Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types

Abstract

Type systems certify program properties in a compositional way. From a bigger program one can abstract out a part and certify the properties of the resulting abstract program by just using the type of the part that was abstracted away. Termination and productivity are non-trivial yet desired program properties, and several type systems have been put forward that guarantee termination, compositionally. These type systems are intimately connected to the definition of least and greatest fixed-points by ordinal iteration. While most type systems use "conventional" iteration, we consider inflationary iteration in this article. We demonstrate how this leads to a more principled type system, with recursion based on well-founded induction. The type system has a prototypical implementation, MiniAgda, and we show in particular how it certifies productivity of corecursive and mixed recursive-corecursive functions.

Wellfounded recursion with copatterns

“…This will put coinduction in these systems on a robust foundation. We have already implemented sizebased type checking for patterns and copatterns in MiniAgda (Abel 2012) which gives us confidence in the approach.…”

“…• Our typing rules are formulated as a bidirectional type-checking algorithm that can be implemented as such. See, e. g., MiniAgda (Abel 2012). • We prove soundness of F cop ω by an untyped term model based on Girard's reducibility candidates.…”

Higher-Order Processes, Functions, and Sessions: A Monadic Integration

“…We could also use call-by-name, but for communication across channels, especially if distributed, one would not want to pass potentially large computations and the data structures they still rely on. If the functional language is overlaid with a termination checker (for examples, along the lines of Abel's proposal [1]), then the two should semantically coincide in any case. Since this is standard, we focus on the interesting new constructs: the monad, and the process expressions contained in them.…”

Up-to techniques using sized types

“…The approach to sized types presented above is based on deflationary iteration [Abel 2012]. For more details about this kind of sized types, including normalisation proofs (in one case sketched), see Abel and Pientka [2016] and Sacchini [2015].…”

“…Some type theories support a more flexible variant of corecursion based on sized types [Hughes et al 1996;Amadio and Coupet-Grimal 1998;Giménez 1998;Xi 2002;Blanqui 2004Blanqui , 2005Barthe et al 2004Barthe et al , 2006Grégoire and Sacchini 2010;Abel 2012;Sacchini 2013Sacchini , 2014Abel and Pientka 2016;Abel et al 2017]. Sized types tend to make the type theory more complicated, but my experienceÐbased on using what is perhaps the most mature implementation of type theory with sized types, Agda [Agda Team 2017]Ðis that they make it much easier to write corecursive programs.…”