Böhm theorem and Böhm trees for the Lambda-mu-calculus

Alexis Saurin 1, 2, *
* Corresponding author
2 PI.R2 - Design, study and implementation of languages for proofs and programs
PPS - Preuves, Programmes et Systèmes, Inria Paris-Rocquencourt, UPD7 - Université Paris Diderot - Paris 7, CNRS - Centre National de la Recherche Scientifique : UMR7126
Abstract : Parigot's λμ-calculus (Parigot, 1992) is now a standard reference about the computational content of classical logic as well as for the formal study of control operators in functional languages. In addition to the fine-grained Curry-Howard correspondence between minimal classical deductions and simply typed λμ-terms and to the ability to encode many usual control operators such as call/cc in the λμ-calculus (in its historical call-by-name presentation or in call-by-value versions), the success of the λμ-calculus comes from its simplicity, its good meta-theoretical properties both as a typed and an untyped calculus (confluence, strong normalization, etc.) as well as the fact that it naturally extends Church's λ-calculus. Though, in 2001, David and Py proved that Böhm's theorem, which is a fundamental result of the untyped λ-calculus, cannot be lifted to Parigot's calculus. In the present article, we exhibit a natural extension to Parigot's calculus, the Λμ-calculus, in which Böhm's property, also known as separation property, can be stated and proved. This is made possible by a careful and detailed analysis of David and Py's proof of non-separability and of the characteristics of the λμ-calculus which break the property: we identify that the crucial point lies in the design of Parigot's λμ-calculus with a twolevel syntax. In addition, we establish a standardization theorem for the extended calculus, deduce a characterization of solvability, describe Λμ-Böhm trees and connect the calculus with stream computing and delimited control.
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Submitted on : Wednesday, May 9, 2012 - 8:37:59 AM
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Alexis Saurin. Böhm theorem and Böhm trees for the Lambda-mu-calculus. Theoretical Computer Science, Elsevier, 2012, 435, pp.106-138. ⟨10.1016/j.tcs.2012.02.027⟩. ⟨hal-00695534⟩



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