International Conference on Interactive Theorem Proving

ITP 2015: Interactive Theorem Proving pp 170-186 | Cite as

Proof-Producing Reflection for HOL

With an Application to Model Polymorphism
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9236)

Abstract

We present a reflection principle of the form “If \(\ulcorner \varphi \urcorner \) is provable, then \(\varphi \)” implemented in the HOL4 theorem prover, assuming the existence of a large cardinal. We use the large-cardinal assumption to construct a model of HOL within HOL, and show how to ensure \(\varphi \) has the same meaning both inside and outside of this model. Soundness of HOL implies that if \(\ulcorner \varphi \urcorner \) is provable, then it is true in this model, and hence \(\varphi \) holds. We additionally show how this reflection principle can be extended, assuming an infinite hierarchy of large cardinals, to implement model polymorphism, a technique designed for verifying systems with self-replacement functionality.

References

  1. 1.
    Allen, S.F., Constable, R.L., Howe, D.J., Aitken, W.E.: The semantics of reflected proof. In: Proceedings of the LICS, pp. 95–105, IEEE Computer Society (1990)Google Scholar
  2. 2.
    Dybjer, P., Setzer, A.: A finite axiomatization of inductive-recursive definitions. In: Girard, J.-Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 129–146. Springer, Heidelberg (1999) CrossRefGoogle Scholar
  3. 3.
    Fallenstein, B., Soares, N.: Vingean reflection. Technical report, Machine Intelligence Research Institute, Berkeley, CA (2015)Google Scholar
  4. 4.
    Feferman, S.: Transfinite recursive progressions of axiomatic theories. J. Symb. Log. 27(3), 259–316 (1962)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Franzén, T.: Transfinite progressions: a second look at completeness. B. Symb. Log. 10(3), 367–389 (2004). http://www.math.ucla.edu/~asl/bsl/1003/1003-003.ps
  6. 6.
    Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte fr Mathematik und Physik 38(1), 173–198 (1931)CrossRefGoogle Scholar
  7. 7.
    Gonthier, G.: The four colour theorem: engineering of a formal proof. In: Kapur, D. (ed.) ASCM 2007. LNCS (LNAI), vol. 5081, pp. 333–333. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  8. 8.
    Gonthier, G., Mahboubi, A.: An introduction to small scale reflection in Coq. J. Form. Reasoning 3(2), 95–152 (2010)MATHMathSciNetGoogle Scholar
  9. 9.
    Gordon, M.: From LCF to HOL: a short history. In: Plotkin, G.D., Stirling, C., Tofte, M. (eds.) Proof, Language, and Interaction, Essays in Honour of Robin Milner, pp. 169–186. The MIT Press, Cambridge (2000)Google Scholar
  10. 10.
    Harrison, J.: Metatheory and reflection in theorem proving: A survey and critique. Technical report CRC-053, SRI, Cambridge, UK (1995). http://www.cl.cam.ac.uk/~jrh13/papers/reflect.dvi.gz
  11. 11.
    Harrison, J.: Towards self-verification of HOL Light. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 177–191. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  12. 12.
    Harrison, J.: HOL Light: an overview. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 60–66. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  13. 13.
    Jech, T.: Set Theory. The Third Millenium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Heidelberg (2003) Google Scholar
  14. 14.
    Klein, G., Gamboa, R. (eds.): Interactive Theorem Proving. Springer, Heidelberg (2014) MATHGoogle Scholar
  15. 15.
    Kumar, R., Arthan, R., Myreen, M.O., Owens, S.: HOL with definitions: semantics, soundness, and a verified implementation. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 308–324. Springer, Heidelberg (2014) Google Scholar
  16. 16.
    Kumar, R., Arthan, R., Myreen, M.O., Owens, S.: Self-formalisation of higher-order logic. J. Autom. Reasoning (2015), submitted. Preprint at https://cakeml.org
  17. 17.
    Mohamed, O.A., Muñoz, C.A., Tahar, S. (eds.): Theorem Proving in Higher Order Logics. Springer, Heidelberg (2008) MATHGoogle Scholar
  18. 18.
    Myreen, M.O., Davis, J.: The reflective Milawa theorem prover is sound - (down to the machine code that runs it). In: Klein and Gamboa [14], pp. 421–436Google Scholar
  19. 19.
    Myreen, M.O., Owens, S.: Proof-producing translation of higher-order logic into pure and stateful ML. J. Funct. Program. 24(2–3), 284–315 (2014)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Norrish, M., Huffman, B.: Ordinals in HOL: transfinite arithmetic up to (and beyond) \(\omega _{1}\). In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 133–146. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  21. 21.
    Slind, K., Norrish, M.: A brief overview of HOL4. In: Mohamed et al. [17], pp. 28–32Google Scholar
  22. 22.
    Turing, A.M.: Systems of logic based on ordinals. Proc. LMS 2(1), 161–228 (1939)MathSciNetGoogle Scholar
  23. 23.
    Wenzel, M., Paulson, L.C., Nipkow, T.: The Isabelle framework. In: Mohamed et al. [17], pp. 33–38Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Machine Intelligence Research InstituteBerkeleyUSA
  2. 2.Computer LaboratoryUniversity of CambridgeCambridgeUK

Personalised recommendations