Weierstrass Approximation Theorem and Łukasiewicz Formulas with one Quantified Variable

  • Stefano Aguzzoli
  • Daniele Mundici
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 114)


The logic ∃Ł of continuous piecewise linear functions with rational coefficients has enough expressive power to formalize Weierstrass approximation theorem. Thus, up to any prescribed error, every continuous (control) function can be approximated by a formula of ∃Ł. As shown in this work, ∃Ł is just infinite-valued Lukasiewicz propositional logic with one quantified propositional variable. We evaluate the computational complexity of the decision problem for ∃Ł. Enough background material is provided for all readers wishing to acquire a deeper understanding of the rapidly growing literature on Łukasiewicz propositional logic and its applications.


Rational Coefficient Simplicial Complex Toric Variety Piecewise Linear Function Propositional Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Aguzzoli, S.: The complexity of McNaughton functions of one variable, Advances in Applied Mathematics, 21 (1998) 58–77MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Aguzzoli, S., Ciabattoni, A.: Finiteness in infinite-valued Lukasiewicz logic. Journal of Logic, Language and Information. Special issue on Logics of Uncertainty Mundici, D. (Ed.), 9 (2000) 5–29Google Scholar
  3. 3.
    Aguzzoli, S., Mundici, D.: An algorithmic desingularization of 3—dimensional toric varieties. Tôhoku Mathematical Journal, 46 (1994) 557–572MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Amato, P., Porto, M.: An algorithm for the automatic generation of a logical formula representing a control law. Neural Network World, 10 (2000) 777–786Google Scholar
  5. 5.
    Baaz, M., Veith, H.: Interpolation in fuzzy logic. Archive for Mathematical Logic, 38 (1999) 461–489MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Berlekamp, E.R.: Block coding for the binary symmetric channel with noiseless, delayless feedback. In: Error-correcting Codes. Mann, H.B. (Ed.) Wiley, New York (1968) 330–335Google Scholar
  7. 7.
    Chang, C.C.: Algebraic analysis of many-valued logics. Transactions of the American Mathematical Society, 88 (1958) 467–490MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Chang, C.C.: A new proof of the completeness of the Lukasiewicz axioms. Transactions of the American Mathematical Society, 93 (1959) 74–90MathSciNetMATHGoogle Scholar
  9. 9.
    Chang, C.C.: The writing of the MV-algebras, Studia Logica, special issue on Many-valued Logics, Mundici, D. (Ed.) 61 (1998) 3–6Google Scholar
  10. 10.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D. Algebraic Foundations of Many-valued Reasoning (Trends in Logic, Studia Logica Library), Kluwer, Dordrecht (2000)CrossRefGoogle Scholar
  11. 11.
    Esteva, F., Godo, L., Montagna, F.: The LIT and Li/1/2 logics: two complete fuzzy systems joining Lukasiewicz and product logic, Archive for Mathematical Logic. 40 (2001) 39–67MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ewald, G.: Combinatorial convexity and algebraic geometry (Graduate Texts in Mathematics, vol. 168 ). Springer-Verlag, Berlin, Heidelberg, New York (1996)Google Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and intractability. A guide to the Theory of NP-completeness. W.H. Freeman and Company, San Francisco (1979)Google Scholar
  14. 14.
    Hâjek, P.: Metamathematics of fuzzy logic. Kluwer, Dordrecht. ( Trends in Logic, Studia Logica Library ) (1998)CrossRefGoogle Scholar
  15. 15.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms (Trends in Logic, Studia Logica Library).Kluwer, Dordrecht (2000)CrossRefGoogle Scholar
  16. 16.
    McNaughton, R.: A theorem about infinite-valued sentential logic. The Journal of Symbolic Logic, 16 (1951) 1–13MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential time. Proc. 13th Ann. Symp. on Switching and Automata Theory, IEEE Computer Society (1972) 125–129Google Scholar
