Advertisement

The undecidability of second order linear affine logic

  • Alexei P. Kopylov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1234)

Abstract

Quantifier-free prepositional linear affine logic (i.e. linear logic with weakening) is decidable [Kop, Laf2]. Lafont and Scedrov proved that the multiplicative fragment of second-order linear logic is undecidable [LS]. In this paper we show that second order linear affine logic is undecidable as well. At the same time it turns out that even its multiplicative fragment is undecidable. Moreover, we obtain a whide class of undecidabile second order logics which lie between the Lambek calculus (LC) and linear affine logic. The proof is based on an encoding of two-counter Minsky machines in second order linear affine logic. The faithfulness of the encoding is proved by means of the phase semantics.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Abr91]
    V.M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear prepositional logic. Jornal of Symbolic logic 56, 1403–1451. 1991.Google Scholar
  2. [Gir]
    J.-Y. Girard. Linear Logic. Theoretical Computer Science, 50, 1–102, 1987.Google Scholar
  3. [Kan1]
    M.I. Kanovich. The Direct Simulation of Minsky machine in Linear logic. To appear in Advances in Linear Logic, edited by J.-Y. Girard, Y. Lafont & L. Regnier, London Mathematical Society Lecture Note Series, Cambridge University Press. 1995.Google Scholar
  4. [Kan2]
    M.I. Kanovich. Second order Lambek is undecidable. Message to LL list. 3 July 1995.Google Scholar
  5. [Kop]
    A.P. Kopylov. Decidability of Linear Affine Logic. 10-th Annual IEEE Symposium on Logic in Computer Science, San Diego, California, IEEE Computer Society Press. 1995.Google Scholar
  6. [Laf1]
    Y. Lafont. The Undecidability of Second Order Linear Logic without Exponentials. To apear in Jornal of Symbolic Logic. Available by anonymous ftp from lmd.univ-mrs.fr as pub/lafont/undecid. [dvi.ps]. Z. 1995.Google Scholar
  7. [Laf2]
    Y. Lafont. The Finite Model Property for Various Fragments of Linear Logic. Submited to publication. Available by anonymous ftp from lmd.univ-mrs.fr as pub/lafont/model.[dvi.ps].Z. 1995.Google Scholar
  8. [LS]
    Y. Lafont, A. Scedrov. The Undecidability of Second Order Multiplicative Linear Logic. To apear in Information and Computation. Available by anonymous ftp from lmd.univ-mrs.fr as pub/lafont/m112. [dvi,ps].Z. 1995.Google Scholar
  9. [LSS]
    P. Lincoln, A. Scedrov, N. Shankar. Decision Problem for Second Order Linear Logic. 10-th Annual IEEE Symposium on Logic in Computer Science, San Diego, California, IEEE Computer Society Press. 1995.Google Scholar
  10. [Lk]
    J. Lambek. How to program an infinite abacus. Canadian Math. Bulletin 4. 295–302. 1961.Google Scholar
  11. [M]
    M. Minsky. Recursive unsolvability of Post's problem of’ tag’ and other topics in the theory of Turing machines. Annals of Mathematics, 74:3:437–455, 1961.Google Scholar
  12. [T]
    A.S. Troelstra. Lectures on Linear Logic. CSLI lecture notes; no.29, 1992Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Alexei P. Kopylov
    • 1
  1. 1.Department of Mathematics and MechanicsMoscow State UniversityMoscowRussia

Personalised recommendations