Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Computing in Łukasiewicz Logic and AF-Algebras

  • Chapter
  • First Online:
The Logic of Software. A Tasting Menu of Formal Methods

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13360))

  • 643 Accesses

Abstract

Since the beginning of his career as a computer scientist, Reiner Hähnle has made important contributions to the proof theory of Łukasiewicz logic Ł\(_\infty \), e.g., applying Mixed Integer Programming techniques to the satisfiability problem for the Łukasiewicz calculus. His work in this area culminated in a monograph on automated deduction in many-valued logic and other key contributions on this vibrating field of research. The present paper discusses recent developments in Łukasiewicz logic Ł\(_\infty \), its associated algebraic semantics given by C.C.Chang MV-algebras, and related computational issues concerning the approximately finite-dimensional (AF) C*-algebras of quantum statistical mechanics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The attentive reader will have noted the key role of (truth-)functionality in this definition, hinging upon the nonambiguity of the syntax of F.

  2. 2.

    Once \(\{0,1\}\) is equipped with the discrete topology, boolean implication is trivially continuous.

  3. 3.

    When A is a boolean algebra, its states are also known as “finitely additive probability measures”.

  4. 4.

    see, e.g., Corollary 1 in T.Tao, https://terrytao.wordpress.com/2009/01/03/.

  5. 5.

    By a traditional abuse of notation, the MV-algebraic operation symbols also denote their corresponding operations.

References

  1. Behncke, H., Leptin, H.: C*-algebras with a two-point dual. J. Funct. Anal. 10(3), 330–335 (1972). https://doi.org/10.1016/0022-1236(72)90031-6

    Article  MathSciNet  MATH  Google Scholar 

  2. Bigard, A., Wolfenstein, S., Keimel, K.: Groupes et Anneaux Réticulés. LNM, vol. 608. Springer, Heidelberg (1977). https://doi.org/10.1007/BFb0067004

    Book  MATH  Google Scholar 

  3. Blackadar, B.: A simple C*-algebra with no nontrivial projections. Proc. Amer. Math. Soc. 78, 504–508 (1980). https://www.jstor.org/stable/2042420

  4. Boca, F.: An AF algebra associated with the Farey tessellation. Canad. J. Math. 60, 975–1000 (2008). https://doi.org/10.4153/CJM-2008-043-1

    Article  MathSciNet  MATH  Google Scholar 

  5. Bratteli, O.: Inductive limits of finite-dimensional C*-algebras. Trans. Amer. Math. Soc. 171 , 195–234 (1972). https://www.jstor.org/stable/1996380

  6. Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88, 467–490 (1958). https://www.jstor.org/stable/1993227

  7. Chang, C.C.: A new proof of the completeness of the Łukasiewicz axioms. Trans. Amer. Math. Soc. 93, 74–90 (1959). https://www.jstor.org/stable/1993423

  8. Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic foundations of many-valued reasoning. Trends in Logic, vol. 7, Kluwer, Dordrecht (2000). Reprinted, Springer (2013). https://doi.org/10.1007/978-94-015-9480-6

  9. Cignoli, R., Elliott, G.A., Mundici, D.: Reconstructing C*-algebras from their Murray von Neumann order. Adv. Math. 101, 166–179 (1993). https://doi.org/10.1016/j.jpaa.2003.10.021

    Article  MathSciNet  MATH  Google Scholar 

  10. Davidson, K.R.: C*-Algebras by Example, Fields Institute Monographs. American Mathematical Society, Providence, vol. 6 (1996). https://doi.org/10.1090/fim/006

  11. Dixmier, J.: C*-algebras, North-Holland Mathematical Library. North-Holland, Amsterdam, vol. 15(1977). https://www.sciencedirect.com/bookseries/north-holland-mathematical-library/vol/15/suppl/C

  12. Effros, E.G.: Dimensions and C*-algebras, CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, vol. 46 (1981). https://bookstore.ams.org/cbms-46/20ISBN:78-0-8218-1697-4

  13. Elliott, G.A.: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra 38, 29–44 (1976). https://doi.org/10.1016/0021-8693(76)90242-8

    Article  MathSciNet  MATH  Google Scholar 

  14. Elliott, G.A.: On totally ordered groups, and \(K_0\). In: Handelman, D., Lawrence, J. (eds.) Ring Theory Waterloo 1978 Proceedings, University of Waterloo, Canada, 12–16 June 1978. LNM, vol. 734, pp. 1–49. Springer, Heidelberg (1979). https://doi.org/10.1007/BFb0103152

    Chapter  Google Scholar 

  15. Hähnle, R.: Tableaux-based theorem proving in multiple-valued logics, Ph.D. thesis, University of Karlsruhe, Department of Computer Science (1992). http://tubiblio.ulb.tu-darmstadt.de/101360/

