Skip to main content
SpringerLink
Log in

(Dual) Hoops Have Unique Halving

  • Chapter
Automated Reasoning and Mathematics

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7788))

Abstract

Continuous logic extends the multi-valued Łukasiewicz logic by adding a halving operator on propositions. This extension is designed to give a more satisfactory model theory for continuous structures. The semantics of these logics can be given using specialisations of algebraic structures known as hoops and coops. As part of an investigation into the metatheory of propositional continuous logic, we were indebted to Prover9 for finding proofs of important algebraic laws.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Anantharaman, S., Bonacina, M.P.: An Application of Automated Equational Reasoning to Many-Valued Logic. In: Okada, M., Kaplan, S. (eds.) CTRS 1990. LNCS, vol. 516, pp. 156–161. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  2. Arthan, R., Oliva, P.: Hoops, coops and the algebraic semantics of continuous logic (2012), http://arXiv.org/abs/1212.2887v1

  3. Ben Yaacov, I., Pedersen, A.P.: A proof of completeness for continuous first-order logic (2009), http://arxiv.org/0903.4051

  4. Bierman, G.M.: On intuitionistic linear logic. PhD thesis, University of Cambridge Computer Laboratory (December 1993)

    Google Scholar 

  5. Blok, W.J., Ferreirim, I.M.A.: On the structure of hoops. Algebra Universalis 43(2-3), 233–257 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonacina, M.P.: Problems in Łukasiewicz logic. Newsletter of the Association for Automated Reasoning 18, 5–12 (1991), http://www.AARInc.org

    Google Scholar 

  7. Büchi, J.R., Owens, T.M.: Complemented monoids and hoops (1975) (Unpublished manuscript)

    Google Scholar 

  8. Chang, C.C.: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88, 467–490 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chang, C.C.: A new proof of the completeness of the Łukasiewicz axioms. Trans. Amer. Math. Soc. 93, 74–80 (1959)

    MathSciNet  MATH  Google Scholar 

  10. Girard, J.-Y.: Linear logic. Theoretical Computer Science 50(1), 1–102 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers (1998)

    Google Scholar 

  12. Harris, K., Fitelson, B.: Distributivity in \({\L}_{\aleph_0}\) and other sentential logics. J. Autom. Reasoning 27, 141–156 (2001)

    Article  MATH  Google Scholar 

  13. Hay, L.S.: Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic 28, 77–86 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  14. Henson, C.W., Iovino, J.: Ultraproducts in analysis. In: Analysis and Logic. London Mathematical Society Lecture Notes, vol. 262, pp. 1–113. Cambridge University Press (2002)

    Google Scholar 

  15. Köhler, P.: Brouwerian semilattices. Trans. Amer. Math. Soc. 268, 103–126 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. C. R. Soc. Sc. Varsovie 23, 30–50 (1930)

    Google Scholar 

  17. McCune, W.: Prover9 and Mace4 (2005-2010), http://www.cs.unm.edu/~mccune/prover9/

  18. Raftery, J.G.: On the variety generated by involutive pocrims. Rep. Math. Logic 42, 71–86 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Slaney, J.K.: More proofs of an axiom of Łukasiewicz. J. Autom. Reasoning 29, 59–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Solovay, R., Arthan, R.D., Harrison, J.: Some new results on decidability for elementary algebra and geometry. Ann. Pure Appl. Logic 163(12), 1765–1802 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sutcliffe, G.: The TPTP Problem Library and Associated Infrastructure: The FOF and CNF Parts, v3.5.0. Journal of Automated Reasoning 43(4), 337–362 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Veroff, R., Spinks, M.: On a homomorphism property of hoops. Bulletin of the Section of Logic 33(3), 135–142 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Wos, L.: New challenge problem in sentential calculus. Newsletter of the Association for Automated Reasoning 16, 7–8 (1990), http://www.AARInc.org

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electronic Engineering and Computer Science, Queen Mary University of London, Mile End Road, London, E1 4NS, UK

    Rob Arthan & Paulo Oliva

Authors
  1. Rob Arthan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paulo Oliva
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Verona, 37134, Verona, Italy

    Maria Paola Bonacina

  2. Artificial Intelligence Center, SRI International, 333 Ravenswood Avenue, 94025, Menlo Park, CA, USA

    Mark E. Stickel

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Arthan, R., Oliva, P. (2013). (Dual) Hoops Have Unique Halving. In: Bonacina, M.P., Stickel, M.E. (eds) Automated Reasoning and Mathematics. Lecture Notes in Computer Science(), vol 7788. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36675-8_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-36675-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36674-1

  • Online ISBN: 978-3-642-36675-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation