Skip to main content
SpringerLink
Log in

Partial Recursive Functions in Higher-Order Logic

  • Conference paper
Book cover Automated Reasoning (IJCAR 2006)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4130))

Included in the following conference series:

Abstract

Based on inductive definitions, we develop an automated tool for defining partial recursive functions in Higher-Order Logic and providing appropriate reasoning tools for them. Our method expresses termination in a uniform manner and includes a very general form of pattern matching, where patterns can be arbitrary expressions. Termination proofs can be deferred, restricted to subsets of arguments and are interchangeable with other proofs about the function. We show that this approach can also facilitate termination arguments for total functions, in particular for nested recursions. We implemented our tool as a definitional specification mechanism for Isabelle/HOL.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barthe, G., Forest, J., Pichardie, D., Rusu, V.: Defining and reasoning about recursive functions: a practical tool for the Coq proof assistant. In: Hagiya, M., Wadler, P. (eds.) FLOPS 2006. LNCS, vol. 3945, Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  2. Berghofer, S., Wenzel, M.: Inductive datatypes in HOL - lessons learned in formal-logic engineering. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 19–36. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  3. Bertot, Y., Castéran, P.: Interactive theorem proving and program development: Coq’Art: the calculus of inductive constructions. Texts in theoretical comp. science. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  4. Bove, A., Capretta, V.: Modelling general recursion in type theory. Mathematical Structures in Computer Science 15(4), 671–708 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Boyer, R.S., Moore, J.S.: A Computational Logic. Academic Press, New York (1979)

    MATH  Google Scholar 

  6. Dubois, C., Donzeau-Gouge, V.: A step towards the mechanization of partial functions: domains as inductive predicates. In: CADE-15 Workshop on mechanization of partial functions (1998)

    Google Scholar 

  7. Giesl, J.: Termination of nested and mutually recursive algorithms. Journal of Automated Reasoning 19(1), 1–29 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Giesl, J., Thiemann, R., Schneider-Kamp, P.: Proving and disproving termination of higher-order functions. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, pp. 216–231. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  9. Gordon, M., Melham, T. (eds.): Introduction to HOL: A theorem proving environment for higher order logic. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  10. Harrison, J.: The HOL Light theorem prover, http://www.cl.cam.ac.uk/users/jrh/hol-light

  11. Krstić, S., Matthews, J.: Inductive invariants for nested recursion. In: Basin, D., Wolff, B. (eds.) TPHOLs 2003. LNCS, vol. 2758, pp. 253–269. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  12. Lee, C.S., Jones, N.D., Ben-Amram, A.M.: The size-change principle for program termination. In: ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 81–92 (2001)

    Google Scholar 

  13. Manolios, P., Moore, J.S.: Partial functions in ACL2. J. Autom. Reasoning 31(2), 107–127 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Müller, O., Slind, K.: Treating partiality in a logic of total functions. The Computer Journal 40(10), 640–652 (1997)

    Article  Google Scholar 

  15. Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)

    Book  MATH  Google Scholar 

  16. Paulson, L.C.: A fixedpoint approach to implementing (co)inductive definitions. In: Bundy, A. (ed.) CADE 1994. LNCS, vol. 814, pp. 148–161. Springer, Heidelberg (1994)

    Google Scholar 

  17. Slind, K.: Function definition in Higher-Order Logic. In: von Wright, J., Harrison, J., Grundy, J. (eds.) TPHOLs 1996. LNCS, vol. 1125, pp. 381–397. Springer, Heidelberg (1996)

    Google Scholar 

  18. Slind, K.: Reasoning About Terminating Functional Programs. PhD thesis, Institut für Informatik, TU München (1999)

    Google Scholar 

  19. Slind, K.: Another look at nested recursion. In: Aagaard, M.D., Harrison, J. (eds.) TPHOLs 2000. LNCS, vol. 1869, pp. 498–518. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  20. Urban, C., Tasson, C.: Nominal techniques in Isabelle/HOL. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 38–53. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  21. Walther, C.: On proving the termination of algorithms by machine. Artif. Intell. 71(1), 101–157 (1994)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Technische Universität München,  

    Alexander Krauss

Authors
  1. Alexander Krauss
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Artificial Intelligence Research Group, University of Koblenz-Landau, Universitätsstr. 1, 56070, Koblenz

    Ulrich Furbach

  2. SRI International, MS EL256,, 333 Ravenswood Avenue, 94025-3493, Menlo Park, CA, USA

    Natarajan Shankar

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Krauss, A. (2006). Partial Recursive Functions in Higher-Order Logic. In: Furbach, U., Shankar, N. (eds) Automated Reasoning. IJCAR 2006. Lecture Notes in Computer Science(), vol 4130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11814771_48

Download citation

  • DOI: https://doi.org/10.1007/11814771_48

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37187-8

  • Online ISBN: 978-3-540-37188-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation