Knowledge on a Budget
For researchers in epistemic logic and resource-aware computing, this work bridges qualitative reasoning and quantitative resource constraints, though it is an incremental extension of TEL.
This paper introduces semiring-annotated topological spaces (seats) to extend Topological Evidence Logic (TEL) with resource budgets for evidence acquisition, enabling reasoning about what is affordable given specific budgets. The framework provides sound and strongly complete axiomatizations for resource-indexed epistemic logics.
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.