Interpreting a Classical Geometric Proof with Interactive Realizability

TL;DR

This paper demonstrates how to extract an effective, adaptive geometric algorithm from a classical proof using interactive realizability, transforming non-constructive reasoning into a practical, learning-based method.

Contribution

It introduces a novel application of interactive realizability to convert classical geometric proofs involving real numbers into effective algorithms with backtracking and learning capabilities.

Findings

The extracted algorithm performs only necessary comparisons, optimizing computational effort.

It successfully converts a non-constructive proof into a constructive, effective algorithm.

The approach handles undecidable real number comparisons through interactive backtracking.

Abstract

We show how to extract a monotonic learning algorithm from a classical proof of a geometric statement by interpreting the proof by means of interactive realizability, a realizability sematics for classical logic. The statement is about the existence of a convex angle including a finite collections of points in the real plane and it is related to the existence of a convex hull. We define real numbers as Cauchy sequences of rational numbers, therefore equality and ordering are not decidable. While the proof looks superficially constructive, it employs classical reasoning to handle undecidable comparisons between real numbers, making the underlying algorithm non-effective. The interactive realizability interpretation transforms the non-effective linear algorithm described by the proof into an effective one that uses backtracking to learn from its mistakes. The effective algorithm exhibits a "smart" behavior, performing comparisons only up to the precision required to prove the final statement. This behavior is not explicitly planned but arises from the interactive interpretation of comparisons between Cauchy sequences.