Resumptions

Roughly speaking, a resumption is a process which can consume a state, and then nondeterministically produce a number of outputs, each of which is either

a new state, or
a new state and another resumption.

Resumptions were introduced in Milner 1973. The first semantic model of resumptions is due to Plotkin 1976; it consists of a domain which satisfies the equation

R = S_\bot \to \mathcal{P}[S_\bot + S_\bot \times R]

where \mathcal{P}(-) is the Plotkin powerdomain construction.

References

Resumptions were introduced under the (awfully generic) name of processes by Robin Milner in a paper presented at the 1973 Logic Colloqium in Bristol:

Milner, Robin. 1975. ‘Processes: A Mathematical Model of Computing Agents’. In Logic Colloquium ’73, 80:157–73. Studies in Logic and the Foundations of Mathematics. Elsevier. https://doi.org/10.1016/S0049-237X(08)71948-7.

@inproceedings{milner_processes_1975,
	series = {Studies in {Logic} and the {Foundations} of {Mathematics}},
	title = {Processes: {A} {Mathematical} {Model} of {Computing} {Agents}},
	volume = {80},
	doi = {10.1016/S0049-237X(08)71948-7},
	booktitle = {Logic Colloquium '73},
	publisher = {Elsevier},
	author = {Milner, Robin},
	year = {1975},
	pages = {157--173}
}

The first domain-theoretic model of resumptions was introduced in

Plotkin, Gordon D. 1976. ‘A Powerdomain Construction’. SIAM Journal on Computing 5 (3): 452–87.

@article{plotkin_1976,
	title = {A powerdomain construction},
	volume = {5},
	number = {3},
	journal = {SIAM Journal on Computing},
	author = {Plotkin, Gordon D.},
	year = {1976},
	pages = {452--487}
}

Uses of a resumption (or resumption-like) monad in the context of pure functional programming appear in

Claessen, Koen. “A Poor Man’s Concurrency Monad.” Journal of Functional Programming 9, no. 3 (May 1999): 313–23. https://doi.org/10.1017/S0956796899003342.

@article{claessen_1999,
	title = {A poor man's concurrency monad},
	volume = {9},
	doi = {10.1017/S0956796899003342},
	pages = {313--323},
	number = {3},
	journaltitle = {Journal of Functional Programming},
	author = {Claessen, Koen},
	date = {1999}
}

Piróg, Maciej, and Jeremy Gibbons. “Tracing Monadic Computations and Representing Effects.” Electronic Proceedings in Theoretical Computer Science 76 (February 11, 2012): 90–111. https://doi.org/10.4204/EPTCS.76.8.

@article{pirog_2012,
	title = {Tracing monadic computations and representing effects},
	volume = {76},
	doi = {10.4204/EPTCS.76.8},
	pages = {90--111},
	journaltitle = {Electronic Proceedings in Theoretical Computer Science},
	author = {Piróg, Maciej and Gibbons, Jeremy},
	date = {2012}
}