Soft Computing

, Volume 21, Issue 1, pp 175–189 | Cite as

Density revisited

Focus

Abstract

In this (part survey) paper, we revisit algebraic and proof-theoretic methods developed by Franco Montagna and his co-authors for proving that the chains (totally ordered members) of certain varieties of semilinear residuated lattices embed into dense chains of these varieties, a key step in establishing standard completeness results for fuzzy logics. Such “densifiable” varieties are precisely the varieties that are generated as quasivarieties by their dense chains. By showing that all dense chains satisfy a certain e-cyclicity equation, we give a short proof that the variety of all semilinear residuated lattices is not densifiable (first proved by Wang and Zhao). We then adapt the Jenei–Montagna standard completeness proof for monoidal t-norm logic to show that any variety of integral semilinear residuated lattices axiomatized by additional lattice-ordered monoid equations is densifiable. We also generalize known results to show that certain varieties of cancellative semilinear residuated lattices are densifiable. Finally, we revisit the Metcalfe–Montagna proof-theoretic approach, which establishes densifiability of a variety via the elimination of a density rule for a suitable hypersequent calculus, focussing on the case of commutative semilinear residuated lattices.

Keywords

Many-valued logics Fuzzy logics Standard completeness Residuated lattices Semilinearity Density rule 

References

  1. Avron A (1991) Hypersequents, logical consequence and intermediate logics for concurrency. Ann Math Artif Intel 4(3–4):225–248CrossRefMATHMathSciNetGoogle Scholar
  2. Baaz M, Zach R (2000) Hypersequents and the proof theory of intuitionistic fuzzy logic. Proceedings of CSL 2000, vol 1862., lecture notes in computer scienceSpringer, Berlin, pp 187–201Google Scholar
  3. Baaz M, Ciabattoni A, Montagna F (2004) Analytic calculi for monoidal t-norm based logic. Fundam Inform 59(4):315–332MATHMathSciNetGoogle Scholar
  4. Baldi P (2014) A note on standard completeness for some extensions of uninorm logic. Soft Comput 18(8):1463–1470CrossRefMATHGoogle Scholar
  5. Baldi P, Ciabattoni A (2015a) Standard completeness for uninorm-based logics. In: Proceedings of ISMVL 2015, IEEE Computer Society Press, pp 78–83Google Scholar
  6. Baldi P, Ciabattoni A (2015b) Uniform proofs of standard completeness for extensions of first-order MTL. Theor Comput Sci 603:43–57CrossRefMATHMathSciNetGoogle Scholar
  7. Baldi P, Terui K (2016) Densification of FL chains via residuated frames. Algebra Univ 75(2):169–195CrossRefMATHMathSciNetGoogle Scholar
  8. Baldi P, Ciabattoni A, Spendier L (2012) Standard completeness for extensions of MTL: an automated approach. Proceedings of WoLLIC 2012, vol 8701, lecture notes in computer science. Springer, Berlin, pp 154–167Google Scholar
  9. Blount K, Tsinakis C (2003) The structure of residuated lattices. Int J Algebra Comput 13(4):437–461CrossRefMATHMathSciNetGoogle Scholar
  10. Botur M, Kühr J, Liu L, Tsinakis C (2015) The Conrad program: from l-groups to algebras of logic. J Algebra 450:173–203CrossRefMATHMathSciNetGoogle Scholar
  11. Chang CC (1958) Algebraic analysis of many-valued logics. Trans Am Math Soc 88:467–490CrossRefMATHMathSciNetGoogle Scholar
  12. Ciabattoni A, Metcalfe G (2008) Density elimination. Theor Comput Sci 403(1–2):328–346CrossRefMATHMathSciNetGoogle Scholar
  13. Ciabattoni A, Esteva F, Godo L (2002) T-norm based logics with n-contraction. Neural Netw World 12(5):441–453Google Scholar
  14. Ciabattoni A, Galatos N, Terui K (2008) From axioms to analytic rules in nonclassical logics. In: Proceedings of LICS 2008, IEEE Computer Society Press, pp 229–240Google Scholar
  15. Ciabattoni A, Galatos N, Terui K (2012) Algebraic proof theory for substructural logics: cut-elimination and completions. Ann Pure Appl Log 163(3):266–290Google Scholar
  16. Ciabattoni A, Galatos N, Terui K (2016) Algebraic proof theory for substructural logics: hypersequents. ManuscriptGoogle Scholar
  17. Cignoli R, D’Ottaviano ID, Mundici D (2000) Algebraic foundations of many-valued reasoning. Trends in logic. Kluwer, DordrechtGoogle Scholar
  18. Cignoli R, Esteva F, Godo L, Torrens A (2000) Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput 4:106–112CrossRefGoogle Scholar
  19. Cintula P, Noguera C (2011) A general framework for mathematical fuzzy logic. In: Cintula P, Hájek P, Noguera C (eds) Handbook of mathematical fuzzy logic, vol 1, studies in logic, mathematical logic and foundations, vol 37. College Publications, pp 103–207Google Scholar
  20. Cintula P, Noguera C (2016) Implicational (semilinear) logics III. ManuscriptGoogle Scholar