  16. Beckert, B., Hähnle, R., Gerberding, S., Kernig, W.: The tableau-based theorem prover \(_3\)T\({^A}\)P for multiple-valued logics. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 758–760. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-55602-8_219

    Chapter  Google Scholar 

  17. Hähnle, R.: Short CNF in finitely-valued logics. In: Komorowski, J., Raś, Z.W. (eds.) ISMIS 1993. LNCS, vol. 689, pp. 49–58. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56804-2_5

    Chapter  Google Scholar 

  18. Hähnle, R.: Automated Deduction in Multiple-Valued Logics. Oxford University Press (1994). https://global.oup.com/academic/product/automated-deduction-in-multiple-valued-logics-9780198539896?lang=en&cc=nl ISBN: 9780198539896

  19. Hähnle, R.: Many-valued logic and mixed integer programming. Ann. Math. Artif. Intell. 12, 231–263 (1994). https://doi.org/10.1007/BF01530787

    Article  MathSciNet  MATH  Google Scholar 

  20. Hähnle, R.: Automated deduction and integer programming. In: Collegium Logicum (Annals of the Kurt-Gödel-Society), vol. 1, pp. 67–77. Springer, Vienna (1995). https://doi.org/10.1007/978-3-7091-9394-5_6

  21. Hähnle, R.: Exploiting data dependencies in many-valued logics. J. Appl. Non-Classical Logics 6(1), 49–69 (1996). https://doi.org/10.1080/11663081.1996.10510866

    Article  MathSciNet  MATH  Google Scholar 

  22. Hähnle, R.: Proof theory of many-valued logic-linear optimization-logic design: connections and interactions. Soft Comput. 1(3), pp. 107–119 (1997). https://link.springer.com/content/pdf/10.1007%2Fs005000050012.pdf

  23. Hähnle, R.: Commodious axiomatization of quantifiers in multiple-valued logic. Stud. Logica 61(1), 101–121 (1998). https://doi.org/10.1023/A:1005086415447

    Article  MathSciNet  MATH  Google Scholar 

  24. Hähnle, R., Bernhard, B., Escalada-Imaz, G.: Simplification of many-valued logic formulas using anti-links. J. Logic Comput. 8(4), 569–588 (1998). https://doi.org/10.1093/logcom/8.4.569

  25. Hähnle, R.: Tableaux for many-valued logics. In: D’Agostino, M., et al. (eds.) Handbook of Tableau Methods, pp. 529–580. Kluwer, Dordrecht (1999). https://link.springer.com/book/10.1007%2F978-94-017-1754-0

  26. Hähnle, R., Beckert, B., Manyá, F.: Transformations between signed and classical clause logic. In: Proceedings, 29th IEEE International Symposium on Multiple-Valued Logic (ISMVL 1999), pp. 248–255. https://doi.org/10.1109/ISMVL.1999.779724

  27. Hähnle, R., Beckert, B., Manyá, F.: The 2-SAT problem of regular signed CNF formulas. In: Proceedings 30th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2000), pp. 331–336. https://doi.org/10.1109/ISMVL.2000.848640

  28. Hähnle, R., Beckert, B., Manyá, F.: The SAT problem of signed CNF formulas. In: Basin, D., et al. (eds.) Labelled Deduction. Applied Logic Series, vol. 17. Springer, Dordrecht (2000). https://doi.org/10.1007/978-94-011-4040-9_3

  29. Hähnle, R.: Short conjunctive normal forms in finitely valued logics. J. Logic Comput. 4, 905–927 (2000). https://doi.org/10.1093/logcom/4.6.905

    Article  MathSciNet  MATH  Google Scholar 

  30. Hähnle, R.: Proof theory of many-valued logic and linear optimization. In: B. Reusch et al. (eds.) Advances in Soft Computing. Computational Intelligence in Theory and Practice. Advances in Soft Computing. Physica, Heidelberg, vol. 8, pp. 15–33 (2001). https://doi.org/10.1007/978-3-7908-1831-4_2

  31. Hähnle, R.: Advanced Many-Valued Logics. In: Gabbay, D.M., et al. (eds.) Handbook of Philosophical Logic, vol. 2, pp. 297–395. Springer, Dordrecht (2001). https://doi.org/10.1007/978-94-017-0452-6_5

  32. Hähnle, R.: Tableaux and related methods. In: Handbook of Automated Reasoning 1, Elsevier Science Publishers B.V., pp. 101–178 (2001). https://doi.org/10.1016/B978-044450813-3/50005-9

  33. Hähnle, R.: Complexity of many-valued logics. In: Fitting, M., et al. (eds.) Beyond Two: Theory and Applications of Multiple-Valued Logic, Studies in Fuzziness and Soft Computing. Physica, Heidelberg, vol. 114, pp. 211–233 (2003). https://doi.org/10.1007/978-3-7908-1769-0_9

