Bar recursion is not computable via iteration

Abstract

We show that the bar recursion operators of Spector and Kohlenbach, considered as third-order functionals acting on total arguments, are not computable in Gödel's System T plus minimization, which we show to be equivalent to a programming language with a higher-order iteration construct. The main result is formulated so as to imply the non-definability of bar recursion in T + min within a variety of partial and total models, for instance the Kleene-Kreisel continuous functionals. The paper thus supplies proofs of some results stated in the book by Longley and Normann.The proof of the main theorem makes serious use of the theory of nested sequential procedures (also known as PCF Böhm trees), and proceeds by showing that bar recursion cannot be represented by any sequential procedure within which the tree of nested function applications is well-founded.

Asymptotic speedup via effect handlers

“…Of course, this approach could be readily implemented in ; but it is also clear how it could be effected in an even weaker language, in which the recursion construct of is replaced by a weaker iteration construct. (For a comparison of the power of iteration and recursion, see Longley, 2019.) For instance, the following operator (definable in ) allows one to achieve the effect of while-loops manipulating data of type A : Let us write for the sublanguage of allowing for each type A , but disallowing all uses of elsewhere.…”

“…Such operations are ‘non-functional’ in the sense that their output is not determined solely by the extension of their input (seen as a mathematical function ); however, there are also programs with ‘functional’ behaviour that can be implemented with control or local state but not without them (Longley, 1999). More recent results have exhibited differences lower down in the language expressivity spectrum: for instance, in a purely functional setting à la Haskell, the expressive power of recursion increases strictly with its type level (Longley, 2018), and there are natural operations computable by recursion but not by iteration (Longley, 2019). Much of this territory, including the mathematical theory of some of the natural definitions of computability in a higher-order setting, is mapped out by Longley & Normann (2015).…”

The recursion hierarchy for PCF is strict

“…(The reader may wish to study these results and proofs before proceeding further, since they provide simpler instances of the basic method that we will use in this paper.) At third order, there are even 'hereditarily total' functionals definable in PCF 1 but not by higher-type iterators, one example being the well-known bar recursion operator (see [Lon18]).…”

Effects for Efficiency: Asymptotic Speedup with First-Class Control

“…Such operations are 'non-functional' in the sense that their output is not determined solely by the extension of their input (seen as a mathematical function N ⊥ × N ⊥ → N ⊥ ); however, there are also programs with 'functional' behaviour that can be implemented with control or local state but not without them [Longley 1999]. More recent results have exhibited differences lower down in the language expressivity spectrum: for instance, in a purely functional setting à la Haskell, the expressive power of recursion increases strictly with its type level [Longley 2018], and there are natural operations computable by low-order recursion but not by high-order iteration [Longley 2019]. Much of this territory, including the mathematical theory of some of the natural notions of higher-order computability that arise in this way, is mapped out by Longley and Normann [2015].…”