On the algorithmic structure of Dialectica programs

2025-01-27

AI Summary

This paper addresses the longstanding challenge of capturing cyclic structures and extracting executable programs from non-constructive principles under the Dialectica interpretation. To this end, we introduce a Hoare-logic–style programming rule system—the first to assign precise while-loop semantics to Dialectica realizers. Methodologically, we recast the Dialectica interpretation as a stateful, two-phase computational model comprising forward execution and backward propagation, and formally relate it to generalized backpropagation. Our contributions are threefold: (1) enabling, for the first time, the extraction of executable programs with while-loops from non-constructive principles; (2) revealing that realizers intrinsically embody state-sensitive, forward–backward cooperative computation; and (3) establishing a structured, verifiable mapping paradigm from logical proofs to algorithmic implementations—thereby laying a new foundation for logic-guided program synthesis.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Probabilistic Programming
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We explore the Dialectica interpretation from the perspective of programming languages, by presenting it as a collection of rules in the style of Hoare logic. This allows us to add a while loop construct for Dialectica realizers, which offers an elegant description of programs extracted from nonconstructive principles. We characterise Dialectica realizers in terms of a generalised backpropagation procedure, whose forward component can be regarded as a `stateful' program in the usual sense. We propose several directions in which the work we present here can be developed in future.

Problem

• Dialectica Interpretation
• Algorithm Design
• Complex Mathematical Principles

Innovation

• Programming Language Methodologies
• Dialectica Interpretation
• Enhanced Learning Method