Fragments of ML Decidable by Nested Data Class Memory Automata

  • Conrad Cotton-Barratt
  • David Hopkins
  • Andrzej S. Murawski
  • C. -H. Luke Ong
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9034)


The call-by-value language RML may be viewed as a canonical restriction of Standard ML to ground-type references, augmented by a “bad variable” construct in the sense of Reynolds. We consider the fragment of (finitary) RML terms of order at most 1 with free variables of order at most 2, and identify two subfragments of this for which we show observational equivalence to be decidable. The first subfragment, RML\(^{\rm P-Str}_{2\vdash 1}\), consists of those terms in which the P-pointers in the game semantic representation are determined by the underlying sequence of moves. The second subfragment consists of terms in which the O-pointers of moves corresponding to free variables in the game semantic representation are determined by the underlying moves. These results are shown using a reduction to a form of automata over data words in which the data values have a tree-structure, reflecting the tree-structure of the threads in the game semantic plays. In addition we show that observational equivalence is undecidable at every third- or higher-order type, every second-order type which takes at least two first-order arguments, and every second-order type (of arity greater than one) that has a first-order argument which is not the final argument.


Type Sequent Free Variable Partial Evaluation Initial Move Nest Data 
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  1. 1.
    Abramsky, S., McCusker, G.: Linearity, sharing and state: a fully abstract game semantics for idealized algol with active expressions. Electr. Notes Theor. Comput. Sci. 3, 2–14 (1996)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Abramsky, S., McCusker, G.: Call-by-value games. In: Nielsen, M., Thomas, W. (eds.) CSL 1997. LNCS, vol. 1414, pp. 1–17. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. 3.
    Björklund, H., Schwentick, T.: On notions of regularity for data languages. Theor. Comput. Sci. 411(4-5), 702–715 (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cotton-Barratt, C., Murawski, A.S., Ong, C.-H.L.: Weak and nested class memory automata. In: Proceedings of LATA 2015 (to appear, 2015)Google Scholar
  5. 5.
    Decker, N., Habermehl, P., Leucker, M., Thoma, D.: Ordered navigation on multi-attributed data words. In: Baldan, P., Gorla, D. (eds.) CONCUR 2014. LNCS, vol. 8704, pp. 497–511. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  6. 6.
    Ghica, D.R., McCusker, G.: The regular-language semantics of second-order idealized algol. Theor. Comput. Sci. 309(1-3), 469–502 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Honda, K., Yoshida, N.: Game-theoretic analysis of call-by-value computation. Theor. Comput. Sci. 221(1-2), 393–456 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Hopkins, D.: Game Semantics Based Equivalence Checking of Higher-Order Programs. PhD thesis, Department of Computer Science, University of Oxford (2012)Google Scholar
  9. 9.
    Hopkins, D., Murawski, A.S., Ong, C.-H.L.: A fragment of ML decidable by visibly pushdown automata. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 149–161. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Hyland, J.M.E., Luke Ong, C.-H.: On full abstraction for PCF: i, ii, and III. Inf. Comput. 163(2), 285–408 (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Murawski, A.S.: On program equivalence in languages with ground-type references. In: 18th IEEE Symposium on Logic in Computer Science (LICS 2003), p. 108. IEEE Computer Society (2003)Google Scholar
  12. 12.
    Murawski, A.S.: Functions with local state: Regularity and undecidability. Theor. Comput. Sci. 338(1-3), 315–349 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Murawski, A.S., Ong, C.-H.L., Walukiewicz, I.: Idealized algol with ground recursion, and DPDA equivalence. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 917–929. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Murawski, A.S., Tzevelekos, N.: Algorithmic nominal game semantics. In: Barthe, G. (ed.) ESOP 2011. LNCS, vol. 6602, pp. 419–438. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Murawski, A.S., Tzevelekos, N.: Algorithmic games for full ground references. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 312–324. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  16. 16.
    Luke Ong, C.-H.: An approach to deciding the observational equivalence of algol-like languages. Ann. Pure Appl. Logic 130(1-3), 125–171 (2004)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Pitts, A.M., Stark, I.D.B.: Operational reasoning for functions with local state. In: Higher Order Operational Techniques in Semantics, pp. 227–273 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Conrad Cotton-Barratt
    • 1
  • David Hopkins
    • 1
  • Andrzej S. Murawski
    • 2
  • C. -H. Luke Ong
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK
  2. 2.Department of Computer ScienceUniversity of WarwickCoventryUK

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