Modified Bar Recursion

Modified bar recursion

We introduce a variant of Spector’s bar recursion (called “modified bar recursion”) in finite types to give a realizability interpretation of the classical axiom of countable choice allowing for the extraction of witnesses from proofs of Σ1 formulas in classical analysis. As a second application of modified bar recursion we present a bar recursive definition of the fan functional. Moreover, we ...

Modified Bar Recursion and Classical Dependent Choice

On Spector's bar recursion

We show that Spector’s “restricted” form of bar recursion is sufficient (over system T ) to define Spector’s search functional. This new result is then used to show that Spector’s restricted form of bar recursion is in fact as general as the supposedly more general form of bar recursion. Given that these two forms of bar recursion correspond to the (explicitly controlled) iterated products of s...

The equivalence of bar recursion and open recursion

Several extensions of Gödel’s system T with new forms of recursion have been designed for the purpose of giving a computational interpretation to classical analysis. One can organise many of these extensions into two groups: those based on bar recursion, which include Spector’s original bar recursion, modified bar recursion and the more recent products of selections functions, or those based on...

A Coalgebraic View of Bar Recursion and Bar Induction

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 nonwellfounded trees satisfying certain properties. Bar recursion, introduced later by Spector, is the corresponding function ...