  21. Cintula P, Esteva F, Gispert J, Godo L, Montagna F, Noguera C (2009) Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann Pure Appl Logic 160(1):53–81CrossRefMATHMathSciNetGoogle Scholar
  22. Czelakowski J, Dziobiak W (1990) Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class. Algebra Univ 27:128–149CrossRefMATHMathSciNetGoogle Scholar
  23. Dummett M (1959) A propositional calculus with denumerable matrix. J Symb Log 24:97–106CrossRefMATHMathSciNetGoogle Scholar
  24. Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124(3):271–288CrossRefMATHMathSciNetGoogle Scholar
  25. Esteva F, Gispert L, Godo L, Montagna F (2002) On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic. Stud Log 71(2):199–226CrossRefMATHMathSciNetGoogle Scholar
  26. Galatos N, Horčik R (2016) Densification via polynomials, languages, and frames. J Pure Appl AlgebraGoogle Scholar
  27. Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics., Studies in logic and the foundations of mathematics. Elsevier, AmsterdamGoogle Scholar
  28. Gödel K (1932) Zum intuitionistischen Aussagenkalkül. Anzeiger der Akademie der Wissenschaftischen in Wien 69:65–66MATHGoogle Scholar
  29. Häjek P (1998) Metamathematics of fuzzy logic, trends in logic, vol 4. Kluwer, DordrechtCrossRefMATHGoogle Scholar
  30. Hájek P, Godo L, Esteva F (1996) A complete many-valued logic with product-conjunction. Arch Math Log 35:191–208CrossRefMATHMathSciNetGoogle Scholar
  31. Horčík R (2005) Standard completeness theorem for \(\varPi \)MTL. Arch Math Log 44(4):413–424CrossRefMATHMathSciNetGoogle Scholar
  32. Horčík R (2011) Algebraic semantics: semilinear FL-algebras. In: Cintula P, Häjek P, Noguera C (eds) Handbook of mathematical fuzzy logic, vol 1, studies in logic, mathematical logic and foundations, vol 37. College Publications, pp 283–353Google Scholar
  33. Horčík R, Montagna F, Noguera R (2006) On weakly cancellative fuzzy logics. J Log Comput 16(4):423–450CrossRefMATHMathSciNetGoogle Scholar
  34. Huss ME (1991) The lex property for varieties of lattice ordered groups. Algebra Univ 28:535–548CrossRefMathSciNetGoogle Scholar
  35. Jenei S, Montagna F (2002) A proof of standard completeness for Esteva and Godo fs logic MTL. Stud Log 70(2):183–192CrossRefMATHGoogle Scholar
  36. Jenei S, Montagna F (2003) A proof of standard completeness for non-commutative monoidal t-norm logic. Neural Netw World 13(5):481–489Google Scholar
  37. Jipsen P, Tsinakis C (2002) A survey of residuated lattices. In: Martinez J (ed) Ordered algebraic structures. Kluwer, Dordrecht, pp 19–56CrossRefGoogle Scholar
  38. Łukasiewicz J, Tarski A (1930) Untersuchungen über den Aussagenkalkül, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie 23(iii):30–50Google Scholar
  39. Metcalfe G (2003) Proof theory for propositional fuzzy logics. Ph.D. thesis, King’s College LondonGoogle Scholar
  40. Metcalfe G (2011) Proof theory of mathematical fuzzy logic. In: Cintula P, Häjek P, Noguera C (eds) Handbook of mathematical fuzzy logic—vol 1. Studies in logic, mathematical logic and foundations, vol 37. College Publications, pp 209–282Google Scholar
  41. Metcalfe G, Montagna F (2007) Substructural fuzzy logics. J Symb Log 72(3):834–864CrossRefGoogle Scholar
  42. Metcalfe G, Olivetti N, Gabbay D (2004) Analytic proof calculi for product logics. Arch Math Log 43(7):859–889CrossRefMATHGoogle Scholar
  43. Metcalfe G, Olivetti N, Gabbay D (2005) Sequent and hypersequent calculi for abelian and Łukasiewicz logics. ACM Trans Comput Log 6(3):578–613CrossRefMATHMathSciNetGoogle Scholar
  44. Metcalfe G, Olivetti N, Gabbay D (2008) Proof theory for fuzzy logics, applied logic, vol 36. Springer, BerlinMATHGoogle Scholar
  45. Metcalfe G, Paoli F, Tsinakis C (2010) Ordered algebras and logic, uncertainty and rationality. In: Hosni H, Montagna F (eds) Publications of the Scuola Normale Superiore di Pisa, vol 10, pp 1–85Google Scholar
  46. Montagna F, Ono H (2002) Kripke semantics, undecidability and standard completeness for Esteva and Godo’s logic MTL\(\forall \). Stud Log 71(2):227–245CrossRefMATHMathSciNetGoogle Scholar
  47. Ono H, Komori Y (1985) Logics without the contraction rule. J Symb Log 50:169–201CrossRefMATHMathSciNetGoogle Scholar
  48. Rose A, Rosser JB (1958) Fragments of many-valued statement calculi. Trans Am Math Soc 87:1–53Google Scholar
  49. Takeuti G, Titani T (1984) Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J Symb Log 49(3):851–866CrossRefMATHMathSciNetGoogle Scholar
  50. Wang S, Zhao B (2009) HpsUL is not the logic of pseudo-uninorms and their residua. Log J IGPL 17(4):413–419CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of BernBernSwitzerland
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA

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