amounts of data, and I suspect that this may require radical changes in the types of neuron we use”. In addition, in another interview (Axion, Sep. 15, 2017), he strongly cast doubts about AI's current methodologies, and said that, “My view is throw it all away and start again”, Similarly, Mr. Suleyman (the head of Applied AI, now at DeepMind/Google) stated in an interview at TechCrunch (Dec. 5, 2016) that he thinks that the “general AI is still a long way off”.

[0195] So, to our knowledge, beyond the futuristic movies, wish-lists, science fiction novels, and generic non-scientific or non-technical articles (which have no basis/reliance/foundation on theory or experiment or proper/complete teachings), nobody has been successful in the application/usage/demonstration of General-AI, yet, in the AI industry or academia around the world. Thus, our demo/ZAC General-AI image Recognition Software Platform here is a very significant breakthrough in the field/science of AI and machine learning technology. (These results had been originally reported in our parent cases, as well.)

[0196] Please note that General-AI is also called/referred to as General Artificial Intelligence (GAI), or Artificial General Intelligence (AGI), or General-Purpose AI, or Strong Artificial Intelligence (AI), or True AI, or as we call it, Thinking-AI, or Reasoning-AI, or Cognition-AI, or Flexible-AI, or Full-Coverage-AI, or Comprehensive-AI, which can perform tasks that was never specifically trained for, e.g., in different context/environment, to recycle/re-use the experience and knowledge, using reasoning and cognition layers, usually in a completely different or unexpected or very new situation/condition/environment (same as what a human can do). Accordingly, we have shown here in this disclosure a new/novel/revolutionary architecture, system, method, algorithm, theory, and technique, to implement General-AI, e.g., for 3-D image/object recognition from any directions and other applications discussed here.

[0197] Our technology here (based on General-AI) is in contrast to (versus) Specific AI (or Vertical or Functional or Narrow or Weak AI) (or as we have coined the phrase, “Dumb-AI”), because, e.g., a Specific AI machine trained for face recognition cannot do any other tasks, e.g., finger-print recognition or medical imaging recognition. That is, the Specific AI machine cannot carry over/learn from any experience or knowledge that it has gained from one domain (face recognition) into another/new domain (finger-print or medical imaging), which it has not seen before (or was not trained for before). So, Specific AI has a very limited scope/“intelligence”/functionality/usage/re-usability/flexibility/usefulness.

[0198] Please note that the conventional/current state-of-the-art technologies in the industry/academia (e.g., Convolutional Neural Nets or Deep Learning) are based on the Specific AI, which has some major/serious theoretical/practical limits. For example, it cannot perform a 3-D image/object recognition from all directions, or cannot carry over/learn from any experience or knowledge in another domain, or requires extremely large number of training samples (which may not be available at all, or is impractical, or is too expensive, or takes too long to gather or train), or requires extremely large neural network (which cannot converge in the training stage, due to too much degree of freedom, or tends to memorize (rather than learn) the patterns (which is not good for out-of-sample recognition accuracy)), or requires extremely large computing power (which is impractical, or is too expensive, or is not available, or still cannot converge in the training stage). So, they have serious theoretical/practical limitations.

[0199] In addition, in Specific AI, if a new class of objects is added/introduced/found to the universe of all objects (e.g., a new animal/species is discovered), the training has to be done from scratch. Otherwise, training on just the last object will bias the whole learning machine, which is not good/accurate for recognition later on. Thus, all weights/biases or parameters in the learning machine must be erased completely, and the whole learning, with the new class added/mixed randomly with previous ones, must be repeated again from scratch, with all parameters erased and re-done/calculated again. So, the solution is not cumulative, or scalable, or practical, at all, e.g., for daily learning or continuous learning, as is the case for most practical situations, or as how the humans or most animals do/learn/recognize. So, they have serious theoretical/practical limitations.

[0200] Furthermore, for Specific AI, the learning phase cannot be mixed with the training phase. That is, they are not simultaneous, in the same period of time. So, during the training phase, the machine is useless or idle for all practical purposes, as it cannot recognize anything properly at that time. This is not how humans learn/recognize on a daily basis. So, they have serious theoretical/practical limitations.

[0201] General-AI solves/overcomes all of the above problems, as shown/discussed here in this disclosure. So, it has a huge advantage, for many reasons, as stated here, over Specific-AI.

[0202] It is also noteworthy that using smaller CPU/GPU power enables easier integration in mobile devices and wearables and IoT and telephones and watches, as an example, which, otherwise, drains the battery very quickly, and thus, requires much bigger battery or frequent recharging, which is not practical for most situations at all.

[0203] The industries/applications for our inventions are, e.g.:

• • [0204] a Mobile devices (e.g., phones, wearable devices, eyeglasses, tablets)
  • [0205] Smart devices & connected/Internet appliances
  • [0206] The Internet of Things (IoT), as the network of physical devices, vehicles, home appliances, wearables, mobile devices, stationary devices, wireless or cellular devices, BlueTooth or WiFi devices, and the like, embedded with electronics, software, sensors, actuators, mechanical parts, switches, and/or connectivity, which enables these objects to connect and exchange data/commands/info/trigger events,
  • [0207] Natural Language Processing
  • [0208] Photo albums & web sites containing pictures
  • [0209] Video libraries & web sites
  • [0210] Image and video search & summarization & directory & archiving & storage
  • [0211] Image & video Big Data analytics
  • [0212] Smart Camera
  • [0213] Smart Scanning Device
  • [0214] Social networks
  • [0215] Dating sites
  • [0216] Tourism
  • [0217] Real estate
  • [0218] Manufacturing
  • [0219] Biometrics
  • [0220] Security
  • [0221] Satellite or aerial images
  • [0222] Medical
  • [0223] Financial forecasting
  • [0224] Robotics vision & control
  • [0225] Control systems & optimization
  • [0226] Autonomous vehicles

