A Model of Countable Nondeterminism in Guarded Type Theory
We show how to construct a logical relation for countable nondeterminism in a guarded type theory, corresponding to the internal logic of the topos Shω1 of sheaves over ω1. 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 Shω1.
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- 2.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
- 4.Birkedal, L., Bizjak, A., Schwinghammer, J.: Step-indexed relational reasoning for countable nondeterminism. Logical Methods in Computer Science 9(4) (2013)Google Scholar
- 5.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)Google Scholar
- 6.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
- 8.Dreyer, D., Ahmed, A., Birkedal, L.: Logical step-indexed logical relations. Logical Methods in Computer Science 7(2) (2011)Google Scholar
- 9.Lambek, J., Scott, P.: Introduction to Higher-Order Categorical Logic. Cambridge Studies in Advanced Mathematics. Cambridge University Press (1988)Google Scholar
- 10.Lassen, S.B.: Relational Reasoning about Functions and Nondeterminism. Ph.D. thesis, University of Aarhus (1998)Google Scholar
- 12.Lassen, S.B., Pitcher, C.: Similarity and bisimilarity for countable non-determinism and higher-order functions. Electronic Notes in Theoretical Computer Science 10 (1997)Google Scholar
- 13.Levy, P.B.: Infinitary Howe’s method. In: Coalgebraic Methods in Computer Science, pp. 85–104 (2006)Google Scholar