Linear Abadi and Plotkin Logic
<|reference_start|>Linear Abadi and Plotkin Logic: We present a formalization of a version of Abadi and   Plotkin's logic for parametricity for a polymorphic dual intuitionistic/linear type theory with fixed points, and show, following Plotkin's suggestions, that it can be used to define a wide collection of types, including existential types, inductive types, coinductive types and general recursive types. We show that the recursive types satisfy a universal property called dinaturality, and we develop reasoning principles for the constructed types. In the case of recursive types, the reasoning principle is a mixed induction/coinduction principle, with the curious property that coinduction holds for general relations, but induction only for a limited collection of ``admissible'' relations. A similar property was observed in Pitts' 1995 analysis of recursive types in domain theory. In a future paper we will develop a category theoretic notion of models of the logic presented here, and show how the results developed in the logic can be transferred to the models.<|reference_end|>
@article{birkedal2006linear,
    title={Linear Abadi and Plotkin Logic},
    author={Lars Birkedal and Rasmus E. M{o}gelberg and Rasmus Lerchedahl
  Petersen},
    journal={Logical Methods in Computer Science, Volume 2, Issue 5 (November
  3, 2006) lmcs:2233},
    year={2006},
    doi={10.2168/LMCS-2(5:2)2006},
    archivePrefix={arXiv},
    eprint={cs/0611004},
    primaryClass={cs.LO}
}