[0227] We have the following usage examples: object/face recognition; rules engines & control modules; Computation with Words & soft boundaries; classification &. search; information web; data search & organizer & data mining & marketing data analysis; search for similar-looking locations or monuments; search for similar-looking properties; defect analysis; fingerprint, iris, and face recognition; Face/emotion/expression recognition, monitoring, tracking; recognition & information extraction, for security & map; diagnosis, using images & rules engines; and Pattern and data analysis & prediction; image ad network; smart cameras and phones; mobile and wearable devices; searchable albums and videos; marketing analytics; social network analytics; dating sites; security; tracking and monitoring; medical records and diagnosis and analysis, based on images; real estate and tourism, based on building, structures, and landmarks; maps and location services and security/intelligence, based on satellite or aerial images; big data analytics; deep image recognition and search platform; deep/detailed machine learning; object recognition (e.g., shoe, bag, clothing, watch, earring, tattoo, pants, hat, cap, jacket, tie, medal, wrist band, necklace, pin, decorative objects, fashion accessories, ring, food, appliances, equipment, tools, machines, cars, electrical devices, electronic devices, office supplies, office objects, factory objects, and the like).

[0228] Here, we also introduce Z-webs, including Z-factors and Z-nodes, for the understanding of relationships between objects, subjects, abstract ideas, concepts, or the like, including face, car, images, people, emotions, mood, text, natural language, voice, music, video, locations, formulas, facts, historical data, landmarks, personalities, ownership, family, friends, love, happiness, social behavior, voting behavior, and the like, to be used for many applications in our life, including on the search engine, analytics, Big Data processing, natural language processing, economy forecasting, face recognition, dealing with reliability and certainty, medical diagnosis, pattern recognition, object recognition, biometrics, security analysis, risk analysis, fraud detection, satellite image analysis, machine generated data analysis, machine learning, training samples, extracting data or patterns (from the video, images, text, or music, and the like), editing video or images, and the like. Z-factors include reliability factor, confidence factor, expertise factor, bias factor, truth factor, trust factor, validity factor, “trustworthiness of speaker”, “sureness of speaker”, “statement helpfulness”, “expertise of speaker”, “speaker's truthfulness”, “perception of speaker (or source of information)”, “apparent confidence of speaker”, “broadness of statement”, and the like, which is associated with each Z-node in the Z-web.

[0229] For one embodiment/example, e.g., we have “Usually, people wear short sleeve and short pants in Summer.”, as a rule number N given by an SME, e.g., human expert. The word “short” is a fuzzy parameter for both instances above. The sentence above is actually expressed as a Z-number, as described before, invented recently by Prof. Lotfi Zadeh, one of our inventors here. The collection of these rules can simplify the recognition of objects in the images, with higher accuracy and speed, e.g., as a hint, e.g., during Summer vacation, the pictures taken probably contain shirts with short sleeves, as a clue to discover or confirm or examine the objects in the pictures, e.g., to recognize or examine the existence of shirts with short sleeves, in the given pictures, taken during the Summer vacation. Having other rules, added in, makes the recognition faster and more accurate, as they can be in the web of relationships connecting concepts together, e.g., using our concept of Z-web, described before, or using semantic web. For example, the relationship between 4th of July and Summer vacation, as well as trip to Florida, plus shirt and short sleeve, in the image or photo, can all be connected through the Z-web, as nodes of the web, with Z numbers or probabilities in between on connecting branches, between each 2 parameters or concepts or nodes, as described before in this disclosure and in our prior parent applications.

[0230] In addition, there are many other embodiments in the current disclosure that deal with other important and innovative topics/subjects, e.g., related to General AI, versus Specific or Vertical or Narrow AI, machine learning, using/requiring only a small number of training samples (same as humans can do), learning one concept and use it in another context or environment (same as humans can do), addition of reasoning and cognitive layers to the learning module (same as humans can do), continuous learning and updating the learning machine continuously (same as humans can do), simultaneous learning and recognition (at the same time) (same as humans can do), and conflict and contradiction resolution (same as humans can do), with application, e.g., for image recognition, application for any pattern recognition, e.g., sound or voice, application for autonomous or driverless cars, application for security and biometrics, e.g., partial or covered or tilted or rotated face recognition, or emotion and feeling detections, application for playing games or strategic scenarios, application for fraud detection or verification/validation, e.g., for banking or cryptocurrency or tracking fund or certificates, application for medical imaging and medical diagnosis and medical procedures and drug developments and genetics, application for control systems and robotics, application for prediction, forecasting, and risk analysis, e.g., for weather forecasting, economy, oil price, interest rate, stock price, insurance premium, and social unrest indicators/parameters, and the like. (These results had been originally reported in our parent cases, as well.)

[0231] In one embodiment, we present a brief description of the basics of stratified programming (SP). SP is a computational system in which the objects of computation are in the main, nested strata of data centering on a target set, T. SP has a potential for significant applications in many fields, among them, robotics, optimal control, planning, multiobjective optimization, exploration, search, and Big Data. In spirit, SP has some similarity to dynamic programing (DP), but conceptually it is much easier to understand and much easier to implement. An interesting question which relates to neuro science is: Is the human brain employ stratification to store information? It will be natural to represent a concept such as a chair as a collection of strata with one or more strata representing a type of chair.

