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International Symposium on Functional and Logic Programming

FLOPS 2016: Functional and Logic Programming pp 109–125Cite as

Executable Relational Specifications of Polymorphic Type Systems Using Prolog

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9613)

Abstract

A concise, declarative, and machine executable specification of the Hindley–Milner type system (HM) can be formulated using logic programming languages such as Prolog. Modern functional language implementations such as the Glasgow Haskell Compiler support more extensive flavors of polymorphism beyond Milner’s theory of type polymorphism in the late 70’s. We progressively extend the HM specification to include more advanced type system features. An interesting development is that extending dimensions of polymorphism beyond HM resulted in a multi-staged solution: resolve the typing relations first, while delaying to resolve kinding relations, and then resolve the delayed kinding relations. Our work demonstrates that logic programing is effective for prototyping polymorphic type systems with rich features of polymorphism, and that logic programming could have been even more effective for specifying type inference if it were equipped with better theories and tools for staged resolution of different relations at different levels.

Keywords

  • Hindley–Milner
  • Functional language
  • Type system
  • Type inference
  • Unification
  • Parametric polymorphism
  • Higher-kinded polymorphism
  • Type constructor polymorphism
  • Kind polymorphism
  • Algebraic datatype
  • Nested datatype
  • Logic programming
  • Prolog
  • Delayed goals

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Notes

  1. 1.

    http://kyagrd.github.io/tiper/.

  2. 2.

    We intentionallay adopted the same symbol as the application operator $ in the Haskell standard library. In contrast to the right-associative operator in Haskell, our operator represents the default function application most often denoted by empty spaces and left-associative by convention.

  3. 3.

    Here, in this subsection, we consider non-recursive bindings only but the specification of HM can be easily modified to support recursive bindings (see Sect. 3.1).

  4. 4.

    See miniml.apl in the examples directory of the \(\alpha \)Prolog version 0.4 or 0.3.

  5. 5.

    Kanren is a phonetic transcription of the Kanji word meaning “relation”. See http://kanren.sourceforge.net/.

  6. 6.

    See http://minikanren.org/.

References

  1. Abel, A., Matthes, R., Uustalu, T.: Generalized iteration and coiteration for higher-order nested datatypes. In: FoSSaCS 2003 (2003)

    Google Scholar 

  2. Ahn, K.Y.: The Nax language. Ph.D. thesis, Department of Computer Science, Portland State University, November 2014

    Google Scholar 

  3. Ahn, K.Y., Sheard, T.: A hierarchy of Mendler-style recursion combinators. In: ICFP 2011. ACM (2011)

    Google Scholar 

  4. Ahn, K.Y., Sheard, T., Fiore, M., Pitts, A.M.: System Fi: a higher-order polymorphic lambda calculus with erasable term indices. In: TLCA 2013 (2013)

    Google Scholar 

  5. Alves, S., Florido, M.: Type inference using Constraint Handling Rules. In: WFLP 2001, vol. 64 of Electronic Notes in TCS, pp. 56–72. Elsevier (2002)

    Google Scholar 

  6. Ancona, D., Lagorio, G.: Coinductive type systems for object-oriented languages. In: Drossopoulou, S. (ed.) ECOOP 2009. LNCS, vol. 5653, pp. 2–26. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  7. Barendregt, H.: Introduction to generalized type systems. J. Funct. Program. 1(2), 125–154 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Bird, R., Meertens, L.: Nested datatypes. In: MPC: 4th International Conference on Mathematics of Program Construction (1998)

    Google Scholar 

  9. Byrd, W.E.: Relational programming in miniKanren: techniques, applications,and implementations. Ph.D. thesis, Indiana University (2009)

    Google Scholar 

  10. Cheney, J., Urban, C.: \(\alpha \)Prolog: A logic programming language with names, binding, and \(\alpha \)-equivalence. In: Demoen, B., Lifschitz, V. (eds.) ICLP 2004. LNCS, vol. 3132, pp. 269–283. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  11. Csorba, J., Zombori, Z., Szeredi, P.: Pros and cons of using CHR for type inference. In: CHR 2012, KU Leuven, Deptarment of CS, Tech-report CW 624 (2012)

    Google Scholar 

  12. Despeyroux, T.: Executable specification of static semantics. In: MacQueen, D.B., Plotkin, G., Kahn, G. (eds.) Semantics of Data Types 1984. LNCS, vol. 173, pp. 215–233. Springer, Heidelberg (1984)

    CrossRef  Google Scholar 

  13. Dubois, C.: Proving ML type soundness within Coq. In: Aagaard, M.D., Harrison, J. (eds.) TPHOLs 2000. LNCS, vol. 1869, pp. 126–144. Springer, Heidelberg (2000)

