Logical Methods in Computer Science Volume 15, Issue 2, 2019, pp. 18:1–18:57 Submitted Apr. 24, 2018 https://lmcs.episciences.org/ Published May 24, 2019 BISIMULATIONS FOR DELIMITED-CONTROL OPERATORS DARIUSZ BIERNACKI a, SERGUE¨I LENGLET b, AND PIOTR POLESIUK a a University of Wroclaw e-mail address: fdabi,
[email protected] b Universit´ede Lorraine e-mail address:
[email protected] Abstract. We present a comprehensive study of the behavioral theory of an untyped λ-calculus extended with the delimited-control operators shift and reset. To that end, we define a contextual equivalence for this calculus, that we then aim to characterize with coinductively defined relations, called bisimilarities. We consider different styles of bisimilarities (namely applicative, normal-form, and environmental) within a unifying framework, and we give several examples to illustrate their respective strengths and weaknesses. We also discuss how to extend this work to other delimited-control operators. 1. Introduction Delimited-control operators. Control operators for delimited continuations enrich a pro- gramming language with the ability to delimit the current continuation, to capture such a delimited continuation, and to compose delimited continuations. Such operators have been originally proposed independently by Felleisen [26] and by Danvy and Filinski [21], with nu- merous variants designed subsequently [34, 66, 32, 25]. The applications of delimited-control operators range from non-deterministic programming [21, 45], partial evaluation [58, 19], and normalization by evaluation [24] to concurrency [34], mobile code [89], linguistics [84], oper- ating systems [43], and probabilistic programming [44]. Several variants of delimited-control operators are nowadays available in mainstream functional languages such as Haskell [25], OCaml [42], Racket [29], and Scala [76].