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The arrow calculus

Published online by Cambridge University Press:  26 January 2010

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We introduce the arrow calculus, a metalanguage for manipulating Hughes's arrows with close relations both to Moggi's metalanguage for monads and to Paterson's arrow notation. Arrows are classically defined by extending lambda calculus with three constructs satisfying nine (somewhat idiosyncratic) laws; in contrast, the arrow calculus adds four constructs satisfying five laws (which fit two well-known patterns). The five laws were previously known to be sound; we show that they are also complete, and hence that the five laws may replace the nine.

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Atkey, R. (2008) What is a categorical model of arrows? In Mathematical Structures in Functional Programming, Capreta, V. & McBride, C. (eds), Electronic Notes in Theoretical Computer Science, Reykjavic, Iceland.Google Scholar
Benton, N. (1995) A mixed linear and non-linear logic: Proofs, terms and models. In Computer Science Logics, Pacholski, L. & Tiuryn, J. (eds), Lecture Notes in Computer Science, vol. 933. Springer–Verlag, Kazimierz, Poland.Google Scholar
Courtney, A. & Elliott, C. (2001) Genuinely Functional User Interfaces. Haskell workshop, 4169.Google Scholar
Hudak, P., Courtney, A., Nilsson, H. & Peterson, J. (2003) Arrows, robots, and functional reactive programming. In Advanced Functional Programming, 4th International School, Jeuring, J. & Jones, S. P. (eds), LNCS, vol. 2638. Springer-Verlag, Oxford, UK.Google Scholar
Hughes, J. (2000) Generalising monads to arrows, Sci Comput Program., 37: 67111.CrossRefGoogle Scholar
Jansson, P. & Jeuring, J. (1999) Polytypic compact printing and parsing. Pages 273–287 of: European Symposium on Programming, LNCS, vol. 1576. Springer-Verlag, Amsterdam, The Netherlands.Google Scholar
Lindley, S., Wadler, P. & Yallop, J. (2008a) The Arrow Calculus. Tech. rept. EDI-INF-RR-1258. School of Informatics, University of Edinburgh.Google Scholar
Lindley, S., Wadler, P. & Yallop, J. (2008b) Idioms are oblivious, arrows are meticulous, monads are promiscuous. In Mathematical Structures in Functional Programming, Capreta, V. & McBride, C. (eds), Electronic Notes in Theoretical Computer Science, Reykjavic, Iceland.Google Scholar
McBride, C. & Paterson, R. (2008) Applicative programming with effects, J. Funct. Program., 18 (1): 113.CrossRefGoogle Scholar
Moggi, E. (1991) Notions of computation and monads, Inf. Comput., 93 (1): 5592.CrossRefGoogle Scholar
Paterson, R. (2001) A new notation for arrows. Pages 229–240 of: International Conference on Functional Programming, ACM Press, Florence, Italy.Google Scholar
Power, J. & Robinson, E. (1997) Premonoidal categories and notions of computation, Math. Struct. Comput. Sci., 7 (5): 453468.CrossRefGoogle Scholar
Power, J. & Thielecke, H. (1999) Closed Freyd- and kappa-categories. In International colloquium on automata, languages, and programming, LNCS, vol. 1644. Springer, Prague, Czech Republic.Google Scholar
Sabry, A. & Felleisen, M. (1993) Reasoning about programs in continuation-passing style, Lisp Symbol. Comput., 6 (3/4): 289360.CrossRefGoogle Scholar
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