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I am looking for an accessible reference that explains the concept of guarded recursion. Some things I have in mind are:

  • How is guarded recursion defined syntactically?
  • Why do the greatest- and least-fixed points coincide for functions that obey guardedness?
  • Are additional polarity related restrictions required in order to ensure that the fixed point exists?
  • What are the connections to modal $\mu$-calculus?

Many of these things seem to be folklore among scholars. For instance, Haskell programmers seem to be aware that Inductive and Co-Inductive types in Haskell coincide. Also, recent papers like [1] and [2] seem to refer to Nakano's guarded recursion theory and ▶ modality [3].


[1] Robert Atkey and Conor McBride. 2013. Productive coprogramming with guarded recursion. SIGPLAN Not. 48, 9 (September 2013), 197–208.

[2] Patrick Bahr, Christian Uldal Graulund, and Rasmus Ejlers Møgelberg. 2021. Diamonds are not forever: liveness in reactive programming with guarded recursion. Proc. ACM Program. Lang. 5, POPL, Article 2 (January 2021), 28 pages.

[3] H. Nakano, "A modality for recursion," Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332), Santa Barbara, CA, USA, 2000, pp. 255-266, doi: 10.1109/LICS.2000.855774.

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