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UNDECIDABILITY OF CONSEQUENCE RELATION IN FULL NON-ASSOCIATIVE LAMBEK CALCULUS

Published online by Cambridge University Press:  22 April 2015

KAREL CHVALOVSKÝ
KAREL CHVALOVSKÝ*
Affiliation:
INSTITUTE OF COMPUTER SCIENCE ACADEMY OF SCIENCES OF THE CZECH REPUBLIC POD VODÁRENSKOUVĚŽÍ 271/2, 182 07 PRAGUE 8, CZECH REPUBLICE-mail:[email protected]: http://karel.chvalovsky.cz
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Abstract

We prove that the consequence relation in the Full Non-associative Lambek Calculus is undecidable. An encoding of the halting problem for 2-tag systems using finitely many sequents in the language {⋅,∨} is presented. Therefore already the consequence relation in this fragment is undecidable. Moreover, the construction works even when the structural rules of exchange and contraction are added.

Keywords

Substructural logicsconsequence relationundecidabilitytag systemsword problemrewriting systems
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 2 , June 2015 , pp. 567 - 586
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

REFERENCES

Blok, Willem J. and van Alten, Clint J., On the finite embeddability property for residuated ordered groupoids. Transactions of the American Mathematical Society, vol. 357 (2004), no. 10, pp. 4141–4157.CrossRefGoogle Scholar
Buszkowski, Wojciech, Some decision problems in the theory of syntactic categories. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 28 (1982), no. 33–38, pp. 539–548.Google Scholar
Buszkowski, Wojciech, Lambek calculus with nonlogical axioms, Language and grammar: Studies in mathematical linguistics and natural language (Casadio, C., Scott, P. J., and Seely, R. A. G., editors), CSLI Lecture Notes, no. 168, CSLI, Stanford, CA, 2005, pp. 77–93.Google Scholar
Buszkowski, Wojciech and Farulewski, Maciej, Non-associative Lambek calculus with additives and context-free languages, Languages: From formal to natural (Grumberg, Orna, Kaminski, Michael, Katz, Shmuel, and Wintner, Shuly, editors), Lecture Notes in Computer Science, no. 5533, Springer, Berlin, Heidelberg, 2009, pp. 45–58.Google Scholar
Cocke, John and Minsky, Marvin Lee, Universality of Tag systems with P=2. Journal of the ACM, vol. 11 (1964), no. 1, pp. 15–20.Google Scholar
Došen, Kosta, Sequent-systems and groupoid models. I. Studia Logica, vol. 47 (1988), no. 4, pp. 353–385 (English).Google Scholar
Farulewski, Maciej, Finite embeddability property for residuated groupoids. Reports on Mathematical Logic, vol. 43 (2008), pp. 25–42.Google Scholar
Galatos, Nikolaos, Jipsen, Peter, Kowalski, Tomasz, and Ono, Hiroakira, Residuated lattices: An algebraic glimpse at substructural logics, Studies in Logic and the Foundations of Mathematics, vol. 151, Elsevier, Amsterdam, 2007.Google Scholar
Galatos, Nikolaos and Ono, Hiroakira, Cut elimination and strong separation for substructural logics: An algebraic approach. Annals of Pure and Applied Logic, vol. 161 (2010), no. 9, pp. 1097–1133.CrossRefGoogle Scholar
Grätzer, George, Lattice theory: Foundation, Birkhäuser, Basel, 2011.CrossRefGoogle Scholar
Haniková, Zuzana and Horčík, Rostislav, Finite embeddability property for residuated groupoids. Algebra Universalis, vol. 72 (2014), no. 1, pp. 1–13. doi:10.1007/s00012-014-0284-1.Google Scholar
Hori, Ryuichi, Ono, Hiroakira, and Schellinx, Harold, Extending intuitionistic linear logic with knotted structural rules. Notre Dame Journal of Formal Logic, vol. 35(1994), no. 2, pp. 219–242.CrossRefGoogle Scholar
Lambek, Joachim, The mathematics of sentence structure. American Mathematical Monthly, vol. 65 (1958), no. 3, pp. 154–170.Google Scholar
Lambek, Joachim, On the calculus of syntactic types, Structure of language and its mathematical aspects (Jakobson, Roman, editor), American Mathematical Society, Providence, RI, 1961, pp. 166–178.Google Scholar
Ono, Hiroakira, Proof-theoretic methods in nonclassical logic: An introduction, Theories of types and proofs (Takahashi, M., Okada, M., and Dezani-Ciancaglini, M., editors), MSJ Memoirs, vol. 2, Mathematical Society of Japan, Tokyo, 1998, pp. 207–254.Google Scholar
Post, Emil Leon, Formal reductions of the general combinatorial decision problem. American Journal of Mathematics, vol. 65 (1943), no. 2, pp. 197–215.CrossRefGoogle Scholar
Troelstra, Anne Sjerp, Lectures on linear logic, CSLI lecture notes, no. 29, Center for the Study of Language and Information, Stanford, CA, 1992.Google Scholar