A new logic is posited for the widely used HOL theorem prover, as an extension of the existing higher order logic of the HOL4 system. The logic is extended to three levels, adding kinds to the existing levels of types and terms. New types include type operator variables and universal types as in System F. Impredicativity is avoided through the stratification of types by ranks according to the depth of universal types. The new system, called HOL-Omega or HOL ω , is a merging of HOL4, HOL2P[11], and major aspects of System F ω from chapter 30 of [10]. This document presents the abstract syntax and semantics for the kinds, types, and terms of the logic, as well as the new fundamental axioms and rules of inference. As the new logic is constructed according to the design principles of the LCF approach, the soundness of the entire system depends critically and solely on the soundness of this core.


Type Variable Type Operator Natural Transformation Abstract Syntax Type Inference 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bird, R., de Moor, O.: Algebra of Programming. Prentice Hall (1997)Google Scholar
  2. 2.
    Coquand, T.: A new paradox in type theory. In: Prawitx, D., Skyrms, B., Westerstahl, D. (eds.) Proceedings 9th Int. Congress of Logic, Methodology and Philosophy of Science, pp. 555–570. North-Holland, Amsterdam (1994)Google Scholar
  3. 3.
    Gordon, M.J.C., Melham, T.F.: Introduction to HOL. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  4. 4.
    Gordon, M.J.C., Pitts, A.M.: The HOL Logic and System. In: Bowen, J. (ed.) Towards Verified Systems, ch. 3, pp. 49–70. Elsevier Science B.V., Amsterdam (1994)CrossRefGoogle Scholar
  5. 5.
    The HOL System DESCRIPTION (Version Kananaskis 4),
  6. 6.
    The HOL System LOGIC (Version Kananaskis 4),
  7. 7.
    Lack, S., Street, R.: The formal theory of monads II. Journal of Pure Applied Algorithms 175, 243–265 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Melham, T.F.: The HOL Logic Extended with Quantification over Type Variables. Formal Methods in System Design 3(1-2), 7–24 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Monk, J.D.: Introduction to Set Theory. McGraw-Hill, New York (1969)zbMATHGoogle Scholar
  10. 10.
    Pierce, B.C.: Types and Programming Languages. MIT Press, Cambridge (2002)zbMATHGoogle Scholar
  11. 11.
    Völker, N.: HOL2P - A System of Classical Higher Order Logic with Second Order Polymorphism. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 334–351. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Wadler, P.: Monads for functional programming. In: Jeuring, J., Meijer, E. (eds.) AFP 1995. LNCS, vol. 925. Springer, Heidelberg (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Peter V. Homeier
    • 1
  1. 1.U. S. Department of DefenseUSA

Personalised recommendations