[0232] Underlining of our approach is a model, call it FSM. FSM is a finite state system. The importance of FSM as a model varies from use of digitalization (granulation, quantization) to almost any kind of system that can be approximated by a finite state system. The most important part is the concept of reachability of a target set in minimum number of steps. The objective of minimum number of steps serves as a basis for verification of the step of FSM state space. A concept which plays a key role in our approach is the target set reachability. Reachability involves moving (transitioning) FSM from a state ω to a state in target state, T, in a minimum number of steps. To this end, the state space, W, is stratified through the use of what is called the incremental enlargement principle. Reachability is also related to the concept of accessibility.

[0233] For the current inventions, we can combine/attach/integrate/connect any and all the systems and methods (or embodiments or steps or sub-components or algorithms or techniques or examples) of our own prior applications/teachings/spec/appendices/FIGS., which we have priority claim for, as mentioned in the current spec/application, to provide very efficient and fast algorithms for image processing, learning machines, NLT, pattern recognition, classification, SVM, deep/detailed analysis/discovery, and the like, for all the applications and usages mentioned here in this disclosure, with all tools, systems, and methods provided here.

[0574] This disclosure has many embodiments, systems, methods, algorithms, inventions, vertical applications, usages, topics, functions, variations, and examples. We divided them into sections for ease of reading, but they are all related and can be combined as one system, or as combination of subsystems and modules, in any combinations or just alone. We start here with the embodiment Z-number, and other inventions/embodiments will follow below after this section.

[0575] A Z-number is an ordered pair of fuzzy numbers, (A, B). For simplicity, in one embodiment, A and B are assumed to be trapezoidal fuzzy numbers. A Z-number is associated with a real-valued uncertain variable, X, with the first component, A, playing the role of a fuzzy restriction, R(X), on the values which X can take, written as X is A, where A is a fuzzy set. What should be noted is that, strictly speaking, the concept of a restriction has greater generality than the concept of a constraint. A probability distribution is a restriction but is not a constraint (see L. A. Zadeh, Calculus of fuzzy restrictions. In: L. A. Zadeh, K. S. Fu, K. Tanaka, and M. Shimura (Eds.), Fuzzy sets and Their Applications to Cognitive and Decision Processes, Academic Press, N.Y., 1975, pp. 1-39). A restriction may be viewed as a generalized constraint (see L. A. Zadeh, Generalized theory of uncertainty (GTU)—principal concepts and ideas, Computational Statistics &. Data Analysis 51, (2006) 15-46). In this embodiment only, the terms restriction and constraint are used interchangeably.

[0576] The restriction

[0577] R(X): X is A,

[0578] is referred to as a possibilistic restriction (constraint), with A playing the role of the possibility distribution of X. More specifically,

R(X): X is A→Poss(X=u)=μ_A(u)

[0579] where μ_A is the membership function of A, and u is a generic value of X. μ_A may be viewed as a constraint which is associated with R(X), meaning that μ_A(u) is the degree to which u satisfies the constraint.

[0580] When X is a random variable, the probability distribution of X plays the role of a probabilistic restriction on X. A probabilistic restriction is expressed as:

[0581] R(X): X isp p

[0582] where p is the probability density function of X. In this case,

R(X): X isp p→Prob(u≤X≤u+du)=p(u)du

[0583] Note Generally, the term “restriction” applies to X is R. Occasionally, “restriction” applies to R. Context serves to disambiguate the meaning of “restriction.”

[0584] The ordered triple (X,A, B) is referred to as a Z-valuation. A Z-valuation is equivalent to an assignment statement, X is (A, B). X is an uncertain variable if A is not a singleton. In a related way, uncertain computation is a system of computation in which the objects of computation are not values of variables but restrictions on values of variables. In this embodiment/section, unless stated to the contrary, X is assumed to be a random variable. For convenience, A is referred to as a value of X, with the understanding that, strictly speaking, A is not a value of X but a restriction on the values which X can take. The second component, B, is referred to as certainty. Certainty concept is related to other concepts, such as sureness, confidence, reliability, strength of belief, probability, possibility, etc. However, there are some differences between these concepts.

[0585] In one embodiment, when X is a random variable, certainty may be equated to probability. Informally, B may be interpreted as a response to the question: How sure are you that X is A? Typically, A and B are perception-based and are described in a natural language. Example: (about 45 minutes, usually.) A collection of Z-valuations is referred to as Z-information. It should he noted that much of everyday reasoning and decision-making is based, in effect, on Z-information. For purposes of computation. when A and B are described in a natural language, the meaning of A and B is precisiated (graduated) through association with membership functions, μ_A and μ_B, respectively, FIG. 1.

[0586] The membership function of A, μ_A, may be elicited by asking a succession of questions of the form: To what degree does the number, a, fit your perception of A? Example: To what degree does 50 minutes fit your perception of about 45 minutes? The same applies to B. The fuzzy set, A, may be interpreted as the possibility distribution of X. The concept of a Z-number may be generalized in various ways. In particular, X may be assumed to take values in Rⁿ, in which case A is a Cartesian product of fuzzy numbers. Simple examples of Z-valuations are:

[0587] (anticipated budget deficit, close to 2 million dollars, very likely)

[0588] (population of Spain, about 45 million, quite sure)

[0589] (degree of Robert's honesty, very high. absolutely)

[0590] (degree of Robert's honesty, high, not sure)

[0591] (travel time by car from Berkeley to San Francisco, about 30 minutes, usually)

[0592] (price of oil in the near future, significantly over 100 dollars/barrel, very likely)

[0593] It is important to note that many propositions in a natural language are expressible as Z-valuations. Example: The proposition, p,

[0594] p: Usually, it takes Robert about an hour to get home from work,

[0595] is expressible as a Z-valuation:

[0596] (Robert's travel time from office to home, about one hour, usually)

[0597] If X is a random variable, then X is A represents a fuzzy event in R, the real line. The probability of this event, p, may be expressed as (see L. A. Zadeh, Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications 23 (2), (1968) 421-427.):

$p=\underset{R}{\int }{\mu }_A\left(u\right)p_X\left(u\right)\mathrm{du},$

[0598] where p_X is the underlying (hidden) probability density of X. In effect, the Z-valuation (X,A, B) may be viewed as a restriction (generalized constraint) on X defined by:

[0599] Prob(X is A) is B.

[0600] What should be underscored is that in a Z-number, (A, B), the underlying probability distribution, p_X, is not known. What is known is a restriction on p_X which may be expressed as:

$\underset{R}{\int }{\mu }_A\left(u\right)p_X\left(u\right)\mathrm{du}\mathrm{is}B$

[0601] Note: In this embodiment only, the term “probability distribution” is not used in its strict technical sense.

[0602] In effect, a Z-number may be viewed as a summary of p_X. It is important to note that in everyday decision-making, most decisions are based on summaries of information. Viewing a Z-number as a summary is consistent with this reality. In applications to decision analysis, a basic problem which arises relates to ranking of Z-numbers. Example: is (approximately 100, likely) greater than (approximately 90, very likely)? Is this a meaningful question? We are going to address these questions below.

[0603] An immediate consequence of the relation between p_X and B is the following. If Z=(A, B) then Z′=(A′,1-B), where A′ is the complement of A and Z′ plays the role of the complement of Z. 1-B is the antonym of B (see, e.g., E. Trillas, C. Moraga, S. Guadarrama, S. Cubillo and E. Castiñeira, Computing with Antonyms. In: M. Nikravesh, J. Kacprzyk and L. A. Zadeh (Eds.). Forging New Frontiers: Fuzzy Pioneers I, Studies in Fuzziness and Soft Computing Vol 217, Springer-Verlag, Berlin Heidelberg 2007, pp. 133-153.).

[0604] An important qualitative attribute of a Z-number is informativeness. Generally, but not always, a Z-number is informative if its value has high specificity, that is, is tightly constrained (see, for example, R. R. Yager. On measures of specificity. In: O. Kaynak, L. A, Zadeh, B, Turksen, I. J. Rudas (Eds.), Computational intelligence: Soft Computing and Fuzzy-Neuro Integration with Applications, Springer-Verlag, Berlin, 1998, pp. 94-113), and its certainty is high. Informativeness is a desideratum when a Z-number is a basis for a decision. It is important to know that if the informativeness of a Z-number is sufficient to serve as a basis for an intelligent decision.

[0605] The concept of a Z-number is after the concept of a fuzzy granule (see, for example, L. A. Zadeh, Fuzzy sets and information granularity. In: M. Gupta, R. Ragade, R. Yager (Eds.), Advances in Fuzzy Set Theory and Applications, North-Holland Publishing Co., Amsterdam, 1979, pp. 3-18. Also, see L. A. Zadeh, Possibility theory and soft data analysis. In: L. Cobb, R. M. Thrall (Eds.), Mathematical Frontiers of the Social and Policy Sciences, Westview Press, Boulder, Colo., 1981, pp, 69-129. Also, see L.A_— Zadeh, Generalized theory of uncertainty (GTU)—principal concepts and ideas, Computational Statistics & Data Analysis 51, (2006) 15-46.). It should be noted that the concept of a Z-number is much more general than the concept of confidence interval in probability theory. There are some links between the concept of a Z-number, the concept of a fuzzy random number and the concept of a fuzzy random variable (see, e.g J. J. Buckley, J. J. Leonard, Chapter 4: Random fuzzy numbers and vectors. In: Monte Carlo Methods in Fuzzy Optimization, Studies in Fuzziness and Soft Computing 222, Springer-Verlag, Heidelberg, Germany, 2008. Also, see A. Kaufman, M. M. Gupta. Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold Company, New York, 1985. Also, see C. V. Negoita, D. A. Ralescu, Applications of Fuzzy. Sets to Systems Analysis, Wiley, N.Y., 1975).

[0606] A concept which is closely related to the concept of a Z-number is the concept of a Z⁺-number. Basically, a Z⁺-number, Z⁺, is a combination of a fuzzy number, A, and a random number, R, written as an ordered pair Z⁺=(A,R). In this pair, A plays the same role as it does in a Z-number, and R is the probability distribution of a random number. Equivalently, R may be viewed as the underlying probability distribution of X in the Z-valuation (X,A, B). Alternatively, a Z⁺-number may be expressed as (A,p_X) or (pA,p_X), where 1.₂k is the membership function of A. A Z⁺-valuation is expressed as (X,A,p_X) or, equivalently, as (X,μ_A,p_X), where p_X is the probability distribution (density) of X. A Z⁺-number is associated with what is referred to as a bimodal distribution, that is, a distribution which combines the possibility and probability distributions of X. Informally, these distributions are compatible if the centroids of μ_A and p_X are coincident, that is,

$\underset{R}{\int }u·p_X\left(u\right)·\mathrm{du}=\frac{\underset{R}{\int }u·{\mu }_A\left(u\right)·\mathrm{du}}{\underset{R}{\int }{\mu }_A\left(u\right)·\mathrm{du}}$

[0607] The scalar product of μ_A and p_X, μ_A·p_X, is the probability measure, P_A, of A. More concretely,

${\mu }_A·p_X=P_A=\underset{R}{\int }{\mu }_A\left(u\right)p_X\left(u\right)\mathrm{du}$

It is this relation that links the concept of a Z-number to that of a Z⁺-number. More concretely,

Z(A, B)=Z⁺(A,μ_A·p_X is B)

[0608] What should be underscored is that in the case of a Z-number what is known is not p_X but a restriction on p_X expressed as: μ_A·p_X is B. B_Y definition, a Z⁺-number carries more information than a Z-number. This is the reason why it is labeled a Z⁺-number. Computation with Z⁺-numbers is a portal to computation with Z-numbers.