  18. 18.
    Montagna, F.: An algebraic approach to propositional fuzzy logic. Journal of Logic, Language and Information. Special issue on Logics of Uncertainty Mundici, D. (Ed.), 9 (2000) 91–124Google Scholar
  19. 19.
    Mundici, D.: Interpretation of AF C’-algebras in Lukasiewicz sentential calculus. Journal of Functional Analysis, 65 (1986) 15–63MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Mundici, D.: Satisfiability in many-valued sentential logic is NP-complete. Theoretical Computer Science, 52 (1987) 145–153MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Mundici, D.: The logic of Ulam’s game with lies. In: Knowledge, Belief and Strategic Interaction. (Cambridge Studies in Probability, Induction and Decision Theory) Bicchieri, C., Dalla Chiara, M.L. (Eds.) Cambridge University Press (1992) 275–284CrossRefGoogle Scholar
  22. 22.
    Mundici, D.: A constructive proof of McNaughton’s Theorem in infinite-valued logic. The Journal of Symbolic Logic, 59 (1994) 596–602MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Mundici, D.: Lukasiewicz normal forms and toric desingularizations. In: Proceedings of Logic Colloquium 1993, Keele, England. Hodges, W. et al. (Eds.) Oxford University Press (1996) 401–423Google Scholar
  24. 24.
    Mundici, D.: Nonboolean partitions and their logic, In: First Springer-Verlag Forum on Soft Computing. Prague, August 1997, Soft Computing 2 (1998) 18–22Google Scholar
  25. 25.
    Mundici, D.: Tensor products and the Loomis-Sikorski theorem for MV-algebras. Advances in Applied Mathematics, 22 (1999) 227–248MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Mundici, D.: Ulam game, the logic of Maxsat, and many-valued partitions. In: Fuzzy Sets, Logics, and Reasoning about Knowledge, Dubois, D., Prade, H., Klement, E.P. (Eds.), Kluwer, Dordrecht (2000) 121–137Google Scholar
  27. 27.
    Mundici, D.: Reasoning on imprecisely defined functions, In: Discovering the World with Fuzzy Logic, Novak, V., Perfilieva, I. (Eds.), Physica-Verlag, Springer, Heidelberg, New York (2000) 331–366Google Scholar
  28. 28.
    Mundici, D., Olivetti, N.: Resolution and model building in the infinite-valued calculus of Lukasiewicz. Theoretical Computer Science, 200 (1998) 335–366MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Mundici, D., Riecan, B.: Probability on MV-algebras, In: Handbook of Measure Theory, Pap, E. (ed.), North-Holland, Amsterdam, to appear (200?)Google Scholar
  30. 30.
    Novak, V., Perfilieva, I., MoCkot, J.: Mathematical Principles of Fuzzy Logic. Kluwer, Dordrecht (1999)MATHCrossRefGoogle Scholar
  31. 31.
    Panti, G.: A geometric proof of the completeness of the Lukasiewicz calculus. Journal of Symbolic Logic, 60 (1995) 563–578MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Panti, G.: Multi-valued Logics. In: Quantified Representation of Uncertainty and Imprecision, vol. 1, Smets, P. (Ed.), Kluwer, Dordrecht (1998) 25–74Google Scholar
  33. 33.
    Pavelka, J.: On Fuzzy Logic I,II,III. Zeitschrift für math. Logik and Grundlagen der Mathematik, 25 (1979) 45–52, 119–134, 447–464Google Scholar
  34. 34.
    RieCan, B., Neubrunn, T.: Integral, Measure, and Ordering. Kluwer, Dordrecht (1997)MATHGoogle Scholar
  35. 35.
    Rose, A., Rosser, J.B.: Fragments of many-valued statement calculi. Transactions of the American Mathematical Society, 87 (1958) 1–53MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Tarski, A.: Logic, Semantics, Metamathematics. Clarendon Press, Oxford ( 1956 ). Reprinted, Hackett, Indianapolis (1983).Google Scholar
  37. 37.
    Verbruggen, H.B., Bruijn, P.M.: Fuzzy control and conventional control: What is (and can be) the real contribution of Fuzzy Systems?, Fuzzy Sets and Systems, 90 (1997) 151–160MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Stefano Aguzzoli
    • 1
  • Daniele Mundici
    • 1
  1. 1.Department of Computer ScienceUniversity of MilanMilanItaly

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