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copied!<p>Since you explicitly asked for the most interesting and obscure ones:</p> <p>You can extend C-H to many interesting logics and formulations of logics to obtain a really wide variety of correspondences. Here I've tried to focus on some of the more interesting ones rather than on the obscure, plus a couple of fundamental ones that haven't come up yet.</p> <pre><code>evaluation | proof normalisation/cut-elimination variable | assumption S K combinators | axiomatic formulation of logic pattern matching | left-sequent rules subtyping | implicit entailment (not reflected in expressions) intersection types | implicit conjunction union types | implicit disjunction open code | temporal next closed code | necessity effects | possibility reachable state | possible world monadic metalanguage | lax logic non-termination | truth in an unobservable possible world distributed programs | modal logic S5/Hybrid logic meta variables | modal assumptions explicit substitutions | contextual modal necessity pi-calculus | linear logic </code></pre> <p>EDIT: A reference I'd recommend to anyone interested in learning more about extensions of C-H:</p> <p>"A Judgmental Reconstruction of Modal Logic" <a href="http://www.cs.cmu.edu/~fp/papers/mscs00.pdf" rel="noreferrer">http://www.cs.cmu.edu/~fp/papers/mscs00.pdf</a> - this is a great place to start because it starts from first principles and much of it is aimed to be accessible to non-logicians/language theorists. (I'm the second author though, so I'm biased.)</p>

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