Information Propagation and Contraction in Functional Interpretations

2026-07-21

AI Summary

This work addresses the entanglement of information propagation and contraction mechanisms in functional interpretations by introducing an algebraic structure called the “information nucleus” to formally disentangle these two aspects. The information nucleus precisely captures affine information propagation, while a formula-indexed contraction structure handles repeated assumptions. This approach unifies and extends classical interpretations such as Dialectica and Herbrand, yielding a functional interpretation framework applicable to both affine and contraction-containing fragments of finite-type arithmetic. The proposed framework further supports the systematic incorporation of auxiliary principles like continuity, enabling precise tracking of informational content and providing extracted realizers with refined mechanisms for handling challenges and choices.

Abstract

This paper separates two components of functional interpretations: affine information propagation and contraction. We introduce information nuclei as an algebraic interface to capture the affine component. An information nucleus specifies what information is associated with finite-type objects, how exact objects are compatible with such information, and how information is propagated through functions. From any information nucleus we obtain a formula translation and a soundness theorem for affine finite-type arithmetic. Extending soundness to finite-type arithmetic with contraction requires one additional ingredient: a formula-indexed contraction structure reducing the challenges generated by duplicated assumptions to a single challenge. Finite collections of candidates with union yield a Herbrand-style interpretation, while exact information with challenge selection yields the usual Dialectica interpretation over an arithmetic system restricted to decidable primitive formulas. The resulting framework provides a uniform method for specifying the information carried by extracted realizers, allowing existing functional interpretations to be systematically enriched with auxiliary data, such as continuity information.

Problem

• information propagation
• contraction
• functional interpretations
• information nucleus
• realizer extraction

Innovation

• information nucleus
• functional interpretation
• affine propagation
• contraction structure
• realizer enrichment