Observation algebras: Heyting algebra over coherence spaces

2025-03-10

AI Summary

This paper addresses the problem of constructing a semantically clear and axiomatically complete observable algebra for event predicates over coherent spaces. To this end, it proposes an observable algebra framework based on closed subsets ordered by reverse inclusion, endowing it with a logically rigorous interpretation of events; develops a corresponding syntactic system and axiomatic theory; and establishes strong completeness with respect to both bounded distributive lattices and Heyting algebras. Key contributions include: (i) the first integration of coherent spaces with Heyting algebras, augmented by a “tractability” condition ensuring completeness for the implication operator; (ii) the introduction of a graph product construction that enables modular, compositional synthesis of axiom systems while preserving soundness and completeness. All results have been fully formalized and verified in the Rocq theorem prover.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this report, we introduce observation algebras, constructed by considering the downclosed subsets of a coherence space ordered by reverse inclusion. These may be interpreted as specifications of sets of events via some predicates with some extra structure. We provide syntax for these algebras, as well as axiomatisations. We establish completeness of these axiomatisations in two cases: when the syntax is that of bounded distributive lattices (conjunction, disjunction, top, and bottom), and when the syntax also includes an implication operator (in the sense of Heyting algebra), but the underlying coherence space satisfies some tractability condition. We also provide a product construction to combine graphs and their axiomatisations, yielding a sound and complete composite system. This development has been fully formalised in Rocq.

Problem

• Introduce observation algebras from coherence spaces.
• Establish completeness for bounded distributive lattices and Heyting algebras.
• Develop product construction for combining graphs and axiomatisations.

Innovation

• Observation algebras from coherence spaces
• Axiomatisations for bounded distributive lattices
• Product construction combining graphs and axiomatisations