[0609] The concept of a bimodal distribution is of interest in its own right. Let X be a real-valued variable taking values in U. For our purposes, it is convenient to assume that U is a finite set, U={u₁, . . . , u_n}. We can associate with X a possibility distribution, μ, and a probability distribution, p, expressed as:

μ=μ₁/u₁+. . . +μ_n/u_n

p=p₁\u₁+. . . +p_n\u_n

[0610] in which μ_i/u_i means that μ_ii=1, . . . n, is the possibility that X=u_i. Similarly, means that p_i is the probability that X=u_i.

[0611] The possibility distribution, μ, may be combined with the probability distribution, p, through what is referred to as confluence. More concretely,

μ:p=(μ₁, p₁)/u₁+. . . +(μ_n, p_n)/u_η

[0612] As was noted earlier, the scalar product, expressed as μ·p, is the probability measure of A. In terms of the bimodal distribution, the Z⁺-valuation and the Z-valuation associated with X may be expressed as:

(X, A, p_X)

(X, A, B), μ_A·p_X is B,

[0613] respectively, with the understanding that B is a possibilistic restriction on μ_A·p_X.

[0614] Both Z and Z⁺ may be viewed as restrictions on the values which X may take, written as: X is Z and X is Z⁺, respectively. Viewing Z and Z⁺ as restrictions on X adds important concepts to representation of information and characterization of dependencies. In this connection, what should be noted is that the concept of a fuzzy if-then rule plays a pivotal role in most applications of fuzzy logic. What follows is a very brief discussion of what are referred to as Z-rules—if-then rules in which the antecedents and/or consequents involve Z-numbers or Z⁺-numbers.

[0615] A basic fuzzy if-then rule may be expressed as: if X is A then Y is B, where A and B are fuzzy numbers. The meaning of such a rule is defined as:

if X is A then Y is B→(X,Y) is A×B

[0616] where A×B is the Cartesian product of A and B. It is convenient to express a generalization of the basic if-then rule to Z-numbers in terms of Z-valuations. More concretely, if (X, A_X, B_X) then (Y, A_Y, B_Y)

• • [0617] if (anticipated budget deficit, about two million dollars, very likely) then (reduction in staff, about ten percent, very likely)
  • [0618] if (degree of Robert's honesty, high, not sure) then (offer a position, not, sure)
  • [0619] if (X, small) then (Y, large, usually.)

[0620] An important question relates to the meaning of Z-rules and Z⁺-rules. The meaning of a Z⁺-rule may be expressed as:

if (X,A_X,p_X) then (Y, A_Y, p_Y)→(X,Y) is (A_X×A_Y,p_Xp_Y)

[0621] where a_X×A_Y is the Cartesian product A_X and A_Y

[0622] Z-rules have the important applications in decision analysis and modeling of complex systems, especially in the realm of economics (for example, stock market and specific stocks) and medicine (e.g. diagnosis and analysis).

[0623] A problem which plays a key role in many applications of fuzzy logic, especially in the realm of fuzzy control, is that of interpolation. More concretely, the problem of interpolation may be formulated as follows. Consider a collection of fuzzy if-then rules of the form:

if X is A_i then Y is B_i, i=1, . . . , n

[0624] where the A_i and B_i are fuzzy sets with specified membership functions, If X is A, where A is not one of the A_i, then what is the restriction on Y?

[0625] The problem of interpolation may be generalized m various ways. A generalization to Z-numbers may be described as follows. Consider a collection Z-rules of the form:

if X is A_i then usually (Y is B_i), i=1,

[0626] where the A_i and B_i are fuzzy sets. Let A be a fuzzy set which is not one of the A_i. What is the restriction on Y expressed as a Z-number? An answer to this question would add a useful formalism to the analysis of complex systems and decision processes.

[0627] Representation of Z-numbers can be facilitated through the use of what is called a Z-mouse. Basically, a Z-mouse is a visual means of entry and retrieval of fuzzy data.

[0628] The cursor of a Z-mouse is a circular fuzzy mark, called an f-mark, with a trapezoidal distribution of light intensity. This distribution is interpreted as a trapezoidal membership function of a fuzzy set. The parameters of the trapezoid are controlled by the user. A fuzzy number such as “approximately 3” is represented as an f-mark on a scale, with 3 being the centroid of the f-mark (FIG. 2a). The size of the f-mark is a measure of the user's uncertainty about the value of the number. As was noted already, the Z-mouse interprets an f-mark as the membership function of a trapezoidal fuzzy set. This membership function serves as an object of computation. A Z-mouse can be used to draw curves and plot functions.

[0629] A key idea which underlies the concept of a Z-mouse is that visual interpretation of uncertainty is much more natural than its description in natural language or as a membership function of a fuzzy set. This idea is closely related to the remarkable human capability to precisiate (graduate) perceptions, that is, to associate perceptions with degrees, As an illustration, if I am asked “What is the probability that Obama will be reelected?” I would find it easy to put an f-mark on a scale from 0 to I. Similarly, I could put an f-mark on a scale from 0 to 1 if I were asked to indicate the degree to which I like my job. It is of interest to note that a Z-mouse could be used as an informative means of polling, making it possible to indicate one's strength of feeling about an issue. Conventional polling techniques do not assess strength of feeling.

[0630] Using a Z-mouse, a Z-number is represented as two f-marks on two different scales (FIG. 2b). The trapezoidal fuzzy sets which are associated with the f-marks serve as objects of computation.