  34. Hähnle, R.: Many-valued logic, partiality, and abstraction in formal specification languages. Logic J. IPGL 13(4), 415–433 (2005). https://doi.org/10.1093/jigpal/jzi032

    Article  MathSciNet  MATH  Google Scholar 

  35. Kolařík, M.: Independence of the axiomatic system for MV-algebras. Math. Slovaca 63, 1–4 (2013). https://doi.org/10.2478/s12175-012-0076-z

    Article  MathSciNet  MATH  Google Scholar 

  36. Kroupa, T.: Every state on a semisimple MV-algebra is integral. Fuzzy Sets Syst. 157, 2771–2782 (2006). https://doi.org/10.1016/j.fss.2006.06.015

    Article  MathSciNet  MATH  Google Scholar 

  37. McNaughton, R.: A theorem about infinite-valued sentential logic. J. Symbolic Logic, 16, 1–13 (1951). https://www.jstor.org/stable/2268660

  38. Mundici, D.: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986). https://core.ac.uk/download/pdf/81941332.pdf

  39. Mundici, D.: Farey stellar subdivisions, ultrasimplicial groups, and \(K_0\) of AF C*-algebras. Adv. Math. 68, 23–39 (1988). https://doi.org/10.1016/0001-8708(88)90006-0

    Article  MathSciNet  MATH  Google Scholar 

  40. Mundici, D.: Recognizing the Farey-Stern-Brocot AF algebra. Rendiconti Lincei Mat. Appl., 20, 327–338 (2009). https://ems.press/journals/rlm/articles/2994, Dedicated to the memory of Renato Caccioppoli

  41. Mundici, D.: Advanced Łukasiewicz calculus and MV-algebras. Trends in Logic, vol. 35. Springer, Berlin (2011). https://link.springer.com/book/10.1007%2F978-94-007-0840-2

  42. Mundici, D.: Word problems in Elliott monoids. Adv. Math. 335, 343–371 (2018). https://doi.org/10.1016/j.aim.2018.07.015

    Article  MathSciNet  MATH  Google Scholar 

  43. Mundici, D.: What the Łukasiewicz axioms mean. J. Symbolic Logic 85, 906–917 (2020). https://doi.org/10.1017/jsl.2020.74

    Article  MathSciNet  MATH  Google Scholar 

  44. Mundici, D.: Bratteli diagrams via the De Concini-Procesi theorem. Commun. Contemp. Math. 23(07), 2050073 (2021). https://doi.org/10.1142/S021919972050073X

    Article  MathSciNet  MATH  Google Scholar 

  45. Mundici, D., Panti, G.: Extending addition in Elliott’s local semigroup. J. Funct. Anal. 117, 461–471 (1993). https://doi.org/10.1006/jfan.1993.1134

    Article  MathSciNet  MATH  Google Scholar 

  46. Panti, G.: Invariant measures in free MV-algebras. Commun. Algebra 36, 2849–2861 (2009). https://doi.org/10.1080/00927870802104394

    Article  MathSciNet  MATH  Google Scholar 

  47. Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967). https://doi.org/10.1016/C2013-0-08107-8

    Book  MATH  Google Scholar 

  48. Rørdam, M., Størmer, E.: Classification of Nuclear C*-Algebras, Entropy in Operator Algebras, Encyclopaedia of Mathematical Sciences, Operator Algebras and Non-Commutative Geometry, vol. 126. Springer-Verlag, Berlin, Heidelberg (2002). https://doi.org/10.1007/978-3-662-04825-2 ISBN 978-3-540-42305-8

  49. Wójcicki, R.: Lectures on Propositional Calculi. Publishing House of the Polish Academy of Sciences, Ossolineum (1984). http://sl.fr.pl/wojcicki/Wojcicki-Lectures.pdf

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science “Ulisse Dini”, University of Florence, Viale Morgagni 67/A, 50134, Florence, Italy

    Daniele Mundici

Authors
  1. Daniele Mundici
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniele Mundici .

Editor information

Editors and Affiliations

  1. Chalmers University of Technology, Gothenburg, Sweden

    Wolfgang Ahrendt

  2. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Bernhard Beckert

  3. TU Darmstadt, Darmstadt, Germany

    Richard Bubel

  4. University of Oslo, Oslo, Norway

    Einar Broch Johnsen

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mundici, D. (2022). Computing in Łukasiewicz Logic and AF-Algebras. In: Ahrendt, W., Beckert, B., Bubel, R., Johnsen, E.B. (eds) The Logic of Software. A Tasting Menu of Formal Methods. Lecture Notes in Computer Science, vol 13360. Springer, Cham. https://doi.org/10.1007/978-3-031-08166-8_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-08166-8_18

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-08165-1

  • Online ISBN: 978-3-031-08166-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation