Support is Search

2026-03-13

AI Summary

This work investigates the computational interpretation of the support relation in intuitionistic propositional logic under a fixed atomic basis. By encoding semantic clauses in continuation-passing style as goals within a second-order hereditary Harrop logic program, the support relation is precisely captured as a proof search process. This approach endows the base-extension semantics with constructivity and computational transparency, revealing that universal quantifiers therein correspond to eigenvariables subject to freshness constraints. The study establishes an exact correspondence between local semantic structures and logic programming, thereby complementing Sandqvist’s global completeness theorem and offering a viable pathway for semantic implementation and modeling applications.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• 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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Sandqvist's base-extension semantics for intuitionistic propositional logic defines a support relation parametrised by atomic bases, with validity identified as support in every base. Sandqvist's completeness theorem answers the global question: which formulae are valid? This paper addresses the local question: given a fixed base, what does support in that base correspond to? We show that support in a fixed base coincides with proof-search in a second-order hereditary Harrop logic program, via an encoding of formulae as logic-programming goals. This encoding proceeds by reading the semantic clauses in continuation-passing style, revealing that the universal quantifiers over base extensions and atoms appearing in those clauses are not domain-ranging quantifiers over a completed totality, but eigenvariables governed by a standard freshness discipline. Base-extension semantics thereby admits a fully constructive and computationally transparent interpretation: support is proof-search. The result complements Sandqvist's global theorem with a local correspondence, vindicates the anti-realist foundations of the framework on its own terms, and opens the way for implementing the semantics in modelling tasks.

Problem

• base-extension semantics
• support relation
• intuitionistic propositional logic
• proof-search
• hereditary Harrop logic

Innovation

• proof-search
• base-extension semantics
• hereditary Harrop logic
• continuation-passing style
• eigenvariables