Sheaves as oracle computations

2026-02-25

AI Summary

This work formalizes oracle computation within type theory and unifies it with sheaf theory and modal logic through a cohesive modeling framework. Oracle computation is characterized via modal operators induced by oracle predicates, establishing an adjoint correspondence between modalities and propositional truncations. Sheaves are interpreted as algebras of the associated monad using quotient inductive types. The core contributions include proving that all modalities are oracle modalities, introducing étale trees as intensional representations of oracle computations, and providing an explicit construction of Lawvere–Tierney topologies within realizability topoi. This approach yields an equivalent characterization of modalities and oracle computation, simplifies the computation of modal suprema, and furnishes realizability topology with a constructive semantic foundation.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

In type theory, an oracle may be specified abstractly by a predicate whose domain is the type of queries asked of the oracle, and whose proofs are the oracle answers. Such a specification induces an oracle modality that captures a computational intuition about oracles: at each step of reasoning we either know the result, or we ask the oracle a query and proceed upon receiving an answer. We characterize an oracle modality as the least one forcing the given predicate. We establish an adjoint retraction between modalities and propositional containers, from which it follows that every modality is an oracle modality. The left adjoint maps sums to suprema, which makes suprema of modalities easy to compute when they are given in terms of oracle modalities. We also study sheaves for oracle modalities. We describe sheafification in terms of a quotient-inductive type of computation trees, and describe sheaves as algebras for the corresponding monad. We also introduce equifoliate trees, an intensional notion of oracle computation given by a (non-propositional) container. Equifoliate trees descend to sheaves, and lift from sheaves in case the container is projective. As an application, we give a concrete description of all Lawvere-Tierney topologies in a realizability topos, closely related to a game-theoretic characterization by Takayuki Kihara.

Problem

• oracle
• modality
• sheaf
• type theory
• Lawvere-Tierney topology

Innovation

• oracle modality
• sheafification
• quotient-inductive type
• equifoliate trees
• Lawvere-Tierney topology