Computation with Z-Numbers:

[0631] What is meant by computation with Z-numbers? Here is a simple example. Suppose that I intend to drive from Berkeley to San Jose via Palo Alto. The perception-based information which I have may be expressed as Z-valuations: (travel time from Berkeley to Palo Alto, about an hour, usually) and (travel time from Palo Alto to San Jose, about twenty-five minutes, usually.) How long will it take me to drive from Berkeley to San Jose? In this case, we are dealing with the sum of two Z-numbers (about an hour, usually) and (about twenty-five minutes, usually.) Another example: What is the square root of (A, B)? Computation with Z-numbers falls within the province of Computing with Words (CW or CWW). Example: What is the square root of a Z-number?

[0632] Computation with Z⁺-numbers is much simpler than computation with Z-numbers. Assume that * is a binary operation whose operands are Z⁺-numbers, Z⁺_X=(A_X,R_X) and Z⁺_Y=(A_Y,R_Y.) By definition,

Z⁺_X*Z⁺_Y=(A_X*A_Y, R_X*R_Y)

[0633] with the understanding that the meaning of * in R_X*R_Y is not the same as the meaning of * in A_X*A_Y. In this expression, the operands of * in A_X*A_Y are fuzzy numbers; the operands of * in R_X*R_Y are probability distributions.

[0634] Example: Assume that * is sum. In this case, A_X+A_Y is defined by:

μ_(AX+Ay)(v)=sup_u(μ_AX(u) {circumflex over ( )}μ_AY(v−u)), {circumflex over ( )}=min

[0635] Similarly, assuming that R_X and R_Y are independent, the probability density function of R_X*R_Y is the convolution, ∘, of the probability density functions of R_X and R_Y. Denoting these probability density fiinctions as p_RX and p_Ry, respectively, we have:

$p_{R_X+R_Y}\left(v\right)=\underset{R}{\int }{p}_{R_X}\left(u\right)p_{R_Y}\left(v-u\right)\mathrm{du}$

[0636] Thus,

Z⁺_X+Z⁺_Y=(A_X+A_Y, P_RX∘P_Ry)

[0637] It should be noted that the assumption that R_X and R_Y are independent implies worst case analysis.

[0638] More generally, to compute Z_X*Z_Y what is needed is the extension principle of fuzzy logic (see, e.g., L. A. Zadeh, Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications 23 (2), (1968) 421-427.). Basically, the extension principle is a rule for evaluating a function when what are known are not the values of arguments but restrictions on the values of arguments. In other words, the rule involves evaluation of the value of a function under less than complete information about the values of arguments.

[0639] Note. Originally, the term “extension principle” was employed to describe a rule which serves to extend the domain of definition of a function from numbers to fuzzy numbers. In this disclosure, the term “extension principle” has a more general meaning which is stated in terms of restrictions. What should be noted is that, more generally, incompleteness of information about the values of arguments applies also to incompleteness of information about functions, in particular, about functions which are described as collections of if-then rules.

[0640] There are many versions of the extension principle. A basic version was given in the article: (L.A, Zadeh, Fuzzy sets. Information and Control 8, (1965) 338-353). In this version, the extension principle may be described as:

$Y=f\left(X\right)$ $\frac{R\left(X\right):X\mathrm{is}A\left(\mathrm{constraint}\mathrm{on}u\mathrm{is}{\mu }_A\left(u\right)\right)}{\begin{array}{cc}R\left(Y\right):{\mu }_Y\left(v\right)={\sup }_u{\mu }_A\left(u\right)& \left(f\left(A\right)=R\left(Y\right)\right)\end{array}}$ $\mathrm{subject}\mathrm{to}$ $v=f\left(u\right)$

[0641] where A is a fuzzy set, _hA is the membership function of A, μ_Y is the membership function of Y, and u and v are generic values of X and Y, respectively.

[0642] A discrete version of this rule is:

$Y=f\left(X\right)$ $\frac{R\left(X\right):X\mathrm{is}\left({\mu }_1/u_1+\dots +{\mu }_n/u_n\right)}{R\left(Y\right):{\mu }_Y\left(v\right)={\sup }_{u_1{\mathrm{\dots u}}_n}{\mu }_i}$ $\mathrm{subject}\mathrm{to}$ $v=f\left(u_i\right)$

[0643] In a more general version, we have

$Y=f\left(X\right)$ $\frac{R\left(X\right):g\left(X\right)\mathrm{is}A\left(\mathrm{constraint}\mathrm{on}u\mathrm{is}{\mu }_{A}g\left(u\right)\right))}{R\left(Y\right):{\mu }_Y\left(v\right)={\sup }_u{\mu }_A\left(g\left(u\right)\right)}$ $\mathrm{subject}\mathrm{to}$ $v=f\left(u\right)$

[0644] For a function with two arguments, the extension principle reads:

$Z=f\left(X,Y\right)$ $\frac{\begin{array}{c}R\left(X\right):g\left(X\right)\mathrm{is}A\left(\mathrm{constraint}\mathrm{on}u\mathrm{is}{\mu }_A\left(g\left(u\right)\right)\right)\\ R\left(Y\right):h\left(Y\right)\mathrm{is}B\left(\mathrm{constraint}\mathrm{on}u\mathrm{is}{\mu }_B\left(h\left(u\right)\right)\right)\end{array}}{R\left(Z\right):{\mu }_X\left(w\right)={\sup }_{u,v}\left({\mu }_X\left(g\left(u\right)\right)^{\mu }_Y\left(h\left(v\right)\right)\right),^=\min }$ $\mathrm{subject}\mathrm{to}$ $w=f\left(u,v\right)$

[0645] In application to probabilistic restrictions, the extension principle leads to results which coincide with standard results which relate to functions of probability distributions. Specifically, for discrete probability distributions, we have:

