A coalgebraic view of bar recursion and bar induction

Capretta, Venanzio; Uustalu, Tarmo

Abstract

We reformulate the bar recursion and induction principles in terms of recursive and wellfounded coalgebras. Bar induction was originally proposed by Brouwer as an axiom to recover certain classically valid theorems in a constructive setting. It is a form of induction on non- wellfounded trees satisfying certain properties. Bar recursion, introduced later by Spector, is the corresponding function defnition principle.
We give a generalization of these principles, by introducing the notion of barred coalgebra: a process with a branching behaviour given by a functor, such that all possible computations terminate.
Coalgebraic bar recursion is the statement that every barred coalgebra is recursive; a recursive coalgebra is one that allows defnition of functions by a coalgebra-to-algebra morphism. It is a framework to characterize valid forms of recursion for terminating functional programs. One application of the principle is the tabulation of continuous functions: Ghani, Hancock and Pattinson defned a type of wellfounded trees that represent continuous functions on streams. Bar recursion allows us to prove that every stably continuous function can be tabulated to such a tree where by stability we mean that the modulus of continuity is also continuous.
Coalgebraic bar induction states that every barred coalgebra is well-founded; a wellfounded coalgebra is one that admits proof by induction.

Citation

Capretta, V., & Uustalu, T. (2016). A coalgebraic view of bar recursion and bar induction. Lecture Notes in Artificial Intelligence, 9634, 91-106. https://doi.org/10.1007/978-3-662-49630-5_6

• Journal Article Type: Article
• Acceptance Date: Dec 18, 2015
• Publication Date: Mar 22, 2016
• Deposit Date: Jun 10, 2016
• Publicly Available Date: Jun 10, 2016
• Journal: Lecture Notes in Computer Science
• Electronic ISSN: 0302-9743
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 9634
• Pages: 91-106
• DOI: https://doi.org/10.1007/978-3-662-49630-5_6
• Keywords: Constructive Setting; Continuity Principle; Initial Algebra; Finite Path; Polynomial Functor
• Public URL: https://nottingham-repository.worktribe.com/output/779599
• Publisher URL: http://link.springer.com/chapter/10.1007/978-3-662-49630-5_6
• Additional Information: The final publication is available at Springer via http://link.springer.com/chapter/10.1007%2F978-3-662-49630-5_6
• Contract Date: Jun 10, 2016