Abstract
We construct a partially-ordered hierarchy of delimited control operators similar to those of the CPS hierarchy of Danvy and Filinski [5]. However, instead of relying on nested CPS translations, these operators are directly interpreted in linear logic extended with subexponentials (i.e., multiple pairs of ! and ?). We construct an independent proof theory for a fragment of this logic based on the principle of focusing. It is then shown that the new constraints placed on the permutation of cuts correspond to multiple levels of delimited control.
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Notes
- 1.
Adding second-order quantifiers will, however, encounter problems similar to those found in polarized settings: how can one enforce the index restrictions on the bound variable when they are instantiated. First-order \(\forall \) can be represented by \(!_i?_{i'}\forall x.!_k?_{k'}A\) where \(i'\le k'\) and \(k\le i\). These restrictions guarantee \(?_{i'}\forall x.?_{k'} A ~\equiv ~ \forall x.?_{k'}A\) and \(!_i\forall x. !_k A ~\equiv ~ !_i \forall x.A\).
- 2.
When restricted to the modalities \(!_i?_{i'}\) and \(?_{i'}!_i?_{i'}\), dereliction can be expressed by the axiom \(!A\rightarrow A\): this is an intuitionistic implication, which surely has proof \(\lambda x.x\).
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Acknowledgments
The authors wish to thank the reviewers of this paper for their comments, and to Danko Ilik for valuable discussion. This work was funded by the ERC Advanced Grant ProofCert.
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Liang, C., Miller, D. (2015). On Subexponentials, Synthetic Connectives, and Multi-level Delimited Control. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_21
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