$Y=f\left(X\right)$ $\frac{R\left(X\right):X\mathrm{isp}p,p=p_1\backslash u_1+\dots p_n\backslash u_n}{\begin{array}{cc}R\left(Y\right):p_Y\left(v\right)={\sum }_i{p}_i& \left(f\left(p\right)=R\left(Y\right)\right)\end{array}}$ $\mathrm{subject}\mathrm{to}$ $v=f\left(u_i\right)$

[0646] For functions with two arguments, we have:

$Z=f\left(X,Y\right)$ $\frac{\begin{array}{c}R\left(X\right):X\mathrm{isp}p,p=p_1\backslash u_1+\dots p_m\backslash u_m\\ R\left(Y\right):Y\mathrm{isp}q=q_1\backslash v_1+\dots q_n\backslash v_n\end{array}}{\begin{array}{cc}R\left(Z\right):p_z\left(w\right)={\sum }_i{p}_i{q}_j& \left(f\left(p,q\right)=R\left(Z\right)\right)\end{array}}$ $\mathrm{subject}\mathrm{to}$ $w=f\left(u_i,v_j\right)$

[0647] For the case where the restrictions are Z⁺-numbers, the extension principle reads:

$Z=f\left(X,Y\right)$ $\frac{\begin{array}{c}R\left(X\right):X\mathrm{is}\left(A_X,p_X\right)\\ R\left(Y\right):Y\mathrm{is}\left(A_Y,p_Y\right)\end{array}}{R\left(Z\right):Z\mathrm{is}\left(f\left(A_X,A_Y\right),f\left(p_X,p_Y\right)\right)}$

[0648] It is this version of the extension principle that is the basis for computation with Z-numbers. Now, one may want to know if f(p_X, p_Y) is compatible with f(A_X, A_Y).

[0649] Turning to computation with Z-numbers, assume for simplicity that *=sum. Assume that Z_X=(A_X, B_X) and Z_Y=(A_Y, B_Y). Our problem is to compute the sum Z=X+Y. Assume that the associated Z-valuations are (X, A_X, B_X), (Y, A_Y, B_Y) and (Z, A_Z, B_Z).

[0650] The first step involves computation of p_Z. To begin with, let us assume that p_X and p_Y are known, and let us proceed as we did in computing the sum of Z⁺-numbers. Then

P_Z=p_X∘P_Y

[0651] or more concretely,

$p_Z\left(v\right)\underset{R}{\int }{p}_X\left(u\right)p_Y\left(v-u\right)\mathrm{du}$

[0652] In the case of Z-numbers what we know are not p_X and p_Y but restrictions on p_X and p_Y

$\underset{R}{\int }{\mu }_{A_X}\left(u\right)p_X\left(u\right)\mathrm{du}\mathrm{is}X$ $\underset{R}{\int }{\mu }_{A_Y}\left(u\right)p_Y\left(u\right)\mathrm{du}\mathrm{is}{B}_Y$

[0653] In terms of the membership functions of B_X and B_Y, these restrictions may be expressed as:

${\mu }_{B_X}\left({\int }_R{\mu }_{A_X}\left(u\right)p_X\left(u\right)\mathrm{du}\right)$ ${\mu }_{B_Y}\left({\int }_R{\mu }_{A_Y}\left(u\right)p_Y\left(u\right)\mathrm{du}\right)$

[0654] Additional restrictions on p_X and p_Y are:

${\int }_R{p}_X\left(u\right)\mathrm{du}=1$ ${\int }_R{p}_Y\left(u\right)\mathrm{du}=1$ ${\int }_R{\mathrm{up}}_X\left(u\right)\mathrm{du}=\frac{{\int }_{R}u{\mu }_{A_X}\left(u\right)\mathrm{du}}{{\int }_R{\mu }_{A_X}\left(u\right)\mathrm{du}}\left(\mathrm{compatibility}\right)$ ${\int }_R{\mathrm{up}}_Y\left(u\right)\mathrm{du}=\frac{{\int }_{R}u{\mu }_{A_Y}\left(u\right)\mathrm{du}}{{\int }_R{\mu }_{A_Y}\left(u\right)\mathrm{du}}\left(\mathrm{compatibility}\right)$

[0655] Applying the extension principle, the membership function of p_Z may be expressed as:

${\mu }_{p_Z}\left(p_Z\right)={\sup }_{p_X,p_Y}\left({\mu }_{B_X}\left({\int }_R^{}{\mu }_{A_X}\left(u\right)p_X\left(u\right)\mathrm{du}\right)\bigwedge {\mu }_{B_Y}\left({\int }_R^{}{\mu }_{A_Y}\left(u\right)p_Y\left(u\right)\mathrm{du}\right)\right)$ $\mathrm{subject}\mathrm{to}$ $p_Z=p_X\circ p_Y$ ${\int }_R{p}_X\left(u\right)\mathrm{du}=1$ ${\int }_R{p}_Y\left(u\right)\mathrm{du}=1$ ${\int }_R{\mathrm{up}}_X\left(u\right)\mathrm{du}=\frac{{\int }_{R}u{\mu }_{A_X}\left(u\right)\mathrm{du}}{{\int }_R{\mu }_{A_X}\left(u\right)\mathrm{du}}$ ${\int }_R{\mathrm{up}}_Y\left(u\right)\mathrm{du}=\frac{{\int }_{R}u{\mu }_{A_Y}\left(u\right)\mathrm{du}}{{\int }_R{\mu }_{A_Y}\left(u\right)\mathrm{du}}$

[0656] In this case, the combined restriction on the arguments is expressed as a conjunction of their restrictions, with {circumflex over ( )} interpreted as min. In effect, application of the extension principle reduces computation of p_Z to a problem in functional optimization. What is important to note is that the solution is not a value of p_Z but a restriction on the values of p_Z, consistent with the restrictions on p_X and p_Y.

