Studia Logica

, Volume 87, Issue 2–3, pp 145–169 | Cite as

Type Logics and Pregroups

Article

Abstract

We discuss the logic of pregroups, introduced by Lambek [34], and its connections with other type logics and formal grammars. The paper contains some new ideas and results: the cut-elimination theorem and a normalization theorem for an extended system of this logic, its P-TIME decidability, its interpretation in L1, and a general construction of (preordered) bilinear algebras and pregroups whose universe is an arbitrary monoid.

Keywords

bilinear algebra pregroup Lambek calculus substructural logics 

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References

  1. 1.
    Abrusci V.M., (1991) ‘Phase semantics and sequent system for pure noncommutative classical propositional logic’. Journal of Symbolic Logic 56:1403–1454CrossRefGoogle Scholar
  2. 2.
    Ajdukiewicz K., (1935) ‘Die syntaktische Konnexität’. Studia Philosophica 1:1–27Google Scholar
  3. 3.
    Bar-Hillel Y., Gaifman C., Shamir E. (1960) ‘On categorial and phrase structure grammars’. Bull. Res Council Israel F9:155–166Google Scholar
  4. 4.
    Béchet, D., ‘Parsing Pregroup Grammars and the Lambek Calculus using Partial Composition’, this issue.Google Scholar
  5. 5.
    van Benthem J., (1986) Essays in Logical Semantics. D. Reidel, DordrechtGoogle Scholar
  6. 6.
    van Benthem J., (1991) Language in Action. Categories, Lambdas and Dynamic Logic. North-Holland, AmsterdamGoogle Scholar
  7. 7.
    van Benthem J., (1996) Exploring Logical Dynamics. CSLI, StanfordGoogle Scholar
  8. 8.
    van Benthem J., Ter Meulen A. (eds) (1997) Handbook of Logic and Language. Elsevier, AmsterdamGoogle Scholar
  9. 9.
    Bulińska, M., ‘P-TIME Decidabilty of NL1 with Assumptions’, Electronic Proc. Formal Grammars 2006, 29–38.Google Scholar
  10. 10.
    Buszkowski W., (1982) ‘Some decision problems in the theory of syntactic categories’. Zeitschrift f. math. Logik und Grundlagen d. Math. 28:539–548CrossRefGoogle Scholar
  11. 11.
    Buszkowski W., (1986) ‘Completeness results for Lambek Syntactic Calculus’. Zeitschrift f. math. Logik und Grundlagen d. Math. 32:13–28CrossRefGoogle Scholar
  12. 12.
    Buszkowski W., (1986) ‘Generative Capacity of Non-Associative Lambek Calculus’. Bulletin of Polish Academy of Sciences. Math. 34:507–516Google Scholar
  13. 13.
    Buszkowski, W., ‘Generative Power of Categorial Grammars’, [40]:69–94.Google Scholar
  14. 14.
    Buszkowski, W., ‘Lambek Grammars Based on Pregroups’, [24]:95–109.Google Scholar
  15. 15.
    Buszkowski W., (2002) ‘Pregroups: Models and Grammars’. Relational Methods in Computer Science, LNCS 2561:35–49CrossRefGoogle Scholar
  16. 16.
    Buszkowski W., (2003) ‘Sequent Systems for Compact Bilinear Logic’. Mathematical Logic Quarterly, 49:467–474CrossRefGoogle Scholar
  17. 17.
    Buszkowski, W., ‘Lambek Calculus with Non-Logical Axioms’, [19]:77–93.Google Scholar
  18. 18.
    Buszkowski W., (2007) ‘On Action Logic: Equational Theories of Action Algebras’. Journal of Logic and Computation 17.1:199–217Google Scholar
  19. 19.
    Casadio, C., P.J. Scott, and R. Seely (eds.), Language and Grammar. Studies in Mathematical Linguistics and Natural Language, CSLI Lecture Notes 168, Stanford, 2005.Google Scholar
  20. 20.
    Fadda, M., ‘Towards flexible pregroup grammars’, New Perspectives in Logic and Formal Linguistics, 95–112, Bulzoni Editore, Roma, 2002.Google Scholar
  21. 21.
    Farulewski, M., ‘Finite Embeddability Property of Residuated Ordered Groupoids’, Reports on Mathematical Logic. To appear.Google Scholar
  22. 22.
    Francez, N. and M. Kaminski, ‘Commutation-Augmented Pregroup Grammars and Mildly Context-Sensitive Languages’, this issue.Google Scholar
  23. 23.
