Abstract
We show how to construct a logical relation for countable nondeterminism in a guarded type theory, corresponding to the internal logic of the topos Sh ω 1 of sheaves over ω 1. In contrast to earlier work on abstract step-indexed models, we not only construct the logical relations in the guarded type theory, but also give an internal proof of the adequacy of the model with respect to standard contextual equivalence. To state and prove adequacy of the logical relation, we introduce a new propositional modality. In connection with this modality we show why it is necessary to work in the logic of bf Shω 1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agha, G., Mason, I.A., Smith, S.F., Talcott, C.L.: A foundation for actor computation. Journal of Functional Programming 7(1), 1–72 (1997)
Ahmed, A.: Step-indexed syntactic logical relations for recursive and quantified types. Tech. rep., Harvard University (2006), http://www.ccs.neu.edu/home/amal/papers/lr-recquant-techrpt.pdf
Apt, K.R., Plotkin, G.D.: Countable nondeterminism and random assignment. Journal of the ACM 33(4), 724–767 (1986)
Birkedal, L., Bizjak, A., Schwinghammer, J.: Step-indexed relational reasoning for countable nondeterminism. Logical Methods in Computer Science 9(4) (2013)
Birkedal, L., Møgelberg, R.E., Schwinghammer, J., Støvring, K.: First steps in synthetic guarded domain theory: step-indexing in the topos of trees. Logical Methods in Computer Science 8(4) (2012)
Bizjak, A., Birkedal, L., Miculan, M.: A model of countable nondetermism in guarded type theory (2014), http://cs.au.dk/~abizjak/documents/trs/cntbl-gtt-tr.pdf
Di Gianantonio, P., Honsell, F., Plotkin, G.D.: Uncountable limits and the lambda calculus. Nordic Journal of Computing 2(2), 126–145 (1995)
Dreyer, D., Ahmed, A., Birkedal, L.: Logical step-indexed logical relations. Logical Methods in Computer Science 7(2) (2011)
Lambek, J., Scott, P.: Introduction to Higher-Order Categorical Logic. Cambridge Studies in Advanced Mathematics. Cambridge University Press (1988)
Lassen, S.B.: Relational Reasoning about Functions and Nondeterminism. Ph.D. thesis, University of Aarhus (1998)
Lassen, S.B., Moran, A.: Unique fixed point induction for McCarthy’s amb. In: Kutyłowski, M., Wierzbicki, T., Pacholski, L. (eds.) MFCS 1999. LNCS, vol. 1672, pp. 198–208. Springer, Heidelberg (1999)
Lassen, S.B., Pitcher, C.: Similarity and bisimilarity for countable non-determinism and higher-order functions. Electronic Notes in Theoretical Computer Science 10 (1997)
Levy, P.B.: Infinitary Howe’s method. In: Coalgebraic Methods in Computer Science, pp. 85–104 (2006)
MacLane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Mathematical Sciences Research Institute Publications. Springer, New York (1992)
Mason, I.A., Talcott, C.L.: Equivalence in functional languages with effects. Journal of Functional Programming 1(3), 287–327 (1991)
Reyes, G., Zolfaghari, H.: Bi-heyting algebras, toposes and modalities. Journal of Philosophical Logic 25(1), 25–43 (1996)
Sabel, D., Schmidt-Schauß, M.: A call-by-need lambda calculus with locally bottom-avoiding choice: context lemma and correctness of transformations. Mathematical Structures in Computer Science 18(3), 501–553 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Bizjak, A., Birkedal, L., Miculan, M. (2014). A Model of Countable Nondeterminism in Guarded Type Theory. In: Dowek, G. (eds) Rewriting and Typed Lambda Calculi. RTA TLCA 2014 2014. Lecture Notes in Computer Science, vol 8560. Springer, Cham. https://doi.org/10.1007/978-3-319-08918-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-08918-8_8
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08917-1
Online ISBN: 978-3-319-08918-8
eBook Packages: Computer ScienceComputer Science (R0)