Skip to main content
SpringerLink
Log in

Complexity issues in basic logic

  • Focus
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

We survey complexity results concerning a family of propositional many-valued logics. In particular, we shall address satisfiability and tautologousness problems for Hájek's Basic Logic BL and for several of its schematic extensions. We shall review complexity bounds obtained from functional representation results, as well as techniques for dealing with non-trivial ordinal sums of continuous t-norms.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Aglianò P, Montagna F (2003) Varieties of BL-algebras I: general properties. J Pure Appl Algebra 181:105–129

    Google Scholar 

  2. Aguzzoli S (1998) Geometric and proof–theoretic issues in Łukasiewicz propositional logics. PhD Thesis, University of Siena, Italy

  3. Aguzzoli S, Ciabattoni A (2000) Finiteness in infinite-valued Łukasiewicz logic. J Logic Lang Inf 9:5–29

    Google Scholar 

  4. Aguzzoli S, Gerla B (2002) Finite-valued reductions of infinite-valued logics. Arch Math Logic 41:361–399

    Google Scholar 

  5. Aguzzoli S, Gerla B (2002) On countermodels in basic logic. Neural Netw World 12:407–420

    Google Scholar 

  6. Baaz M, Hájek P, Krajíček J, švejda D (1998) Embedding logics into product logic. Studia Logica 61:35–42

    Google Scholar 

  7. Baaz M, Hájek P, Montagna F, Veith H (2002) Complexity of t-tautologies. Ann Pure Appl Logic 113:3–11

    Google Scholar 

  8. Burris S, Sankappanavar HP (1981) A course in universal algebra, Springer, Berlin Heidelberg New York

  9. Cignoli R, D'Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning, trends in logic, vol 7. Kluwer, Dordrecht

  10. Cignoli R, Esteva F, Godo L, Torrens A (2000) Basic logic is the logic of continuous t-norms and their residua. Soft Comput 4:106–112

    Google Scholar 

  11. Cintula P, Gerla B Semi-normal forms and functional representation of product fuzzy logic. Fuzzy Sets Syst (to appear)

  12. Esteva F, Godo L, Hájek P, Navara M (2000) Residuated fuzzy logics with an involutive negation. Arch Math Logic 39:106–112

    Google Scholar 

  13. Esteva F, Godo L, Montagna F (2003) Equational characterization of the subvarieties of BL generated by t-norm algebras, (preprint)

  14. Ewald G (1996) Combinatorial convexity and algebraic geometry, graduate texts in mathematics. vol 168 Springer, Berlin Heidelberg New York

  15. Gerla B (2000) A note on functions associated with Gödel formulas. Soft Comput 4:206–209

    Google Scholar 

  16. Gerla B (2000) Functional representation of many-valued logics based on continuous t-norms. PhD Thesis, University of Milano, Italy

  17. Gödel K (1933) Zum intuitionistischen Aussagenkalkül, Anzeiger Akademie der Wissenschaften Wien, Math.-naturwissensch. Klasse 69, (1932), 65–66. Also in Ergebnisse eines matematischen Kolloquiums 4:40

  18. Hähnle R (1993) Automated deduction in multiple-valued logics. Oxford University Press, Oxford

  19. Hähnle R (1994) Many-valued logic and mixed integer programming. Ann Math Artif Intell 12:231–264

    Google Scholar 

  20. Hähnle R (1997) Proof theory of many-valued logic – linear optimization – logic design: connections and interactions. Soft Comput 1:107–119

    Google Scholar 

  21. Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht

  22. Hájek P (1998) Basic fuzzy logic and BL-algebras. Soft Comput 2:124–128

    Google Scholar 

  23. Haniková Z (2001) Standard algebras for fuzzy propositional calculi. Fuzzy Sets Syst 124:309–320

    Google Scholar 

  24. Haniková Z (2002) A note on the complexity of propositional tautologies of individual t-algebras. Neural Netw World 12:453–460

    Google Scholar 

  25. Haniková Z (2003) Mathematical and metamathematical properties of fuzzy logic. PhD Thesis, Charles University in Prague

  26. McNaughton R (1951) A theorem about infinite-valued sentential logic. J Symbolic Logic 16:1–13

    Google Scholar 

  27. Montagna F (2000) The free BL-algebra on one generator. Neural Netw World 5:837–844

    Google Scholar 

  28. Montagna F (2001) Free BLΔ Algebras. In: Antonio Di Nola, Giangiacomo Gerla (eds) Lectures on soft computing and fuzzy logic, pp 159–171

  29. Mostert PS, Shields AL (1957) On the structure of semigroups on a compact manifold with boundary. Ann Math 65:117–143

    Google Scholar 

  30. Mundici D (1987) Satisfiability in many-valued sentential logic is NP-complete. Theor Comput Sci 52:145–153

    Google Scholar 

  31. Mundici D (1994) A constructive proof of McNaughton's theorem in infinite-valued logics. J Symbolic Logic 59:596–602

    Google Scholar 

  32. Novák V, Perfilieva I, Močkoř J (1999) Mathematical principles of fuzzy logic. Kluwer, Boston/Dordrecht/London

  33. Rose A, Rosser JB (1958) Fragments of many-valued statement calculi. Trans Am Math Soc 87:1–53

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DSI, University of Milano, Italy

    S. Aguzzoli

  2. DMI, University of Salerno, Italy

    B. Gerla

  3. ICS, Academy of Sciences of the Czech Republic,  

    Z. Haniková

Authors
  1. S. Aguzzoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Gerla
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. Haniková
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Aguzzoli.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aguzzoli, S., Gerla, B. & Haniková, Z. Complexity issues in basic logic. Soft Comput 9, 919–934 (2005). https://doi.org/10.1007/s00500-004-0443-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-004-0443-y

Keywords

Search

Navigation