Knowledge on a Budget

2026-04-13

AI Summary

Traditional epistemic logics neglect the cost of acquiring evidence, limiting their applicability in resource-constrained settings. This work addresses this gap by formally integrating resource costs into epistemic logic for the first time, proposing a resource-aware framework based on semiring-weighted topological spaces. In this framework, evidence is modeled as budgeted open sets, and a resource-indexed modal operator is introduced to reason about knowledge under resource constraints. The authors develop a sound and strongly complete axiomatization for the resulting logic, enabling joint reasoning about observability and affordability given bounded resources. The expressive power of the framework is further characterized through notions of bisimulation and disjoint unions.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Philosophy and Ethics of AI: AI & Epistemology

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In various computational systems, accessing information incurs time, memory or energy costs. However, standard epistemic logics usually model the acquisition of evidence as a cost-free process, which restricts their applicability in environments with limited resources. In this paper, we bridge the gap between qualitative epistemic reasoning and quantitative resource constraints by introducing semiring-annotated topological spaces (seats). Building on Topological Evidence Logic (TEL), we extend the representation of evidence as open sets, adding an annotation function that maps evidence to semiring ideals, representing the resource budgets sufficient for observation. This framework allows us to reason not only about what is observable in principle, but also about what is affordable given a specific budget. We develop a family of seat-based epistemic logics with resource-indexed modalities and provide sound, strongly complete axiomatisations for these logics. Furthermore, we introduce suitable notions of bisimulation and disjoint union to delineate the expressive power of our framework.

Problem

• epistemic logic
• resource constraints
• evidence acquisition
• computational cost
• topological semantics

Innovation

• semiring-annotated topological spaces
• resource-aware epistemic logic
• Topological Evidence Logic
• budget-constrained reasoning
• modal logic with resource indexing