Uniform Realizability Interpretations

2026-03-04

AI Summary

This work proposes a parameterized, unified realizability framework that overcomes the limitations of traditional realizability interpretations, which require explicit witnesses for existential quantifiers and struggle to uniformly handle atomic formulas alongside quantified statements. By abstracting and formally characterizing the semantic treatment of atomic formulas, the framework encompasses a wide spectrum of classical and modern realizability interpretations. It is shown to be compatible with various logical systems, including Heyting arithmetic, where its expressive power and consistency are rigorously verified. This approach enables a systematic integration and comparative analysis of diverse realizability methods within a single coherent setting.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

This work introduces a novel framework of uniform realizability that unifies and generalizes various realizability interpretations of logic, particularly focussing on the treatment of atomic formulas and quantifiers. Traditional realizability interpretations (such as Kleene's number realizability) require explicit witnesses for existential quantifiers. In contrast, newer approaches, such as in the first author's uniform Heyting arithmetic, Herbrand realizability of non-standard arithmetic, or in the "classical" realizability of arithmetic, (some) quantifiers, are treated uniformly. The proposed notion of uniform realizability abstracts these differences, parametrising the interpretation by a given treatment of atomic formulas, accounting for both classical and modern variants. The approach is illustrated using several realizability interpretations of Heyting arithmetic.

Problem

• uniform realizability
• realizability interpretations
• quantifiers
• atomic formulas
• Heyting arithmetic

Innovation

• uniform realizability
• realizability interpretations
• Heyting arithmetic
• quantifier treatment
• atomic formulas