    CrossRef  Google Scholar 

  14. Friedman, D.P., Byrd, W.E., Kiselyov, O.: The Reasoned Schemer. MIT Press, Cambridge (2005). ISBN 978-0-262-56214-0

    Google Scholar 

  15. Fu, P., Komendantskaya, E.: A type theoretic approach to structural resolution. In: Pre-Proceedings of LOPSTR 2015 (2015)

    Google Scholar 

  16. Gaster, B.R., Jones, M.P.: A polymorphic type system for extensible records and variants. Technical report (1996)

    Google Scholar 

  17. J.-Y. Girard. Interprétation fonctionelle et élimination des coupures de l’arithmétique d’ordre supérieur. Ph.D. thesis, Université Paris VII (1972)

    Google Scholar 

  18. Hemann, J., Friedman, D.P.: \(\mu \)Kanren: A minimal functional core for relational programming. In: Scheme 2013 (2013)

    Google Scholar 

  19. Johann, P., Komendantskaya, E., Komendantskiy, V.: Structural resolution for logic programming. In: Techincal Communications of ICLP (2015)

    Google Scholar 

  20. Jones, M.P.: Typing Haskell in Haskell. In: Haskell 1999, October 1999

    Google Scholar 

  21. Matthes, R.: Extensions of System F by Iteration and Primitive Recursion on Monotone Inductive Types. Ph.D. thesis, Ludwig-Maximilians University (1998)

    Google Scholar 

  22. McBride, C.: First-order unification by structural recursion. J. Func. Program. 13, 1061–1075 (2003). ISSN 1469–7653

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Mitchell, D.J., Nadathur, G.: System description: teyjus - a compiler and abstract machine based implementation of \(\lambda \)prolog. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 287–291. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  24. Odersky, M., Sulzmann, M., Wehr, M.: Type inference with constrained types. Theor. Pract. Object Syst. 5(1), 35–55 (1999). ISSN 1074–3227

    CrossRef  Google Scholar 

  25. Pereira, F.C.N., Warren, D.H.D.: Definite clause grammars for language analysis. Artif. Intell. 13, 231–278 (1980)

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Poole, D., Mackworth, A.K.: Artificial Intelligence - Foundations of Computational Agents. Cambridge University Press, Cambridge (2010)

    CrossRef  MATH  Google Scholar 

  27. Russo, C.V., Vytiniotis, D.: Qml: explicit first-class polymorphism for ml. In: ML 2009, pp. 3–14. ACM, New York (2009)

    Google Scholar 

  28. Schrijvers, T., Costa, V.S., Wielemaker, J., Demoen, B.: Towards typed prolog. In: Garcia de la Banda, M., Pontelli, E. (eds.) ICLP 2008. LNCS, vol. 5366, pp. 693–697. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  29. Schrijvers, T., Demoen, B., Desouter, B., Wielemaker, J.: Delimited continuations for prolog. In: Proceedings of ICLP 2013 TPLP (2013)

    Google Scholar 

  30. Scott, D.S.: A type-theoretic alternative to CUCH ISWIM OWHY. Manuscript (1969)

    Google Scholar 

  31. SWI-Prolog team. SWI-Prolog reference manual (section 4.2) (2005)

    Google Scholar 

  32. Urban, C., Nipkow, T.: Nominal verification of algorithm W. In: From Semantics to Computer Science, pp. 363–382. Cambridge University Press (2009)

    Google Scholar 

  33. Vene, V.: Categorical Programming with Inductive and Coinductive Types. Ph.D. thesis, Department of Computer Science, University of Tartu, August 2000

    Google Scholar 

  34. Yorgey, B.A., Weirich, S., Cretin, J., Peyton Jones, S., Vytiniotis, D., Magalhães, J.P.: Giving Haskell a promotion. In: TLDI 2012. ACM (2012)

    Google Scholar 

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Acknowledgements

Thanks to Patricia Johann for helping us clarify the specification for Mendler-style iteration, Ekaterina Komendantskaya and Frantisek Farka for the discussions on S-Resolution, Peng Fu for pointers to Kanren, Chris Warburton for careful proofreading, and FLOPS’16 reviewers for their feedback.

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Authors and Affiliations

  1. Portland State University, Portland, OR, USA

    Ki Yung Ahn

  2. Chalmers University of Technology, Gothenburg, Sweden

    Andrea Vezzosi

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  1. Ki Yung Ahn
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Correspondence to Ki Yung Ahn .

Editor information

Editors and Affiliations

  1. Tohoku University, Sendai, Japan

    Oleg Kiselyov

  2. School of Computing, University of Kent, Canterbury, United Kingdom

    Andy King

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Ahn, K.Y., Vezzosi, A. (2016). Executable Relational Specifications of Polymorphic Type Systems Using Prolog. In: Kiselyov, O., King, A. (eds) Functional and Logic Programming. FLOPS 2016. Lecture Notes in Computer Science(), vol 9613. Springer, Cham. https://doi.org/10.1007/978-3-319-29604-3_8

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  • DOI: https://doi.org/10.1007/978-3-319-29604-3_8

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