Abstract
It has been shown by Escardó and the first author that a functional interpretation of proofs in analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the computation of optimal strategies in sequential games. We argue that this result has genuine practical value by interpreting some well-known theorems of mathematics and demonstrating that the product gives these theorems a natural computational interpretation that can be clearly understood in game theoretic terms.
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Notes
- 1.
As in [18] we adopt Kuroda’s variant of the negative translation.
- 2.
In these papers DC is given in a slightly different form to ours.
- 3.
This is a stronger notion than the one introduced in [11] for the more general case where the quantifiers are not necessarily attainable.
- 4.
Assuming stability of atomic formulas.
- 5.
Note that we are replacing the standard conclusion \(\varphi _{\omega f}(f(\omega f),qf)\) with a stronger variant \(\forall i \leq \omega f\,\varphi _{i}(fi,qf)\). This is not essential as one can, given an \(\omega\), define \(\tilde{\omega }(f) =\mu i \leq \omega (f)\neg \varphi _{i}(fi,qf)\) so that \(\varphi _{\tilde{\omega }f}(f(\tilde{\omega }f),qf)\) implies \(\forall i \leq \omega f\,\varphi _{i}(fi,qf)\). We prefer the version \(\forall i \leq \omega f\,\varphi _{i}(fi,qf)\) since (1) we can directly realise, and (2) it makes the interpretation more intuitive by viewing ω f as a bound up to which the play f is required to be “optimal”.
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Acknowledgements
The first author gratefully acknowledges support of the Royal Society (grant 516002.K501/RH/kk). The second author acknowledges the support of an EPSRC doctoral training grant.
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Oliva, P., Powell, T. (2015). A Game-Theoretic Computational Interpretation of Proofs in Classical Analysis. In: Kahle, R., Rathjen, M. (eds) Gentzen's Centenary. Springer, Cham. https://doi.org/10.1007/978-3-319-10103-3_18
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