[0657] At this point it is helpful to pause and summarize where we stand. Proceeding as if we are dealing with Z⁺-numbers, we arrive at an expression for p_Z as a function of p_X and p_Y. Using this expression and applying the extension principle we can compute the restriction on p_Z which is induced by the restrictions on p_X and p_Y. The allowed values of p_Z consist of those values of p_Z which are consistent with the given information, with the understanding that consistency is a matter of degree.

[0658] The second step involves computation of the probability of the fuzzy event, Z is A_Z, given p_Z. As was noted earlier, in fuzzy logic the probability measure of the fuzzy event X is A, where A is a fuzzy set and X is a random variable with probability density p_X, is defined as:

${\int }_R^{}{\mu }_A\left(u\right)p_X\left(u\right)\mathrm{du}$

[0659] Using this expression, the probability measure of A_Z may be expressed as:

$B_Z={\int }_R^{}{\mu }_{A_Z}\left(u\right)p_Z\left(u\right)\mathrm{du},\mathrm{where}$ ${\mu }_{A_Z}\left(u\right)={\sup }_v\left({\mu }_{A_X}\left(v\right)\bigwedge {\mu }_{A_Y}\left(u-v\right)\right)$

[0660] It should be noted that B_Z is a number when p_Z is a known probability density function. Since 3what we know about p_Z is its possibility distribution, μ_pZ(pz), B_Z is a fuzzy set with membership function μ_BZ. Applying the extension principle, we arrive at an expression for μ_BZ. More specifically,

${\mu }_{B_Z}\left(\omega \right)={\sup }_{p_Z}{\mu }_{p_Z}\left(p_Z\right)$ $\mathrm{subject}\mathrm{to}$ $w={\int }_R^{}{\mu }_{A_Z}\left(u\right)p_Z\left(u\right)\mathrm{du}$

where μ_pZ(pz) is the result of the first step. In principle, this completes computation of the sum of Z-numbers, Z_Xand Z_Y.

[0661] In a similar way, we can compute various functions of Z-numbers. The basic idea which underlies these computations may be summarized as follows, Suppose that our problem is that of computing f(Z_X, Z_Y), where Z_Xand Z_Y are Z-numbers, Z_X=(A_X, B_X) and Z_Y=(A_Y, B_Y), respectively, and f(Z_X, Z_Y)=(A_Z, B_Z). We begin by assuming that the underlying probability distributions p_X and p_Y are known. This assumption reduces the computation of f(Z_X, Z_Y) to computation of f(Z_X⁺, Z_Y⁺), which can be carried out through the use of the version of the extension principle which applies to restrictions which are Z⁺-numbers. At this point, we recognize that what we know are not p_X and p_Y but restrictions on p_X and p_Y. Applying the version of the extension principle which relates to probabilistic restrictions, we are led to f(Z_X, Z_Y). We can compute the restriction, B_Z, of the scalar product of f(A_X, A_Y) and f(p_X, p_Y). Since A_Z=f(A_X, A_Y), computation of B_Z completes the computation of f(Z_X, Z_Y).

[0662] It is helpful to express the summary as a version of the extension principle. More concretely, we can write:

$Z=f\left(X,Y\right)$ $X\mathrm{is}\left(A_X,B_X\right)\left(\mathrm{restriction}\mathrm{on}X\right)$ $Y\mathrm{is}\left(A_Y,B_Y\right)\left(\mathrm{restriction}\mathrm{on}Y\right)$ $Z\mathrm{is}\left(A_Z,B_Z\right)\left(\mathrm{induced}\mathrm{restriction}\mathrm{on}Z\right)$ $\stackrel{\_}{A_Z=f\left(A_X,A_Y\right)\left(\mathrm{application}\mathrm{of}\mathrm{extension}\mathrm{principle}\mathrm{for}\mathrm{fuzzy}\mathrm{numbers}\right)}$ $B_Z={\mu }_{A_Z}·f\left(p_X,p_Y\right)$

[0663] where p_X and p_Y are constrained by:

${\int }_R^{}{\mu }_{A_X}\left(u\right)p_X\left(u\right)\mathrm{du}\mathrm{is}{B}_X$ ${\int }_R^{}{\mu }_{A_Y}\left(u\right)p_Y\left(u\right)\mathrm{du}\mathrm{is}{B}_Y$

[0664] In terms of the membership functions of B_X and B_Y, these restrictions may be expressed as:

${{\mu }_B}_X\left({\int }_R^{}{\mu }_{A_X}\left(u\right)p_X\left(u\right)\mathrm{du}\right)$ ${{\mu }_B}_Y\left({\int }_R^{}{\mu }_{A_Y}\left(u\right)p_Y\left(u\right)\mathrm{du}\right)$

[0665] Additional restrictions on p_X and p_Y are:

${\int }_R{p}_X\left(u\right)\mathrm{du}=1$ ${\int }_R{p}_Y\left(u\right)\mathrm{du}=1$ ${\int }_R{\mathrm{up}}_X\left(u\right)\mathrm{du}=\frac{{\int }_{R}u{\mu }_{A_X}\left(u\right)\mathrm{du}}{{\int }_R{\mu }_{A_X}\left(u\right)\mathrm{du}}\left(\mathrm{compatibility}\right)$ ${\int }_R{\mathrm{up}}_Y\left(u\right)\mathrm{du}=\frac{{\int }_{R}u{\mu }_{A_Y}\left(u\right)\mathrm{du}}{{\int }_R{\mu }_{A_Y}\left(u\right)\mathrm{du}}\left(\mathrm{compatibility}\right)$