Pedagogical second-order λ -calculus

• Published in: Theoretical computer science Vol. 410; no. 42; pp. 4190 - 4203
• Main Authors: Colson, Loïc, Michel, David
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier B.V 28.09.2009; Elsevier
• Subjects: Applied sciences; Artificial intelligence; Computer science; control theory; systems; Constructive mathematics; Exact sciences and technology; Learning and adaptive systems; Logic and foundations; Mathematical logic; Mathematical logic, foundations, set theory; Mathematics; Miscellaneous; Natural deduction; Negationless mathematics; Proof theory and constructive mathematics; Sciences and techniques of general use; Theoretical computing; Typed [formula omitted]-calculus; Mathematical logic; Constructive mathematics; Natural deduction; Negationless mathematics; Typed λ -calculus; Second order; Translation; Computer theory; Negation; Constraint; Propositional calculus; Hypothesis; Substitution; Proof; Typed λ-calculus
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2009.04.020

The present work introduces the notion of pedagogical natural deduction systems, i.e. natural deduction systems for which hypotheses occurring in a proof must be motivated by an example. In formal terms, we replace the rule (Hyp): [Display omitted] by the rule (P-Hyp): [Display omitted] where σ denotes a substitution which replaces all variables of Γ with an example. This substitution is called the motivation of Γ . These systems are in essence negationless. In the present paper, we study second order propositional calculus, since it is a simple non-trivial natural deduction system in which negation can be defined. We present a Curry–Howard version of pedagogical second-order propositional calculus, i.e. λ -calculus bearing the pedagogical constraint. We establish that this system has the usual properties of typed calculi (subject reduction and strong normalization) and that it can type all second-order terms of λ μ -calculus using a CPS translation. Furthermore, the main novelty is that all functions are useful: each polymorphic function has an instance which can be applied to a closed term.