Intelligence, Creativity and Fantasy

UID/HIS/04666/2019 This is the 2nd volume of PHI series, published by CRC Press, the 4th published by CRC Press and the 5th volume of PHI proceedings.The texts presented in Proportion Harmonies and Identities (PHI) - INTELLIGENCE, CREATIVITY AND FANTASY were compiled with the intent to establish a multidisciplinary platform for the presentation, interaction and dissemination of research. The aim is also to foster the awareness and discussion on the topics of Harmony and Proportion with a focus on different visions relevant to Architecture, Arts and Humanities, Design, Engineering, Social and Natural Sciences, and their importance and benefits for the sense of both individual and community identity. The idea of modernity has been a significant motor for development since the Western Early Modern Age. Its theoretical and practical foundations have become the working tools of scientists, philosophers, and artists, who seek strategies and policies to accelerate the development process in different contexts.authorsversionpublishe

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

International audienceParigot'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