Abstract
We make an argument that, for any study involving computational effects such as divergence or continuations, the traditional syntax of simply typed lambda-calculus cannot be regarded as canonical, because standard arguments for canonicity rely on isomorphisms that may not exist in an effectful setting. To remedy this, we define a “jumbo lambda-calculus” that fuses the traditional connectives together into more general ones, so-called “jumbo connectives”. We provide two pieces of evidence for our thesis that the jumbo formulation is advantageous.
Firstly, we show that the jumbo lambda-calculus provides a “complete” range of connectives, in the sense of including every possible connective that, within the beta-eta theory, possesses a reversible rule.
Secondly, in the presence of effects, we show that there is no decomposition of jumbo connectives into non-jumbo ones that is valid in both call-by-value and call-by-name.
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References
Girard, J.-Y.: Locus solum: From the rules of logic to the logic of rules. Mathematical Structures in Computer Science 11(3), 301–506 (2001)
Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. In: Cambridge Tracts in Theoretical Computer Science 7, Cambridge University Press, Cambridge (1988)
Lambek, J., Scott, P.: Introduction to Higher Order Categorical Logic. Cambridge University Press, Cambridge (1986)
McCusker, G.: Full abstraction by translation. In: Proc, 3rd Workshop in Theory and Formal Methods, Imperial College, London (1996)
McCusker, G.: Games and Full Abstraction for a Functional Metalanguage with Recursive Types. PhD thesis, University of London (1996)
Moggi, E.: Computational lambda-calculus and monads. In: LICS 1989, Proc. 4th Ann. Symp. on Logic in Comp. Sci, pp. 14–23. IEEE, Los Alamitos (1989)
Pitts, A.M.: Categorical logic. In: Handbook of Logic in Computer Science, vol. 5, Oxford University Press, Oxford (2000)
Winskel, G.: Formal Semantics of Programming Languages. MIT Press, Cambridge (1993)
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© 2006 Springer-Verlag Berlin Heidelberg
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Levy, P.B. (2006). Jumbo λ-Calculus. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4052. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11787006_38
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DOI: https://doi.org/10.1007/11787006_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35907-4
Online ISBN: 978-3-540-35908-1
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