    Girard J.-Y., (1987) ‘Linear logics’. Theoretical Computer Science 50:1–102CrossRefGoogle Scholar
  24. 24.
    de Groote, P., G. Morrill, and C. Retoré (eds.), Logical Aspects of Computational Linguistics, LNAI 2099, Springer, 2001.Google Scholar
  25. 25.
    Jäger G., (2004) ‘Residuation, structural rules and context-freeness’. Journal of Logic, Language and Information 13:47–59CrossRefGoogle Scholar
  26. 26.
    Jipsen P., (2004) ‘From Semirings to Residuated Kleene Algebras’. Studia Logica 76:291–303CrossRefGoogle Scholar
  27. 27.
    Kanazawa M., (1992) ‘The Lambek Calculus Enriched with Additional Connectives’. Journal of Logic, Language and Information 1.2:141–171CrossRefGoogle Scholar
  28. 28.
    Kandulski M., (1988) ‘The equivalence of nonassociative Lambek categorial grammars and context-free grammars’. Zeitschrift f. math. Logik und Grundlagen d. Math. 34:41–52CrossRefGoogle Scholar
  29. 29.
    Kandulski M., (1993) ‘Normal Form of Derivations for the Nonassociative and Commutative Lambek Calculus with Product’. Mathematical Logic Quarterly 39:103–114CrossRefGoogle Scholar
  30. 30.
    Kiślak-Malinowska, A., ‘On the logic of β pregroups’, this issue.Google Scholar
  31. 31.
    Lambek J., (1958) ‘The mathematics of sentence structure’. American Mathematical Monthly 65:154–170CrossRefGoogle Scholar
  32. 32.
    Lambek J., ‘On the calculus of syntactic types’, Structure of Language and Its Mathematical Aspects. Proc. Symp. Appl. Math., AMS, Providence, 166–178, 1961Google Scholar
  33. 33.
    Lambek, J., ‘From categorial grammar to bilinear logic’, [47]:207–237.Google Scholar
  34. 34.
    Lambek, J., ‘Type Grammars Revisited’, [37]:1–27.Google Scholar
  35. 35.
    Lambek J., (2001) ‘Type Grammars as Pregroups’. Grammars 4:21–39CrossRefGoogle Scholar
  36. 36.
    Lambek, J., ‘Should Pregroup Grammars be Adorned with Additional Operations?’, this issue.Google Scholar
  37. 37.
    Lecomte, A., F. Lamarche, and G. Perrier (eds.), Logical Aspects of Computational Linguistics, LNAI 1582, Springer, 1999.Google Scholar
  38. 38.
    Moortgat, M., ‘Categorial Type Logic’, [8]:93–177.Google Scholar
  39. 39.
    Morrill G., (1994) Type Logical Grammar. Kluwer, DordrechtGoogle Scholar
  40. 40.
    Oehrle, R. T., E. Bach, and D. Wheeler (eds.), Categorial Grammars and Natural Language Structures, D. Reidel, Dordrecht, 1988.Google Scholar
  41. 41.
    Ono, H., ‘Semantics of Substructural Logics’, [47]:259–291.Google Scholar
  42. 42.
    Pentus, M., ‘Lambek Grammars are Context-Free’, Proc. 8th IEEE Symp. Logic in Computer Scie., 429–433, 1993.Google Scholar
  43. 43.
    Pentus M., (1995) ‘Models for the Lambek Calculus’. Annals of Pure and Applied Logic 75:179–213CrossRefGoogle Scholar
  44. 44.
    Pentus M., (2006) ‘Lambek calculus is NP-complete’. Theoretical Computer Science 357:186–201CrossRefGoogle Scholar
  45. 45.
    Preller, A., ‘Category Theoretical Semantics for Pregroup Grammars’, Logical Aspects of Computational Linguistics, LNAI 3492, Springer, Berlin, 2005, pp. 254–270.Google Scholar
  46. 46.
    Preller A., Lambek J., (2007) ‘Free compact 2-categories’. Mathematical Structures in Computer Science 17:309–340CrossRefGoogle Scholar
  47. 47.
    Schroeder-Heister P., Dosen K. (eds) (1993) Substructural Logics. Clarendon Press, OxfordGoogle Scholar
  48. 48.
    Yetter D.N. (1990) ‘Quantales and (Non-Commutative) Linear Logic’. Journal of Symbolic Logic 55:41-64CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Research Group on Mathematical LinguisticsRovira i Virgili University in TarragonaTarragonaSpain
  2. 2.Faculty of Mathematics and Computer ScienceAdam Mickiewicz University in PoznańPoznanPoland
  3. 3.Faculty of Mathematics and Computer ScienceUniversity of Warmia and Mazury in OlsztynOlsztynPoland

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